diff --git a/404.html b/404.html index 53c06d7c..c01e2e78 100644 --- a/404.html +++ b/404.html @@ -32,7 +32,7 @@
Toshiki's Note

Appearance

404

PAGE NOT FOUND

But if you don't change your direction, and if you keep looking, you may end up where you are heading.
Take me home
- + \ No newline at end of file diff --git a/academic/chemistry/index.html b/academic/chemistry/index.html index 74adf42f..20797ebf 100644 --- a/academic/chemistry/index.html +++ b/academic/chemistry/index.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Welcome to Chemistry ​

Author:Anda Toshiki
Updated:a minute ago
Words:3
Reading:1 min
- +
Toshiki's Note

Appearance

Welcome to Chemistry ​

Author:Anda Toshiki
Updated:a minute ago
Words:3
Reading:1 min
+ \ No newline at end of file diff --git a/academic/chemistry/notes/12-5.html b/academic/chemistry/notes/12-5.html index 10680e65..666fdd25 100644 --- a/academic/chemistry/notes/12-5.html +++ b/academic/chemistry/notes/12-5.html @@ -13,7 +13,7 @@ - + @@ -37,7 +37,7 @@
Toshiki's Note

Appearance

12-5: Reaction Mechanism ​

Author:Anda Toshiki
Updated:a minute ago
Words:1.9k
Reading:11 min

12-5-1: Learning Objectives ​

Learning Objectives

One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.

In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:

2C8H18(l)+25O2( g)⟶16CO2( g)+18H2O(g)2 \mathrm{C}_{8} \mathrm{H}_{18}(\mathrm{l})+25 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 16 \mathrm{CO}_{2}(\mathrm{~g})+18 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})

For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.

The overall sequence of elementary reactions is the mechanism of the reaction.

12-5-2: Molecularity and the Rate-Determining Step ​

To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.

2CO+2NO2→2CO2+N2O4\mathrm{2CO + 2NO_2 \rightarrow 2CO_2 + N_2O_4}

From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of NO2\mathrm{NO}_{2} with a molecule of CO\mathrm{CO} that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:

 rate =k[NO2]2\text { rate }=k\left[\mathrm{NO}_{2}\right]^{2}

The fact that the reaction is second order in [NO2]\left[\mathrm{NO}_{2}\right] and independent of [CO][\mathrm{CO}] tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be

 rate =k[NO2][CO].\text { rate }=k\left[\mathrm{NO}_{2}\right][\mathrm{CO}] .

The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:

15-2-1: Two-Step Mechanism ​

StepsReactionReaction Type
step 1NO2+NO2⟶ slow NO3+NO\mathrm{NO}_{2}+\mathrm{NO}_{2} \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}_{3}+\mathrm{NO}elementary reaction
step 2NO3+CO→NO2+CO2‾\underline{\mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}elementary reaction
sumNO2+CO→NO+CO2\mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{NO}+\mathrm{CO}_{2}overall reaction

According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The NO3\mathrm{NO_3} molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.

The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.

12-5-3: Using Molecularity to Describe a Rate Law ​

The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)

About the image

The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)

Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( A→\mathrm{A} \rightarrow products) is

 rate =k[A].\text { rate }=k[A] .

For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form A+B→\mathrm{A}+\mathrm{B} \rightarrow products, the general rate law is

 rate =k[A][B].\text { rate }=k[A][B] .

Elementary ReactionMolecularityRateReaction Order
A→ products \mathrm{A} \rightarrow \text { products }Unimolecular rate =k[ A]\text { rate }=k[\mathrm{~A}]first
2 A→ products 2\mathrm{~A} \rightarrow \text { products }Bimolecular rate =k[ A]2\text { rate }=k[\mathrm{~A}]^2second
A+B→ products \mathrm{A}+\mathrm{B} \rightarrow \text { products }Bimolecular rate =k[ A][B]\text { rate }=k[\mathrm{~A}][\mathrm{B}]second
2 A+B→ products 2 \mathrm{~A}+\mathrm{B} \rightarrow \text { products }Termolecular rate =k[ A]2[ B]\text { rate }=k[\mathrm{~A}]^2[\mathrm{~B}]third
A+B+C→ products \mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow \text { products }Termolecular rate =k[ A][B][C]\text { rate }=k[\mathrm{~A}][\mathrm{B}][\mathrm{C}]third

For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).

12-5-4: Identifying the Rate-Determining Step ​

Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.

Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.

Rate laws for each elementary reaction in our example as well as for the overall reaction

StepsReactionRate
step 1NO2+NO2→k1NO3+NO\mathrm{NO}_2+\mathrm{NO}_2 \stackrel{\mathrm{k}_1}{\rightarrow} \mathrm{NO}_3+\mathrm{NO} rate =k1[NO2]2( predicted) \text { rate }=k_1\left[\mathrm{NO}_2\right]^2(\text { predicted) }
step 2NO3+CO→k2NO2+CO2‾\underline{\mathrm{NO}_3+\mathrm{CO} \stackrel{k_2}{\rightarrow} \mathrm{NO}_2+\mathrm{CO}_2} rate =k2[NO3][CO]( predicted )\text { rate }=k_2\left[\mathrm{NO}_3\right][\mathrm{CO}](\text { predicted })
step 3NO2+CO→kNO+CO2\mathrm{NO}_2+\mathrm{CO} \stackrel{k}{\rightarrow} \mathrm{NO}+\mathrm{CO}_2 rate =k[NO2]2( observed) \text { rate }=k\left[\mathrm{NO}_2\right]^2(\text { observed) }

The experimentally determined rate law for the reaction of NO2\mathrm{NO}_{2} with COC O is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so kk for the overall reaction must equal k1k_{1}. That is, NO3\mathrm{NO}_{3} is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.

Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.

12-5-4-1: Example-A Reaction with an Intermediate ​

In an alternative mechanism for the reaction of NO2\mathrm{NO}_{2} with CO\mathrm{CO} with N2O4\mathrm{N}_{2} \mathrm{O}_{4} appearing as an intermediate.

alternative mechanism for the reaction of NO2\mathrm{NO}_{2} with CO\mathrm{CO} with N2O4\mathrm{N}_{2} \mathrm{O}_{4} appearing as an intermediate.

Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate =k[NO2]2t=\mathrm{k[{NO}_{2}]^{2}t})

Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law

Strategy ​

  • Determine the rate law for each elementary reaction in the reaction.

  • Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.

Solution ​

View solution

A The rate law for step 1 is rate =k1[NO2]2=k_{1}\left[\mathrm{NO}_{2}\right]^{2}; for step 2 , it is rate =k2[ N2O4][CO]=k_{2}\left[\mathrm{~N}_{2} \mathrm{O}_{4}\right][\mathrm{CO}].

B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate == k1[NO2]2k_{1}\left[\mathrm{NO}_{2}\right]^{2}. This is the same as the experimentally determined rate law. Hence this mechanism, with N2O4\mathrm{N}_{2} \mathrm{O}_{4} as an intermediate, and the one described previously, with NO3\mathrm{NO}_{3} as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, NO3\mathrm{NO}_{3} and N2O4\mathrm{N}_{2} \mathrm{O}_{4}, directly.

- + \ No newline at end of file diff --git a/academic/chemistry/problems/02-20.html b/academic/chemistry/problems/02-20.html index 22f9f019..75bdc314 100644 --- a/academic/chemistry/problems/02-20.html +++ b/academic/chemistry/problems/02-20.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Presentation problem: 02-20 ​

Author:Anda Toshiki
Updated:a minute ago
Words:412
Reading:2 min

Question ​

At 500 K500 \mathrm{~K} in the presence of a copper surface, ethanol decomposes according to the equation

C2H5OH(g)⟶CH3CHO(g)+H2(g)\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{g}) \longrightarrow \mathrm{CH}_3 \mathrm{CHO}(g)+\mathrm{H}_2(g)

The pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} was measured as a function of time and the following data were obtained:

Time (s)PC2H5OH (torr) P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }
0250.
100.237
200.224
300.211
400.198
500.185

Since the pressure of a gas is directly proportional to the concentration of gas, we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} after 900.s900 . \mathrm{s} from the start of the reaction. (Hint: To determine the order of the reaction with respect to C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}, compare how the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} decreases with each time listing.)

Solution ​

solution graph
◎

Due to the fact that the graph of p[O2]\mathrm{p}\left[\mathrm{O}_2\right] over time is showing R2\mathrm{R}^2 value of 1 we know we have a zero-order reaction. Therefore:

k=p0[C2H5OH]−p[C2H5OH]t=250 torr −185 torr 500 s=0.13 torr s −1k=0.13 torr s s−1×1 atm760 torr =1.71×10−4 atm s−1 rate =kp(C2H5OH)=−kt+p0(C2H5OH)p(C2H5OH)=−(1.71×10−4 atm s−1)×900 s+0.33 atmp(C2H5OH)=0.176 atm s−1\begin{aligned} & \mathrm{k}=\frac{\mathrm{p}_0\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]-\mathrm{p}\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]}{\mathrm{t}}=\frac{250 \text { torr }-185 \text { torr }}{500 \mathrm{~s}}=0.13 \text { torr s }^{-1} \\ & \mathrm{k}=0.13 \text { torr s } \mathrm{s}^{-1} \times \frac{1 \mathrm{~atm}}{760 \text { torr }}=1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1} \\ & \text { rate }=\mathbf{k} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\mathrm{kt}+\mathrm{p}_0\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right) \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\left(1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1}\right) \times 900 \mathrm{~s}+0.33 \mathrm{~atm} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=0.176 \mathrm{~atm} \mathrm{~s}^{-1} \\ & \end{aligned}

- +
Toshiki's Note

Appearance

Presentation problem: 02-20 ​

Author:Anda Toshiki
Updated:a minute ago
Words:412
Reading:2 min

Question ​

At 500 K500 \mathrm{~K} in the presence of a copper surface, ethanol decomposes according to the equation

C2H5OH(g)⟶CH3CHO(g)+H2(g)\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{g}) \longrightarrow \mathrm{CH}_3 \mathrm{CHO}(g)+\mathrm{H}_2(g)

The pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} was measured as a function of time and the following data were obtained:

Time (s)PC2H5OH (torr) P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }
0250.
100.237
200.224
300.211
400.198
500.185

Since the pressure of a gas is directly proportional to the concentration of gas, we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} after 900.s900 . \mathrm{s} from the start of the reaction. (Hint: To determine the order of the reaction with respect to C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}, compare how the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} decreases with each time listing.)

Solution ​

solution graph
◎

Due to the fact that the graph of p[O2]\mathrm{p}\left[\mathrm{O}_2\right] over time is showing R2\mathrm{R}^2 value of 1 we know we have a zero-order reaction. Therefore:

k=p0[C2H5OH]−p[C2H5OH]t=250 torr −185 torr 500 s=0.13 torr s −1k=0.13 torr s s−1×1 atm760 torr =1.71×10−4 atm s−1 rate =kp(C2H5OH)=−kt+p0(C2H5OH)p(C2H5OH)=−(1.71×10−4 atm s−1)×900 s+0.33 atmp(C2H5OH)=0.176 atm s−1\begin{aligned} & \mathrm{k}=\frac{\mathrm{p}_0\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]-\mathrm{p}\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]}{\mathrm{t}}=\frac{250 \text { torr }-185 \text { torr }}{500 \mathrm{~s}}=0.13 \text { torr s }^{-1} \\ & \mathrm{k}=0.13 \text { torr s } \mathrm{s}^{-1} \times \frac{1 \mathrm{~atm}}{760 \text { torr }}=1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1} \\ & \text { rate }=\mathbf{k} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\mathrm{kt}+\mathrm{p}_0\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right) \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\left(1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1}\right) \times 900 \mathrm{~s}+0.33 \mathrm{~atm} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=0.176 \mathrm{~atm} \mathrm{~s}^{-1} \\ & \end{aligned}

+ \ No newline at end of file diff --git a/academic/chemistry/problems/03-02-1.html b/academic/chemistry/problems/03-02-1.html index 2992e795..cf5956c6 100644 --- a/academic/chemistry/problems/03-02-1.html +++ b/academic/chemistry/problems/03-02-1.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Problem: 03-02-1 ​

Author:Anda Toshiki
Updated:a minute ago
Words:404
Reading:2 min

Question ​

2 N2O5(g)→4NO2(g)+O2(g)2 \mathrm{~N}_2 \mathrm{O}_5(g) \rightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g)

The decomposition of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is represented by the equation above. A sample of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is monitored as it decomposes, and the concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 as a function of time is recorded. The results are shown in the table below.

Time (s)[N2O5]\mathrm{[N_2O_5]}
01.000
25.00.801
50.00.642
75.00.515

Calculate the average rate of the reaction between 50.050.0 and 75.075.0 seconds.

Solution ​

a A+b B→cC+dDa \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D}

the rate of reaction is defined as

 rate =−1aΔ[A]Δt=−1bΔ[B]Δt=+1cΔ[C]Δt=+1dΔ[D]Δt\color{cyan}\text { rate }=-\frac{1}{a} \frac{\Delta[\mathrm{A}]}{\Delta t}=-\frac{1}{b} \frac{\Delta[\mathrm{B}]}{\Delta t}=+\frac{1}{c} \frac{\Delta[\mathrm{C}]}{\Delta t}=+\frac{1}{d} \frac{\Delta[\mathrm{D}]}{\Delta t}

Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( aa, b,cb, c, or dd ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was N2O5\mathrm{N}_2 \mathrm{O}_5. With that in mind, let's write the reaction rate in terms of the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 :

 rate =−12Δ[N2O5]Δt\text { rate }=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta t}

Since the coefficient for N2O5\mathrm{N}_2 \mathrm{O}_5 in the balanced equation is 2 , we divided the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 by 2 . Additionally, since N2O5\mathrm{N}_2 \mathrm{O}_5 is being consumed in the reaction, we included a negative sign in front of the expression.

Now, let's plug in the information from the table to calculate the average reaction rate between 50.050.0 and 75.075.0 seconds:

 rate =−12(0.515M−0.642M)(75.0 s−50.0 s)=2.54×10−3M s−1\begin{aligned} \text { rate } & =-\frac{1}{2} \frac{(0.515 M-0.642 M)}{(75.0 \mathrm{~s}-50.0 \mathrm{~s})} \\ & =2.54 \times 10^{-3} M \mathrm{~s}^{-1} \end{aligned}

So, the average rate of the reaction between 50.050.0 and 75.075.0 seconds is 2.54×10−3Ms−1\bold{2.54 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}}.

- +
Toshiki's Note

Appearance

Problem: 03-02-1 ​

Author:Anda Toshiki
Updated:a minute ago
Words:404
Reading:2 min

Question ​

2 N2O5(g)→4NO2(g)+O2(g)2 \mathrm{~N}_2 \mathrm{O}_5(g) \rightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g)

The decomposition of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is represented by the equation above. A sample of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is monitored as it decomposes, and the concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 as a function of time is recorded. The results are shown in the table below.

Time (s)[N2O5]\mathrm{[N_2O_5]}
01.000
25.00.801
50.00.642
75.00.515

Calculate the average rate of the reaction between 50.050.0 and 75.075.0 seconds.

Solution ​

a A+b B→cC+dDa \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D}

the rate of reaction is defined as

 rate =−1aΔ[A]Δt=−1bΔ[B]Δt=+1cΔ[C]Δt=+1dΔ[D]Δt\color{cyan}\text { rate }=-\frac{1}{a} \frac{\Delta[\mathrm{A}]}{\Delta t}=-\frac{1}{b} \frac{\Delta[\mathrm{B}]}{\Delta t}=+\frac{1}{c} \frac{\Delta[\mathrm{C}]}{\Delta t}=+\frac{1}{d} \frac{\Delta[\mathrm{D}]}{\Delta t}

Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( aa, b,cb, c, or dd ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was N2O5\mathrm{N}_2 \mathrm{O}_5. With that in mind, let's write the reaction rate in terms of the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 :

 rate =−12Δ[N2O5]Δt\text { rate }=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta t}

Since the coefficient for N2O5\mathrm{N}_2 \mathrm{O}_5 in the balanced equation is 2 , we divided the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 by 2 . Additionally, since N2O5\mathrm{N}_2 \mathrm{O}_5 is being consumed in the reaction, we included a negative sign in front of the expression.

Now, let's plug in the information from the table to calculate the average reaction rate between 50.050.0 and 75.075.0 seconds:

 rate =−12(0.515M−0.642M)(75.0 s−50.0 s)=2.54×10−3M s−1\begin{aligned} \text { rate } & =-\frac{1}{2} \frac{(0.515 M-0.642 M)}{(75.0 \mathrm{~s}-50.0 \mathrm{~s})} \\ & =2.54 \times 10^{-3} M \mathrm{~s}^{-1} \end{aligned}

So, the average rate of the reaction between 50.050.0 and 75.075.0 seconds is 2.54×10−3Ms−1\bold{2.54 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}}.

+ \ No newline at end of file diff --git a/academic/chemistry/problems/03-02-2.html b/academic/chemistry/problems/03-02-2.html index 602c1c36..245dcbb9 100644 --- a/academic/chemistry/problems/03-02-2.html +++ b/academic/chemistry/problems/03-02-2.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Problem: 03-02-2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:205
Reading:1 min

Question ​

The rate law for a particular reaction is rate =k[XY]2=k[\mathrm{XY}]^2. In an experiment, the initial rate of the reaction is determined to be 0.16 mol/(L⋅s)0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s}) when the initial concentration of XY\mathrm{XY} is 0.40 mol/L0.40 \mathrm{~mol} / \mathrm{L}.

  • What is the value of the rate constant, kk, for the reaction?
    • 0.10 L/(mol⋅s)0.10 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 0.40 L/(mol⋅s)0.40 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 2.5 L/(mol⋅s)2.5 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

Solution ​

lo tind the value of the rate constant for the reaction, let's first solve the rate law for kk :

k= rate [XY]2k=\frac{\text { rate }}{[\mathrm{XY}]^2}

Next, let's plug in the initial rate and concentration given in the text:

k=0.16 mol/(L⋅s)(0.40 mol/L)2=0.16 mol/(L⋅s)0.16 mol2/L2=1.0 L/(mol⋅s)\begin{aligned} k & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{(0.40 \mathrm{~mol} / \mathrm{L})^2} \\ & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{0.16 \mathrm{~mol}^2 / \mathrm{L}^2} \\ & =1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) \end{aligned}

So, the value of the rate constant for the reaction is 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

- +
Toshiki's Note

Appearance

Problem: 03-02-2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:205
Reading:1 min

Question ​

The rate law for a particular reaction is rate =k[XY]2=k[\mathrm{XY}]^2. In an experiment, the initial rate of the reaction is determined to be 0.16 mol/(L⋅s)0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s}) when the initial concentration of XY\mathrm{XY} is 0.40 mol/L0.40 \mathrm{~mol} / \mathrm{L}.

  • What is the value of the rate constant, kk, for the reaction?
    • 0.10 L/(mol⋅s)0.10 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 0.40 L/(mol⋅s)0.40 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 2.5 L/(mol⋅s)2.5 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

Solution ​

lo tind the value of the rate constant for the reaction, let's first solve the rate law for kk :

k= rate [XY]2k=\frac{\text { rate }}{[\mathrm{XY}]^2}

Next, let's plug in the initial rate and concentration given in the text:

k=0.16 mol/(L⋅s)(0.40 mol/L)2=0.16 mol/(L⋅s)0.16 mol2/L2=1.0 L/(mol⋅s)\begin{aligned} k & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{(0.40 \mathrm{~mol} / \mathrm{L})^2} \\ & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{0.16 \mathrm{~mol}^2 / \mathrm{L}^2} \\ & =1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) \end{aligned}

So, the value of the rate constant for the reaction is 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

+ \ No newline at end of file diff --git a/academic/chemistry/problems/03-02-3.html b/academic/chemistry/problems/03-02-3.html index 2ab155f3..08846820 100644 --- a/academic/chemistry/problems/03-02-3.html +++ b/academic/chemistry/problems/03-02-3.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Problem 03-02-3 ​

Author:Anda Toshiki
Updated:a minute ago
Words:246
Reading:1 min

Question ​

2X(g)+2Y(g)→Q(g)+2R(g)2 \mathrm{X}(g)+2 \mathrm{Y}(g) \rightarrow \mathrm{Q}(g)+2 \mathrm{R}(g)

The reaction represented above is found to be second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}.

What happens to the rate of the reaction when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled?

  • It increases by a factor of 4.
  • It decreases by a factor of 2.
  • It decreases by a factor of 4.
  • It does not change.

Solution ​

To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}, so the rate law is rate =k[X]2[Y]=k[\mathrm{X}]^2[\mathrm{Y}]

Now, let's think about how the rate changes when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled. Since [X][\mathrm{X}] is raised to the second power in the rate law, halving [X][\mathrm{X}] decreases the reaction rate by a factor of 4 . Similarly, since [Y][\mathrm{Y}] is raised to the first power, doubling [Y][\mathrm{Y}] increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled, the rate of the reaction decreases by a factor of 2 .

- +
Toshiki's Note

Appearance

Problem 03-02-3 ​

Author:Anda Toshiki
Updated:a minute ago
Words:246
Reading:1 min

Question ​

2X(g)+2Y(g)→Q(g)+2R(g)2 \mathrm{X}(g)+2 \mathrm{Y}(g) \rightarrow \mathrm{Q}(g)+2 \mathrm{R}(g)

The reaction represented above is found to be second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}.

What happens to the rate of the reaction when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled?

  • It increases by a factor of 4.
  • It decreases by a factor of 2.
  • It decreases by a factor of 4.
  • It does not change.

Solution ​

To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}, so the rate law is rate =k[X]2[Y]=k[\mathrm{X}]^2[\mathrm{Y}]

Now, let's think about how the rate changes when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled. Since [X][\mathrm{X}] is raised to the second power in the rate law, halving [X][\mathrm{X}] decreases the reaction rate by a factor of 4 . Similarly, since [Y][\mathrm{Y}] is raised to the first power, doubling [Y][\mathrm{Y}] increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled, the rate of the reaction decreases by a factor of 2 .

+ \ No newline at end of file diff --git a/academic/cis105/cis105-l1-lecture-note.html b/academic/cis105/cis105-l1-lecture-note.html index add60e84..e7927d99 100644 --- a/academic/cis105/cis105-l1-lecture-note.html +++ b/academic/cis105/cis105-l1-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,7 +36,7 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lec. 1. ​

Author:Anda Toshiki
Updated:a minute ago
Words:971
Reading:6 min

Chapter 1: Everything Changes ​

1.1: The Historic Timelines ​

  • 1903: Wright bros, invented the airplane.

  • 1969: Armstrong walks on the moon.

  • 2004: SpaceX Falcon 9 lands upright.

  • 2015: Electreic plane crosses English Channel

  • 20xx: Google Live goes online.

  • Boeing 777, commercial airline airplanes normally cost approximately $200,000,000,000\$ 200,000,000,000 for a singular aircraft.

1.2: Analytical Engne ​

A mechanical computing device, was a special-purpose machine designed to tabulate logarithms and trigonometric functions by evaluating finite differences to create approximating polynomials.

1.3: Ada Lovelace ​

It tok the powerful insights of a mathematicial named Ada Lovelace to realize the true potentilal of the analytical engine. She was the first person to recognize that the machine could be used for more than pure calculations. She developed the first algorithm for the engine. It was the very first example of computer programming.

1.4: Information Technology is for People ​

  • Information technology (IT): the study, design, development, implementation, support, and management of computer-baed information systems, particularly software hardware.
    • People are the most important component in an information system because it is only a human who can conceive information from data. Computer system is only a machinery device and human is expected to operate towards aspects on how to perform, what to perform, what to outcome.
  • End-users (people) are what make computers start.
  • Attitude will be the defining factor in your success when it comes to computer competency.
  • Best practice: a management process, technique, or method that is most effective at arriving at desired outcome or better outcome than any other process, technique, or method.
    • The best and effective way of practice is through persistent and consistent training and reviewing.

1.5: Moore's Law ​

  • Gordon Moore: One of the founding fathers of the Intel.
  • Computer power doubles every eighteen months.
    • This essentially means that, if a business purchases a computer today, in eighteen months there will be a computer available that is twice as fast.
  • Being a Leader vs. Follower controversy

1.6: The Cuff Smartwatch? ​

US rapper/producer/entrepreneur Will.i.Am announced his foray into the world of wearable tech in 2014, proclaiming to have created a device so life-changing and futuristic it'd blow our archaic mind.

1.7: Types of Computer ​

  • Mainframe: Banks, Tech Companies
  • Midrange: Small to Mid-sized business, not as powerful.
  • Microcomputer: Laptops, Desktops, Tablet & Handled.

1.8: What is System Software? ​

  • System software vs. Operating system

  • System software vs. Application software

  • Operating system: Manages resources. Responsible for reading and writing data.

  • Graphical User Interface (GUI) & Command line interface.

    • The level of an operational system layers by interface visibility.

      • mermaid
        graph LR;
+    
        Toshiki's Note
        Appearance
        CIS105: Computer Applications & Information Systems Lec. 1. ​
         Author:Anda Toshiki
         Updated:a minute ago
         Words:971
         Reading:6 min
        Chapter 1: Everything Changes ​
        1.1: The Historic Timelines ​
        • 1903: Wright bros, invented the airplane.
        • 1969: Armstrong walks on the moon.
        • 2004: SpaceX Falcon 9 lands upright.
        • 2015: Electreic plane crosses English Channel
        • 20xx: Google Live goes online.
        • Boeing 777, commercial airline airplanes normally cost approximately $200,000,000,000\$ 200,000,000,000 for a singular aircraft.
        1.2: Analytical Engne ​
        A mechanical computing device, was a special-purpose machine designed to tabulate logarithms and trigonometric functions by evaluating finite differences to create approximating polynomials.
        1.3: Ada Lovelace ​
        It tok the powerful insights of a mathematicial named Ada Lovelace to realize the true potentilal of the analytical engine. She was the first person to recognize that the machine could be used for more than pure calculations. She developed the first algorithm for the engine. It was the very first example of computer programming.
        1.4: Information Technology is for People ​
        • Information technology (IT): the study, design, development, implementation, support, and management of computer-baed information systems, particularly software hardware. 
          • People are the most important component in an information system because it is only a human who can conceive information from data. Computer system is only a machinery device and human is expected to operate towards aspects on how to perform, what to perform, what to outcome.
        • End-users (people) are what make computers start.
        • Attitude will be the defining factor in your success when it comes to computer competency.
        • Best practice: a management process, technique, or method that is most effective at arriving at desired outcome or better outcome than any other process, technique, or method. 
          • The best and effective way of practice is through persistent and consistent training and reviewing.
        1.5: Moore's Law ​
        • Gordon Moore: One of the founding fathers of the Intel.
        • Computer power doubles every eighteen months. 
          • This essentially means that, if a business purchases a computer today, in eighteen months there will be a computer available that is twice as fast.
        • Being a Leader vs. Follower controversy
        1.6: The Cuff Smartwatch? ​
        US rapper/producer/entrepreneur Will.i.Am announced his foray into the world of wearable tech in 2014, proclaiming to have created a device so life-changing and futuristic it'd blow our archaic mind.
        1.7: Types of Computer ​
        • Mainframe: Banks, Tech Companies
        • Midrange: Small to Mid-sized business, not as powerful.
        • Microcomputer: Laptops, Desktops, Tablet & Handled.
        1.8: What is System Software? ​
        • System software vs. Operating system
        • System software vs. Application software
        • Operating system: Manages resources. Responsible for reading and writing data.
        • Graphical User Interface (GUI) & Command line interface.
          • The level of an operational system layers by interface visibility.
            • mermaid
              graph LR;
 a[GUI] --> b[Application software];
 b --> c[Operating system];
 c --> d[System software];
@@ -44,8 +44,8 @@
 a[GUI] --> b[Application software];
 b --> c[Operating system];
 c --> d[System software];
-d --> e[CLI]
        1.9: Operating Systems/Platforms ​
        • A collection of computer programs that work together to manage hardware and software to ensure that they are working properly.
          • Memory allocation
          • Administer input and output of information
          • File management
        • Platform (OS): Microsoft Windows, macOS, iOS, Linux, UNIX.
        2.0: Processes/Multitasking ​
        • Process: Any task a computer performs.
        • Multitasking: the execution of multiple processes over a certain period of time.
        2.1: Memory/Disk management ​
        • Operating systems: Cache, random access memory (ram), registers, virtual memory.
        • Primary storage vs. Secondary Storage: RAM, hard drive, file allocation table (FAT).
          • The table of contents of a hard drive, or the file system directory structure of a system.
          • Primary storage is accessed randomly which indicating the storage is only temporary without preserved.
          • Secondary storage is the permanent storage on a specific device while user information is stored on the machine within the physical drive storage.
        2.2: Business Computing Software ​
        • Software suite: Spreadsheets, word processors, database, and presentation software (aka application suite or productivity suite).
          • Sharing information between these applications is te mot important aspect of a software suite.
        • Hot keys/shortcuts: Ctrl+ C & Ctrl + V on Windows; Command+ C & Command + V on Windows.
        • Object linking and embedding (OLE) (Static)
          • In static linking, the system linker copies the dependencies into the final executable. At the time of linking an external library, the linker finds all dependencies that are defined in that library. And it replaces them with the corresponding functions from the library to resolve dependencies in our code. Afterward, the linker generates the final executable file that we can execute on the underlying machine.
            For example, let’s say our application calls the function print() from an external library named Library. The assembler generates the object file with all native symbols resolved to their memory addresses. The external reference print() cannot be resolved. The linker loads this library and finds the definition of print() in it. Then, it maps to print() to a memory location and thus resolves the dependency:
            Static Linking
            So, a statically linked file contains our program’s code as well as the code of all the libraries it invokes. Since we copy complete libraries, we need space on both the disk and in the main memory because the resulting file may be very large.
        • Object linking (Dynamic)
          • In dynamic linking, we copy the names of the external libraries into our final executable as unresolved symbols. We do the actual linking of these unresolved symbols only at runtime. How? When encountering an unresolved symbol, we query RAM for it. If the corresponding library isn’t loaded, the operating system loads it in the memory. So, the operating system performs dynamic linking for us by resolving each external symbol on the first muss. As a result, we load only a single copy of a library in memory and all processes use it.
        
-    
+d --> e[CLI]

1.9: Operating Systems/Platforms ​

  • A collection of computer programs that work together to manage hardware and software to ensure that they are working properly.
    • Memory allocation
    • Administer input and output of information
    • File management
  • Platform (OS): Microsoft Windows, macOS, iOS, Linux, UNIX.

2.0: Processes/Multitasking ​

  • Process: Any task a computer performs.
  • Multitasking: the execution of multiple processes over a certain period of time.

2.1: Memory/Disk management ​

  • Operating systems: Cache, random access memory (ram), registers, virtual memory.

  • Primary storage vs. Secondary Storage: RAM, hard drive, file allocation table (FAT).

    • The table of contents of a hard drive, or the file system directory structure of a system.

    • Primary storage is accessed randomly which indicating the storage is only temporary without preserved.

    • Secondary storage is the permanent storage on a specific device while user information is stored on the machine within the physical drive storage.

2.2: Business Computing Software ​

  • Software suite: Spreadsheets, word processors, database, and presentation software (aka application suite or productivity suite).

    • Sharing information between these applications is te mot important aspect of a software suite.

  • Hot keys/shortcuts: Ctrl+ C & Ctrl + V on Windows; Command+ C & Command + V on Windows.

  • Object linking and embedding (OLE) (Static)

    • In static linking, the system linker copies the dependencies into the final executable. At the time of linking an external library, the linker finds all dependencies that are defined in that library. And it replaces them with the corresponding functions from the library to resolve dependencies in our code. Afterward, the linker generates the final executable file that we can execute on the underlying machine.

      For example, let’s say our application calls the function print() from an external library named Library. The assembler generates the object file with all native symbols resolved to their memory addresses. The external reference print() cannot be resolved. The linker loads this library and finds the definition of print() in it. Then, it maps to print() to a memory location and thus resolves the dependency:

      Static Linking

      So, a statically linked file contains our program’s code as well as the code of all the libraries it invokes. Since we copy complete libraries, we need space on both the disk and in the main memory because the resulting file may be very large.

  • Object linking (Dynamic)

    • In dynamic linking, we copy the names of the external libraries into our final executable as unresolved symbols. We do the actual linking of these unresolved symbols only at runtime. How? When encountering an unresolved symbol, we query RAM for it. If the corresponding library isn’t loaded, the operating system loads it in the memory. So, the operating system performs dynamic linking for us by resolving each external symbol on the first muss. As a result, we load only a single copy of a library in memory and all processes use it.
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l10-lecture-note.html b/academic/cis105/cis105-l10-lecture-note.html index 5616e1a0..a5cfa72e 100644 --- a/academic/cis105/cis105-l10-lecture-note.html +++ b/academic/cis105/cis105-l10-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,10 +36,10 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 10 ​

Author:Anda Toshiki
Updated:a minute ago
Words:474
Reading:2 min

Chapter 10: Business Websites ​

10.1: Business Website Alternative ​

  • Web development platforms (WDP): Website creation programs that help non-web developers develop a website for a relatively low price.
    • Web.com, Wix.com, Squarespace.com
    • A great option for business owners
    • Search Engine Optimization

10.2: Early Website Technology ​

  • Who created Hypertext Markup Language (HTML)?
  • HTML is platform-neutral
  • What happens when you use the internet:
    • You don't visit websites, websites visit you
    • Copies of request kept in client's cache
mermaid
graph LR
+    
Toshiki's Note
Appearance
CIS105: Computer Applications & Information Systems Lect. 10 ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:474
 Reading:2 min
Chapter 10: Business Websites ​
10.1: Business Website Alternative ​
  • Web development platforms (WDP): Website creation programs that help non-web developers develop a website for a relatively low price. 
    • Web.com, Wix.com, Squarespace.com
    • A great option for business owners
    • Search Engine Optimization
10.2: Early Website Technology ​
  • Who created Hypertext Markup Language (HTML)?
  • HTML is platform-neutral
  • What happens when you use the internet: 
    • You don't visit websites, websites visit you
    • Copies of request kept in client's cache
mermaid
graph LR
 A[You click on hyperlink] --> B[Web server ses HTTP to send copy of request to client]
graph LR
-A[You click on hyperlink] --> B[Web server ses HTTP to send copy of request to client]
10.3: Hypertext Markup Language ​
  • The principal programming language for browsers
  • Elements: HTML represented in the form of tags that are enclosed by brackets: </>
  • <meta></meta>: Used to help search engines 
    • Will include info like web page's author, descriptions, and keywords
    • Called meta keywords or meta-words.
  • Link element: <a href="www.youtube.com">Youtube</a>
  • Image element: <img src="meadow.gif" alt="meadow"></img>
  • JavaScript: A client-side programming language used to build websites 
    • Requires a Java-enabled browser to work
    • Ex: Countdown timer
10.4: Websites Styles & Categories ​
  • Website categories: Personal, Commercial, Governmental, and intranet
    • Static Website (AKA Brochure websites): Content typically designed with HTML; never changes
    • Dynamic Websites: Generate web pages directly from the server depending on the context of the user 
      • Ex: Social Media
      • Much more difficult to program, but are much easier to maintain
10.5: Types of Websites ​
  • Blog: An online diary, originally called a web-log
  • Content Website: typically dispense original content and often supported by advertising
  • Corporate Website: Distribute info about a business
  • Commerce Website (AKA E-Commerce site): to sell goods and services online
  • Search Engine Website: Offers general information and intended to be the entryway to other websites
  • Gaming Website: Typically, games unto themselves; require plug-ins like Adobe Flash, JavaScript, and ActiveX
  • News Websites: Specifically devoted to providing news and commentary
  • School Websites: Typically used by teachers, students, parents, and administrators.
  • Video-sharing websites: Allow users to upload videos for a wider audience 
    • Utilizes Users Datagram Protocol (UDP)
10.6: Website Online Software ​
  • HTML is not a powerful programming language
  • Fat Client: Provides rich functionality on the client side; independent of a central server 
    • Ex: Microsoft programs
    • Pros: More functionality, very powerful
    • Cons: Expensive. Requires additional software and set up, requires more processing power
  • Thin Client: A client that depends on a server for computing processing 
    • Ex: Web browsers
    • Pros: Platform neutral, cheaper to maintain, better distribution of info
    • Cons: Not as powerful, less functionality, requires refreshing from the user
10.7: Online Advertising ​
  • Online Advertising: A form of sponsorship that uses the WWW for delivering business-marketing strategies
  • Examples: 
    • Search Engine Results
    • Native Advertising
    • Social Network Advertising
    • Banner advertisement
  • Purchasing Online Advertising 
    • Cost Per Click (CPC)
    • Cost Per Thousand Impressions (CPM)
    • Cost Per Action (CPA)

-    
+A[You click on hyperlink] --> B[Web server ses HTTP to send copy of request to client]

10.3: Hypertext Markup Language ​

  • The principal programming language for browsers
  • Elements: HTML represented in the form of tags that are enclosed by brackets: </>
  • <meta></meta>: Used to help search engines
    • Will include info like web page's author, descriptions, and keywords
    • Called meta keywords or meta-words.
  • Link element: <a href="www.youtube.com">Youtube</a>
  • Image element: <img src="meadow.gif" alt="meadow"></img>
  • JavaScript: A client-side programming language used to build websites
    • Requires a Java-enabled browser to work
    • Ex: Countdown timer

10.4: Websites Styles & Categories ​

  • Website categories: Personal, Commercial, Governmental, and intranet
    • Static Website (AKA Brochure websites): Content typically designed with HTML; never changes
    • Dynamic Websites: Generate web pages directly from the server depending on the context of the user
      • Ex: Social Media
      • Much more difficult to program, but are much easier to maintain

10.5: Types of Websites ​

  • Blog: An online diary, originally called a web-log
  • Content Website: typically dispense original content and often supported by advertising
  • Corporate Website: Distribute info about a business
  • Commerce Website (AKA E-Commerce site): to sell goods and services online
  • Search Engine Website: Offers general information and intended to be the entryway to other websites
  • Gaming Website: Typically, games unto themselves; require plug-ins like Adobe Flash, JavaScript, and ActiveX
  • News Websites: Specifically devoted to providing news and commentary
  • School Websites: Typically used by teachers, students, parents, and administrators.
  • Video-sharing websites: Allow users to upload videos for a wider audience
    • Utilizes Users Datagram Protocol (UDP)

10.6: Website Online Software ​

  • HTML is not a powerful programming language
  • Fat Client: Provides rich functionality on the client side; independent of a central server
    • Ex: Microsoft programs
    • Pros: More functionality, very powerful
    • Cons: Expensive. Requires additional software and set up, requires more processing power
  • Thin Client: A client that depends on a server for computing processing
    • Ex: Web browsers
    • Pros: Platform neutral, cheaper to maintain, better distribution of info
    • Cons: Not as powerful, less functionality, requires refreshing from the user

10.7: Online Advertising ​

  • Online Advertising: A form of sponsorship that uses the WWW for delivering business-marketing strategies
  • Examples:
    • Search Engine Results
    • Native Advertising
    • Social Network Advertising
    • Banner advertisement
  • Purchasing Online Advertising
    • Cost Per Click (CPC)
    • Cost Per Thousand Impressions (CPM)
    • Cost Per Action (CPA)
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l11-lecture-note.html b/academic/cis105/cis105-l11-lecture-note.html index 06fb5108..fe42faed 100644 --- a/academic/cis105/cis105-l11-lecture-note.html +++ b/academic/cis105/cis105-l11-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 11 ​

Author:Anda Toshiki
Updated:a minute ago
Words:406
Reading:2 min

Chapter 11: Computer Security ​

Midterm reminder: March 28th - 29th

  • Exam Review after Chapter 11KC
  • 40 new questions, 10 KC questions
    • Same format as last time

11.1: System Security & Computer Privacy ​

  • An unsecured computer is extremely easy to breach by any seasoned hacker
  • Computer Security: Concerned with risk management, confidentiality, integrity, and the availability of the electronic info processed and stored within a computing system
  • Risk Management: Includes the recognition, consequences, and assessment of risk to a computer's assets, and developing strategies to manage and protect them

11.2: Business System Threats ​

  • Intranet: A businesses' private version of the internet
  • The key aspect of an intranet is privacy
  • Firewall: Hardware and software working together to ensure that only authorized personnel can access the business' intranet

11.3: Firewall ​

  • Without a firewall, an intranet is just another part of the internet
  • Proxy Server: A special security computer that allows communication to flow in and out of an intranet to check for external threats
  • Two standard security practices for firewalls:
    • Default-Deny
    • Default-Allow

11.4: Malware Threats ​

  • The term "virus" comes from biology
  • Malware Motivations:
    • Experimental Malware
    • Worms
    • Trojan Horses
    • Spyware/Keylogging
    • Adware
    • Spamming
    • Denial-of-Service Attach (DoS)
      • The deadliest
    • Reverse Phishing - "Spear Phishing"

11.5: Malware Solutions ​

  • Antivirus Software
    • Norton AntiVirus
    • McAfee's VirusScan
  • System patches
  • Data Backup
  • Employee training

11.6: Internet Fraud ​

  • Internet fraud: Any fraudulent activity in an online setting
  • Google AdSense, Google's advertising network; the middleman between the advertiser and the website owner
  • Click Fraud: a program that automatically clicks on Ads
  • Purchase Scam: scammer asks seller to pay for shipping via credit card, cancels credit card after order has shipped
  • Most uncomplicated type of internet fraud
  • Phishing: Uses social engineering techniques to gain private information

11.7: Computer Privacy ​

  • Privacy: The ability of an individual to keep their personal information out of public view
  • Anonymous: One's personal identity Is not be known
  • Tracking Cookie: A small text file that tracks your interaction on a website
  • History files: A list of stored or cached websites on a user's computer

50% of the time an unsecured computer can be hacked, and a password can be guessed

11.8: Identity Theft ​

  • Identity theft: A crime concerning the unlawful practice of assuming another individual's identity
  • Ways to acquire personal information:
    • Shoulder Surfing
    • Dumpster Diving
  • SCAM
    • Stingy
    • Check
    • Ask
    • Maintain
  • ASU's Recommended Password Manager: Stache
    • Bitwarden, 1Password, Dashlane, Keeper, KeePass
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 11 ​

Author:Anda Toshiki
Updated:a minute ago
Words:406
Reading:2 min

Chapter 11: Computer Security ​

Midterm reminder: March 28th - 29th

  • Exam Review after Chapter 11KC
  • 40 new questions, 10 KC questions
    • Same format as last time

11.1: System Security & Computer Privacy ​

  • An unsecured computer is extremely easy to breach by any seasoned hacker
  • Computer Security: Concerned with risk management, confidentiality, integrity, and the availability of the electronic info processed and stored within a computing system
  • Risk Management: Includes the recognition, consequences, and assessment of risk to a computer's assets, and developing strategies to manage and protect them

11.2: Business System Threats ​

  • Intranet: A businesses' private version of the internet
  • The key aspect of an intranet is privacy
  • Firewall: Hardware and software working together to ensure that only authorized personnel can access the business' intranet

11.3: Firewall ​

  • Without a firewall, an intranet is just another part of the internet
  • Proxy Server: A special security computer that allows communication to flow in and out of an intranet to check for external threats
  • Two standard security practices for firewalls:
    • Default-Deny
    • Default-Allow

11.4: Malware Threats ​

  • The term "virus" comes from biology
  • Malware Motivations:
    • Experimental Malware
    • Worms
    • Trojan Horses
    • Spyware/Keylogging
    • Adware
    • Spamming
    • Denial-of-Service Attach (DoS)
      • The deadliest
    • Reverse Phishing - "Spear Phishing"

11.5: Malware Solutions ​

  • Antivirus Software
    • Norton AntiVirus
    • McAfee's VirusScan
  • System patches
  • Data Backup
  • Employee training

11.6: Internet Fraud ​

  • Internet fraud: Any fraudulent activity in an online setting
  • Google AdSense, Google's advertising network; the middleman between the advertiser and the website owner
  • Click Fraud: a program that automatically clicks on Ads
  • Purchase Scam: scammer asks seller to pay for shipping via credit card, cancels credit card after order has shipped
  • Most uncomplicated type of internet fraud
  • Phishing: Uses social engineering techniques to gain private information

11.7: Computer Privacy ​

  • Privacy: The ability of an individual to keep their personal information out of public view
  • Anonymous: One's personal identity Is not be known
  • Tracking Cookie: A small text file that tracks your interaction on a website
  • History files: A list of stored or cached websites on a user's computer

50% of the time an unsecured computer can be hacked, and a password can be guessed

11.8: Identity Theft ​

  • Identity theft: A crime concerning the unlawful practice of assuming another individual's identity
  • Ways to acquire personal information:
    • Shoulder Surfing
    • Dumpster Diving
  • SCAM
    • Stingy
    • Check
    • Ask
    • Maintain
  • ASU's Recommended Password Manager: Stache
    • Bitwarden, 1Password, Dashlane, Keeper, KeePass
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l12-lecture-note.html b/academic/cis105/cis105-l12-lecture-note.html index 285699ae..4af6a42b 100644 --- a/academic/cis105/cis105-l12-lecture-note.html +++ b/academic/cis105/cis105-l12-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 12 ​

Author:Anda Toshiki
Updated:a minute ago
Words:150
Reading:1 min

Chapter 12: Introduction to SQL ​

12.1: SQL Getting Started ​

  • Capture the data
    • Point of Sale, Scanning, etc
  • Get the data into a database
    • Import from other systems
    • Direct input from forms
  • Extract the data
    • SQL
  • Use the data
    • Analysis systems
    • Reporting systems
  • SQL: The language of relational databases
    • Structured query Language

If you plan to analyze data for a business, there is a good chance it's stored in a data warehouse, and you will need to learn how to retrieve it yourself.

12.2: SQL Statement ​

  • SELECT: What fields (column) you want
    • fieldname: If the name is used in only one table
  • FROM: What table or tables contain the fields
  • ORDER BY: Sorting
  • DISTINCT: Use after SELECT when you only want to show each vale of the selected field(s) once
  • LIMIT: Use when you want to limit the number of records produced
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 12 ​

Author:Anda Toshiki
Updated:a minute ago
Words:150
Reading:1 min

Chapter 12: Introduction to SQL ​

12.1: SQL Getting Started ​

  • Capture the data
    • Point of Sale, Scanning, etc
  • Get the data into a database
    • Import from other systems
    • Direct input from forms
  • Extract the data
    • SQL
  • Use the data
    • Analysis systems
    • Reporting systems
  • SQL: The language of relational databases
    • Structured query Language

If you plan to analyze data for a business, there is a good chance it's stored in a data warehouse, and you will need to learn how to retrieve it yourself.

12.2: SQL Statement ​

  • SELECT: What fields (column) you want
    • fieldname: If the name is used in only one table
  • FROM: What table or tables contain the fields
  • ORDER BY: Sorting
  • DISTINCT: Use after SELECT when you only want to show each vale of the selected field(s) once
  • LIMIT: Use when you want to limit the number of records produced
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l13-lecture-note.html b/academic/cis105/cis105-l13-lecture-note.html index 1ddbbc68..0ee32f85 100644 --- a/academic/cis105/cis105-l13-lecture-note.html +++ b/academic/cis105/cis105-l13-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 13 ​

Author:Anda Toshiki
Updated:a minute ago
Words:168
Reading:1 min

Chapter 13: Information Systems in Business ​

13.1: Business Functions ​

  • What is the definition of a business
    • Accounting
    • Human Resources
    • Marketing
    • Research and Development (R&D)
    • Production
    • Collaboration is key

13.2: The Accounting Department ​

  • "The mathematics of business"
  • Generally Accepted Accounting Principles (GAAP)
  • Sarbanes-Oxley Act of 2002 (SOX)

13.3: The Human Resources Department ​

  • Management of employees
  • What is the most important asset of any business?

13.4: The Marketing Department ​

  • Promotion of a business' products
  • The Four P's of Marketing - E. Jerome McCarthy
    • Products
    • Pricing
    • Promotion
    • Placement

13.5: The Research & Development Department ​

  • If a business isn't growing, it's dying
  • Creative and systematic approach
    • AKA Brainstorming
  • Is this idea feasible?

13.6: The Production Department ​

  • The makers of the business
  • Tangible vs. Intangible

13.7: Information System Collaboration ​

  • Electronic Computing Tols
    • Discussion Boards
    • Instant Messaging
    • Texting
  • Video Conferencing Software
    • Webinar
  • Project Management Software
    • Microsoft Project
    • Gantt Chart
    • System Development Life Cycle (SDLC)
    • Unified Modeling Language (UML)
  • Knowledge Management Systems
  • Online Application Software
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 13 ​

Author:Anda Toshiki
Updated:a minute ago
Words:168
Reading:1 min

Chapter 13: Information Systems in Business ​

13.1: Business Functions ​

  • What is the definition of a business
    • Accounting
    • Human Resources
    • Marketing
    • Research and Development (R&D)
    • Production
    • Collaboration is key

13.2: The Accounting Department ​

  • "The mathematics of business"
  • Generally Accepted Accounting Principles (GAAP)
  • Sarbanes-Oxley Act of 2002 (SOX)

13.3: The Human Resources Department ​

  • Management of employees
  • What is the most important asset of any business?

13.4: The Marketing Department ​

  • Promotion of a business' products
  • The Four P's of Marketing - E. Jerome McCarthy
    • Products
    • Pricing
    • Promotion
    • Placement

13.5: The Research & Development Department ​

  • If a business isn't growing, it's dying
  • Creative and systematic approach
    • AKA Brainstorming
  • Is this idea feasible?

13.6: The Production Department ​

  • The makers of the business
  • Tangible vs. Intangible

13.7: Information System Collaboration ​

  • Electronic Computing Tols
    • Discussion Boards
    • Instant Messaging
    • Texting
  • Video Conferencing Software
    • Webinar
  • Project Management Software
    • Microsoft Project
    • Gantt Chart
    • System Development Life Cycle (SDLC)
    • Unified Modeling Language (UML)
  • Knowledge Management Systems
  • Online Application Software
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l14-lecture-note.html b/academic/cis105/cis105-l14-lecture-note.html index d5ebd27a..eedb2117 100644 --- a/academic/cis105/cis105-l14-lecture-note.html +++ b/academic/cis105/cis105-l14-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 14 ​

Author:Anda Toshiki
Updated:a minute ago
Words:370
Reading:2 min

Chapter 14: More SQL Statement ​

14.1: Statement Review ​

  • SELECT: What fields (column) you want
    • fieldname: If the name is used in only one table
  • FROM: What table or tables contain the fields
  • ORDER BY: Sorting
  • DISTINCT: Use after SELECT when you only want to show each vale of the selected field(s) once
  • LIMIT: Use when you want to limit the number of records produced

14.2: New SQL Statements ​

  • WHERE: Applies conditions, filters

    • WHERE clause conditions follow a pattern: field + comparison operator + value

    • sql
      WHERE crime.iucr_no = 420
      WHERE crime.iucr_no = 420

  • Comparison Operators:

    • > : Returns all records larger than the specified value
    • >=: Returns all records that are at least as large as the specified value (including the value)
    • < : Returns all records smaller than the specified value
    • <=: Returns all records that are at least as small as the specified value (including the value)
    • = : Returns all records that are exactly equal to the specified value
    • <>: Returns all records that are not equal to the specified value

  • NULL: An empty value

  • NOT: Can be used to create a condition that reverses the logic of any condition

    • Similar logic to the NOT function in Excel

  • AND: Both conditions must be met for row to be included in query

  • OR: Used to combine criteria when we want our results to mach any criteria

  • BETWEEN: Shorthand way to include values in a range

  • IN: Shorthand way to include values in a range

  • LIKE: Use this statement when you aren't sure of the values in a record

  • %: any combination of characters

    • %Keyword: ends with keyword
    • %Keyword%: looks for %keyword% anywhere in the record
    • Keyword%: starts with keyword

  • - substitutes for any single character

    • 't_p' would return 'tip', 'tap', or 'top''; but not 'stop'

    • 'p_st" would return 'past"', but not 'paste'

    • 'h__s_n' would return 'hanson', 'hensen', or 'hansen'; but not 'harrison' (note that 'h%s_n' would return 'harrison')

14.3: Rules to Remember ​

  • To combine AND and OR statements, you must follow the order of operations (AND 1st, OR 2nd)
    • If OR must be first, use ()
  • All words must be in between ('_'), ("_")
    • If your value contains a quotation, use double quotation or a /
    • All #s is left as-is
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 14 ​

Author:Anda Toshiki
Updated:a minute ago
Words:370
Reading:2 min

Chapter 14: More SQL Statement ​

14.1: Statement Review ​

  • SELECT: What fields (column) you want
    • fieldname: If the name is used in only one table
  • FROM: What table or tables contain the fields
  • ORDER BY: Sorting
  • DISTINCT: Use after SELECT when you only want to show each vale of the selected field(s) once
  • LIMIT: Use when you want to limit the number of records produced

14.2: New SQL Statements ​

  • WHERE: Applies conditions, filters

    • WHERE clause conditions follow a pattern: field + comparison operator + value

    • sql
      WHERE crime.iucr_no = 420
      WHERE crime.iucr_no = 420

  • Comparison Operators:

    • > : Returns all records larger than the specified value
    • >=: Returns all records that are at least as large as the specified value (including the value)
    • < : Returns all records smaller than the specified value
    • <=: Returns all records that are at least as small as the specified value (including the value)
    • = : Returns all records that are exactly equal to the specified value
    • <>: Returns all records that are not equal to the specified value

  • NULL: An empty value

  • NOT: Can be used to create a condition that reverses the logic of any condition

    • Similar logic to the NOT function in Excel

  • AND: Both conditions must be met for row to be included in query

  • OR: Used to combine criteria when we want our results to mach any criteria

  • BETWEEN: Shorthand way to include values in a range

  • IN: Shorthand way to include values in a range

  • LIKE: Use this statement when you aren't sure of the values in a record

  • %: any combination of characters

    • %Keyword: ends with keyword
    • %Keyword%: looks for %keyword% anywhere in the record
    • Keyword%: starts with keyword

  • - substitutes for any single character

    • 't_p' would return 'tip', 'tap', or 'top''; but not 'stop'

    • 'p_st" would return 'past"', but not 'paste'

    • 'h__s_n' would return 'hanson', 'hensen', or 'hansen'; but not 'harrison' (note that 'h%s_n' would return 'harrison')

14.3: Rules to Remember ​

  • To combine AND and OR statements, you must follow the order of operations (AND 1st, OR 2nd)
    • If OR must be first, use ()
  • All words must be in between ('_'), ("_")
    • If your value contains a quotation, use double quotation or a /
    • All #s is left as-is
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l15-lecture-note.html b/academic/cis105/cis105-l15-lecture-note.html index fd608f97..2b3e239a 100644 --- a/academic/cis105/cis105-l15-lecture-note.html +++ b/academic/cis105/cis105-l15-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 15 ​

Author:Anda Toshiki
Updated:a minute ago
Words:169
Reading:1 min

Chapter 15: Business System Reporting ​

Final months schedule

  • Today 4/9: Chapter 15/16
  • Thursday 4/11: Final SQL Assessment
  • Tuesday 4/16; Final Lecture and Final Exam Review
  • Thursday 4/18: MKT
  • 4/23 & 4/26: Zoom Office Hours
  • Final Exam: 4/25 - 4/26
  • Final day to complete all missing assignments: 4/28

15.1: Business System Reporting ​

  • Defined as informaitonal output from an information system to facilitate decision-making.
  • Report are periodic in nature
    • Daily, weekly, monthly, etc
    • Demand / Ad Hoc Reports
  • Strategic vs. Tactical decisions
  • Three main aspects of reporting:
    • Reporting summarization levels
    • Analysis and predictive reporting
    • Management Level Reporting
  • Reporting Summarization Levels: The level of detail or summarization in the report
  • Transaction Processing System (TPS)
    • Uses databases to store data
    • Groundwork for all other reports
    • Highest level of detail
  • Analysis and Predictive Reporting
    • Use TPS as a foundation
    • Trend reporting
    • Graphical repors
    • Budget, forecast, and variance reports
  • Management Level reporting
    • Top-level executive reporting
    • Management reporting
    • Supervisory reporting

​

- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 15 ​

Author:Anda Toshiki
Updated:a minute ago
Words:169
Reading:1 min

Chapter 15: Business System Reporting ​

Final months schedule

  • Today 4/9: Chapter 15/16
  • Thursday 4/11: Final SQL Assessment
  • Tuesday 4/16; Final Lecture and Final Exam Review
  • Thursday 4/18: MKT
  • 4/23 & 4/26: Zoom Office Hours
  • Final Exam: 4/25 - 4/26
  • Final day to complete all missing assignments: 4/28

15.1: Business System Reporting ​

  • Defined as informaitonal output from an information system to facilitate decision-making.
  • Report are periodic in nature
    • Daily, weekly, monthly, etc
    • Demand / Ad Hoc Reports
  • Strategic vs. Tactical decisions
  • Three main aspects of reporting:
    • Reporting summarization levels
    • Analysis and predictive reporting
    • Management Level Reporting
  • Reporting Summarization Levels: The level of detail or summarization in the report
  • Transaction Processing System (TPS)
    • Uses databases to store data
    • Groundwork for all other reports
    • Highest level of detail
  • Analysis and Predictive Reporting
    • Use TPS as a foundation
    • Trend reporting
    • Graphical repors
    • Budget, forecast, and variance reports
  • Management Level reporting
    • Top-level executive reporting
    • Management reporting
    • Supervisory reporting

​

+ \ No newline at end of file diff --git a/academic/cis105/cis105-l16-lecture-note.html b/academic/cis105/cis105-l16-lecture-note.html index d15d935e..15262282 100644 --- a/academic/cis105/cis105-l16-lecture-note.html +++ b/academic/cis105/cis105-l16-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 16 ​

Author:Anda Toshiki
Updated:a minute ago
Words:396
Reading:2 min

Chapter 16: Information Technology Careers ​

16.1: IT Careers Intro ​

  • Information Technology (IT) is everywhere
  • A common axiom for IT professionals is "know the business"
  • Value-Added: Refers to the added value of a product or service over the cost of products and services.

16.2: Business IT Perception ​

  • IT department are responsible for servicing and enhancing all computing, technical assets, and intellectual property of a business organization
    • Service IT
    • IT consulting
  • IT is the glue that holds organizations together

16.3: Information Technology Jobs ​

  • Chief Information Officer (CIO): Te board-level head of Information Technology departments
    • AKA Chief Technology Officer (CTO)
    • Replaced 'IT Director' title
    • Highest paid technology position
  • Head Application Developers In charge of software application development teams that develop, design, program and test software applications.
  • System Development Life Cycle (SDLC): A process for creating and altering computer applications through an approach that includes planning, system analysis, design, implementation, and system maintenance
  • Unified Modeling Language (UML): set of graphical notation techniques to create computer models and applications. It could be said that UML is a "blueprint" for developing applications
    • Application Architects: design different parts of applications
    • Middleware
    • Programming Infrastructure

16.4: Information Technology Jobs Contd. ​

  • Database Administrators (DBAs): the wizards of databases and data requirements
  • Network Administrators
  • Senior Web Developers
  • Business Intelligence Analyst
  • Information Technology Staff Consultants
  • IT Auditors

All IT jobs require experience, a degree, or both*

16.5: Enterprise Resource Planning Solution ​

  • Enterprise Resource Planning (ERP): an overall software solution that attempts to get departments on the same page, allowing them to share information and more easily collaborate with each other

    • Advantages

      • Increase production, collab, & efficiency

      • Revenue cycles and order tracking more easily available; enhances decision making

    • Disadvantages

      • Must re-engineer own processes to fit ERP solutions
      • Expensive
      • Must have process for tracking new entries to prevent inaccurate data from entering the database
      • Difficult to modify

16.6: Unified Modeling Language ​

  • Use case scenarios: "Who" can do "what"
  • Object-Oriented Analysis (OOA): Is focused on what a computer system does
  • Object Oriented Design (OOD): Focused on how a computer system what it does
  • Sequence Diagrams: Illustrations that sow how computer processes related and interact with each other and in what order.

16.7: System Development Methodologies ​

  • System Development Life Cycle
    • Initiation
    • Concept Development
    • Planing
    • Requirements
    • Design
    • Development
    • Testing
    • Implementation
  • Rapid Application Development (RAD)
    • Develope as you go
  • Software Prototyping
    • Prototypes AKA vaporware
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 16 ​

Author:Anda Toshiki
Updated:a minute ago
Words:396
Reading:2 min

Chapter 16: Information Technology Careers ​

16.1: IT Careers Intro ​

  • Information Technology (IT) is everywhere
  • A common axiom for IT professionals is "know the business"
  • Value-Added: Refers to the added value of a product or service over the cost of products and services.

16.2: Business IT Perception ​

  • IT department are responsible for servicing and enhancing all computing, technical assets, and intellectual property of a business organization
    • Service IT
    • IT consulting
  • IT is the glue that holds organizations together

16.3: Information Technology Jobs ​

  • Chief Information Officer (CIO): Te board-level head of Information Technology departments
    • AKA Chief Technology Officer (CTO)
    • Replaced 'IT Director' title
    • Highest paid technology position
  • Head Application Developers In charge of software application development teams that develop, design, program and test software applications.
  • System Development Life Cycle (SDLC): A process for creating and altering computer applications through an approach that includes planning, system analysis, design, implementation, and system maintenance
  • Unified Modeling Language (UML): set of graphical notation techniques to create computer models and applications. It could be said that UML is a "blueprint" for developing applications
    • Application Architects: design different parts of applications
    • Middleware
    • Programming Infrastructure

16.4: Information Technology Jobs Contd. ​

  • Database Administrators (DBAs): the wizards of databases and data requirements
  • Network Administrators
  • Senior Web Developers
  • Business Intelligence Analyst
  • Information Technology Staff Consultants
  • IT Auditors

All IT jobs require experience, a degree, or both*

16.5: Enterprise Resource Planning Solution ​

  • Enterprise Resource Planning (ERP): an overall software solution that attempts to get departments on the same page, allowing them to share information and more easily collaborate with each other

    • Advantages

      • Increase production, collab, & efficiency

      • Revenue cycles and order tracking more easily available; enhances decision making

    • Disadvantages

      • Must re-engineer own processes to fit ERP solutions
      • Expensive
      • Must have process for tracking new entries to prevent inaccurate data from entering the database
      • Difficult to modify

16.6: Unified Modeling Language ​

  • Use case scenarios: "Who" can do "what"
  • Object-Oriented Analysis (OOA): Is focused on what a computer system does
  • Object Oriented Design (OOD): Focused on how a computer system what it does
  • Sequence Diagrams: Illustrations that sow how computer processes related and interact with each other and in what order.

16.7: System Development Methodologies ​

  • System Development Life Cycle
    • Initiation
    • Concept Development
    • Planing
    • Requirements
    • Design
    • Development
    • Testing
    • Implementation
  • Rapid Application Development (RAD)
    • Develope as you go
  • Software Prototyping
    • Prototypes AKA vaporware
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l17-lecture-note.html b/academic/cis105/cis105-l17-lecture-note.html new file mode 100644 index 00000000..32c6328d --- /dev/null +++ b/academic/cis105/cis105-l17-lecture-note.html @@ -0,0 +1,49 @@ + + + + + + CIS105: Computer Applications & Information Systems Lect. 17 | Toshiki's Note + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 17 ​

Author:Anda Toshiki
Updated:a minute ago
Words:131
Reading:1 min

Chapter 17: SQL Clauses: JOIN Query ​

17.1: JOIN & ON Clauses ​

  • JOIN: Join tables together to connect multiple fields into 1 query.
  • ON: Linking the primary and foreign key
  • Linking formula: FROM table1 + JOIN table2 ON table1.primarykey = table2.foreignkey
sql
SELECT first_name, last_name, link_to_major, major_id, major_name
+	FROM member
+	JOIN major
+		ON member. link_to_major = major.major_id
SELECT first_name, last_name, link_to_major, major_id, major_name
+	FROM member
+	JOIN major
+		ON member. link_to_major = major.major_id

17.2: SQL Statements: Primary vs. Foreign Keys ​

  • Primary key: A field on a database table that uniquely identifies each row of the table.
  • Foreign key: is used to combine the rows from one table wit the rows of another table.

17.3: SQL Statement: Date & Time Formatting ​

  • The common conventional format of SQL date & time searching is as followed, in form of YEAR-MONTH-DATT12:00:00

    • sql
      '2022-04-17T12:00:00'
      '2022-04-17T12:00:00'
+ + + + \ No newline at end of file diff --git a/academic/cis105/cis105-l2-lecture-note.html b/academic/cis105/cis105-l2-lecture-note.html index b7c474a7..6a8e7c17 100644 --- a/academic/cis105/cis105-l2-lecture-note.html +++ b/academic/cis105/cis105-l2-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:336
Reading:2 min

Chapter 2: Application Software ​

  • An application program is a computer program designed to carry out a specific task other than one relating to the operation of the computer itself.

2.1: Word Processing Software ​

  • For creating, updating and editing documents
  • Can create a Table of Contents
  • Saved to secondary memory
  • Often underestimated
  • Microsoft Word: Most popular word processor in the world
  • Adobe Acrobat: Most portable document format (pdf)
    • Platform Neutral
  • Corel WordPerfect: most dominant in early 80s (still used in some industries like law)
    • Created and Brigham Young university

2.2: Spreadsheets ​

Microsoft Excel: Most popular spreadsheet application (provides "what-if" analysis)

  • Interface of information in a grid form
  • Separated by columns and rows
    • Intersection of both called a cell
  • Often for financial calculations
  • Supports graphing
  • Replaced Lotus 1-2-3 as industry standard in 1993

2.3: Database Software ​

Database: The heavy-lifting application

  • Well thought out collection of files
  • Consists of records (rows)
  • Separated by fields (columns)
  • Can be queried
  • Often called a Database Management System or simply referred as DBMS
  • One-to-many relationship

2.4: Presentation Software ​

  • Displays information
  • Often a slide show and on-screen
  • Easy to build
  • Popularized by Business Intelligence
  • Allows users to:
    • Edit, update, insert, and delete text
    • Include graphics, video, and hypertext

2.5: Browser Software ​

  • User interface software that allows the user to display web pages found on the World Wide Web
  • Browsers display hyperlinks that are clickable navigation elements
  • Microsoft Edge, Google Chroe, Mozilla Firefox, and Apple's Safari
  • The first browser was called WorldWideWeb (no spaces)
  • Sir Tim Berners-Lee was the inventor of the first web browser

2.6: Networks ​

  • A computer network is two or more computers connected together for resource sharing and communication.
  • Resources: Computer files, folder, software
  • Peripheral hardware: Printers, scanners, webcam, etc.

The advantages of a computer network over a stand-alone computer are so significant that business cannot compete effectively in the marketplace without a network of some kind, even if the business is a sole proprietorship.

- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:336
Reading:2 min

Chapter 2: Application Software ​

  • An application program is a computer program designed to carry out a specific task other than one relating to the operation of the computer itself.

2.1: Word Processing Software ​

  • For creating, updating and editing documents
  • Can create a Table of Contents
  • Saved to secondary memory
  • Often underestimated
  • Microsoft Word: Most popular word processor in the world
  • Adobe Acrobat: Most portable document format (pdf)
    • Platform Neutral
  • Corel WordPerfect: most dominant in early 80s (still used in some industries like law)
    • Created and Brigham Young university

2.2: Spreadsheets ​

Microsoft Excel: Most popular spreadsheet application (provides "what-if" analysis)

  • Interface of information in a grid form
  • Separated by columns and rows
    • Intersection of both called a cell
  • Often for financial calculations
  • Supports graphing
  • Replaced Lotus 1-2-3 as industry standard in 1993

2.3: Database Software ​

Database: The heavy-lifting application

  • Well thought out collection of files
  • Consists of records (rows)
  • Separated by fields (columns)
  • Can be queried
  • Often called a Database Management System or simply referred as DBMS
  • One-to-many relationship

2.4: Presentation Software ​

  • Displays information
  • Often a slide show and on-screen
  • Easy to build
  • Popularized by Business Intelligence
  • Allows users to:
    • Edit, update, insert, and delete text
    • Include graphics, video, and hypertext

2.5: Browser Software ​

  • User interface software that allows the user to display web pages found on the World Wide Web
  • Browsers display hyperlinks that are clickable navigation elements
  • Microsoft Edge, Google Chroe, Mozilla Firefox, and Apple's Safari
  • The first browser was called WorldWideWeb (no spaces)
  • Sir Tim Berners-Lee was the inventor of the first web browser

2.6: Networks ​

  • A computer network is two or more computers connected together for resource sharing and communication.
  • Resources: Computer files, folder, software
  • Peripheral hardware: Printers, scanners, webcam, etc.

The advantages of a computer network over a stand-alone computer are so significant that business cannot compete effectively in the marketplace without a network of some kind, even if the business is a sole proprietorship.

+ \ No newline at end of file diff --git a/academic/cis105/cis105-l3-lecture-note.html b/academic/cis105/cis105-l3-lecture-note.html index 3ed1872c..77b29a28 100644 --- a/academic/cis105/cis105-l3-lecture-note.html +++ b/academic/cis105/cis105-l3-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 3 ​

Author:Anda Toshiki
Updated:a minute ago
Words:582
Reading:3 min

Chapter 3: Computer Hardware ​

3.1: Technical Terminologies ​

  • System Unit: Main body of the computer that contains a motherboard.
  • Motherboard (AKA Circuit Board): Main component of a system unit; a compleex array of electronics that connect and help different components of the computer communicate with each other.
    • PC: Motherboards, Mac; Logic Boards
  • Chassis (AKA Case or Box): Case to enclose the main components of a computer
  • Microprocessor: The brains of the computer
    • Central processing unit (CPU): Interprets program instructions and processes data by performing arithmetic and logical operations.

3.2: Central Processing Unit (CPU) ​

  • Speed is directly, but not solely, related to the CPU
  • Measured in Clock Rate
    • The number of cycles per second, that a computer can perform its most basic task
  • RSIC (Reduced Instruction Set Computer): Many chips encased into one chip
  • Bus Lines; Pathways that transfer data and power between components inside of a computer.
Architecture Diagram of CPUActual CPU Image
Architecture of the central processing unit (CPU) - Computer ...What Is a CPU? a Guide to Your Computer's 'Brain'

3.3 Power Supply Unit (PSU) ​

  • Supplies electricity
  • Converts 100-120 volts or 220-240 volts of alternating current (AC) to a lower voltage direct current (DC) that can be used by the internal components of the system unit
  • Different currents used in different parts of the world.

3.4: Primary and Secondary Storage ​

  • Primary: The workbench
    • Random Access Memory (RAM): Primary storage
  • Secondary: The storage for all your tools and supplies
    • Hard Drive: Secondary storage
  • ROM Chips (Read-Only Memory): Preprogrammed chips that serve specialized internal tasks. No human intervention
    • AKA Firmware

3.4.1: Secondary Storage Contd. ​

  • Internal vs. External Storage
  • Hard Drivers vs. Solid State Drivers (SSDs)
  • Impractical for a computer to be stand-alone
  • Important to consider business needs
    • Bad IT can make or break a business

3.5: Binary Number System ​

  • Computer only understands one language: Machine code or machine language

  • 1 or 0 (on or off)

  • 1s or 0s are referred to as bits (short for binary digits)

  • 8 bits become a byte

    • Byte 8 bits
    • Kilob

  • American Standard Code for Information Interchange (ASCII): The coding scheme that most microcomputer use to represent bytes.

    • NameEqual ToSize (In Bytes)
      Bit1 Bit1/8
      Nibble4 Bits1/2 (rare)
      Byte8 Bits1
      Kilobyte1024 Bytes1024
      Megabyte1, 024 Kilobytes1, 048, 576
      Gigabyte1, 024 Megabytes1, 073, 741, 824
      Terrabyte1, 024 Gigabytes1, 099, 511, 627, 776
      Petabyte1, 024 Terabytes1, 125, 899, 906, 842, 624
      Exabyte1, 024 Petabytes1, 152, 921, 504, 606, 846, 976
      Zettabyte1, 024 Exabytes1, 180, 591, 620, 717, 411, 303, 424
      Yottabyte1, 024 Zettabytes1, 208, 925, 819, 614, 629, 174, 706, 176

3.6: Peripheral Devices ​

  • Input v. Output
  • Input device: keywords, mice, touchpad, stylus, speakers, microphone, digital camera, etc/
  • Optical Character Recognition (OCR): Converting printed text to digital text
  • Radio Frequency Identification (RFID): An input mechanism that can be used to label a product for identification and have the product's information transmitted through radio waves.

3.7: Output Devices Contd. ​

  • Monitor: A series of transistors that translate machine code into text and images./
  • Pixels: A single dot on a graphic or text image
  • Resolution: The number of pixels inside a defined dimension on a monitor, commonly referred to as dots per inch (dpi).
    • Resolution is the most important feature of a monitor.
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 3 ​

Author:Anda Toshiki
Updated:a minute ago
Words:582
Reading:3 min

Chapter 3: Computer Hardware ​

3.1: Technical Terminologies ​

  • System Unit: Main body of the computer that contains a motherboard.
  • Motherboard (AKA Circuit Board): Main component of a system unit; a compleex array of electronics that connect and help different components of the computer communicate with each other.
    • PC: Motherboards, Mac; Logic Boards
  • Chassis (AKA Case or Box): Case to enclose the main components of a computer
  • Microprocessor: The brains of the computer
    • Central processing unit (CPU): Interprets program instructions and processes data by performing arithmetic and logical operations.

3.2: Central Processing Unit (CPU) ​

  • Speed is directly, but not solely, related to the CPU
  • Measured in Clock Rate
    • The number of cycles per second, that a computer can perform its most basic task
  • RSIC (Reduced Instruction Set Computer): Many chips encased into one chip
  • Bus Lines; Pathways that transfer data and power between components inside of a computer.
Architecture Diagram of CPUActual CPU Image
Architecture of the central processing unit (CPU) - Computer ...What Is a CPU? a Guide to Your Computer's 'Brain'

3.3 Power Supply Unit (PSU) ​

  • Supplies electricity
  • Converts 100-120 volts or 220-240 volts of alternating current (AC) to a lower voltage direct current (DC) that can be used by the internal components of the system unit
  • Different currents used in different parts of the world.

3.4: Primary and Secondary Storage ​

  • Primary: The workbench
    • Random Access Memory (RAM): Primary storage
  • Secondary: The storage for all your tools and supplies
    • Hard Drive: Secondary storage
  • ROM Chips (Read-Only Memory): Preprogrammed chips that serve specialized internal tasks. No human intervention
    • AKA Firmware

3.4.1: Secondary Storage Contd. ​

  • Internal vs. External Storage
  • Hard Drivers vs. Solid State Drivers (SSDs)
  • Impractical for a computer to be stand-alone
  • Important to consider business needs
    • Bad IT can make or break a business

3.5: Binary Number System ​

  • Computer only understands one language: Machine code or machine language

  • 1 or 0 (on or off)

  • 1s or 0s are referred to as bits (short for binary digits)

  • 8 bits become a byte

    • Byte 8 bits
    • Kilob

  • American Standard Code for Information Interchange (ASCII): The coding scheme that most microcomputer use to represent bytes.

    • NameEqual ToSize (In Bytes)
      Bit1 Bit1/8
      Nibble4 Bits1/2 (rare)
      Byte8 Bits1
      Kilobyte1024 Bytes1024
      Megabyte1, 024 Kilobytes1, 048, 576
      Gigabyte1, 024 Megabytes1, 073, 741, 824
      Terrabyte1, 024 Gigabytes1, 099, 511, 627, 776
      Petabyte1, 024 Terabytes1, 125, 899, 906, 842, 624
      Exabyte1, 024 Petabytes1, 152, 921, 504, 606, 846, 976
      Zettabyte1, 024 Exabytes1, 180, 591, 620, 717, 411, 303, 424
      Yottabyte1, 024 Zettabytes1, 208, 925, 819, 614, 629, 174, 706, 176

3.6: Peripheral Devices ​

  • Input v. Output
  • Input device: keywords, mice, touchpad, stylus, speakers, microphone, digital camera, etc/
  • Optical Character Recognition (OCR): Converting printed text to digital text
  • Radio Frequency Identification (RFID): An input mechanism that can be used to label a product for identification and have the product's information transmitted through radio waves.

3.7: Output Devices Contd. ​

  • Monitor: A series of transistors that translate machine code into text and images./
  • Pixels: A single dot on a graphic or text image
  • Resolution: The number of pixels inside a defined dimension on a monitor, commonly referred to as dots per inch (dpi).
    • Resolution is the most important feature of a monitor.
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l4-lecture-note.html b/academic/cis105/cis105-l4-lecture-note.html index 968ff277..43bc2b24 100644 --- a/academic/cis105/cis105-l4-lecture-note.html +++ b/academic/cis105/cis105-l4-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 4 ​

Author:Anda Toshiki
Updated:a minute ago
Words:643
Reading:4 min

Chapter 4: Formulas and Functions ​

4.1: References and Calculations ​

  • Calculations
    • Add (+), Subtract (-), Multiply (*), Divide (/)
    • Add: A1+A2
    • Subtract: A1-A2
    • Multiply: A1*A2
    • Divide: A1/A2
  • Always end typing by pressing Enter
  • Al formulas start with an equals (=) sign

4.2: References and Calculations ​

  • References: When we perform calculations in Excel, we often reference the value stored in other cells in our worksheet

  • Three ays to reference a cell:

    • Relative reference - A1
    • Mixed reference - $A1 or A$1
    • Absolute reference - $A$1

  • Relative Reference: An address or pointer that changes when the target item is moved or the relationship to it has changed.

    • RELATIVE REFERENCE IS THE DEFALT REFERENCE IN EXCEL.
    • The reference changes when the formula is copied elsewhere.
    • Excel sees the location of the cells relative to the location of the formula.
    • Click Fn + F4 as hotkeys to construct value as relative value

  • Absolute Reference: Will make either the row or column "constant" in other words, the ABSOLUTE cell location is LOCKED

    • To create an absolute cell reference, put a dollar sign ``$` symbol in front of the part of the reference that you want to remain constant.
  • ReferenceComments
    A1Both the column and row references are "relative" and will change when the reference is copied and pasted to other cell.
    $A1The column reference is "absolute" and will remain constant when copied and pasted to other cells. The row reference is "relative" and will change when copied and pasted to cells in other rows of the worksheet
    A$1The column reference is "relative" and will change when copied and pasted to cells in other columns in the worksheet. The row reference is "absolute" and will remain constant when copied and pasted to other cells.
    $A$1Both the column and row references are "absolute" and will remain constant when the reference is copied and pated to other cells.

4.3: Summary Statistics Functions ​

  • Formula vs. Function
    • Formula: Any calculation in excel
    • Function: A pre-defined calculation
  • To perform calculations in Excel, we often reference the values stored in other cells in our worksheets. We reference the cell location, not the value in the cell.
  • COUNT(value1, [value2]), ...): Counts the number of cells in that contain numbers.
  • COUNTA(value1, [value2], ...): Counts the number of cells in a range of cells that are not blank.
  • AVERAGE(number1, [number2], ...): Calculates the simple average of a set of numbers.
  • MAX(number1, [number2], ...): Returns the largest value in a set of numbers.
  • MIN(number1, [number2], ...): Returns the smallest value in a set of numbers.

4.4: Financial Function ​

  • RATE(nper, pmt, pv, [fv], [type], [guess]) : calculates the interest rate earned for an investment given the number of payments made as part of the investment, the payment amount, and the current value of the investment.
  • EFFECT(nominal _rate, pery): calculates the annual percentage rate for an interest rate given the number of times per year that interest is charged.
  • NPER(rate, pmt, pv, [fv], [type]): calculates the number of payments that will be made to pay off a loan given the interest rate, payment amount, and original loan amount.
  • PMT(rate, per, pv, [fv], [type]): calculates the payment amount for a loan given the interest rate, number of payments to be made to pay off the loan, and the original loan amount.
  • PV(rate, per, pmt, [fv], (type)): calculates the current value (accounting for compounding interest) of an investment given the interest rate, number of payments to be made, and the amount of the payment.
  • FV(rate, per, pmt, [pv], [type]): calculates the future value of an investment given the interest rate, number of payments to be made, and the amount of the payment.
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 4 ​

Author:Anda Toshiki
Updated:a minute ago
Words:643
Reading:4 min

Chapter 4: Formulas and Functions ​

4.1: References and Calculations ​

  • Calculations
    • Add (+), Subtract (-), Multiply (*), Divide (/)
    • Add: A1+A2
    • Subtract: A1-A2
    • Multiply: A1*A2
    • Divide: A1/A2
  • Always end typing by pressing Enter
  • Al formulas start with an equals (=) sign

4.2: References and Calculations ​

  • References: When we perform calculations in Excel, we often reference the value stored in other cells in our worksheet

  • Three ays to reference a cell:

    • Relative reference - A1
    • Mixed reference - $A1 or A$1
    • Absolute reference - $A$1

  • Relative Reference: An address or pointer that changes when the target item is moved or the relationship to it has changed.

    • RELATIVE REFERENCE IS THE DEFALT REFERENCE IN EXCEL.
    • The reference changes when the formula is copied elsewhere.
    • Excel sees the location of the cells relative to the location of the formula.
    • Click Fn + F4 as hotkeys to construct value as relative value

  • Absolute Reference: Will make either the row or column "constant" in other words, the ABSOLUTE cell location is LOCKED

    • To create an absolute cell reference, put a dollar sign ``$` symbol in front of the part of the reference that you want to remain constant.
  • ReferenceComments
    A1Both the column and row references are "relative" and will change when the reference is copied and pasted to other cell.
    $A1The column reference is "absolute" and will remain constant when copied and pasted to other cells. The row reference is "relative" and will change when copied and pasted to cells in other rows of the worksheet
    A$1The column reference is "relative" and will change when copied and pasted to cells in other columns in the worksheet. The row reference is "absolute" and will remain constant when copied and pasted to other cells.
    $A$1Both the column and row references are "absolute" and will remain constant when the reference is copied and pated to other cells.

4.3: Summary Statistics Functions ​

  • Formula vs. Function
    • Formula: Any calculation in excel
    • Function: A pre-defined calculation
  • To perform calculations in Excel, we often reference the values stored in other cells in our worksheets. We reference the cell location, not the value in the cell.
  • COUNT(value1, [value2]), ...): Counts the number of cells in that contain numbers.
  • COUNTA(value1, [value2], ...): Counts the number of cells in a range of cells that are not blank.
  • AVERAGE(number1, [number2], ...): Calculates the simple average of a set of numbers.
  • MAX(number1, [number2], ...): Returns the largest value in a set of numbers.
  • MIN(number1, [number2], ...): Returns the smallest value in a set of numbers.

4.4: Financial Function ​

  • RATE(nper, pmt, pv, [fv], [type], [guess]) : calculates the interest rate earned for an investment given the number of payments made as part of the investment, the payment amount, and the current value of the investment.
  • EFFECT(nominal _rate, pery): calculates the annual percentage rate for an interest rate given the number of times per year that interest is charged.
  • NPER(rate, pmt, pv, [fv], [type]): calculates the number of payments that will be made to pay off a loan given the interest rate, payment amount, and original loan amount.
  • PMT(rate, per, pv, [fv], [type]): calculates the payment amount for a loan given the interest rate, number of payments to be made to pay off the loan, and the original loan amount.
  • PV(rate, per, pmt, [fv], (type)): calculates the current value (accounting for compounding interest) of an investment given the interest rate, number of payments to be made, and the amount of the payment.
  • FV(rate, per, pmt, [pv], [type]): calculates the future value of an investment given the interest rate, number of payments to be made, and the amount of the payment.
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l5-lecture-note.html b/academic/cis105/cis105-l5-lecture-note.html index 6d91b560..4ede389d 100644 --- a/academic/cis105/cis105-l5-lecture-note.html +++ b/academic/cis105/cis105-l5-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 5 ​

Author:Anda Toshiki
Updated:a minute ago
Words:602
Reading:3 min

Chapter 5: Operating System ​

5.1: Operating System Fundamentals ​

  • 1980: Bill Gates and Paul Allen leased IBM a Personal Computer (PC) called MS-DOS (Microsoft Disk Operating System) for a one-time fee of $50,000\$50,000.
    • Gates Kept the copyright for MS-DOS
  • 1984: Steve Jobs and Steve Wozniak created Mac OS
    • Advertised on "More user-friendly"

5.2: Understanding Operating Systems ​

  • System software: Software that automatically runs to configure a computer (no human intervention)
  • Operating system (AKA Platform): A collection of computer programs that administer the hardware and software of a computer so that they work properly
    • Microsoft Windows, Mac OS, UNIX, Linux
  • Application software cannot run or be installed without a operating system.

5.3: Memory Management ​

  • Operating system coordinate: Computers memory

    • Cache, Random Access Memory (RAM), registers, and virtual memory
    • Disk Storage

  • CPU Cache: A small, very quick memory that stores copies of information and data from the most regularly-used parts of main memory

    • Helps decrease the time it takes to access memory.

  • Random Access Memory (RAM): Volatile memory

  • Disk Storage: Non-volatile memory

    • volatile memory stores data when a computer is on but erases it as soon as the computer is switched off, whereas non-volatile memory remains in a computer even after the system shuts off.

5.4: Memory Management Contd. ​

  • Registers: Small memory locations used to quicken the implementation of computer programs
    • Customization towards the device including wallpaper setting which is referred the process of registering.
  • Virtual Memory: Memory the OS uses to coordinate, track and efficiently allocate the use of CPU cache, registers, RAM and disk storage.
    • The messenger between volatile and non-volatile storage.

5.5: Disk Management ​

  • OS is responsible for saving information as efficiently as possible
    • Creates a table of contents of FAT (File Allocation Table)
  • Defragmentation: A program the reorganizes the hard drive
    • Note required for SSD
  • Hard Disk Drive (HDD)
    • Pros: More affordable, good for standard business procedures
    • Cons: more prone to damage
  • Solid State Drives (SSD)
    • Pros: N moving parts, more reliable
    • Cons: More expensive.

5.6: File Management ​

  • File manager are made up of Hierarchy of Directories, more commonly referred to as folders
    • Can be compared to a physical storage

5.7: Networks ​

  • Network: 2 or more computers connected together
  • Transmission control protocol and internet protocol (TCP/IP): A set of rules for transferring information from one computer to another
    • Transfers info by breaking up files into packates
  • The internet is the world's largest network

5.8: Device Drivers ​

  • Device Driver: Allows the OS to communicate with peripheral devices
  • Modern OS have plug-and-play capability
  • 1996: USB developed and considered industry standard

5.9: embedded Operating Systems ​

  • Embedded Operating System: OS installed on devices and appliances that rely on computing
    • Ex; Cash registers, automatic teller machines (ATM), microwaves, and specialized handled devices
    • Embedded OS cannot be modified in most cases
    • Embedded OS used on most cell pones
      • Google, Android, and iPhone

5.10: Mac or PC? ​

  • The answer is baed on the needs of the business
  • Ask yourself: Which one are your end-users comfortable or have experiencing using? Which one gives you a clear competitive advantage?

5.11: Major Operating Systems ​

  • Microsoft Windows: Windows 10
    • Windows 11 available for download
  • Mac Operating System (AKA Mac OS)
    • Version 14: Sonoma
  • Linux: An open-source OS that can be modified, used, and redistributed by anyone freely
    • Created by Linux Torvalds
  • UNIX: Popular for large processing applications like credit card transactions and approvals.

5.12: Open-Source OS ​

  • Proprietary software vs. open-source
    • Why doesn't everyone use open-source
  • Diffusion: A product or software is widely accepted
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 5 ​

Author:Anda Toshiki
Updated:a minute ago
Words:602
Reading:3 min

Chapter 5: Operating System ​

5.1: Operating System Fundamentals ​

  • 1980: Bill Gates and Paul Allen leased IBM a Personal Computer (PC) called MS-DOS (Microsoft Disk Operating System) for a one-time fee of $50,000\$50,000.
    • Gates Kept the copyright for MS-DOS
  • 1984: Steve Jobs and Steve Wozniak created Mac OS
    • Advertised on "More user-friendly"

5.2: Understanding Operating Systems ​

  • System software: Software that automatically runs to configure a computer (no human intervention)
  • Operating system (AKA Platform): A collection of computer programs that administer the hardware and software of a computer so that they work properly
    • Microsoft Windows, Mac OS, UNIX, Linux
  • Application software cannot run or be installed without a operating system.

5.3: Memory Management ​

  • Operating system coordinate: Computers memory

    • Cache, Random Access Memory (RAM), registers, and virtual memory
    • Disk Storage

  • CPU Cache: A small, very quick memory that stores copies of information and data from the most regularly-used parts of main memory

    • Helps decrease the time it takes to access memory.

  • Random Access Memory (RAM): Volatile memory

  • Disk Storage: Non-volatile memory

    • volatile memory stores data when a computer is on but erases it as soon as the computer is switched off, whereas non-volatile memory remains in a computer even after the system shuts off.

5.4: Memory Management Contd. ​

  • Registers: Small memory locations used to quicken the implementation of computer programs
    • Customization towards the device including wallpaper setting which is referred the process of registering.
  • Virtual Memory: Memory the OS uses to coordinate, track and efficiently allocate the use of CPU cache, registers, RAM and disk storage.
    • The messenger between volatile and non-volatile storage.

5.5: Disk Management ​

  • OS is responsible for saving information as efficiently as possible
    • Creates a table of contents of FAT (File Allocation Table)
  • Defragmentation: A program the reorganizes the hard drive
    • Note required for SSD
  • Hard Disk Drive (HDD)
    • Pros: More affordable, good for standard business procedures
    • Cons: more prone to damage
  • Solid State Drives (SSD)
    • Pros: N moving parts, more reliable
    • Cons: More expensive.

5.6: File Management ​

  • File manager are made up of Hierarchy of Directories, more commonly referred to as folders
    • Can be compared to a physical storage

5.7: Networks ​

  • Network: 2 or more computers connected together
  • Transmission control protocol and internet protocol (TCP/IP): A set of rules for transferring information from one computer to another
    • Transfers info by breaking up files into packates
  • The internet is the world's largest network

5.8: Device Drivers ​

  • Device Driver: Allows the OS to communicate with peripheral devices
  • Modern OS have plug-and-play capability
  • 1996: USB developed and considered industry standard

5.9: embedded Operating Systems ​

  • Embedded Operating System: OS installed on devices and appliances that rely on computing
    • Ex; Cash registers, automatic teller machines (ATM), microwaves, and specialized handled devices
    • Embedded OS cannot be modified in most cases
    • Embedded OS used on most cell pones
      • Google, Android, and iPhone

5.10: Mac or PC? ​

  • The answer is baed on the needs of the business
  • Ask yourself: Which one are your end-users comfortable or have experiencing using? Which one gives you a clear competitive advantage?

5.11: Major Operating Systems ​

  • Microsoft Windows: Windows 10
    • Windows 11 available for download
  • Mac Operating System (AKA Mac OS)
    • Version 14: Sonoma
  • Linux: An open-source OS that can be modified, used, and redistributed by anyone freely
    • Created by Linux Torvalds
  • UNIX: Popular for large processing applications like credit card transactions and approvals.

5.12: Open-Source OS ​

  • Proprietary software vs. open-source
    • Why doesn't everyone use open-source
  • Diffusion: A product or software is widely accepted
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l6-pt1-lecture-note.html b/academic/cis105/cis105-l6-pt1-lecture-note.html index 73398beb..6e56e5a4 100644 --- a/academic/cis105/cis105-l6-pt1-lecture-note.html +++ b/academic/cis105/cis105-l6-pt1-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 6 Pt. 1 ​

Author:Anda Toshiki
Updated:a minute ago
Words:529
Reading:3 min

Chapter 6: System Software ​

6.1: System Software Overview ​

System Software:

  • Enables the Operating System
  • Controls memory, input and output, and manages filing system
  • Runs essential processes automatically
  • Controls the physical hardware so that applications software can work
  • If your computer is running, your system software is working

6.2: Starting the Computer ​

What happens when you press the power button:

  • System searches for Basic Input/Output System (BIOS)
  • Firmware, locates computer hardware
  • SS loads the platform (OS) into the computer’s memory: Booting
  • Powering up the computer for the first time: Cold Boot
  • Restarting a computer: Warm Boot
  • Computer will ask for login and load your profile
  • Preferences, themes, styles, and types of resources set up by the user
  • Registry loads settings made by the user
  • Screen savers, appearance, and personalizations

6.3: Administering Application Software ​

  • When a user clicks on an application, the application info moves from the hard drive to RAM (Random Access Memory)
  • SS and OS make sure application software works correctly

6.4: Memory Management ​

  • Computer memory allocation can be compared to a farmer allocating water for crops
  • Virtual memory: The technique of breaking large files down from the hard drive into smaller, more manageable files that fit in RAM
  • Smaller files called Pages
  • If the pages exceed RAM storage, OS will store temporary copies of pages to hard drive
  • Swap Files: When pages exceed RAM storage, OS swaps pages between hard drive and RAM Ask yourself: Does your business need new computers or new computer parts?
  • New computer: $800-1500
  • Upgraded RAM: $50-$200
  • Upgraded Hard Drive:$40-$100

6.5: Interfacing and Utilities ​

SS in the form of an OS must serve as a translator for both the computer and end user

  • Monitor: displays and renders output an end user can understand
  • GUI
  • Icon: A small pictogram on the desktop that a user clicks with a mouse
  • Window: A GUI unto its own
  • Device Drivers: allow input and output devices to operate properly
  • The background displayed on the monitor is considered the desktop environment, or "desktop" for short
  • Active Window: the window that the user is currently using

6.6: File Management Systems ​

Based on a hierarchy of folders that contain computer files. File management systems are searchable and provide information on files:

  • File Name
  • File Creation Date
  • File Modification Date
  • Location of the file
  • Who created or modified a file
  • File size
  • In business, it is important to establish your file management system early

6.7: System Software Utilities ​

  • Tracks: concentric circular bands
  • Sectors: Tracks that are separated into wedges
  • Defragmentation: Organizes + and – charges to the innermost tracks of the disk to reduce the amount of time it takes the R/W arm to find the info

6.8: Antivirus Utilities ​

Antivirus Utilities: System software that identifies known viruses to block them from entering and infecting a computer system

  • Utilizes a database of known viruses to cross reference against each incoming file
  • Subscription service: Norton Antivirus and McAfee VirusScan

6.9: System Updates ​

  • To compensate for OS inadequacies, OS offer a system software utility called system update
  • Patches sent from Microsoft (or Mac) via the Internet to a user’s computer
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 6 Pt. 1 ​

Author:Anda Toshiki
Updated:a minute ago
Words:529
Reading:3 min

Chapter 6: System Software ​

6.1: System Software Overview ​

System Software:

  • Enables the Operating System
  • Controls memory, input and output, and manages filing system
  • Runs essential processes automatically
  • Controls the physical hardware so that applications software can work
  • If your computer is running, your system software is working

6.2: Starting the Computer ​

What happens when you press the power button:

  • System searches for Basic Input/Output System (BIOS)
  • Firmware, locates computer hardware
  • SS loads the platform (OS) into the computer’s memory: Booting
  • Powering up the computer for the first time: Cold Boot
  • Restarting a computer: Warm Boot
  • Computer will ask for login and load your profile
  • Preferences, themes, styles, and types of resources set up by the user
  • Registry loads settings made by the user
  • Screen savers, appearance, and personalizations

6.3: Administering Application Software ​

  • When a user clicks on an application, the application info moves from the hard drive to RAM (Random Access Memory)
  • SS and OS make sure application software works correctly

6.4: Memory Management ​

  • Computer memory allocation can be compared to a farmer allocating water for crops
  • Virtual memory: The technique of breaking large files down from the hard drive into smaller, more manageable files that fit in RAM
  • Smaller files called Pages
  • If the pages exceed RAM storage, OS will store temporary copies of pages to hard drive
  • Swap Files: When pages exceed RAM storage, OS swaps pages between hard drive and RAM Ask yourself: Does your business need new computers or new computer parts?
  • New computer: $800-1500
  • Upgraded RAM: $50-$200
  • Upgraded Hard Drive:$40-$100

6.5: Interfacing and Utilities ​

SS in the form of an OS must serve as a translator for both the computer and end user

  • Monitor: displays and renders output an end user can understand
  • GUI
  • Icon: A small pictogram on the desktop that a user clicks with a mouse
  • Window: A GUI unto its own
  • Device Drivers: allow input and output devices to operate properly
  • The background displayed on the monitor is considered the desktop environment, or "desktop" for short
  • Active Window: the window that the user is currently using

6.6: File Management Systems ​

Based on a hierarchy of folders that contain computer files. File management systems are searchable and provide information on files:

  • File Name
  • File Creation Date
  • File Modification Date
  • Location of the file
  • Who created or modified a file
  • File size
  • In business, it is important to establish your file management system early

6.7: System Software Utilities ​

  • Tracks: concentric circular bands
  • Sectors: Tracks that are separated into wedges
  • Defragmentation: Organizes + and – charges to the innermost tracks of the disk to reduce the amount of time it takes the R/W arm to find the info

6.8: Antivirus Utilities ​

Antivirus Utilities: System software that identifies known viruses to block them from entering and infecting a computer system

  • Utilizes a database of known viruses to cross reference against each incoming file
  • Subscription service: Norton Antivirus and McAfee VirusScan

6.9: System Updates ​

  • To compensate for OS inadequacies, OS offer a system software utility called system update
  • Patches sent from Microsoft (or Mac) via the Internet to a user’s computer
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l6-pt2-lecture-note.html b/academic/cis105/cis105-l6-pt2-lecture-note.html index 383b685f..fbbc857d 100644 --- a/academic/cis105/cis105-l6-pt2-lecture-note.html +++ b/academic/cis105/cis105-l6-pt2-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 6 Pt. 2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:249
Reading:1 min

Chapter 6: Logical Functions ​

6.1: New Functions ​

  • AND: Checks whether all arguments are TRUE, and returns TRUE if all arguments are TRUE.

  • OR: Checks whether any of the arguments are TRUE, and returns TRUE or FALSE.

  • NOT: Changes TRUE to FALSE, or FALSE to TRUE.

  • IF: Checks whether a condition is met, and returns one value if TRUE, and another value if FALSE.

    • Nested IF: Using multiple IF statements in one functions to account for multiple conditions.

  • VLOOKUP: Looks for a value in a table or a range veryically

    • VLOOKUP What you want to look up, where you want to look for it, the column number in the range containing the value to return, return an approximate or exact match: Indicates as 1/TRUE, or 0/FALSE.

  • HLOOKUP: Looks for a value in a table or a range horizontally.

  • COUNTIF: Counts the number of cells within a range that meet the given condition.

    • =COUNTIF(Where do you want to look? What do you want to look for?)

  • SUMIF: Adds the cells specified by a given condition or criteria.

  • AVERAGEIF: s an Excel Statistical function, which calculates the average of a given range of cells by a specific criterion.

6.2: Comparison Operators ​

  • >: More than
  • <: Less than
  • >=: More than or equal to
  • <=: Less that or equal to

6.3: Reminder ​

  • Excel does not recognize tex
    • When searching for keywords, put keyword between "" delimiters.
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 6 Pt. 2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:249
Reading:1 min

Chapter 6: Logical Functions ​

6.1: New Functions ​

  • AND: Checks whether all arguments are TRUE, and returns TRUE if all arguments are TRUE.

  • OR: Checks whether any of the arguments are TRUE, and returns TRUE or FALSE.

  • NOT: Changes TRUE to FALSE, or FALSE to TRUE.

  • IF: Checks whether a condition is met, and returns one value if TRUE, and another value if FALSE.

    • Nested IF: Using multiple IF statements in one functions to account for multiple conditions.

  • VLOOKUP: Looks for a value in a table or a range veryically

    • VLOOKUP What you want to look up, where you want to look for it, the column number in the range containing the value to return, return an approximate or exact match: Indicates as 1/TRUE, or 0/FALSE.

  • HLOOKUP: Looks for a value in a table or a range horizontally.

  • COUNTIF: Counts the number of cells within a range that meet the given condition.

    • =COUNTIF(Where do you want to look? What do you want to look for?)

  • SUMIF: Adds the cells specified by a given condition or criteria.

  • AVERAGEIF: s an Excel Statistical function, which calculates the average of a given range of cells by a specific criterion.

6.2: Comparison Operators ​

  • >: More than
  • <: Less than
  • >=: More than or equal to
  • <=: Less that or equal to

6.3: Reminder ​

  • Excel does not recognize tex
    • When searching for keywords, put keyword between "" delimiters.
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l7-lecture-note.html b/academic/cis105/cis105-l7-lecture-note.html index e096097c..c573c51b 100644 --- a/academic/cis105/cis105-l7-lecture-note.html +++ b/academic/cis105/cis105-l7-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 7 ​

Author:Anda Toshiki
Updated:a minute ago
Words:737
Reading:4 min

Chapter 7: Green Business Computing ​

7.1: Green Business Computing ​

  • Burning fossil fules creates carbon dioxide which builds up and slows the escape of heat into space, called the "greenhouse effect".
  • NASA predicts Earth's surface temperature will increase by 2.5 to 10.4 Fahrenheit by the year 2100. The concern is that human societies and natural ecosystems may not adapt quickly enough.
  • The National Center for Atmospheric Research (NCAR) in charge of creating and maintaining climate computer models that are used to predict weather admit they are flawed and unreliable.
  • Who's right, ASA or NCAR?

7.2: Climate Change ​

  • Whether global warming exists is a discussion that requires critical thoughtfullnesswith a minimum of passion.
  • It is virtually impossible to not come across terminologies like "green" or challenges, and potential sustainable solutions have gained traction and must be dealt with.
  • Regardless of differing views, climate change has gained diffusion and acceptance as an acknowledged problem that will drive business for many years to come. Environmental responsibility and global governance is simply the right things to do.

7.3: IT Challenges and Opportunities ​

  • Chief technology officers (in charge of all business computing assets) meed to be concerned with overall energy costs associated with computing systems.
  • Electronic Product Environmental Assessment Tool (EPEAT)
    • Started by the Environmental Protection Agency in 2006.
    • Managed by the Green Electronics Council (GEC)
    • Easy to use online tool

7.4: EPEAT Criteria ​

EPEAT evaluates electronic products relative to 51 environmental criteria, 23 required and 28 optional. Computer manufacturers that meet a certain criteria are awarded a gold, silver, or bronze star allowing the consumer to differentiate products.

  • Reduction or elimination of environmentally sensitive materials. Computers contain potentially harmful chemicals like cadmium, mercury, and chromium. EPEAT Encourages manufacturers to reduce or eliminate them altogether.

  • Materials Selection informs the consumer of the percentage of recycled plastic and offers a declaration of percentage of renewable/bio-based plastic material content.

  • Design for End of Life informs consumers about special handling needs for disposing of their product when it no longer useful.

  • Product Longevity/Life Cycle Extension encourages manufacturers to make extended warranties available so their product lasts longer or available to be upgraded.

  • End of Life Management asks manufacturers to take their product back at the end of the product’s life, at no cost to the consumer.

  • Corporate Performance means a manufacturer demonstrates a sound corporate environmental policy throughout their business.

  • Packaging criteria requires manufacturers to reduce or eliminate intentionally added toxins in packaging.

  • Energy Conservation asks manufacturers to adopt the government’s Energy Star® specifications.

EPEAT criteria seems difficult and expensive for a gold star, so why would manufacturers be willing to comply? Had you ever heard of EPEAT? Did you base your last computer purchase on EPEAT criteria? If you were aware of EPEAT, would you pay more just to be green? What if your business pays significantly more for EPEAT computers and your competitor doesn’t, did they get an edge?

7.5: Energy Star® Specifications ​

  • Started in 1772 as a voluntary labeling program to identify and promote energy efficient products.

  • Energy Star® claims that if every computer in the United States met Energy Star® requirements, the savings in energy costs alone will grow to about $2 billion each year and greenhouse gas emissions would be reduced by the the equivalent of that from 2 million cars.

  • In many instances, Energy Star® computers are more expensive to purchase but are far more efficient and cost effective in the long run

7.6: The Green PC ​

A Green PC is a personal computer that is environmentally friendly and includes the following:

  • Energy-efficient power supplies

  • Includes processors that consume minimum amounts of electricity

  • Attempt to do away with or minimize environmentally harmful components

  • Makes use of aluminum or plastic parts from recycled materials

The most important factor in a Green PC is energy efficiency

7.7: Modes ​

  • Active Use: The computer is on and being use

  • Sleep Mode: Low power after a designated period of time

  • Standby Mode (Off): Turns off after designated period of time

7.8: Green Business Computing Plan ​

Based on the idea that businesses can be environmentally friendly, more profitable, and socially responsible.

  • Organizational policies and procedures

  • Best Practice considerations

  • Long term, sustainable considerations

  • Reduction of power usage and paper consumption

  • Recycling policies and procedures

  • Used computer software and hardware disposal

  • EPEAT and Energy Star® compliance

  • Green PC purchasing recommendations

- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 7 ​

Author:Anda Toshiki
Updated:a minute ago
Words:737
Reading:4 min

Chapter 7: Green Business Computing ​

7.1: Green Business Computing ​

  • Burning fossil fules creates carbon dioxide which builds up and slows the escape of heat into space, called the "greenhouse effect".
  • NASA predicts Earth's surface temperature will increase by 2.5 to 10.4 Fahrenheit by the year 2100. The concern is that human societies and natural ecosystems may not adapt quickly enough.
  • The National Center for Atmospheric Research (NCAR) in charge of creating and maintaining climate computer models that are used to predict weather admit they are flawed and unreliable.
  • Who's right, ASA or NCAR?

7.2: Climate Change ​

  • Whether global warming exists is a discussion that requires critical thoughtfullnesswith a minimum of passion.
  • It is virtually impossible to not come across terminologies like "green" or challenges, and potential sustainable solutions have gained traction and must be dealt with.
  • Regardless of differing views, climate change has gained diffusion and acceptance as an acknowledged problem that will drive business for many years to come. Environmental responsibility and global governance is simply the right things to do.

7.3: IT Challenges and Opportunities ​

  • Chief technology officers (in charge of all business computing assets) meed to be concerned with overall energy costs associated with computing systems.
  • Electronic Product Environmental Assessment Tool (EPEAT)
    • Started by the Environmental Protection Agency in 2006.
    • Managed by the Green Electronics Council (GEC)
    • Easy to use online tool

7.4: EPEAT Criteria ​

EPEAT evaluates electronic products relative to 51 environmental criteria, 23 required and 28 optional. Computer manufacturers that meet a certain criteria are awarded a gold, silver, or bronze star allowing the consumer to differentiate products.

  • Reduction or elimination of environmentally sensitive materials. Computers contain potentially harmful chemicals like cadmium, mercury, and chromium. EPEAT Encourages manufacturers to reduce or eliminate them altogether.

  • Materials Selection informs the consumer of the percentage of recycled plastic and offers a declaration of percentage of renewable/bio-based plastic material content.

  • Design for End of Life informs consumers about special handling needs for disposing of their product when it no longer useful.

  • Product Longevity/Life Cycle Extension encourages manufacturers to make extended warranties available so their product lasts longer or available to be upgraded.

  • End of Life Management asks manufacturers to take their product back at the end of the product’s life, at no cost to the consumer.

  • Corporate Performance means a manufacturer demonstrates a sound corporate environmental policy throughout their business.

  • Packaging criteria requires manufacturers to reduce or eliminate intentionally added toxins in packaging.

  • Energy Conservation asks manufacturers to adopt the government’s Energy Star® specifications.

EPEAT criteria seems difficult and expensive for a gold star, so why would manufacturers be willing to comply? Had you ever heard of EPEAT? Did you base your last computer purchase on EPEAT criteria? If you were aware of EPEAT, would you pay more just to be green? What if your business pays significantly more for EPEAT computers and your competitor doesn’t, did they get an edge?

7.5: Energy Star® Specifications ​

  • Started in 1772 as a voluntary labeling program to identify and promote energy efficient products.

  • Energy Star® claims that if every computer in the United States met Energy Star® requirements, the savings in energy costs alone will grow to about $2 billion each year and greenhouse gas emissions would be reduced by the the equivalent of that from 2 million cars.

  • In many instances, Energy Star® computers are more expensive to purchase but are far more efficient and cost effective in the long run

7.6: The Green PC ​

A Green PC is a personal computer that is environmentally friendly and includes the following:

  • Energy-efficient power supplies

  • Includes processors that consume minimum amounts of electricity

  • Attempt to do away with or minimize environmentally harmful components

  • Makes use of aluminum or plastic parts from recycled materials

The most important factor in a Green PC is energy efficiency

7.7: Modes ​

  • Active Use: The computer is on and being use

  • Sleep Mode: Low power after a designated period of time

  • Standby Mode (Off): Turns off after designated period of time

7.8: Green Business Computing Plan ​

Based on the idea that businesses can be environmentally friendly, more profitable, and socially responsible.

  • Organizational policies and procedures

  • Best Practice considerations

  • Long term, sustainable considerations

  • Reduction of power usage and paper consumption

  • Recycling policies and procedures

  • Used computer software and hardware disposal

  • EPEAT and Energy Star® compliance

  • Green PC purchasing recommendations

+ \ No newline at end of file diff --git a/academic/cis105/cis105-l8-lecture-note.html b/academic/cis105/cis105-l8-lecture-note.html index e9906518..44158175 100644 --- a/academic/cis105/cis105-l8-lecture-note.html +++ b/academic/cis105/cis105-l8-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 8 ​

Author:Anda Toshiki
Updated:a minute ago
Words:754
Reading:4 min

Chapter 8: Computer Networks ​

8.1: Computer Networks ​

  • Two or more computers connected together to share resources like:
  • Folders and files
  • Software
  • Hardware
  • Scanners, Printers, Webcams, etc. Consider a network made up of three computers (Clients) connected to one computer called a Server. The server is connected to a printer and a scanner (peripheral devices) enabling all three clients to use them. Any device connected to a network is called a node.

8.2: Building a Computer Network ​

All networks require four basic components

  1. Network Interface Controller Card (NIC): Plugged into the motherboard to interface and connect with other computers.
  2. Protocol: TCP/IP controls how data and information is shared between computers.
  3. Cables: Cables connect servers to clients which can sometimes be replaced through wireless connections.
  4. Hub: Typically, the central location of a network (server) that serves to run the network smoothly

8.3: Network Protocols ​

Provides a common language and rules for communication between computers. A protocol is a convention or standard that controls or allows communication and data transfer between two computers.

  • Transmission Control Protocol / Internet Protocol (TCP/IP): When sending a file from one computer to another, the file is broken down into smaller files called “packets”. The packets arrive at the receiving computer and must be reassembled correctly to guarantee the receiving computer gets a file that will work (not corrupted).
  • User Datagram Packets: Break large files into packets, but unlike TCP/IP, allows a user to use the incoming file.

8.4: Network Operating Systems ​

A Network Operating System (NOS) is similar to a stand-alone platform but responsible for much more:

  • Automatic hardware detection. The NOS recognizes it when it becomes part of the network.
  • Supports multi-processing which allows several processes to occur at one time.
  • Recognizes who is using the network.
  • Provides security

8.5: Network Administrator ​

  • Responsible for smooth network operations
  • Network performance
  • Implementation of new nodes and users
  • Strong business background

8.6: Network Topology ​

Refers to the mapping of a physical network and logical interconnections between nodes. Basically, the way a network is arranged and laid out.

  • Bus Topology: Sometimes called a linear bus topology where the network’s nodes are connected to a common linear backbone with tw endpoints. If the backbone fails, the whole network fails.
  • Star Topology: One central server in which clients connect in a spoke- like fashion, or star. If the server fails, the network fails. (Most common topology)
  • Ring Topology: Data is transferred between computers in a circular manner.

8.7: Types of Computer Networks ​

  • Local Area Network (LAN): A small network typically confined by a single building. Often used by smaller businesses.
  • Campus Area Network (CAN): Connects two or more LANs which are located on universities or large businesses.
  • Metropolitan Area Network (MAN): A computer network that connects CANs and LANs into a larger network like a city.
  • Wide Area Network (WAN): Connects LANs, CANs, and MANs and can cover an entire country
  • Peer to Peer (P2P): Generally, connect computers to each other.
  • Internet: The world’s largest network interconnecting public, private, commercial, and governmental networks. Network Security Starts with a solid understanding of network threats. Network administrators must adopt policies which define rules for network access.
  • Passwords: A secret word or combination of keyboard characters typed in by the user to authenticate their identity to a network and gain appropriate access.
  • Strong Passwords: Same as passwords, but requires a user to use upper case characters, lower case characters, numbers, and special characters.
  • Passphrase: Same as a Strong Password, but a phrase that is memorable.

8.8: Network Security ​

  • Web Browsing on a Network: Part of network security includes policies and procedures regarding employee’s web browsing. Should a business let employees browse on the Web?
  • Email / Instant Messaging: On a business network, consider that all email and instant message is recorded and saved, therefore, policies and procedures must be in place regarding appropriate usage.
  • Permissions: Network access is called “permissions”, which grants and denies access to a network depending on the user.
  • Firewall: Hardware and software on a proxy server that determines who is allowed on a network, and who is not.

8.9: Green Business Computing Plan ​

Based on the idea that businesses can be environmentally friendly, more profitable, and socially responsible.

  • Organizational policies and procedures
  • Best Practice considerations
  • Long term, sustainable considerations
  • Reduction of power usage and paper consumption
  • Recycling policies and procedures
  • Used computer software and hardware disposal
  • EPEAT and Energy Star® compliance
  • Green PC purchasing recommendations
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 8 ​

Author:Anda Toshiki
Updated:a minute ago
Words:754
Reading:4 min

Chapter 8: Computer Networks ​

8.1: Computer Networks ​

  • Two or more computers connected together to share resources like:
  • Folders and files
  • Software
  • Hardware
  • Scanners, Printers, Webcams, etc. Consider a network made up of three computers (Clients) connected to one computer called a Server. The server is connected to a printer and a scanner (peripheral devices) enabling all three clients to use them. Any device connected to a network is called a node.

8.2: Building a Computer Network ​

All networks require four basic components

  1. Network Interface Controller Card (NIC): Plugged into the motherboard to interface and connect with other computers.
  2. Protocol: TCP/IP controls how data and information is shared between computers.
  3. Cables: Cables connect servers to clients which can sometimes be replaced through wireless connections.
  4. Hub: Typically, the central location of a network (server) that serves to run the network smoothly

8.3: Network Protocols ​

Provides a common language and rules for communication between computers. A protocol is a convention or standard that controls or allows communication and data transfer between two computers.

  • Transmission Control Protocol / Internet Protocol (TCP/IP): When sending a file from one computer to another, the file is broken down into smaller files called “packets”. The packets arrive at the receiving computer and must be reassembled correctly to guarantee the receiving computer gets a file that will work (not corrupted).
  • User Datagram Packets: Break large files into packets, but unlike TCP/IP, allows a user to use the incoming file.

8.4: Network Operating Systems ​

A Network Operating System (NOS) is similar to a stand-alone platform but responsible for much more:

  • Automatic hardware detection. The NOS recognizes it when it becomes part of the network.
  • Supports multi-processing which allows several processes to occur at one time.
  • Recognizes who is using the network.
  • Provides security

8.5: Network Administrator ​

  • Responsible for smooth network operations
  • Network performance
  • Implementation of new nodes and users
  • Strong business background

8.6: Network Topology ​

Refers to the mapping of a physical network and logical interconnections between nodes. Basically, the way a network is arranged and laid out.

  • Bus Topology: Sometimes called a linear bus topology where the network’s nodes are connected to a common linear backbone with tw endpoints. If the backbone fails, the whole network fails.
  • Star Topology: One central server in which clients connect in a spoke- like fashion, or star. If the server fails, the network fails. (Most common topology)
  • Ring Topology: Data is transferred between computers in a circular manner.

8.7: Types of Computer Networks ​

  • Local Area Network (LAN): A small network typically confined by a single building. Often used by smaller businesses.
  • Campus Area Network (CAN): Connects two or more LANs which are located on universities or large businesses.
  • Metropolitan Area Network (MAN): A computer network that connects CANs and LANs into a larger network like a city.
  • Wide Area Network (WAN): Connects LANs, CANs, and MANs and can cover an entire country
  • Peer to Peer (P2P): Generally, connect computers to each other.
  • Internet: The world’s largest network interconnecting public, private, commercial, and governmental networks. Network Security Starts with a solid understanding of network threats. Network administrators must adopt policies which define rules for network access.
  • Passwords: A secret word or combination of keyboard characters typed in by the user to authenticate their identity to a network and gain appropriate access.
  • Strong Passwords: Same as passwords, but requires a user to use upper case characters, lower case characters, numbers, and special characters.
  • Passphrase: Same as a Strong Password, but a phrase that is memorable.

8.8: Network Security ​

  • Web Browsing on a Network: Part of network security includes policies and procedures regarding employee’s web browsing. Should a business let employees browse on the Web?
  • Email / Instant Messaging: On a business network, consider that all email and instant message is recorded and saved, therefore, policies and procedures must be in place regarding appropriate usage.
  • Permissions: Network access is called “permissions”, which grants and denies access to a network depending on the user.
  • Firewall: Hardware and software on a proxy server that determines who is allowed on a network, and who is not.

8.9: Green Business Computing Plan ​

Based on the idea that businesses can be environmentally friendly, more profitable, and socially responsible.

  • Organizational policies and procedures
  • Best Practice considerations
  • Long term, sustainable considerations
  • Reduction of power usage and paper consumption
  • Recycling policies and procedures
  • Used computer software and hardware disposal
  • EPEAT and Energy Star® compliance
  • Green PC purchasing recommendations
+ \ No newline at end of file diff --git a/academic/cis105/cis105-l9-lecture-note.html b/academic/cis105/cis105-l9-lecture-note.html index a3c4f5bc..cd0f1aae 100644 --- a/academic/cis105/cis105-l9-lecture-note.html +++ b/academic/cis105/cis105-l9-lecture-note.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 9 ​

Author:Anda Toshiki
Updated:a minute ago
Words:426
Reading:2 min

Chapter 9: Internet ​

9.1: Internet History ​

  • 1969: University of California Los Angeles (UCLA) created the Advanced Research Project Agency Network (ARPANET)
  • 1989: Hypertext Transfer Protocol (HTTP) / World Wide Web (WWW)
  • 1993: Hypertext Markup Langauge (HTML)

9.2: Internet and WWW Structure ​

  • he internet is a network called the Client/Server Model
  • The browser application software interprets HTML and HTTP to create web pages
  • Everything on the internet lives somewhere physically; the trick is using computer addresses to send info from one place to another

9.3: Internet Addresses ​

  • Internet Protocol Address (IP): An address system used to identify and communicate through the network
  • Uniform Resource Locators (URLs)
    • https://asu.edu
    • https: refers to standard protocol
    • www: refers to the world wide web
    • Canvas.asu: domain name
    • .edu: a top-level domain code or extension

9.4/9.5: Websites/Browsers ​

  • Website: A collection of web pages on a server that exists on the web that can be requested by a client and displayed with a browser
  • Hyperlink: Clickable navigation element
  • Browser: An application software that provides an interface for the WWW
  • Aspects of a good website:
    • Content
    • Interactivity
    • Design

9.6: Browser Features ​

  • Common Browser Features
    • Bookmarks: AKA favorites
    • Caching: Just like a CPU cache on the internet
    • Cookies: Text files used to track info about users
    • Graphics File (GIF, JPEG): Images and videos
    • Javascript: A programming language
      • Used to make websites dynamics; Ex: Countdown timer
  • Browser Considerations:
    • Security
    • Popup Blockers
    • Usability

9.7: Internet Service Providers (ISP) ​

  • Connectivity: When one computer connects to another and shares info & resources
  • Internet Service Providers (ISP)
    • Dial-up & Digital Subscriber Line (DSL)
    • Broadband Internet Access
    • Web Hosting

9.8: Search Engines ​

  • Search engine: A website that lets a user type in specific key words or phrases and then return a list of hyperlinks that best match the search criteria
    • Google, Bing, Yahoo
    • Directory searches
    • Metasearch Engines
      • Ex: Dogpile

9.9: E-Commerce ​

  • Electronic Commerce (E-Commerce): The distribution, buying, selling, advertising, and marketing of goods and services over the web
    • Business-to-Business (B2B)
    • Business-to-Consumer (B2C)
    • Consumer-to-Consumer (C2C)
  • The rules of business still apply to online environments

9.10: Security ​

  • Malware: Software designed to damage a computer's system without the owner's knowledge
    • Virus
    • Worms
    • Trojan Horse
    • Spyware
    • Adware

9.11/9.12: Email/Email Security ​

  • Email: "A store and forward" system of composing, sending, storing, and receiving message over electronic communication systems.
  • Never send an email you wouldn't be willing to say in public
  • Phishing: An email that looks official but is malicious in nature
  • Spam: Unwanted mail
- +
Toshiki's Note

Appearance

CIS105: Computer Applications & Information Systems Lect. 9 ​

Author:Anda Toshiki
Updated:a minute ago
Words:426
Reading:2 min

Chapter 9: Internet ​

9.1: Internet History ​

  • 1969: University of California Los Angeles (UCLA) created the Advanced Research Project Agency Network (ARPANET)
  • 1989: Hypertext Transfer Protocol (HTTP) / World Wide Web (WWW)
  • 1993: Hypertext Markup Langauge (HTML)

9.2: Internet and WWW Structure ​

  • he internet is a network called the Client/Server Model
  • The browser application software interprets HTML and HTTP to create web pages
  • Everything on the internet lives somewhere physically; the trick is using computer addresses to send info from one place to another

9.3: Internet Addresses ​

  • Internet Protocol Address (IP): An address system used to identify and communicate through the network
  • Uniform Resource Locators (URLs)
    • https://asu.edu
    • https: refers to standard protocol
    • www: refers to the world wide web
    • Canvas.asu: domain name
    • .edu: a top-level domain code or extension

9.4/9.5: Websites/Browsers ​

  • Website: A collection of web pages on a server that exists on the web that can be requested by a client and displayed with a browser
  • Hyperlink: Clickable navigation element
  • Browser: An application software that provides an interface for the WWW
  • Aspects of a good website:
    • Content
    • Interactivity
    • Design

9.6: Browser Features ​

  • Common Browser Features
    • Bookmarks: AKA favorites
    • Caching: Just like a CPU cache on the internet
    • Cookies: Text files used to track info about users
    • Graphics File (GIF, JPEG): Images and videos
    • Javascript: A programming language
      • Used to make websites dynamics; Ex: Countdown timer
  • Browser Considerations:
    • Security
    • Popup Blockers
    • Usability

9.7: Internet Service Providers (ISP) ​

  • Connectivity: When one computer connects to another and shares info & resources
  • Internet Service Providers (ISP)
    • Dial-up & Digital Subscriber Line (DSL)
    • Broadband Internet Access
    • Web Hosting

9.8: Search Engines ​

  • Search engine: A website that lets a user type in specific key words or phrases and then return a list of hyperlinks that best match the search criteria
    • Google, Bing, Yahoo
    • Directory searches
    • Metasearch Engines
      • Ex: Dogpile

9.9: E-Commerce ​

  • Electronic Commerce (E-Commerce): The distribution, buying, selling, advertising, and marketing of goods and services over the web
    • Business-to-Business (B2B)
    • Business-to-Consumer (B2C)
    • Consumer-to-Consumer (C2C)
  • The rules of business still apply to online environments

9.10: Security ​

  • Malware: Software designed to damage a computer's system without the owner's knowledge
    • Virus
    • Worms
    • Trojan Horse
    • Spyware
    • Adware

9.11/9.12: Email/Email Security ​

  • Email: "A store and forward" system of composing, sending, storing, and receiving message over electronic communication systems.
  • Never send an email you wouldn't be willing to say in public
  • Phishing: An email that looks official but is malicious in nature
  • Spam: Unwanted mail
+ \ No newline at end of file diff --git a/academic/cis105/index.html b/academic/cis105/index.html index 7163a1dd..eb21de29 100644 --- a/academic/cis105/index.html +++ b/academic/cis105/index.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

CIS105: Computer Appls&Info Technology (2024 Spring) ​

Author:Anda Toshiki
Updated:a minute ago
Words:4.2k
Reading:26 min

Courseload Overview ​

REQUIRED

TEXTBOOKS:

Week 1

Week 2

 
 

Week 3

Week 4

 

Week 5

 

Week 6

 

Exam 1: February 22nd - 23rd

 

Week 7

Week 8

 

Mar 4th - 10th

SPRING BREAK

Week 9

 

Week 10

 

Week 11

 
 Exam 2

Week 12

 

Week 13

 
 
 
  • Lecture - Thur, Apr 18th
  • Final Exam Review
 Final Exam
- +
Toshiki's Note

Appearance

CIS105: Computer Appls&Info Technology (2024 Spring) ​

Author:Anda Toshiki
Updated:a minute ago
Words:4.2k
Reading:26 min

Courseload Overview ​

REQUIRED

TEXTBOOKS:

Week 1

Week 2

 
 

Week 3

Week 4

 

Week 5

 

Week 6

 

Exam 1: February 22nd - 23rd

 

Week 7

Week 8

 

Mar 4th - 10th

SPRING BREAK

Week 9

 

Week 10

 

Week 11

 
 Exam 2

Week 12

 

Week 13

 
 
 
  • Lecture - Thur, Apr 18th
  • Final Exam Review
 Final Exam
+ \ No newline at end of file diff --git a/academic/literature/index.html b/academic/literature/index.html index 13af0b96..c9ac150a 100644 --- a/academic/literature/index.html +++ b/academic/literature/index.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Welcome to Literature ​

Author:Anda Toshiki
Updated:a minute ago
Words:3
Reading:1 min
- +
Toshiki's Note

Appearance

Welcome to Literature ​

Author:Anda Toshiki
Updated:a minute ago
Words:3
Reading:1 min
+ \ No newline at end of file diff --git a/academic/literature/writing/methods-of-development.html b/academic/literature/writing/methods-of-development.html index ad669096..b8297a79 100644 --- a/academic/literature/writing/methods-of-development.html +++ b/academic/literature/writing/methods-of-development.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Patterns of Organization and Methods of Development ​

Author:Anda Toshiki
Updated:a minute ago
Words:2.2k
Reading:13 min

Patterns of organization can help your readers follow the ideas within your essay and your paragraphs, but they can also work as methods of development to help you recognize and further develop ideas and relationships in your writing. Here are some strategies that can help you with both organization and development in your essays.

Major Patterns of Organization ​

A fruit pie.

Read the following sentences:

  • Now take the pie out of the oven and let it cool on the stovetop.
  • Mix the dry ingredients with the liquid ingredients.
  • Set the pie crust aside while you make the filling.

How did it feel to read the above list? A bit confusing, I would guess. That’s because the steps for making a pie were not well organized, and the steps don’t include enough detail for us to know exactly what we should do. (Like what are the dry and liquid ingredients?) We all know that starting instructions from the beginning and giving each detailed step in the order it should happen is vital to having a good outcome, in this case a yummy pie! But it’s not always so simple to know how to organize or develop ideas, and sometimes there’s more than one way, which complicates things even further.

First, let’s take a look at a couple of ways to think about organization.

General to Specific or Specific to General ​

It might be useful to think about organizing your topic like a triangle:

Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General.

The first triangle represents starting with the most general, big picture information first, moving then to more detailed and often more personal information later in the paper. The second triangle represents an organizational structure that starts with the specific, small scale information first and then moves to the more global, big picture stuff.

For example, if your topic is traffic in Vancouver, British Columbia, an essay that uses the general-to-specific organizational structure might begin this way:

Many people consider Vancouver, British Columbia, to be a relaxed place to live. They would be shocked to know how bad the traffic is traveling major arteries into the city and even driving around the city itself.

An essay that uses the specific-to-general structure might start like this:

Transit is crowded, parking is expensive, and vehicles stop and go through the main streets of the city of Vancouver, British Columbia, and that is just once travelers brave the crowded arteries to enter the city; Vancouver’s traffic problem does not lend itself to the relaxed atmosphere many believe the city to have.

What’s the difference between these two introductions? And how might they appeal to the intended audience for this essay in different ways? The first introduction is looking at the big picture of the problem and mentions pollution’s impact on all citizens in Portland, while the second introduction focuses on one specific family. The first helps readers see how vast the problem really is, and the second helps connect readers to a real family, making an emotional appeal from the very beginning. Neither introduction is necessarily better. You’ll choose one over the other based on the kind of tone you’d like to create and how you’d like to affect your audience. It’s completely up to you to make this decision.

Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? ​

The triangle is kind of a general guide, meaning you’re allowed to move around within it all you want. For example, it’s possible that each of your paragraphs will be its own triangle, starting with the general or specific and moving out or in. However, if you begin very broadly, it might be effective to end your essay in a more specific, personal way. And if you begin with a personal story, consider ending your essay by touching on the global impact and importance of your topic.

Are There Other Ways to Think about Organizing My Ideas? ​

Yes! Rather than thinking about which of your ideas are most specific or personal or which are more broad or universal, you might consider one of the following ways of organizing your ideas:

  • Most important information first (consider what you want readers to focus on first)
  • Chronological order (the order in time that events take place)
  • Compare and contrast (ideas are organized together because of their relationship to each other)

The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.

Choose one of the following topics, and practice writing a few opening sentences like we did above, once using the general-to-specific format and once using the specific-to-general. Which do you like better? What audience would be attracted to which one? Share with peers to see how others tackled this challenge. How would you rewrite their sentences? Why? Discuss your changes and listen to how your peers have revised your sentences. Taking in other people’s ideas will help you see new ways to approach your own writing and thinking.
Topics:

  • Facing fears
  • Safety in sports
  • Community policing
  • Educating prisoners
  • Sex education

Methods of Development ​

The methods of development covered here are best used as ways to look at what’s already happening in your draft and to consider how you might emphasize or expand on any existing patterns. You might already be familiar with some of these patterns because teachers will sometimes assign them as the purpose for writing an essay. For example, you might have been asked to write a cause-and-effect essay or a comparison-and-contrast essay.

It’s important to emphasize here that patterns of organization or methods of developing content usually happen naturally as a consequence of the way the writer engages with and organizes information while writing. That is to say, most writers don’t sit down and say, “I think I’ll write a cause-and-effect essay today.” Instead, a writer might be more likely to be interested in a topic, say, the state of drinking water in the local community, and as the writer begins to explore the topic, certain cause-and-effect relationships between environmental pollutants and the community water supply may begin to emerge.

So if these patterns just occur naturally in writing, what’s the use in knowing about them? Well, sometimes you might be revising a draft and notice that some of your paragraphs are a bit underdeveloped. Maybe they lack a clear topic, or maybe they lack support. In either case, you can look to these common methods of development to find ways to sharpen those vague topics or to add support where needed. Do you have a clear cause statement somewhere but you haven’t explored the effects? Are you lacking detail somewhere where a narrative story or historical chronology can help build reader interest and add support? Are you struggling to define an idea that might benefit from some comparison or contrast? Read on to consider some of the ways that these strategies can help you in revision.

Cause and Effect (or Effect and Cause) ​

Do you see a potential cause-and-effect relationship developing in your draft? The cause-and-effect pattern may be used to identify one or more causes followed by one or more effects or results. Or you may reverse this sequence and describe effects first and then the cause or causes. For example, the causes of water pollution might be followed by its effects on both humans and animals. You may use obvious transitions to clarify cause and effect, such as “What are the results? Here are some of them…” or you might simply use the words cause, effect, and result, to cue the reader about your about the relationships that you’re establishing.

Problem-Solution ​

At some point does your essay explore a problem or suggest a solution? The problem-solution pattern is commonly used in identifying something that’s wrong and in contemplating what might be done to remedy the situation. There are probably more ways to organize a problem-solution approach, but but here are three possibilities:

  • Describe the problem, followed by the solution.
  • Propose the solution first and then describe the problems that motivated it.
  • Or a problem may be followed by several solutions, one of which is selected as the best.

When the solution is stated at the end of the paper, the pattern is sometimes called the delayed proposal. For a hostile audience, it may be effective to describe the problem, show why other solutions do not work, and finally suggest the favored solution. You can emphasize the words problem and solution to signal these sections of your paper for your reader.

Chronology or Narrative ​

Do you need to develop support for a topic where telling a story can illustrate some important concept for your readers? Material arranged chronologically is explained as it occurs in time. A chronological or narrative method of development might help you find a way to add both interest and content to your essay. Material arranged chronologically is explained as it occurs in time. This pattern may be used to establish what has happened. Chronology or narrative can be a great way to introduce your essay by providing a background or history behind your topic. Or you may want to tell a story to develop one or more points in the body of your essay. You can use transitional words like then, next, and finally to make the parts of the chronology clear.

Comparison and Contrast ​

Are you trying to define something? Do you need your readers to understand what something is and what it is not? The comparison-and-contrast method of development is particularly useful in extending a definition, or anywhere you need to show how a subject is like or unlike another subject. For example, the statement is often made that drug abuse is a medical problem instead of a criminal justice issue. An author might attempt to prove this point by comparing drug addiction to AIDS, cancer, or heart disease to redefine the term “addiction” as a medical problem. A statement in opposition to this idea could just as easily establish contrast by explaining all the ways that addiction is different from what we traditionally understand as an illness. In seeking to establish comparison or contrast in your writing, some words or terms that might be useful are by contrast, in comparison, while, some, and others.

Summary

These four methods of development—cause and effect, problem-solution, chronology or narrative, and comparison and contrast—are just a few ways to organize and develop ideas and content in your essays. It’s important to note that they should not be a starting point for writers who want to write something authentic—something that they care deeply about. Instead, they can be a great way to help you look for what’s already happening with your topic or in a draft, to help you to write more, or to help you reorganize some parts of an essay that seem to lack connection or feel disjointed. Look for organizational patterns when you’re reading work by professional writers. Notice where they combine strategies (e.g., a problem-solution pattern that uses cause-and-effect organization, or a comparison-contrast pattern that uses narrative or chronology to develop similarities or differences). Pay attention to how different writers emphasize and develop their main ideas, and use what you find to inspire you in your own writing. Better yet, work on developing completely new patterns of your own.

Reference

  • This chapter was adapted from “Patterns of Organization and Methods of Development” in The Word on College Reading and Writing by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a CC BY-NC 4.0 Licence. Adapted by Allison Kilgannon.

  • “Peach and lavender pie” by Heather Joan is licensed under a CC BY-NC-ND 2.0 Licence.

  • “General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a CC BY-NC 4.0 Licence.

  • Kilgannon, Allison. “Patterns of Organization and Methods of Development.” Opentextbc.ca, 20 Aug. 2021, opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development.

- +
Toshiki's Note

Appearance

Patterns of Organization and Methods of Development ​

Author:Anda Toshiki
Updated:a minute ago
Words:2.2k
Reading:13 min

Patterns of organization can help your readers follow the ideas within your essay and your paragraphs, but they can also work as methods of development to help you recognize and further develop ideas and relationships in your writing. Here are some strategies that can help you with both organization and development in your essays.

Major Patterns of Organization ​

A fruit pie.

Read the following sentences:

  • Now take the pie out of the oven and let it cool on the stovetop.
  • Mix the dry ingredients with the liquid ingredients.
  • Set the pie crust aside while you make the filling.

How did it feel to read the above list? A bit confusing, I would guess. That’s because the steps for making a pie were not well organized, and the steps don’t include enough detail for us to know exactly what we should do. (Like what are the dry and liquid ingredients?) We all know that starting instructions from the beginning and giving each detailed step in the order it should happen is vital to having a good outcome, in this case a yummy pie! But it’s not always so simple to know how to organize or develop ideas, and sometimes there’s more than one way, which complicates things even further.

First, let’s take a look at a couple of ways to think about organization.

General to Specific or Specific to General ​

It might be useful to think about organizing your topic like a triangle:

Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General.

The first triangle represents starting with the most general, big picture information first, moving then to more detailed and often more personal information later in the paper. The second triangle represents an organizational structure that starts with the specific, small scale information first and then moves to the more global, big picture stuff.

For example, if your topic is traffic in Vancouver, British Columbia, an essay that uses the general-to-specific organizational structure might begin this way:

Many people consider Vancouver, British Columbia, to be a relaxed place to live. They would be shocked to know how bad the traffic is traveling major arteries into the city and even driving around the city itself.

An essay that uses the specific-to-general structure might start like this:

Transit is crowded, parking is expensive, and vehicles stop and go through the main streets of the city of Vancouver, British Columbia, and that is just once travelers brave the crowded arteries to enter the city; Vancouver’s traffic problem does not lend itself to the relaxed atmosphere many believe the city to have.

What’s the difference between these two introductions? And how might they appeal to the intended audience for this essay in different ways? The first introduction is looking at the big picture of the problem and mentions pollution’s impact on all citizens in Portland, while the second introduction focuses on one specific family. The first helps readers see how vast the problem really is, and the second helps connect readers to a real family, making an emotional appeal from the very beginning. Neither introduction is necessarily better. You’ll choose one over the other based on the kind of tone you’d like to create and how you’d like to affect your audience. It’s completely up to you to make this decision.

Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? ​

The triangle is kind of a general guide, meaning you’re allowed to move around within it all you want. For example, it’s possible that each of your paragraphs will be its own triangle, starting with the general or specific and moving out or in. However, if you begin very broadly, it might be effective to end your essay in a more specific, personal way. And if you begin with a personal story, consider ending your essay by touching on the global impact and importance of your topic.

Are There Other Ways to Think about Organizing My Ideas? ​

Yes! Rather than thinking about which of your ideas are most specific or personal or which are more broad or universal, you might consider one of the following ways of organizing your ideas:

  • Most important information first (consider what you want readers to focus on first)
  • Chronological order (the order in time that events take place)
  • Compare and contrast (ideas are organized together because of their relationship to each other)

The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.

Choose one of the following topics, and practice writing a few opening sentences like we did above, once using the general-to-specific format and once using the specific-to-general. Which do you like better? What audience would be attracted to which one? Share with peers to see how others tackled this challenge. How would you rewrite their sentences? Why? Discuss your changes and listen to how your peers have revised your sentences. Taking in other people’s ideas will help you see new ways to approach your own writing and thinking.
Topics:

  • Facing fears
  • Safety in sports
  • Community policing
  • Educating prisoners
  • Sex education

Methods of Development ​

The methods of development covered here are best used as ways to look at what’s already happening in your draft and to consider how you might emphasize or expand on any existing patterns. You might already be familiar with some of these patterns because teachers will sometimes assign them as the purpose for writing an essay. For example, you might have been asked to write a cause-and-effect essay or a comparison-and-contrast essay.

It’s important to emphasize here that patterns of organization or methods of developing content usually happen naturally as a consequence of the way the writer engages with and organizes information while writing. That is to say, most writers don’t sit down and say, “I think I’ll write a cause-and-effect essay today.” Instead, a writer might be more likely to be interested in a topic, say, the state of drinking water in the local community, and as the writer begins to explore the topic, certain cause-and-effect relationships between environmental pollutants and the community water supply may begin to emerge.

So if these patterns just occur naturally in writing, what’s the use in knowing about them? Well, sometimes you might be revising a draft and notice that some of your paragraphs are a bit underdeveloped. Maybe they lack a clear topic, or maybe they lack support. In either case, you can look to these common methods of development to find ways to sharpen those vague topics or to add support where needed. Do you have a clear cause statement somewhere but you haven’t explored the effects? Are you lacking detail somewhere where a narrative story or historical chronology can help build reader interest and add support? Are you struggling to define an idea that might benefit from some comparison or contrast? Read on to consider some of the ways that these strategies can help you in revision.

Cause and Effect (or Effect and Cause) ​

Do you see a potential cause-and-effect relationship developing in your draft? The cause-and-effect pattern may be used to identify one or more causes followed by one or more effects or results. Or you may reverse this sequence and describe effects first and then the cause or causes. For example, the causes of water pollution might be followed by its effects on both humans and animals. You may use obvious transitions to clarify cause and effect, such as “What are the results? Here are some of them…” or you might simply use the words cause, effect, and result, to cue the reader about your about the relationships that you’re establishing.

Problem-Solution ​

At some point does your essay explore a problem or suggest a solution? The problem-solution pattern is commonly used in identifying something that’s wrong and in contemplating what might be done to remedy the situation. There are probably more ways to organize a problem-solution approach, but but here are three possibilities:

  • Describe the problem, followed by the solution.
  • Propose the solution first and then describe the problems that motivated it.
  • Or a problem may be followed by several solutions, one of which is selected as the best.

When the solution is stated at the end of the paper, the pattern is sometimes called the delayed proposal. For a hostile audience, it may be effective to describe the problem, show why other solutions do not work, and finally suggest the favored solution. You can emphasize the words problem and solution to signal these sections of your paper for your reader.

Chronology or Narrative ​

Do you need to develop support for a topic where telling a story can illustrate some important concept for your readers? Material arranged chronologically is explained as it occurs in time. A chronological or narrative method of development might help you find a way to add both interest and content to your essay. Material arranged chronologically is explained as it occurs in time. This pattern may be used to establish what has happened. Chronology or narrative can be a great way to introduce your essay by providing a background or history behind your topic. Or you may want to tell a story to develop one or more points in the body of your essay. You can use transitional words like then, next, and finally to make the parts of the chronology clear.

Comparison and Contrast ​

Are you trying to define something? Do you need your readers to understand what something is and what it is not? The comparison-and-contrast method of development is particularly useful in extending a definition, or anywhere you need to show how a subject is like or unlike another subject. For example, the statement is often made that drug abuse is a medical problem instead of a criminal justice issue. An author might attempt to prove this point by comparing drug addiction to AIDS, cancer, or heart disease to redefine the term “addiction” as a medical problem. A statement in opposition to this idea could just as easily establish contrast by explaining all the ways that addiction is different from what we traditionally understand as an illness. In seeking to establish comparison or contrast in your writing, some words or terms that might be useful are by contrast, in comparison, while, some, and others.

Summary

These four methods of development—cause and effect, problem-solution, chronology or narrative, and comparison and contrast—are just a few ways to organize and develop ideas and content in your essays. It’s important to note that they should not be a starting point for writers who want to write something authentic—something that they care deeply about. Instead, they can be a great way to help you look for what’s already happening with your topic or in a draft, to help you to write more, or to help you reorganize some parts of an essay that seem to lack connection or feel disjointed. Look for organizational patterns when you’re reading work by professional writers. Notice where they combine strategies (e.g., a problem-solution pattern that uses cause-and-effect organization, or a comparison-contrast pattern that uses narrative or chronology to develop similarities or differences). Pay attention to how different writers emphasize and develop their main ideas, and use what you find to inspire you in your own writing. Better yet, work on developing completely new patterns of your own.

Reference

  • This chapter was adapted from “Patterns of Organization and Methods of Development” in The Word on College Reading and Writing by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a CC BY-NC 4.0 Licence. Adapted by Allison Kilgannon.

  • “Peach and lavender pie” by Heather Joan is licensed under a CC BY-NC-ND 2.0 Licence.

  • “General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a CC BY-NC 4.0 Licence.

  • Kilgannon, Allison. “Patterns of Organization and Methods of Development.” Opentextbc.ca, 20 Aug. 2021, opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development.

+ \ No newline at end of file diff --git a/academic/physics/index.html b/academic/physics/index.html index d571f66b..683bf797 100644 --- a/academic/physics/index.html +++ b/academic/physics/index.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Welcome to Physics ​

Author:Anda Toshiki
Updated:a minute ago
Words:3
Reading:1 min
- +
Toshiki's Note

Appearance

Welcome to Physics ​

Author:Anda Toshiki
Updated:a minute ago
Words:3
Reading:1 min
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/1.html b/academic/physics/ipho-formulas-jpn/1.html index 4e7a097b..ff197a2d 100644 --- a/academic/physics/ipho-formulas-jpn/1.html +++ b/academic/physics/ipho-formulas-jpn/1.html @@ -13,7 +13,7 @@ - + @@ -56,8 +56,8 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​​

1.13: 積分 ​

  1. 積分：微分の公式の左辺と右辺を入れ替えたものと同 じ（逆演算）.例えば,

    ∫xn dx=xn+1/(n+1).\int x^n \mathrm{~d} x=x^{n+1} /(n+1) .

    置換積分の特別な場合 :

    ∫f(ax+b)dx=F(ax+b)/a.\int f(a x+b) \mathrm{d} x=F(a x+b) / a .

1.14: 円錐曲線 ​

  1. 円錐曲線: a11x2+2a12xy+a22y2+a1x+a2y+a0=a_{11} x^2+2 a_{12} x y+a_{22} y^2+a_1 x+a_2 y+a_0= 0 で, a11=a22a_{11}=a_{22} ならば円, a11(a11a22−a122)>0a_{11}\left(a_{11} a_{22}-a_{12}^2\right)>0 ならば楕円, ⋯<0\cdots<0 ならば双曲線, a11a22−a122=0a_{11} a_{22}-a_{12}^2=0 ならば放物線. 楕円 : l1+l2=2a,α1=α2l_1+l_2=2 a, \alpha_1=\alpha_2 [訳 者注 : 焦点と曲線上の点を結ぶ直線と接線とのなす角 ], A=πabA=\pi a b. 双曲線 : l1−l2=2a,α1+α2=0l_1-l_2=2 a, \alpha_1+\alpha_2=0. 放物線 :l+h=: l+h= const., α1=α2\alpha_1=\alpha_2.

1.15: 数值計算 & 台形規則 ​

  1. 数值計算. f(x)=0f(x)=0 の解を求めるニュートン法 :

    xn+1=xn−f(xn)/f′(xn)x_{n+1}=x_n-f\left(x_n\right) / f^{\prime}\left(x_n\right)

    近似積分の台形規則：

    ∫abf(x)dx≈b−a2n[f(x0)+2{f(x1)+⋯ +f(xn−1)}+f(xn)]\begin{array}{r} \int_a^b f(x) \mathrm{d} x \approx \frac{b-a}{2 n}\left[f\left(x_0\right)+2\left\{f\left(x_1\right)+\cdots\right.\right. \left.\left.+f\left(x_{n-1}\right)\right\}+f\left(x_n\right)\right] \end{array}

1.16: ベクトルの微分 & 積分 ​

  1. ベクトルの微分と積分：成分ごとに微分/積分する．あるいは無限に近い22つのベクトルの差を求めることで 微分す.
- +M1001 80h400000v40h-400000z">​​

1.13: 積分 ​

  1. 積分：微分の公式の左辺と右辺を入れ替えたものと同 じ（逆演算）.例えば,

    ∫xn dx=xn+1/(n+1).\int x^n \mathrm{~d} x=x^{n+1} /(n+1) .

    置換積分の特別な場合 :

    ∫f(ax+b)dx=F(ax+b)/a.\int f(a x+b) \mathrm{d} x=F(a x+b) / a .

1.14: 円錐曲線 ​

  1. 円錐曲線: a11x2+2a12xy+a22y2+a1x+a2y+a0=a_{11} x^2+2 a_{12} x y+a_{22} y^2+a_1 x+a_2 y+a_0= 0 で, a11=a22a_{11}=a_{22} ならば円, a11(a11a22−a122)>0a_{11}\left(a_{11} a_{22}-a_{12}^2\right)>0 ならば楕円, ⋯<0\cdots<0 ならば双曲線, a11a22−a122=0a_{11} a_{22}-a_{12}^2=0 ならば放物線. 楕円 : l1+l2=2a,α1=α2l_1+l_2=2 a, \alpha_1=\alpha_2 [訳 者注 : 焦点と曲線上の点を結ぶ直線と接線とのなす角 ], A=πabA=\pi a b. 双曲線 : l1−l2=2a,α1+α2=0l_1-l_2=2 a, \alpha_1+\alpha_2=0. 放物線 :l+h=: l+h= const., α1=α2\alpha_1=\alpha_2.

1.15: 数值計算 & 台形規則 ​

  1. 数值計算. f(x)=0f(x)=0 の解を求めるニュートン法 :

    xn+1=xn−f(xn)/f′(xn)x_{n+1}=x_n-f\left(x_n\right) / f^{\prime}\left(x_n\right)

    近似積分の台形規則：

    ∫abf(x)dx≈b−a2n[f(x0)+2{f(x1)+⋯ +f(xn−1)}+f(xn)]\begin{array}{r} \int_a^b f(x) \mathrm{d} x \approx \frac{b-a}{2 n}\left[f\left(x_0\right)+2\left\{f\left(x_1\right)+\cdots\right.\right. \left.\left.+f\left(x_{n-1}\right)\right\}+f\left(x_n\right)\right] \end{array}

1.16: ベクトルの微分 & 積分 ​

  1. ベクトルの微分と積分：成分ごとに微分/積分する．あるいは無限に近い22つのベクトルの差を求めることで 微分す.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/10.html b/academic/physics/ipho-formulas-jpn/10.html index 315f4b17..3f0c6ecb 100644 --- a/academic/physics/ipho-formulas-jpn/10.html +++ b/academic/physics/ipho-formulas-jpn/10.html @@ -13,7 +13,7 @@ - + @@ -55,8 +55,8 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​,ν=vnS.

10.10: Carnot サイクル ​

  1. Carnot サイクル : 断熱過程 2 つと等温過程 2 つ. S−TS-T 座標を用いることにより η=(T1−T2)/T1\eta=\left(T_1-T_2\right) / T_1 を得る.

10.11: ヒートポンプ ​

  1. ヒートポンプ: Carnot サイクルの逆. η=T1T1−T2\eta=\frac{T_1}{T_1-T_2}.

10.12: エントロピー ​

  1. エントロピー :dS=dQ/T: \mathrm{d} S=\mathrm{d} Q / T.

10.13: 熱力学第一法則 ​

  1. 熱力学第一法則 : d′U=d′A+d′Q\mathrm{d}^{\prime} U=\mathrm{d}^{\prime} A+\mathrm{d}^{\prime} Q

10.14: 熱力学第二法則 ​

  1. 熱力学第二法則 : ΔS≥0\Delta S \geq 0 (また ηreal ≤ηCarnot )\left.\eta_{\text {real }} \leq \eta_{\text {Carnot }}\right).

10.15: 気体のする仕事 ​

  1. 気体のする仕事（ポイント 10 も参照）:

    A=∫p dV, 断熱過程: A=i2Δ(pV)A=\int p \mathrm{~d} V, \quad \text { 断熱過程: } A=\frac{i}{2} \Delta(p V)

10.16: Dalton の法則 ​

  1. Dalton の法則: p=p= ∑pi\sum p_i

    訳者注

    理想気体のみ成立

10.17: 沸騰 ​

  1. 沸騰: 飽和蒸気の圧力 pv=p0.2p_v=p_0 .2 液の界面では pv1+pv2=p0p_{v 1}+p_{v 2}=p_0.

10.18: 熱流 ​

  1. 熱流: P=kSΔT/lP=k S \Delta T / l ( kk は熱伝導率). 直流回路に似て いる (P↔I,ΔT↔V,k↔1/ρ)(P \leftrightarrow I, \Delta T \leftrightarrow V, k \leftrightarrow 1 / \rho).

10.19: 熱容量 ​

  1. 熱容量 : Q=∫c(T)dTQ=\int c(T) \mathrm{d} T. 固体では低温で c∝T3c \propto T^3, 高温で c=3NkBc=3 N k_B （Dulong-Petit の法則. ここで NN は結晶中の原子数）

10.20: 表面張力 ​

  1. 表面張力 :

    U=Sσ,F=lσ,p=2σ/RU=S \sigma, F=l \sigma, p=2 \sigma / R

10.21: Stefan-Boltzmann の法則 (灰色体) ​

  1. Stefan-Boltzmann の法則 (灰色体) : P=εσAT4P=\varepsilon \sigma A T^4.

10.22: Wien の変位則 ​

  1. Wien の変位則: νmax⁡=AkBT/h(A≈\nu_{\max }=A k_B T / h(A \approx 2.8), λmax⁡=hc/A′kBT(A′≈5)\lambda_{\max }=h c / A^{\prime} k_B T\left(A^{\prime} \approx 5\right).
- +M1001 80h400000v40h-400000z">​,ν=vnS.

10.10: Carnot サイクル ​

  1. Carnot サイクル : 断熱過程 2 つと等温過程 2 つ. S−TS-T 座標を用いることにより η=(T1−T2)/T1\eta=\left(T_1-T_2\right) / T_1 を得る.

10.11: ヒートポンプ ​

  1. ヒートポンプ: Carnot サイクルの逆. η=T1T1−T2\eta=\frac{T_1}{T_1-T_2}.

10.12: エントロピー ​

  1. エントロピー :dS=dQ/T: \mathrm{d} S=\mathrm{d} Q / T.

10.13: 熱力学第一法則 ​

  1. 熱力学第一法則 : d′U=d′A+d′Q\mathrm{d}^{\prime} U=\mathrm{d}^{\prime} A+\mathrm{d}^{\prime} Q

10.14: 熱力学第二法則 ​

  1. 熱力学第二法則 : ΔS≥0\Delta S \geq 0 (また ηreal ≤ηCarnot )\left.\eta_{\text {real }} \leq \eta_{\text {Carnot }}\right).

10.15: 気体のする仕事 ​

  1. 気体のする仕事（ポイント 10 も参照）:

    A=∫p dV, 断熱過程: A=i2Δ(pV)A=\int p \mathrm{~d} V, \quad \text { 断熱過程: } A=\frac{i}{2} \Delta(p V)

10.16: Dalton の法則 ​

  1. Dalton の法則: p=p= ∑pi\sum p_i

    訳者注

    理想気体のみ成立

10.17: 沸騰 ​

  1. 沸騰: 飽和蒸気の圧力 pv=p0.2p_v=p_0 .2 液の界面では pv1+pv2=p0p_{v 1}+p_{v 2}=p_0.

10.18: 熱流 ​

  1. 熱流: P=kSΔT/lP=k S \Delta T / l ( kk は熱伝導率). 直流回路に似て いる (P↔I,ΔT↔V,k↔1/ρ)(P \leftrightarrow I, \Delta T \leftrightarrow V, k \leftrightarrow 1 / \rho).

10.19: 熱容量 ​

  1. 熱容量 : Q=∫c(T)dTQ=\int c(T) \mathrm{d} T. 固体では低温で c∝T3c \propto T^3, 高温で c=3NkBc=3 N k_B （Dulong-Petit の法則. ここで NN は結晶中の原子数）

10.20: 表面張力 ​

  1. 表面張力 :

    U=Sσ,F=lσ,p=2σ/RU=S \sigma, F=l \sigma, p=2 \sigma / R

10.21: Stefan-Boltzmann の法則 (灰色体) ​

  1. Stefan-Boltzmann の法則 (灰色体) : P=εσAT4P=\varepsilon \sigma A T^4.

10.22: Wien の変位則 ​

  1. Wien の変位則: νmax⁡=AkBT/h(A≈\nu_{\max }=A k_B T / h(A \approx 2.8), λmax⁡=hc/A′kBT(A′≈5)\lambda_{\max }=h c / A^{\prime} k_B T\left(A^{\prime} \approx 5\right).
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/11.html b/academic/physics/ipho-formulas-jpn/11.html index e20d307d..eb895560 100644 --- a/academic/physics/ipho-formulas-jpn/11.html +++ b/academic/physics/ipho-formulas-jpn/11.html @@ -13,7 +13,7 @@ - + @@ -46,8 +46,8 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​) であれば容易に透過する.

11.7: Bohr モデル ​

  1. Bohr モデル : En∝−1/n2E_n \propto-1 / n^2. (古典的に計算される) 円軌道では, 軌道の長さが波長 λ=h/mv\lambda=h / m v の整数倍.

11.8: Compton 効果 ​

  1. Compton 効果: 光子が電子から散乱されると, 光子の Δλ=λC(1−cos⁡θ)\Delta \lambda=\lambda_C(1-\cos \theta)

11.9: 光電効果 ​

  1. 光電効果: W+mvmax⁡2/2=hνW+m v_{\max }^2 / 2=h \nu ( WW は仕事関数 )). I−VI-V グラフ：光電流は阻止電圧 V=−(hν−W)/eV=-(h \nu-W) / e で始まり, 正方向に電圧が大きくなると緩和する.

11.10: Stefan-Boltzmann の法則 ​

  1. Stefan-Boltzmann の法則 : P=σAT4P=\sigma A T^4
- +M1001 80h400000v40h-400000z">​) であれば容易に透過する.

11.7: Bohr モデル ​

  1. Bohr モデル : En∝−1/n2E_n \propto-1 / n^2. (古典的に計算される) 円軌道では, 軌道の長さが波長 λ=h/mv\lambda=h / m v の整数倍.

11.8: Compton 効果 ​

  1. Compton 効果: 光子が電子から散乱されると, 光子の Δλ=λC(1−cos⁡θ)\Delta \lambda=\lambda_C(1-\cos \theta)

11.9: 光電効果 ​

  1. 光電効果: W+mvmax⁡2/2=hνW+m v_{\max }^2 / 2=h \nu ( WW は仕事関数 )). I−VI-V グラフ：光電流は阻止電圧 V=−(hν−W)/eV=-(h \nu-W) / e で始まり, 正方向に電圧が大きくなると緩和する.

11.10: Stefan-Boltzmann の法則 ​

  1. Stefan-Boltzmann の法則 : P=σAT4P=\sigma A T^4
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/12.html b/academic/physics/ipho-formulas-jpn/12.html index d471dfc2..5d625705 100644 --- a/academic/physics/ipho-formulas-jpn/12.html +++ b/academic/physics/ipho-formulas-jpn/12.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 12 ​

Author:Anda Toshiki
Updated:a minute ago
Words:332
Reading:1 min

12: Kepler の法則 ​

12.1: F & U ​

  1. F=GMm/r2,U=−GMm/r. F=G M m / r^2, U=-G M m / r \text {. }

12.2: Kepler の第一法則 ​

  1. Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).

12.3: Kepler の第二法則 ​

  1. Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.

12.4: Kepler の第三法則 ​

  1. Kepler の第三法則 :r−2: r^{-2} に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の 32\frac{3}{2} 乗 に比例する :

    T12/T22=a13/a23T_1^2 / T_2^2=a_1^3 / a_2^3

12.5: 楕円軌道 ​

  1. 重力場中で楕円軌道を描く質点の全エネルギー (K+U)(K+U) :

    E=−GMm/2aE=-G M m / 2 a

12.6: 楕円率 ​

  1. 楕円率が ε=d/a≪1\varepsilon=d / a \ll 1 の場合, 軌道は焦点をずらし た円形をしていると考えられる.

12.7: 楕円の性質 ​

  1. 楕円の性質 :l1+l2=2a（l1: l_1+l_2=2 a （ l_1 と l2l_2 は焦点までの距 離). α1=α2\alpha_1=\alpha_2 (一方の焦点から出た光は他方の焦点に 反射する). S=πabS=\pi a b.

12.8: 円の中心 ​

  1. 円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.

12.9: nge-Lenz ベクトル ​

  1. Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 ee に一致する.]：

    e=v×MGMm−er= const. \boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. }

- +
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 12 ​

Author:Anda Toshiki
Updated:a minute ago
Words:332
Reading:1 min

12: Kepler の法則 ​

12.1: F & U ​

  1. F=GMm/r2,U=−GMm/r. F=G M m / r^2, U=-G M m / r \text {. }

12.2: Kepler の第一法則 ​

  1. Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).

12.3: Kepler の第二法則 ​

  1. Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.

12.4: Kepler の第三法則 ​

  1. Kepler の第三法則 :r−2: r^{-2} に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の 32\frac{3}{2} 乗 に比例する :

    T12/T22=a13/a23T_1^2 / T_2^2=a_1^3 / a_2^3

12.5: 楕円軌道 ​

  1. 重力場中で楕円軌道を描く質点の全エネルギー (K+U)(K+U) :

    E=−GMm/2aE=-G M m / 2 a

12.6: 楕円率 ​

  1. 楕円率が ε=d/a≪1\varepsilon=d / a \ll 1 の場合, 軌道は焦点をずらし た円形をしていると考えられる.

12.7: 楕円の性質 ​

  1. 楕円の性質 :l1+l2=2a（l1: l_1+l_2=2 a （ l_1 と l2l_2 は焦点までの距 離). α1=α2\alpha_1=\alpha_2 (一方の焦点から出た光は他方の焦点に 反射する). S=πabS=\pi a b.

12.8: 円の中心 ​

  1. 円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.

12.9: nge-Lenz ベクトル ​

  1. Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 ee に一致する.]：

    e=v×MGMm−er= const. \boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. }

+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/13.html b/academic/physics/ipho-formulas-jpn/13.html index 02641dc1..3bc22a76 100644 --- a/academic/physics/ipho-formulas-jpn/13.html +++ b/academic/physics/ipho-formulas-jpn/13.html @@ -13,7 +13,7 @@ - + @@ -96,8 +96,8 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​

13.11: 電場と磁場の Lorentz 変換 ​

  1. 電場と磁場の Lorentz 変換 : E∥′=E∥,B∥′=B∥\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|},

    E⊥′/c=γ(E⊥/c+v/c×B⊥),B⊥′=γ(B⊥−v/c×E⊥/c)\begin{gathered} \boldsymbol{E}_{\perp}^{\prime} / c=\gamma\left(\boldsymbol{E}_{\perp} / c+\boldsymbol{v} / c \times \boldsymbol{B}_{\perp}\right), \\ \boldsymbol{B}_{\perp}^{\prime}=\gamma\left(\boldsymbol{B}_{\perp}-\boldsymbol{v} / c \times \boldsymbol{E}_{\perp} / c\right) \end{gathered}

- +M1001 80h400000v40h-400000z">​

13.11: 電場と磁場の Lorentz 変換 ​

  1. 電場と磁場の Lorentz 変換 : E∥′=E∥,B∥′=B∥\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|},

    E⊥′/c=γ(E⊥/c+v/c×B⊥),B⊥′=γ(B⊥−v/c×E⊥/c)\begin{gathered} \boldsymbol{E}_{\perp}^{\prime} / c=\gamma\left(\boldsymbol{E}_{\perp} / c+\boldsymbol{v} / c \times \boldsymbol{B}_{\perp}\right), \\ \boldsymbol{B}_{\perp}^{\prime}=\gamma\left(\boldsymbol{B}_{\perp}-\boldsymbol{v} / c \times \boldsymbol{E}_{\perp} / c\right) \end{gathered}

+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/2.html b/academic/physics/ipho-formulas-jpn/2.html index 370f3a69..d4c0a7aa 100644 --- a/academic/physics/ipho-formulas-jpn/2.html +++ b/academic/physics/ipho-formulas-jpn/2.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:210
Reading:1 min

2: 一般的な推奨事 ​

  1. 全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または ∞\infty ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.
  2. 問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む.
  3. 長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする.
  4. 絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける.
  5. 実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.
- +
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 2 ​

Author:Anda Toshiki
Updated:a minute ago
Words:210
Reading:1 min

2: 一般的な推奨事 ​

  1. 全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または ∞\infty ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.
  2. 問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む.
  3. 長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする.
  4. 絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける.
  5. 実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/3.html b/academic/physics/ipho-formulas-jpn/3.html index cb59587c..89e4a7b3 100644 --- a/academic/physics/ipho-formulas-jpn/3.html +++ b/academic/physics/ipho-formulas-jpn/3.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 3 ​

Author:Anda Toshiki
Updated:a minute ago
Words:473
Reading:2 min

3: 運動学 ​

3.1: 質点 ​

  1. 質点または剛体の並進運動の場合（積分 → グラフの下 の面積）：

    v=dxdt,x=∫vdt(x=∫vx dt など )a=dvdt=d2xdt2,v=∫adtt=∫vx−1 dx=∫ax−1 dvx,x=∫vxax dvx\begin{gathered} \boldsymbol{v}=\frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d} t}, \boldsymbol{x}=\int \boldsymbol{v} \mathrm{d} t\left(x=\int v_x \mathrm{~d} t \text { など }\right) \\ \boldsymbol{a}=\frac{\mathrm{d} \boldsymbol{v}}{\mathrm{d} t}=\frac{\mathrm{d}^2 \boldsymbol{x}}{\mathrm{d} t^2}, \boldsymbol{v}=\int \boldsymbol{a} \mathrm{d} t \\ t=\int v_x^{-1} \mathrm{~d} x=\int a_x^{-1} \mathrm{~d} v_x, x=\int \frac{v_x}{a_x} \mathrm{~d} v_x \end{gathered}

    もし aa が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば

    x=v0t+at2/2=(v2−v02)/2a. x=v_0 t+a t^2 / 2=\left(v^2-v_0^2\right) / 2 a \text {. }

3.2: 回転運動 ​

  1. 回転運動は, 並進運動と似ていて：

    ω=dφ/dt,ε=dω/dta=τdv/dt+nv2/R\begin{aligned} \omega & =\mathrm{d} \varphi / \mathrm{d} t, \varepsilon=\mathrm{d} \omega / \mathrm{d} t \\ \boldsymbol{a} & =\boldsymbol{\tau} \mathrm{d} v / \mathrm{d} t+\boldsymbol{n} v^2 / R \end{aligned}

3.3: 曲線運動 ​

  1. 曲線運動は，ポイント 1 と同じだが，ベクトルは線速 度，加速度，経路長に置き換える.

3.4: 剛体の運動 ​

  1. 剛体の運動:
    • vAcos⁡α=vBcos⁡βv_A \cos \alpha=v_B \cos \beta ここで, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B は剛体上の点 AA と BB の速度, α\alpha と β\beta は vA\boldsymbol{v}_A と vB\boldsymbol{v}_B が直線 ABA B となす角.
    • 瞬間回転中心 (#質点の軌道 の曲率中心）は, a\boldsymbol{a} と b\boldsymbol{b} に下ろした垂線の交点. 又は もし vA,vB⊥AB\boldsymbol{v}_A, \boldsymbol{v}_B \perp A B ならば, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B の先端を結ぶ 直線と ABA B の交点.

3.5: 非慣性系 ​

  1. 非慣性系：

    v2=v0+v1,a2=a0+a1+ω2R+aCor ここで, aCor⊥v1. もし v1=0 なら aCor=0.\begin{array}{r} \quad \boldsymbol{v}_2=\boldsymbol{v}_0+\boldsymbol{v}_1, \boldsymbol{a}_2=\boldsymbol{a}_0+\boldsymbol{a}_1+\omega^2 \boldsymbol{R}+\boldsymbol{a}_{C o r} \\ \text { ここで, } \boldsymbol{a}_{C o r} \perp \boldsymbol{v}_1 . \text { もし } \boldsymbol{v}_1=0 \text { なら } \boldsymbol{a}_{C o r}=0 . \end{array}

3.6: 弾道問題 ​

  1. 弾道問題：到達可能な範囲は

    y≤v02/(2g)−gx2/(2v02)y \leq v_0^2 /(2 g)-g x^2 /\left(2 v_0^2\right)

    最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.

3.7: 最短経路 ​

  1. 最短経路を求めるには，Fermat と Huygens の原理が 使える.

3.8: ベクトル ​

  1. ベクトル（速度，加速度）を求めるには，その向きと （場合によっては傾いた）ある軸への射影を求めれば 充分.
- +
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 3 ​

Author:Anda Toshiki
Updated:a minute ago
Words:473
Reading:2 min

3: 運動学 ​

3.1: 質点 ​

  1. 質点または剛体の並進運動の場合（積分 → グラフの下 の面積）：

    v=dxdt,x=∫vdt(x=∫vx dt など )a=dvdt=d2xdt2,v=∫adtt=∫vx−1 dx=∫ax−1 dvx,x=∫vxax dvx\begin{gathered} \boldsymbol{v}=\frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d} t}, \boldsymbol{x}=\int \boldsymbol{v} \mathrm{d} t\left(x=\int v_x \mathrm{~d} t \text { など }\right) \\ \boldsymbol{a}=\frac{\mathrm{d} \boldsymbol{v}}{\mathrm{d} t}=\frac{\mathrm{d}^2 \boldsymbol{x}}{\mathrm{d} t^2}, \boldsymbol{v}=\int \boldsymbol{a} \mathrm{d} t \\ t=\int v_x^{-1} \mathrm{~d} x=\int a_x^{-1} \mathrm{~d} v_x, x=\int \frac{v_x}{a_x} \mathrm{~d} v_x \end{gathered}

    もし aa が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば

    x=v0t+at2/2=(v2−v02)/2a. x=v_0 t+a t^2 / 2=\left(v^2-v_0^2\right) / 2 a \text {. }

3.2: 回転運動 ​

  1. 回転運動は, 並進運動と似ていて：

    ω=dφ/dt,ε=dω/dta=τdv/dt+nv2/R\begin{aligned} \omega & =\mathrm{d} \varphi / \mathrm{d} t, \varepsilon=\mathrm{d} \omega / \mathrm{d} t \\ \boldsymbol{a} & =\boldsymbol{\tau} \mathrm{d} v / \mathrm{d} t+\boldsymbol{n} v^2 / R \end{aligned}

3.3: 曲線運動 ​

  1. 曲線運動は，ポイント 1 と同じだが，ベクトルは線速 度，加速度，経路長に置き換える.

3.4: 剛体の運動 ​

  1. 剛体の運動:
    • vAcos⁡α=vBcos⁡βv_A \cos \alpha=v_B \cos \beta ここで, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B は剛体上の点 AA と BB の速度, α\alpha と β\beta は vA\boldsymbol{v}_A と vB\boldsymbol{v}_B が直線 ABA B となす角.
    • 瞬間回転中心 (#質点の軌道 の曲率中心）は, a\boldsymbol{a} と b\boldsymbol{b} に下ろした垂線の交点. 又は もし vA,vB⊥AB\boldsymbol{v}_A, \boldsymbol{v}_B \perp A B ならば, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B の先端を結ぶ 直線と ABA B の交点.

3.5: 非慣性系 ​

  1. 非慣性系：

    v2=v0+v1,a2=a0+a1+ω2R+aCor ここで, aCor⊥v1. もし v1=0 なら aCor=0.\begin{array}{r} \quad \boldsymbol{v}_2=\boldsymbol{v}_0+\boldsymbol{v}_1, \boldsymbol{a}_2=\boldsymbol{a}_0+\boldsymbol{a}_1+\omega^2 \boldsymbol{R}+\boldsymbol{a}_{C o r} \\ \text { ここで, } \boldsymbol{a}_{C o r} \perp \boldsymbol{v}_1 . \text { もし } \boldsymbol{v}_1=0 \text { なら } \boldsymbol{a}_{C o r}=0 . \end{array}

3.6: 弾道問題 ​

  1. 弾道問題：到達可能な範囲は

    y≤v02/(2g)−gx2/(2v02)y \leq v_0^2 /(2 g)-g x^2 /\left(2 v_0^2\right)

    最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.

3.7: 最短経路 ​

  1. 最短経路を求めるには，Fermat と Huygens の原理が 使える.

3.8: ベクトル ​

  1. ベクトル（速度，加速度）を求めるには，その向きと （場合によっては傾いた）ある軸への射影を求めれば 充分.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/4.html b/academic/physics/ipho-formulas-jpn/4.html index 0b638fbc..66224631 100644 --- a/academic/physics/ipho-formulas-jpn/4.html +++ b/academic/physics/ipho-formulas-jpn/4.html @@ -13,7 +13,7 @@ - + @@ -45,8 +45,8 @@ c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z -M1001 80h400000v40h-400000z">​,l~=l+Ic​/ml​

4.12: 慣性モーメントの係数 ​

  1. 慣性モーメントの係数 : 円柱 12\frac{1}{2}, 球 25\frac{2}{5}, 球殼 23\frac{2}{3}, 棒 112\frac{1}{12} （端に対しては 13\frac{1}{3} ), 正方形 16\frac{1}{6}.

4.13: よく使われる保存則 ​

  1. よく使われる保存則：エネルギー（弾性衝突，摩擦な し), 運動量（正味の外力なし, 各方向について成立), 角運動量（正味の外トルクなし, 例えば, 外力のうでの 長さが 0 (これが 2 又は 3 点のまわりに成り立てば運 動量保存で代用できる))

4.14: 非慣性系における見かけの力 ​

  1. 非慣性系における見かけの力 : 慣性力 −ma-m a, 遠心力 mω2Rm \omega^2 \boldsymbol{R}, Coriolis 力 2mv×Ω2 m \boldsymbol{v} \times \Omega (避けた方がよい. 速 度に垂直なので仕事はしない).

4.15: 傾いた座標 ​

  1. 傾いた座標：斜面上での運動については, 斜面に平行 と垂直な方向に軸をとるのがよい。このとき重力加速 度は xx 成分と yy 成分をもつ. 軸は斜交することもある が, v=vxex+vyey\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y のとき vxv_x は v\boldsymbol{v} の xx 軸への射 影ではない.

4.16: 2 つの物体の衝突 ​

  1. 2 つの物体の衝突 : 保存されるのは, a) 全運動量, b) 全角運動量，c) 一方の物体の衝突点に関する角運動量, d) 全エネルギー（弾性衝突の場合， 摩擦がある場合 は, 摩擦力に垂直な方向の運動エネルギーが保存される. e) 衝突中に滑りが止まったならば，接触点の最終 速度は接触面上にある. f) 滑りが止まらなかったなら ば, 一方の物体から他方に伝わる運動量は, 接触面の 法線と arctan⁡μ\arctan \mu の角度をなす.

4.17: 剛体のすべての運動 ​

  1. 剛体のすべての運動は (物体の各点の速度を見ると) 瞬 間回転中心 CC まわりの回転として表せる. 物体上の点 PP の CC からの距離は PP の軌跡の曲率半径とは異なる ことに注意せよ.

4.18: 紐の張力 ​

  1. 紐の張力：重さのある吊り紐では, 張力の水平成分は 一定で垂直成分は下にある紐の重さにより変わる. 滑 らかな面の上の紐による（単位長さあたりの）力は，そ の曲率半径と張力で決まり, N=T/RN=T / R. 似た場合と して, 表面張力による圧力は p=2σ/Rp=2 \sigma / R. 導出には直 径に沿った圧力を調ベる.

4.19: 液体の表面 ​

  1. 液体の表面は（表面張力を無視すれば）等ポテンシャ ル面になる. 非圧縮性流体では, ww をポテンシャルエ ネルギーの体積密度として, p=P0−wp=P_0-w.

4.20: 非圧縮性流体に対する Bernoulli の法則 ​

  1. 非圧縮性流体に対する Bernoulli の法則：

    p+12ρv2+ρϕ= const. p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. }

    一様な重力場では ϕ=gh\phi=g h. 比熱が cp[ J/kg]c_p[\mathrm{~J} / \mathrm{kg}] である気 体では,

    12v2+cpT= const. \frac{1}{2} v^2+c_p T=\text { const. }

4.21: 直線的な流線 ​

  1. 直線的な流線に沾う運動量の連続性 : p+ρv2=p+\rho v^2= const.

4.22: 断熱不変量 ​

  1. 断熱不変量 : 振動する系の 1 周期の間のパラメータの 相対的な変化が小さければ，位相空間（ x−px-p 座標で表さ れる）上に書かれるループの面積は非常に高い精度で 保存される.

4.23: 安定性 ​

  1. 安定性を調ベるには, a) ポテンシャルエネルギー最小 の原理，又は b) 仮想仕事の原理を用いる.

4.24: 空間的に有限な運動に対する Virial 定理 ​

  1. 空間的に有限な運動に対する Virial 定理：a) もし F∝∣r∣F \propto|\boldsymbol{r}| ならば ⟨K⟩=⟨U⟩\langle K\rangle=\langle U\rangle (時間平均). b) もし F∝∣r∣−2F \propto|\boldsymbol{r}|^{-2} ならば 2⟨K⟩=−⟨U⟩2\langle K\rangle=-\langle U\rangle.

4.25: Tsiolkovsky 公式 ​

  1. Tsiolkovsky の公式（ロケット）: Δv=uln⁡Mm\Delta v=u \ln \frac{M}{m}
- +M1001 80h400000v40h-400000z">​,l~=l+Ic​/ml​

4.12: 慣性モーメントの係数 ​

  1. 慣性モーメントの係数 : 円柱 12\frac{1}{2}, 球 25\frac{2}{5}, 球殼 23\frac{2}{3}, 棒 112\frac{1}{12} （端に対しては 13\frac{1}{3} ), 正方形 16\frac{1}{6}.

4.13: よく使われる保存則 ​

  1. よく使われる保存則：エネルギー（弾性衝突，摩擦な し), 運動量（正味の外力なし, 各方向について成立), 角運動量（正味の外トルクなし, 例えば, 外力のうでの 長さが 0 (これが 2 又は 3 点のまわりに成り立てば運 動量保存で代用できる))

4.14: 非慣性系における見かけの力 ​

  1. 非慣性系における見かけの力 : 慣性力 −ma-m a, 遠心力 mω2Rm \omega^2 \boldsymbol{R}, Coriolis 力 2mv×Ω2 m \boldsymbol{v} \times \Omega (避けた方がよい. 速 度に垂直なので仕事はしない).

4.15: 傾いた座標 ​

  1. 傾いた座標：斜面上での運動については, 斜面に平行 と垂直な方向に軸をとるのがよい。このとき重力加速 度は xx 成分と yy 成分をもつ. 軸は斜交することもある が, v=vxex+vyey\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y のとき vxv_x は v\boldsymbol{v} の xx 軸への射 影ではない.

4.16: 2 つの物体の衝突 ​

  1. 2 つの物体の衝突 : 保存されるのは, a) 全運動量, b) 全角運動量，c) 一方の物体の衝突点に関する角運動量, d) 全エネルギー（弾性衝突の場合， 摩擦がある場合 は, 摩擦力に垂直な方向の運動エネルギーが保存される. e) 衝突中に滑りが止まったならば，接触点の最終 速度は接触面上にある. f) 滑りが止まらなかったなら ば, 一方の物体から他方に伝わる運動量は, 接触面の 法線と arctan⁡μ\arctan \mu の角度をなす.

4.17: 剛体のすべての運動 ​

  1. 剛体のすべての運動は (物体の各点の速度を見ると) 瞬 間回転中心 CC まわりの回転として表せる. 物体上の点 PP の CC からの距離は PP の軌跡の曲率半径とは異なる ことに注意せよ.

4.18: 紐の張力 ​

  1. 紐の張力：重さのある吊り紐では, 張力の水平成分は 一定で垂直成分は下にある紐の重さにより変わる. 滑 らかな面の上の紐による（単位長さあたりの）力は，そ の曲率半径と張力で決まり, N=T/RN=T / R. 似た場合と して, 表面張力による圧力は p=2σ/Rp=2 \sigma / R. 導出には直 径に沿った圧力を調ベる.

4.19: 液体の表面 ​

  1. 液体の表面は（表面張力を無視すれば）等ポテンシャ ル面になる. 非圧縮性流体では, ww をポテンシャルエ ネルギーの体積密度として, p=P0−wp=P_0-w.

4.20: 非圧縮性流体に対する Bernoulli の法則 ​

  1. 非圧縮性流体に対する Bernoulli の法則：

    p+12ρv2+ρϕ= const. p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. }

    一様な重力場では ϕ=gh\phi=g h. 比熱が cp[ J/kg]c_p[\mathrm{~J} / \mathrm{kg}] である気 体では,

    12v2+cpT= const. \frac{1}{2} v^2+c_p T=\text { const. }

4.21: 直線的な流線 ​

  1. 直線的な流線に沾う運動量の連続性 : p+ρv2=p+\rho v^2= const.

4.22: 断熱不変量 ​

  1. 断熱不変量 : 振動する系の 1 周期の間のパラメータの 相対的な変化が小さければ，位相空間（ x−px-p 座標で表さ れる）上に書かれるループの面積は非常に高い精度で 保存される.

4.23: 安定性 ​

  1. 安定性を調ベるには, a) ポテンシャルエネルギー最小 の原理，又は b) 仮想仕事の原理を用いる.

4.24: 空間的に有限な運動に対する Virial 定理 ​

  1. 空間的に有限な運動に対する Virial 定理：a) もし F∝∣r∣F \propto|\boldsymbol{r}| ならば ⟨K⟩=⟨U⟩\langle K\rangle=\langle U\rangle (時間平均). b) もし F∝∣r∣−2F \propto|\boldsymbol{r}|^{-2} ならば 2⟨K⟩=−⟨U⟩2\langle K\rangle=-\langle U\rangle.

4.25: Tsiolkovsky 公式 ​

  1. Tsiolkovsky の公式（ロケット）: Δv=uln⁡Mm\Delta v=u \ln \frac{M}{m}
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/5.html b/academic/physics/ipho-formulas-jpn/5.html index 5febae35..a2ed1ddb 100644 --- a/academic/physics/ipho-formulas-jpn/5.html +++ b/academic/physics/ipho-formulas-jpn/5.html @@ -13,7 +13,7 @@ - + @@ -105,8 +105,8 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​.

5.10: Doppler 効果 ​

  1. Doppler 効果 : ν=ν01+v∥/cs1−u∥/cs\nu=\nu_0 \frac{1+v_{\|} / c_s}{1-u_{\|} / c_s}.

5.11: Huygens の原理 ​

  1. Huygens の原理 : 波面は段階的に構成される. 過去 の波面のすべての点に仮想的な波源を置く. 結果は距 離 Δx=csΔt\Delta x=c_s \Delta t で区切られた曲線（ここで Δt\Delta t は時間 間隔， csc_s は与えられた点の速度). 波は波面に垂直に 進む.
- +M1001 80h400000v40h-400000z">​.

5.10: Doppler 効果 ​

  1. Doppler 効果 : ν=ν01+v∥/cs1−u∥/cs\nu=\nu_0 \frac{1+v_{\|} / c_s}{1-u_{\|} / c_s}.

5.11: Huygens の原理 ​

  1. Huygens の原理 : 波面は段階的に構成される. 過去 の波面のすべての点に仮想的な波源を置く. 結果は距 離 Δx=csΔt\Delta x=c_s \Delta t で区切られた曲線（ここで Δt\Delta t は時間 間隔， csc_s は与えられた点の速度). 波は波面に垂直に 進む.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/6.html b/academic/physics/ipho-formulas-jpn/6.html index 0e745b99..8ece0a9b 100644 --- a/academic/physics/ipho-formulas-jpn/6.html +++ b/academic/physics/ipho-formulas-jpn/6.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 6 ​

Author:Anda Toshiki
Updated:a minute ago
Words:428
Reading:1 min

6: 幾何光学，測光 ​

6.1: Fermat 原理 ​

  1. Fermat の原理 : 点 AA から BB への波の経路は波の移動 時間が最も短いもの.

6.2: Snell 法則 ​

  1. Snell の法則 :

sin⁡α1/sin⁡α2=n2/n1=v1/v2.\sin \alpha_1 / \sin \alpha_2=n_2 / n_1=v_1 / v_2 .

6.3: 屈折率 ​

  1. 屈折率が連続的に変化するならば，媒質を屈折率が nn で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, n′=n/r(rn^{\prime}=n / r \quad(r は曲率半径 ))

6.4: 屈折率な座標 ​

  1. 屈折率が z\mathrm{z} 座標にのみ依存するならば, 光子の運動量 px,pyp_x, p_y とエネルギーは保存される:

kx,ky= const., ∣k∣/n= const. k_x, k_y=\text { const., }|\boldsymbol{k}| / n=\text { const. }

6.5:薄いレンズの式 ​

  1. 薄いレンズの式（符号に注意する）：

1/a+1/b=1/f≡D1 / a+1 / b=1 / f \equiv D

6.6: Newton の式 ​

  1. Newton の式 : 物体側焦点から物体までの距離を x1x_1, 像側焦点から像までの距離を x2x_2 とすると, x1x2=f2x_1 x_2=f^2

6.7: 像の位置を求める視差法 ​

  1. 像の位置を求める視差法 : 目の位置と垂直に動かした ときに，鉛筆の先が像に対してずれないような位置を 探す.

6.8: レンズを通る光線の経路の幾何学的な描き方 ​

  1. レンズを通る光線の経路の幾何学的な描き方：a) レン ズの中心を通る光線は屈折しない。b) 光軸に平行な光 線は焦点を通る，c) 屈折後, 初めに平行だった光線どうしは焦点面（焦点を通り光軸に垂直な平面）上で集 まる．d) 平面の像は平面であり，この 2 つの平面はレ ンズの平面上で交わる．

6.9: 光束 ​

  1. 光束 Φ\Phi [単位: lumen (lm⁡)(\operatorname{lm})] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, I=Φ/ΩI=\Phi / \Omega. 照度 [lux⁡(lx)][\operatorname{lux}(\mathrm{lx})] は（面に入射する） 面積あたりの光束で, E=Φ/SE=\Phi / S.

6.10: Gauss 定理 ​

  1. 光束についての Gauss の定理 : 光度 IiI_i の点光源を囲 む閉曲面を通って外に出る光束は, Φ=4π∑Ii\Phi=4 \pi \sum I_i. 光 源が 1 つで距離が rr のとき E=I/r2E=I / r^2

6.11: 実験のヒント ​

  1. 実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.
- +
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 6 ​

Author:Anda Toshiki
Updated:a minute ago
Words:428
Reading:1 min

6: 幾何光学，測光 ​

6.1: Fermat 原理 ​

  1. Fermat の原理 : 点 AA から BB への波の経路は波の移動 時間が最も短いもの.

6.2: Snell 法則 ​

  1. Snell の法則 :

sin⁡α1/sin⁡α2=n2/n1=v1/v2.\sin \alpha_1 / \sin \alpha_2=n_2 / n_1=v_1 / v_2 .

6.3: 屈折率 ​

  1. 屈折率が連続的に変化するならば，媒質を屈折率が nn で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, n′=n/r(rn^{\prime}=n / r \quad(r は曲率半径 ))

6.4: 屈折率な座標 ​

  1. 屈折率が z\mathrm{z} 座標にのみ依存するならば, 光子の運動量 px,pyp_x, p_y とエネルギーは保存される:

kx,ky= const., ∣k∣/n= const. k_x, k_y=\text { const., }|\boldsymbol{k}| / n=\text { const. }

6.5:薄いレンズの式 ​

  1. 薄いレンズの式（符号に注意する）：

1/a+1/b=1/f≡D1 / a+1 / b=1 / f \equiv D

6.6: Newton の式 ​

  1. Newton の式 : 物体側焦点から物体までの距離を x1x_1, 像側焦点から像までの距離を x2x_2 とすると, x1x2=f2x_1 x_2=f^2

6.7: 像の位置を求める視差法 ​

  1. 像の位置を求める視差法 : 目の位置と垂直に動かした ときに，鉛筆の先が像に対してずれないような位置を 探す.

6.8: レンズを通る光線の経路の幾何学的な描き方 ​

  1. レンズを通る光線の経路の幾何学的な描き方：a) レン ズの中心を通る光線は屈折しない。b) 光軸に平行な光 線は焦点を通る，c) 屈折後, 初めに平行だった光線どうしは焦点面（焦点を通り光軸に垂直な平面）上で集 まる．d) 平面の像は平面であり，この 2 つの平面はレ ンズの平面上で交わる．

6.9: 光束 ​

  1. 光束 Φ\Phi [単位: lumen (lm⁡)(\operatorname{lm})] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, I=Φ/ΩI=\Phi / \Omega. 照度 [lux⁡(lx)][\operatorname{lux}(\mathrm{lx})] は（面に入射する） 面積あたりの光束で, E=Φ/SE=\Phi / S.

6.10: Gauss 定理 ​

  1. 光束についての Gauss の定理 : 光度 IiI_i の点光源を囲 む閉曲面を通って外に出る光束は, Φ=4π∑Ii\Phi=4 \pi \sum I_i. 光 源が 1 つで距離が rr のとき E=I/r2E=I / r^2

6.11: 実験のヒント ​

  1. 実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/7.html b/academic/physics/ipho-formulas-jpn/7.html index 8ec4dce7..d591433c 100644 --- a/academic/physics/ipho-formulas-jpn/7.html +++ b/academic/physics/ipho-formulas-jpn/7.html @@ -13,7 +13,7 @@ - + @@ -46,8 +46,8 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​ (普通 μ≈1\mu \approx 1 ) であることに注意せよ. エネ ルギー流密度（単位面積を通過する単位時間あたりの エネルギー）: I=cnε0E2=cnμ0B2(EI=c n \varepsilon_0 E^2=\frac{c}{n \mu_0} B^2(E と BB は実 効值)

7.12: Malus の法則 ​

  1. Malus の法則 : 直線偏光が角度 φ\varphi で偏光板を通過する と I=I0cos⁡2φI=I_0 \cos ^2 \varphi

7.13: 1/4 波長版 ​

  1. 1/41 / 4 波長版 : 直線偏光成分間の位相が π/2\pi / 2 ずれる.

7.14: Brewster 角 ​

  1. Brewster 角: 入射角が tan⁡φ=n\tan \varphi=n を満たすとき, 反 射波と屈折波が垂直になり反射波は直線偏光となる.

7.15: 光学素子による回折 ​

  1. 光学素子による回折 : レンズやプリズムなどを通る光 の光学距離を計算する必要はなく, 図形的に考える. 例えば，双プリズムは二重スリットによる回折と同じ 回折をする.

7.16: 光ファイバー ​

  1. 光ファイバー：Mach-Zehnder 干渉計は二重スリッ トによる干渉と, 円形共振器は Fabry-Pérot 干渉計 と似ている. Bragg フィルターは X\mathrm{X} 線の場合と同 じように働く. シングルモードの光ファイバーでは, Δn/n≈12(λ/d)2\Delta n / n \approx \frac{1}{2}(\lambda / d)^2.
- +M1001 80h400000v40h-400000z">​ (普通 μ≈1\mu \approx 1 ) であることに注意せよ. エネ ルギー流密度（単位面積を通過する単位時間あたりの エネルギー）: I=cnε0E2=cnμ0B2(EI=c n \varepsilon_0 E^2=\frac{c}{n \mu_0} B^2(E と BB は実 効值)

7.12: Malus の法則 ​

  1. Malus の法則 : 直線偏光が角度 φ\varphi で偏光板を通過する と I=I0cos⁡2φI=I_0 \cos ^2 \varphi

7.13: 1/4 波長版 ​

  1. 1/41 / 4 波長版 : 直線偏光成分間の位相が π/2\pi / 2 ずれる.

7.14: Brewster 角 ​

  1. Brewster 角: 入射角が tan⁡φ=n\tan \varphi=n を満たすとき, 反 射波と屈折波が垂直になり反射波は直線偏光となる.

7.15: 光学素子による回折 ​

  1. 光学素子による回折 : レンズやプリズムなどを通る光 の光学距離を計算する必要はなく, 図形的に考える. 例えば，双プリズムは二重スリットによる回折と同じ 回折をする.

7.16: 光ファイバー ​

  1. 光ファイバー：Mach-Zehnder 干渉計は二重スリッ トによる干渉と, 円形共振器は Fabry-Pérot 干渉計 と似ている. Bragg フィルターは X\mathrm{X} 線の場合と同 じように働く. シングルモードの光ファイバーでは, Δn/n≈12(λ/d)2\Delta n / n \approx \frac{1}{2}(\lambda / d)^2.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/8.html b/academic/physics/ipho-formulas-jpn/8.html index 7ec58071..bc755d51 100644 --- a/academic/physics/ipho-formulas-jpn/8.html +++ b/academic/physics/ipho-formulas-jpn/8.html @@ -13,7 +13,7 @@ - + @@ -58,8 +58,8 @@ c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z -M834 80h400000v40h-400000z">​ が成立.
- +M834 80h400000v40h-400000z">​ が成立.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/9.html b/academic/physics/ipho-formulas-jpn/9.html index 3ef3c022..9e1e2450 100644 --- a/academic/physics/ipho-formulas-jpn/9.html +++ b/academic/physics/ipho-formulas-jpn/9.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 9 ​

Author:Anda Toshiki
Updated:a minute ago
Words:950
Reading:4 min

9: 電磁気学 ​

9.1: Coulomb の法則 ​

  1. F=kq1q2/r2,U=kq1q2/rF=k q_1 q_2 / r^2, U=k q_1 q_2 / r で, Kepler の法則が 使える (Section 12 参照).

9.2: Gauss の法則 ​

  1. Gauss の法則 : ∮B⋅dS=0\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0,

    ∮εE⋅dS=Q,∮g⋅dS=−4πGM\oint \varepsilon \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{S}=Q, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{S}=-4 \pi G M

9.3: 循環定理 ​

  1. 循環定理 :

    ∮E⋅dl=0(=Φ˙),∮B⋅dlμ=I,∮g⋅dl=0\oint \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{l}=0(=\dot{\Phi}), \oint \frac{\boldsymbol{B} \cdot \mathrm{d} \boldsymbol{l}}{\mu}=I, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{l}=0

9.4: 電流素片により生じる磁束密度 ​

  1. 電流素片により生じる磁束密度 :

    dB=μI4πdl×err2.\mathrm{d} \boldsymbol{B}=\frac{\mu I}{4 \pi} \frac{\mathrm{d} \boldsymbol{l} \times \boldsymbol{e}_r}{r^2} .

    したがって電流 II が流れる円形回路の中心では B=μ0I2rB=\frac{\mu_0 I}{2 r}.

9.5: ローレンツ力 ​

  1. F=e(E+v×B),F=I×Bl\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l.

9.6: Gauss の定理と循環定理より ​

  1. Gauss の定理と循環定理より：帯電した導線について E=σ2πε0rE=\frac{\sigma}{2 \pi \varepsilon_0 r}, 電流が流れる導線について B=μ0I2πrB=\frac{\mu_0 I}{2 \pi r}. 帯電した面について E=σ2ε0E=\frac{\sigma}{2 \varepsilon_0}, 電流が流れる面につい て B=μ0i2B=\frac{\mu_0 i}{2}. 一様に帯電した球殼（又は無限に長い円 筒）の内部で E=0E=0, 軸に沿って表面に電流が流れる 円筒の内部で B=0B=0. 密度 ρ\rho で一様に帯電, 又は一様 な電流 i\boldsymbol{i} が流れる, 球 (d=3)/(d=3) / 円柱 (d−2)/(d-2) / 平面 (d=1)(d=1) の内部で,

    E=ρεdr,B=1μdi×r\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r}

9.7: 長いソレノイド ​

  1. 長いソレノイド: 内部で B=μnIB=\mu n I, 外部で B=0B=0. 磁束 Φ=NBS(n=Nl)\Phi=N B S\left(n=\frac{N}{l}\right). インダクタンス L=L= Φ/I=μn2V\Phi / I=\mu n^2 V. 短いソレノイド :B∥=μnIΩ4π(Ω: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega は 立体角).

9.8: 磁場を小型コイルや衝撃検流計で測定する ​

  1. 磁場を小型コイルや衝撃検流計で測定する： q=q= ∫VR dt=NSΔB/R\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R.

9.9: 静電場のエネルギー ​

  1. 静電場のエネルギー：

    U=k∑i<jqiqjrij=12∫ϕ(r)dq, dq=ρ(r)dVU=k \sum_{i<j} \frac{q_i q_j}{r_{i j}}=\frac{1}{2} \int \phi(\boldsymbol{r}) \mathrm{d} q, \mathrm{~d} q=\rho(\boldsymbol{r}) \mathrm{d} V

9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 ​

  1. 一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.

9.11: 全ての電荷 ​

  1. 全ての電荷が距離 rr にある場合（例えば，不均一に帯 電した球やリングの中心） ϕϕ=kQ/r\phi \phi=k Q / r

9.12: 外部電荷 ​

  1. 外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.

9.14: 導体 ​

  1. 導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも QQ という 電荷を持った導電性の球があるように見える).

9.15: 静電容量 ​

  1. 静電容量: C=εS/dC=\varepsilon S / d (平板), 4πεr4 \pi \varepsilon r (球), 2πεl(ln⁡R/r)−12 \pi \varepsilon l(\ln R / r)^{-1} (同軸円筒).

9.16: 双極子モーメント ​

  1. 双極子モーメント：

    pe=∑qiri=qd,pμ=IS\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S}

9.17: 双極子場 ​

  1. 双極子場 : ϕ=kp⋅er/r2,E,B∝r−3\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}

9.18: 双極子に働く力 ​

  1. 双極子に働く力 : F=(pe⋅E)′,F=(pμ⋅B)′F=\left(\boldsymbol{p}_e \cdot \boldsymbol{E}\right)^{\prime}, F=\left(\boldsymbol{p}_\mu \cdot \boldsymbol{B}\right)^{\prime} [訳 者注 : ここの微分はむしろ grad⁡\operatorname{grad} である]. 2 つの双極 子間の相互作用 :F∝r−4: F \propto r^{-4}.

9.19: 磁気双極子としての点電荷 ​

  1. 磁気双極子としての点電荷 : pμ∝Φ∝v⊥2/Bp_\mu \propto \Phi \propto v_{\perp}^2 / B は断熱 不変量 (Section 4: #22 参照).

9.20: 鏡像法 ​

  1. 鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.

9.21: 一様（電）場中の球 (円柱) の分極 ​

  1. 一様（電）場中の球 (円柱) の分極 : (+ρ(+\rho と −ρ-\rho に一 様に帯電した球 (円柱) の重ね合わせで， d∝Ed \propto E.

9.22: 渦電流 ​

  1. 渦電流: 電流損失密度 ≈B2v2/ρ.1\approx B^2 v^2 / \rho .1 回の通過で与え られる運動量 : Fτ≈B2a3d/ρF \tau \approx B^2 a^3 d / \rho (ここで dd は厚さ, aa は大きさ).
- +
Toshiki's Note

Appearance

Formulas for IPhO 日本語版: Section 9 ​

Author:Anda Toshiki
Updated:a minute ago
Words:950
Reading:4 min

9: 電磁気学 ​

9.1: Coulomb の法則 ​

  1. F=kq1q2/r2,U=kq1q2/rF=k q_1 q_2 / r^2, U=k q_1 q_2 / r で, Kepler の法則が 使える (Section 12 参照).

9.2: Gauss の法則 ​

  1. Gauss の法則 : ∮B⋅dS=0\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0,

    ∮εE⋅dS=Q,∮g⋅dS=−4πGM\oint \varepsilon \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{S}=Q, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{S}=-4 \pi G M

9.3: 循環定理 ​

  1. 循環定理 :

    ∮E⋅dl=0(=Φ˙),∮B⋅dlμ=I,∮g⋅dl=0\oint \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{l}=0(=\dot{\Phi}), \oint \frac{\boldsymbol{B} \cdot \mathrm{d} \boldsymbol{l}}{\mu}=I, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{l}=0

9.4: 電流素片により生じる磁束密度 ​

  1. 電流素片により生じる磁束密度 :

    dB=μI4πdl×err2.\mathrm{d} \boldsymbol{B}=\frac{\mu I}{4 \pi} \frac{\mathrm{d} \boldsymbol{l} \times \boldsymbol{e}_r}{r^2} .

    したがって電流 II が流れる円形回路の中心では B=μ0I2rB=\frac{\mu_0 I}{2 r}.

9.5: ローレンツ力 ​

  1. F=e(E+v×B),F=I×Bl\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l.

9.6: Gauss の定理と循環定理より ​

  1. Gauss の定理と循環定理より：帯電した導線について E=σ2πε0rE=\frac{\sigma}{2 \pi \varepsilon_0 r}, 電流が流れる導線について B=μ0I2πrB=\frac{\mu_0 I}{2 \pi r}. 帯電した面について E=σ2ε0E=\frac{\sigma}{2 \varepsilon_0}, 電流が流れる面につい て B=μ0i2B=\frac{\mu_0 i}{2}. 一様に帯電した球殼（又は無限に長い円 筒）の内部で E=0E=0, 軸に沿って表面に電流が流れる 円筒の内部で B=0B=0. 密度 ρ\rho で一様に帯電, 又は一様 な電流 i\boldsymbol{i} が流れる, 球 (d=3)/(d=3) / 円柱 (d−2)/(d-2) / 平面 (d=1)(d=1) の内部で,

    E=ρεdr,B=1μdi×r\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r}

9.7: 長いソレノイド ​

  1. 長いソレノイド: 内部で B=μnIB=\mu n I, 外部で B=0B=0. 磁束 Φ=NBS(n=Nl)\Phi=N B S\left(n=\frac{N}{l}\right). インダクタンス L=L= Φ/I=μn2V\Phi / I=\mu n^2 V. 短いソレノイド :B∥=μnIΩ4π(Ω: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega は 立体角).

9.8: 磁場を小型コイルや衝撃検流計で測定する ​

  1. 磁場を小型コイルや衝撃検流計で測定する： q=q= ∫VR dt=NSΔB/R\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R.

9.9: 静電場のエネルギー ​

  1. 静電場のエネルギー：

    U=k∑i<jqiqjrij=12∫ϕ(r)dq, dq=ρ(r)dVU=k \sum_{i<j} \frac{q_i q_j}{r_{i j}}=\frac{1}{2} \int \phi(\boldsymbol{r}) \mathrm{d} q, \mathrm{~d} q=\rho(\boldsymbol{r}) \mathrm{d} V

9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 ​

  1. 一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.

9.11: 全ての電荷 ​

  1. 全ての電荷が距離 rr にある場合（例えば，不均一に帯 電した球やリングの中心） ϕϕ=kQ/r\phi \phi=k Q / r

9.12: 外部電荷 ​

  1. 外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.

9.14: 導体 ​

  1. 導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも QQ という 電荷を持った導電性の球があるように見える).

9.15: 静電容量 ​

  1. 静電容量: C=εS/dC=\varepsilon S / d (平板), 4πεr4 \pi \varepsilon r (球), 2πεl(ln⁡R/r)−12 \pi \varepsilon l(\ln R / r)^{-1} (同軸円筒).

9.16: 双極子モーメント ​

  1. 双極子モーメント：

    pe=∑qiri=qd,pμ=IS\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S}

9.17: 双極子場 ​

  1. 双極子場 : ϕ=kp⋅er/r2,E,B∝r−3\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}

9.18: 双極子に働く力 ​

  1. 双極子に働く力 : F=(pe⋅E)′,F=(pμ⋅B)′F=\left(\boldsymbol{p}_e \cdot \boldsymbol{E}\right)^{\prime}, F=\left(\boldsymbol{p}_\mu \cdot \boldsymbol{B}\right)^{\prime} [訳 者注 : ここの微分はむしろ grad⁡\operatorname{grad} である]. 2 つの双極 子間の相互作用 :F∝r−4: F \propto r^{-4}.

9.19: 磁気双極子としての点電荷 ​

  1. 磁気双極子としての点電荷 : pμ∝Φ∝v⊥2/Bp_\mu \propto \Phi \propto v_{\perp}^2 / B は断熱 不変量 (Section 4: #22 参照).

9.20: 鏡像法 ​

  1. 鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.

9.21: 一様（電）場中の球 (円柱) の分極 ​

  1. 一様（電）場中の球 (円柱) の分極 : (+ρ(+\rho と −ρ-\rho に一 様に帯電した球 (円柱) の重ね合わせで， d∝Ed \propto E.

9.22: 渦電流 ​

  1. 渦電流: 電流損失密度 ≈B2v2/ρ.1\approx B^2 v^2 / \rho .1 回の通過で与え られる運動量 : Fτ≈B2a3d/ρF \tau \approx B^2 a^3 d / \rho (ここで dd は厚さ, aa は大きさ).
+ \ No newline at end of file diff --git a/academic/vocabulary/2023/02/2023-02-27.html b/academic/vocabulary/2023/02/2023-02-27.html index 0e157721..a5048637 100644 --- a/academic/vocabulary/2023/02/2023-02-27.html +++ b/academic/vocabulary/2023/02/2023-02-27.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

2023-2-27: Vocabulary ​

Author:Anda Toshiki
Updated:a minute ago
Words:299
Reading:1 min

This table of vocabularies are from "Sadlier Vocabulary Workshop-Level G (Unit 4)" with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.

VocabularyDefinition
Atrophy(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away
BastionA fortified place, stronghold
ConcordA state of agreement, harmony, unanimity; a treaty, pact, covenant
Consummate(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection
Disarray(n.) disorder, confusion; (v.) to throw into disorder
ExigencyUrgency, pressure; urgent demand, pressing need; an emergency
FlotsamFloating debris; homeless, impoverished people
FreneticFrenzied, highly agitated
GleanTo gather bit by bit; to gather small quantities of grain left in a field by the reapers.
Grouse(n.) A type of game bird; a complaint; (v.) to complain, grumble
IncarcerateTo imprison, confine, jail
Incumbent(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of
JocularHumorous, jesting, jolly, joking
LudicrousRidiculous, laughable, absurd
MordantBiting or caustic in thought, manner, or style; sharply or bitterly harsh.
Nettle(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely
PecuniaryConsisting of or measured in money; of or related to money
PusillanimousContemptibly cowardly or mean-spirited
RecumbentIn a reclining position, lying down, in the posture of one sleeping or resting.
StratagemA scheme to outwit or deceive an opponent or to gain and end.

Reference

- +
Toshiki's Note

Appearance

2023-2-27: Vocabulary ​

Author:Anda Toshiki
Updated:a minute ago
Words:299
Reading:1 min

This table of vocabularies are from "Sadlier Vocabulary Workshop-Level G (Unit 4)" with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.

VocabularyDefinition
Atrophy(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away
BastionA fortified place, stronghold
ConcordA state of agreement, harmony, unanimity; a treaty, pact, covenant
Consummate(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection
Disarray(n.) disorder, confusion; (v.) to throw into disorder
ExigencyUrgency, pressure; urgent demand, pressing need; an emergency
FlotsamFloating debris; homeless, impoverished people
FreneticFrenzied, highly agitated
GleanTo gather bit by bit; to gather small quantities of grain left in a field by the reapers.
Grouse(n.) A type of game bird; a complaint; (v.) to complain, grumble
IncarcerateTo imprison, confine, jail
Incumbent(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of
JocularHumorous, jesting, jolly, joking
LudicrousRidiculous, laughable, absurd
MordantBiting or caustic in thought, manner, or style; sharply or bitterly harsh.
Nettle(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely
PecuniaryConsisting of or measured in money; of or related to money
PusillanimousContemptibly cowardly or mean-spirited
RecumbentIn a reclining position, lying down, in the posture of one sleeping or resting.
StratagemA scheme to outwit or deceive an opponent or to gain and end.

Reference

+ \ No newline at end of file diff --git a/academic/vocabulary/index.html b/academic/vocabulary/index.html index 9bc00be6..4d9f5b1e 100644 --- a/academic/vocabulary/index.html +++ b/academic/vocabulary/index.html @@ -13,7 +13,7 @@ - + @@ -36,8 +36,8 @@ -
Toshiki's Note

Appearance

Welcome to My Vocabulary List! ​

Author:Anda Toshiki
Updated:a minute ago
Words:5
Reading:1 min
- +
Toshiki's Note

Appearance

Welcome to My Vocabulary List! ​

Author:Anda Toshiki
Updated:a minute ago
Words:5
Reading:1 min
+ \ No newline at end of file diff --git a/application/markdown-it-katex/how-to-use.html b/application/markdown-it-katex/how-to-use.html index 0af4b148..d7f3643e 100644 --- a/application/markdown-it-katex/how-to-use.html +++ b/application/markdown-it-katex/how-to-use.html @@ -13,7 +13,7 @@ - + @@ -36,13 +36,13 @@ -
Toshiki's Note

Appearance

@andatoshiki/markdown-it-katex ​

Author:Anda Toshiki
Updated:a minute ago
Words:1.1k
Reading:7 min

Add graceful KaTeX\KaTeX rendering to your Markdown like a charm with markdown-it plugin.

1: Installation ​

Before you start using this plugin, make sure you have already installed the default markdown-it parser; if not, please run the following command or refer to the official markdown-it documentation.

sh
$ npm install markdown-it --save
$ npm install markdown-it --save

First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing </body> for markdown-it-katex's core utils to be loaded for the static page.

sh
$ npm install -D @andatoshiki/markdown-it-katex
$ npm install -D @andatoshiki/markdown-it-katex
sh
$ yarn add --dev @andatoshiki/markdown-it-katex
$ yarn add --dev @andatoshiki/markdown-it-katex
sh
$ pnpm add -D @andatoshiki/markdown-it-katex
$ pnpm add -D @andatoshiki/markdown-it-katex
html
<!-- your other body contents ... -->
+    
Toshiki's Note
Appearance
@andatoshiki/markdown-it-katex ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:1.1k
 Reading:7 min
Add graceful KaTeX\KaTeX rendering to your Markdown like a charm with markdown-it plugin.
1: Installation ​
Before you start using this plugin, make sure you have already installed the default markdown-it parser; if not, please run the following command or refer to the official markdown-it documentation.
sh
$ npm install markdown-it --save
$ npm install markdown-it --save
First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing </body> for markdown-it-katex's core utils to be loaded for the static page.
sh
$ npm install -D @andatoshiki/markdown-it-katex
$ npm install -D @andatoshiki/markdown-it-katex
sh
$ yarn add --dev @andatoshiki/markdown-it-katex
$ yarn add --dev @andatoshiki/markdown-it-katex
sh
$ pnpm add -D @andatoshiki/markdown-it-katex
$ pnpm add -D @andatoshiki/markdown-it-katex
html
<!-- your other body contents ... -->
     <script src="https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js"></script>
 </body>
<!-- your other body contents ... -->
     <script src="https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js"></script>
-</body>
Including KaTeX CSS is necessary in the way you are convenient with, either link the stylesheet from a third party CDN into the local HTML <head> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,
Or, you could clone or download the entire repository source of KaTeX and self load the fonts/styles/scripts locally, but if you prefer loading from third-party CDN with faster load speed when deployed to save your server resources, the following CDN links might be your choice, you can always switch to other platforms based on your need.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
html
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
scss
// ... your other styles
+</body>
Including KaTeX CSS is necessary in the way you are convenient with, either link the stylesheet from a third party CDN into the local HTML <head> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,
Or, you could clone or download the entire repository source of KaTeX and self load the fonts/styles/scripts locally, but if you prefer loading from third-party CDN with faster load speed when deployed to save your server resources, the following CDN links might be your choice, you can always switch to other platforms based on your need.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
html
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
scss
// ... your other styles
 @import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
// ... your other styles
-@import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (github-markdown-css) for styling your HTML output with a similar style replica of GitHub's markdown styling to your familiarity.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
This forked project maintained by Anda Toshiki comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version 16.0, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, 16.0 may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.
Warning
Since this project is a fork of the original markdown-it-katex project that hasn't been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around ver. 0.9.0 which clearly is outdated and results in broken styles with overflowing and other potential bug presents.
2: Usage ​
RTo render equations, you need to include the markdown-it-katex plugin in the markdown-it components in your JavaScript or TypeScript file as follows,
js
var md = require('markdown-it')(),
+@import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (github-markdown-css) for styling your HTML output with a similar style replica of GitHub's markdown styling to your familiarity.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
This forked project maintained by Anda Toshiki comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version 16.0, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, 16.0 may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.
Warning
Since this project is a fork of the original markdown-it-katex project that hasn't been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around ver. 0.9.0 which clearly is outdated and results in broken styles with overflowing and other potential bug presents.
2: Usage ​
RTo render equations, you need to include the markdown-it-katex plugin in the markdown-it components in your JavaScript or TypeScript file as follows,
js
var md = require('markdown-it')(),
     mk = require('andatoshiki/markdown-it-katex')
 
 md.use(mk)
@@ -85,8 +85,8 @@ M834 80h400000v40h-400000z">$$
$$
 \frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right)
 = \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^ Ir \cdots (r-i+1) (\log y)^{ri}} {\omega^i} \right\}
-$$
the above block equation renders the LaTeX equation as a block with output as follows,
∂r∂ωr(yωω)=(yωω){(log⁡y)r+∑i=1r(−1)Ir⋯(r−i+1)(log⁡y)riωi}\frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right) = \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^ Ir \cdots (r-i+1) (\log y)^{ri}} {\omega^i} \right\} 
5: Syntax ​
Math parsing in markdown is designed to comply with the "latex-in-markdown" conventions set by Pandoc,
Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.
  • Pandoc. “Pandoc - Pandoc User’s Guide.” Pandoc.org, pandoc.org/MANUAL.html#math. Accessed 2 Mar. 2023.
6: Supported math syntax ​
KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. It is designed to be easy to use, and to provide high-quality mathematical typesetting for web applications, refer to the following pages of the document for the full list of function support in KaTeX.
Note
Due to the large number of equations rendered on the next following pages, it might takes time to entirely load the webpage, please be patient if the webpage seems to not respond; or it could be a result of slow network connection to correctly render all the equation that causes broken formulas, refresh the page to proceed.

-    
+$$

the above block equation renders the LaTeX equation as a block with output as follows,

∂r∂ωr(yωω)=(yωω){(log⁡y)r+∑i=1r(−1)Ir⋯(r−i+1)(log⁡y)riωi}\frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right) = \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^ Ir \cdots (r-i+1) (\log y)^{ri}} {\omega^i} \right\}

5: Syntax ​

Math parsing in markdown is designed to comply with the "latex-in-markdown" conventions set by Pandoc,

Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.

  • Pandoc. “Pandoc - Pandoc User’s Guide.” Pandoc.org, pandoc.org/MANUAL.html#math. Accessed 2 Mar. 2023.

6: Supported math syntax ​

KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. It is designed to be easy to use, and to provide high-quality mathematical typesetting for web applications, refer to the following pages of the document for the full list of function support in KaTeX.

Note

Due to the large number of equations rendered on the next following pages, it might takes time to entirely load the webpage, please be patient if the webpage seems to not respond; or it could be a result of slow network connection to correctly render all the equation that causes broken formulas, refresh the page to proceed.

+ \ No newline at end of file diff --git a/application/markdown-it-katex/support-function.html b/application/markdown-it-katex/support-function.html index 00f3dbeb..b934a6ef 100644 --- a/application/markdown-it-katex/support-function.html +++ b/application/markdown-it-katex/support-function.html @@ -13,7 +13,7 @@ - + @@ -334,8 +334,8 @@ M40 281 V448H0V74H40V241H400000v40z">​ \xmapsto{abc}

|=abc\xlongequal{abc} \xlongequal{abc}

Extensible arrows all can take an optional argument in the same manner, as \xrightarrow[under]{over}.

11: Style, Color, Size, and Font ​

Class Assignment

Color

F=ma\color{blue} F=ma \color{blue} F=ma

Note that KaTeX \color acts like a switch. This aligns with LaTeX and differs from MathJax. Other KaTeX color functions expect the content to be a function argument:

For color definition, KaTeX color functions will accept the standard HTML predefined color names. They will also accept an RGB argument in CSS hexa­decimal style. The "#" is optional before a six-digit specification.

Font

Ab0\mathrm{Ab0} \mathrm{Ab0}Ab0\mathbf{Ab0} \mathbf{Ab0}Ab\mathit{Ab} \mathit{Ab}
Ab0\mathnormal{Ab0} \mathnormal{Ab0}Ab0\textbf{Ab0} \textbf{Ab0}Ab\textit{Ab} \textit{Ab}
Ab0\textrm{Ab0} \textrm{Ab0}Ab0\bf Ab0 \bf Ab0Ab\it Ab \it Ab
Ab0\rm Ab0 \rm Ab0Ab0\bold{Ab0} \bold{Ab0}AB\Bbb{AB} \Bbb{AB}
Ab0\textnormal{Ab0} \textnormal{Ab0}Ab\boldsymbol{Ab} \boldsymbol{Ab}AB\mathbb{AB} \mathbb{AB}
Ab0\text{Ab0} \text{Ab0}Ab\bm{Ab} \bm{Ab}Ab0\frak{Ab0} \frak{Ab0}
Ab0\mathsf{Ab0} \mathsf{Ab0}Ab0\mathtt{Ab0} \mathtt{Ab0}Ab0\mathfrak{Ab0} \mathfrak{Ab0}
Ab0\textsf{Ab0} \textsf{Ab0}Ab0\texttt{Ab0} \texttt{Ab0}AB0\mathcal{AB0} \mathcal{AB0}
Ab0\sf Ab0 \sf Ab0Ab0\tt Ab0 \tt Ab0AB\mathscr{AB} \mathscr{AB}

One can stack font family, font weight, and font shape by using the \textXX versions of the font functions. So \textsf{\textbf{H}} will produce H\textsf{\textbf{H}}. The other versions do not stack, e.g., \mathsf{\mathbf{H}} will produce H\mathsf{\mathbf{H}}.

In cases where KaTeX fonts do not have a bold glyph, \pmb can simulate one. For example, \pmb{\mu} renders as : μ\pmb{\mu}

Size

AB\Huge AB \Huge ABAB\normalsize AB \normalsize AB
AB\huge AB \huge ABAB\small AB \small AB
AB\LARGE AB \LARGE ABAB\footnotesize AB \footnotesize AB
AB\Large AB \Large ABAB\scriptsize AB \scriptsize AB
AB\large AB \large ABAB\tiny AB \tiny AB

Style

|∑i=1n\displaystyle\sum_{i=1}^n \displaystyle\sum_{i=1}^n |∑i=1n\textstyle\sum_{i=1}^n \textstyle\sum_{i=1}^n |x\scriptstyle x \scriptstyle x         (The size of a first sub/superscript) |x\scriptscriptstyle x \scriptscriptstyle x (The size of subsequent sub/superscripts) |lim⁡x\lim\limits_x \lim\limits_x |lim⁡x\lim\nolimits_x \lim\nolimits_x |x^2\verb!x^2! \verb!x^2!

\text{…} will accept nested $…$ fragments and render them in math mode.

12: Symbols and Punctuation ​

% comment…\dots \dotsKaTeX\KaTeX \KaTeX
%\% \%⋯\cdots \cdotsLaTeX\LaTeX \LaTeX
#\# \#⋱\ddots \ddotsTeX\TeX \TeX
&\& \&…\ldots \ldots∇\nabla \nabla
_\_ \_⋮\vdots \vdots∞\infty \infty
_\text{\textunderscore} \text{\textunderscore}⋯\dotsb \dotsb∞\infin \infin
–\text{--} \text{--}…\dotsc \dotsc✓\checkmark \checkmark
–\text{\textendash} \text{\textendash} ⁣⋯\dotsi \dotsi†\dag \dag
—\text{---} \text{---}⋯\dotsm \dotsm†\dagger \dagger
—\text{\textemdash} \text{\textemdash}…\dotso \dotso†\text{\textdagger} \text{\textdagger}
~\text{\textasciitilde} \text{\textasciitilde}⋅\sdot \sdot‡\ddag \ddag
‘` `…\mathellipsis \mathellipsis‡\ddagger \ddagger
‘\text{\textquoteleft} text{\textquoteleft}…\text{\textellipsis} \text{\textellipsis}‡\text{\textdaggerdbl} \text{\textdaggerdbl}
‘\lq \lq□\Box \Box‡\Dagger \Dagger
’\text{\textquoteright} \text{\textquoteright}□\square \square∠\angle \angle
′\rq \rq■\blacksquare \blacksquare∡\measuredangle \measuredangle
“\text{\textquotedblleft} \text{\textquotedblleft}△\triangle \triangle∢\sphericalangle \sphericalangle
"" "▽\triangledown \triangledown⊤\top \top
”\text{\textquotedblright} \text{\textquotedblright}◃\triangleleft \triangleleft⊥\bot \bot
 ⁣:\colon \colon▹\triangleright \triangleright$\$ \$
‵\backprime \backprime▽\bigtriangledown \bigtriangledown$\text{\textdollar} \text{\textdollar}
′\prime \prime△\bigtriangleup \bigtriangleup£\pounds \pounds
<\text{\textless} \text{\textless}▲\blacktriangle \blacktriangle£\mathsterling \mathsterling
>\text{\textgreater} \text{\textgreater}▼\blacktriangledown \blacktriangledown£\text{\textsterling} \text{\textsterling}
|\text{\textbar} \text{\textbar}◀\blacktriangleleft \blacktriangleleft¥\yen \yen
∥\text{\textbardbl} \text{\textbardbl}▶\blacktriangleright \blacktriangleright√\surd \surd
{\text{\textbraceleft} \text{\textbraceleft}⋄\diamond \diamond°\degree \degree
}\text{\textbraceright} \text{\textbraceright}◊\Diamond \Diamond°\text{\textdegree} \text{\textdegree}
\text{\P} \text{\P}◊\lozenge \lozenge℧\mho \mho
§\text{\S} \text{\S}⧫\blacklozenge \blacklozenge╲\diagdown \diagdown
§\text{\sect} \text{\sect}⋆\star \star╱\diagup \diagup
©\copyright \copyright★\bigstar \bigstar♭\flat \flat
®\circledR \circledR♣\clubsuit \clubsuit♮\natural \natural
®\text{\textregistered} \text{\textregistered}♣\clubs \clubs♯\sharp \sharp
Ⓢ\circledS \circledS♢\diamondsuit \diamondsuit♡\heartsuit \heartsuit
a◯\text{\textcircled a} \text{\textcircled a}♢\diamonds \diamonds♡\hearts \hearts
✠\maltese \maltese♠\spadesuit \spadesuit♠\spades \spades

Direct Input: £¥∇∞⋅∠∡∢♠♡♢♣♭♮♯✓…⋮⋯⋱!£ ¥ ∇ ∞ · ∠ ∡ ∢ ♠ ♡ ♢ ♣ ♭ ♮ ♯ ✓ … ⋮ ⋯ ⋱ ! ‼

13: Units ​

In KaTeX, units are proportioned as they are in TeX. KaTeX units are different than CSS units.

KaTeX UnitValueKaTeX UnitValue
emCSS embp1/72​ inch × F × G
exCSS expc12 KaTeX pt
mu1/18 CSS emdd1238/1157​ KaTeX pt
pt1/72.27 inch × F × Gcc14856/1157 KaTeX pt
mm1 mm × F × Gnd685/642 KaTeX pt
cm1 cm × F × Gnc1370/107​ KaTeX pt
in1 inch × F × Gsp1/65536 KaTeX pt

where:

The effect of style and size:

Unittextstylescriptscripthuge
em or ex\rule{1em}{1em}\scriptscriptstyle\rule{1em}{1em}\huge\rule{1em}{1em}
mu\rule{18mu}{18mu}\scriptscriptstyle\rule{18mu}{18mu}\huge\rule{18mu}{18mu}
others\rule{10pt}{10pt}\scriptscriptstyle\rule{10pt}{10pt}\huge\rule{10pt}{10pt}
- +M0 50 h400000 v40H0z m0 194h40000v40H0z"> \xlongequal{abc}

Extensible arrows all can take an optional argument in the same manner, as \xrightarrow[under]{over}.

11: Style, Color, Size, and Font ​

Class Assignment

Color

F=ma\color{blue} F=ma \color{blue} F=ma

Note that KaTeX \color acts like a switch. This aligns with LaTeX and differs from MathJax. Other KaTeX color functions expect the content to be a function argument:

For color definition, KaTeX color functions will accept the standard HTML predefined color names. They will also accept an RGB argument in CSS hexa­decimal style. The "#" is optional before a six-digit specification.

Font

Ab0\mathrm{Ab0} \mathrm{Ab0}Ab0\mathbf{Ab0} \mathbf{Ab0}Ab\mathit{Ab} \mathit{Ab}
Ab0\mathnormal{Ab0} \mathnormal{Ab0}Ab0\textbf{Ab0} \textbf{Ab0}Ab\textit{Ab} \textit{Ab}
Ab0\textrm{Ab0} \textrm{Ab0}Ab0\bf Ab0 \bf Ab0Ab\it Ab \it Ab
Ab0\rm Ab0 \rm Ab0Ab0\bold{Ab0} \bold{Ab0}AB\Bbb{AB} \Bbb{AB}
Ab0\textnormal{Ab0} \textnormal{Ab0}Ab\boldsymbol{Ab} \boldsymbol{Ab}AB\mathbb{AB} \mathbb{AB}
Ab0\text{Ab0} \text{Ab0}Ab\bm{Ab} \bm{Ab}Ab0\frak{Ab0} \frak{Ab0}
Ab0\mathsf{Ab0} \mathsf{Ab0}Ab0\mathtt{Ab0} \mathtt{Ab0}Ab0\mathfrak{Ab0} \mathfrak{Ab0}
Ab0\textsf{Ab0} \textsf{Ab0}Ab0\texttt{Ab0} \texttt{Ab0}AB0\mathcal{AB0} \mathcal{AB0}
Ab0\sf Ab0 \sf Ab0Ab0\tt Ab0 \tt Ab0AB\mathscr{AB} \mathscr{AB}

One can stack font family, font weight, and font shape by using the \textXX versions of the font functions. So \textsf{\textbf{H}} will produce H\textsf{\textbf{H}}. The other versions do not stack, e.g., \mathsf{\mathbf{H}} will produce H\mathsf{\mathbf{H}}.

In cases where KaTeX fonts do not have a bold glyph, \pmb can simulate one. For example, \pmb{\mu} renders as : μ\pmb{\mu}

Size

AB\Huge AB \Huge ABAB\normalsize AB \normalsize AB
AB\huge AB \huge ABAB\small AB \small AB
AB\LARGE AB \LARGE ABAB\footnotesize AB \footnotesize AB
AB\Large AB \Large ABAB\scriptsize AB \scriptsize AB
AB\large AB \large ABAB\tiny AB \tiny AB

Style

|∑i=1n\displaystyle\sum_{i=1}^n \displaystyle\sum_{i=1}^n |∑i=1n\textstyle\sum_{i=1}^n \textstyle\sum_{i=1}^n |x\scriptstyle x \scriptstyle x         (The size of a first sub/superscript) |x\scriptscriptstyle x \scriptscriptstyle x (The size of subsequent sub/superscripts) |lim⁡x\lim\limits_x \lim\limits_x |lim⁡x\lim\nolimits_x \lim\nolimits_x |x^2\verb!x^2! \verb!x^2!

\text{…} will accept nested $…$ fragments and render them in math mode.

12: Symbols and Punctuation ​

% comment…\dots \dotsKaTeX\KaTeX \KaTeX
%\% \%⋯\cdots \cdotsLaTeX\LaTeX \LaTeX
#\# \#⋱\ddots \ddotsTeX\TeX \TeX
&\& \&…\ldots \ldots∇\nabla \nabla
_\_ \_⋮\vdots \vdots∞\infty \infty
_\text{\textunderscore} \text{\textunderscore}⋯\dotsb \dotsb∞\infin \infin
–\text{--} \text{--}…\dotsc \dotsc✓\checkmark \checkmark
–\text{\textendash} \text{\textendash} ⁣⋯\dotsi \dotsi†\dag \dag
—\text{---} \text{---}⋯\dotsm \dotsm†\dagger \dagger
—\text{\textemdash} \text{\textemdash}…\dotso \dotso†\text{\textdagger} \text{\textdagger}
~\text{\textasciitilde} \text{\textasciitilde}⋅\sdot \sdot‡\ddag \ddag
‘` `…\mathellipsis \mathellipsis‡\ddagger \ddagger
‘\text{\textquoteleft} text{\textquoteleft}…\text{\textellipsis} \text{\textellipsis}‡\text{\textdaggerdbl} \text{\textdaggerdbl}
‘\lq \lq□\Box \Box‡\Dagger \Dagger
’\text{\textquoteright} \text{\textquoteright}□\square \square∠\angle \angle
′\rq \rq■\blacksquare \blacksquare∡\measuredangle \measuredangle
“\text{\textquotedblleft} \text{\textquotedblleft}△\triangle \triangle∢\sphericalangle \sphericalangle
"" "▽\triangledown \triangledown⊤\top \top
”\text{\textquotedblright} \text{\textquotedblright}◃\triangleleft \triangleleft⊥\bot \bot
 ⁣:\colon \colon▹\triangleright \triangleright$\$ \$
‵\backprime \backprime▽\bigtriangledown \bigtriangledown$\text{\textdollar} \text{\textdollar}
′\prime \prime△\bigtriangleup \bigtriangleup£\pounds \pounds
<\text{\textless} \text{\textless}▲\blacktriangle \blacktriangle£\mathsterling \mathsterling
>\text{\textgreater} \text{\textgreater}▼\blacktriangledown \blacktriangledown£\text{\textsterling} \text{\textsterling}
|\text{\textbar} \text{\textbar}◀\blacktriangleleft \blacktriangleleft¥\yen \yen
∥\text{\textbardbl} \text{\textbardbl}▶\blacktriangleright \blacktriangleright√\surd \surd
{\text{\textbraceleft} \text{\textbraceleft}⋄\diamond \diamond°\degree \degree
}\text{\textbraceright} \text{\textbraceright}◊\Diamond \Diamond°\text{\textdegree} \text{\textdegree}
\text{\P} \text{\P}◊\lozenge \lozenge℧\mho \mho
§\text{\S} \text{\S}⧫\blacklozenge \blacklozenge╲\diagdown \diagdown
§\text{\sect} \text{\sect}⋆\star \star╱\diagup \diagup
©\copyright \copyright★\bigstar \bigstar♭\flat \flat
®\circledR \circledR♣\clubsuit \clubsuit♮\natural \natural
®\text{\textregistered} \text{\textregistered}♣\clubs \clubs♯\sharp \sharp
Ⓢ\circledS \circledS♢\diamondsuit \diamondsuit♡\heartsuit \heartsuit
a◯\text{\textcircled a} \text{\textcircled a}♢\diamonds \diamonds♡\hearts \hearts
✠\maltese \maltese♠\spadesuit \spadesuit♠\spades \spades

Direct Input: £¥∇∞⋅∠∡∢♠♡♢♣♭♮♯✓…⋮⋯⋱!£ ¥ ∇ ∞ · ∠ ∡ ∢ ♠ ♡ ♢ ♣ ♭ ♮ ♯ ✓ … ⋮ ⋯ ⋱ ! ‼

13: Units ​

In KaTeX, units are proportioned as they are in TeX. KaTeX units are different than CSS units.

KaTeX UnitValueKaTeX UnitValue
emCSS embp1/72​ inch × F × G
exCSS expc12 KaTeX pt
mu1/18 CSS emdd1238/1157​ KaTeX pt
pt1/72.27 inch × F × Gcc14856/1157 KaTeX pt
mm1 mm × F × Gnd685/642 KaTeX pt
cm1 cm × F × Gnc1370/107​ KaTeX pt
in1 inch × F × Gsp1/65536 KaTeX pt

where:

The effect of style and size:

Unittextstylescriptscripthuge
em or ex\rule{1em}{1em}\scriptscriptstyle\rule{1em}{1em}\huge\rule{1em}{1em}
mu\rule{18mu}{18mu}\scriptscriptstyle\rule{18mu}{18mu}\huge\rule{18mu}{18mu}
others\rule{10pt}{10pt}\scriptscriptstyle\rule{10pt}{10pt}\huge\rule{10pt}{10pt}
+ \ No newline at end of file diff --git a/application/markdown-it-katex/support-table.html b/application/markdown-it-katex/support-table.html index 91fa8e85..e25d6486 100644 --- a/application/markdown-it-katex/support-table.html +++ b/application/markdown-it-katex/support-table.html @@ -13,7 +13,7 @@ - + @@ -308,8 +308,8 @@ c-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42 18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333 -19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70 - 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z">\xtwoheadrightarrow{abc}

YZ ​

Symbol/FunctionRenderedSource or Comment
\yen¥\yen
\ZZ\Z
\ZetaZ\Zeta
\zetaζ\zeta
- + 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z">\xtwoheadrightarrow{abc}

YZ ​

Symbol/FunctionRenderedSource or Comment
\yen¥\yen
\ZZ\Z
\ZetaZ\Zeta
\zetaζ\zeta
+ \ No newline at end of file diff --git a/application/markdown-it-katex/tips.html b/application/markdown-it-katex/tips.html index 0156590d..574ecd42 100644 --- a/application/markdown-it-katex/tips.html +++ b/application/markdown-it-katex/tips.html @@ -13,7 +13,7 @@ - + @@ -36,7 +36,7 @@ -
Toshiki's Note

Appearance

Tips & Hacks ​

Author:Anda Toshiki
Updated:a minute ago
Words:1.2k
Reading:7 min

1: Responsive KaTeX Styling ​

1.1: Issue background ​

KaTeX works out of the box on large screen devices such as laptops and desktop computers. But as KaTeX's built-in does not support responsiveness on it's default stylings, hence KaTeX equations might overflow out of the default width of the default application containers on small-screened mobile devices as you can see in the following image in the dev tool of chrome, when the dimension of the webpage is set to responsive or under a certain fixed device dimension, the equations rendered in KaTeX overflows out of the viewport as inspected in blue hightlight.

katex overflowing on small screen devices
◎ katex overflowing on small screen devices

This inevitably causes the viewport to break and extends the default width for KaTeX equations <div> to fit, users will be required to manually scroll right in order to view the full equation consequently as follows,

katex overflowing with labels annotated
◎ katex overflowing with labels annotated

The above situation is undoubtedly annoying for user experiences while reading documentation, consider at what scenarios users have to scroll, scroll and scroll only for viewing a single long-blocked equation, who would want to read such a text, right? But don't worry we have a simple fix for this situation via by several line of css.

1.2: Temporary solution ​

If you happened to search over KaTeX's official repo issue tracker on GitHub, there are several user-made css tweaks hack already, the fix is simple by adjusting the overflows of both x and y axes of the KaTeX render <div> blocks. The katex-display > .katex selector targets the child element of the .katex-display class that has the .katex class. This is the element that contains the KaTeX math expression. The first block of styles is mostly concerned with making sure that the KaTeX expression doesn't overflow its container and can be scrolled horizontally if needed. The second block of styles sets the font and line-height for the KaTeX expression and makes sure that its text is properly indented.

@preview

But, the issue here is, the overflowing issue is resolved on the webpage, but the style itself left with a slight whitish "box" at the crossing corner of both horizontal as well as vertical scrollbar tracks, this might not be so explicit in the light mode of the webpage, but when it turns to dark mode, the box become annoying. Some people might say it's an easy tweak via setting the display property of the "box" element to display: none; to directly remove the box out of the page, this is a smart approach; however, while the box is gone, the two crossing tracks is going to form an untouched invisible box again against two bars without color. Thus, neither ways seems to perfectly solve the problem.

1.3: Finding solution ​

After running a quick research on Google, I found a simple hack tweak used in a theme of VuePress, vuepress-theme-hope on GitHub, the theme both integrate with KaTeX and MathJax for math supports.

@preview

Under the styles directory of the repository, several line of scss styles came across my eyes,

KaTeX tweak fix styles in theme
◎ KaTeX tweak fix styles in theme

this scss lines only add a horizontal trackbar at the bottom of each overflowing equation while maximizing the vertical height of the equation, when user is on a small screened device as follows,

Horizontal scrollbar of equations on small screens
◎ Horizontal scrollbar of equations on small screen

Consequently, the fix is easy.

1.4: Solution ​

If you are running a documentation site like this version controlled via Node, especially VitePress without native scss supports, your fix would be installing scss support globally across the project, then add the scss styles and finally import the stylesheets to take effect globally.

Install global scss support using your favored package manager.

sh
$ npm install -D scss
$ npm install -D scss
sh
$ yarn add --dev scss
$ yarn add --dev scss
sh
$ pnpm add -D scss
$ pnpm add -D scss

Then in your scss stylesheets, add the following and link or import them to your global styles, you should be off to go with a complete fix-up for KaTeX.

scss
...
+    
Toshiki's Note
Appearance
Tips & Hacks ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:1.2k
 Reading:7 min
1: Responsive KaTeX Styling ​
1.1: Issue background ​
KaTeX works out of the box on large screen devices such as laptops and desktop computers. But as KaTeX's built-in does not support responsiveness on it's default stylings, hence KaTeX equations might overflow out of the default width of the default application containers on small-screened mobile devices as you can see in the following image in the dev tool of chrome, when the dimension of the webpage is set to responsive or under a certain fixed device dimension, the equations rendered in KaTeX overflows out of the viewport as inspected in blue hightlight.
katex overflowing on small screen devices
◎ katex overflowing on small screen devices
This inevitably causes the viewport to break and extends the default width for KaTeX equations <div> to fit, users will be required to manually scroll right in order to view the full equation consequently as follows,
katex overflowing with labels annotated
◎ katex overflowing with labels annotated
The above situation is undoubtedly annoying for user experiences while reading documentation, consider at what scenarios users have to scroll, scroll and scroll only for viewing a single long-blocked equation, who would want to read such a text, right? But don't worry we have a simple fix for this situation via by several line of css.
1.2: Temporary solution ​
If you happened to search over KaTeX's official repo issue tracker on GitHub, there are several user-made css tweaks hack already, the fix is simple by adjusting the overflows of both x and y axes of the KaTeX render <div> blocks. The katex-display > .katex selector targets the child element of the .katex-display class that has the .katex class. This is the element that contains the KaTeX math expression. The first block of styles is mostly concerned with making sure that the KaTeX expression doesn't overflow its container and can be scrolled horizontally if needed. The second block of styles sets the font and line-height for the KaTeX expression and makes sure that its text is properly indented.
@preview
But, the issue here is, the overflowing issue is resolved on the webpage, but the style itself left with a slight whitish "box" at the crossing corner of both horizontal as well as vertical scrollbar tracks, this might not be so explicit in the light mode of the webpage, but when it turns to dark mode, the box become annoying. Some people might say it's an easy tweak via setting the display property of the "box" element to display: none; to directly remove the box out of the page, this is a smart approach; however, while the box is gone, the two crossing tracks is going to form an untouched invisible box again against two bars without color. Thus, neither ways seems to perfectly solve the problem.
1.3: Finding solution ​
After running a quick research on Google, I found a simple hack tweak used in a theme of VuePress, vuepress-theme-hope on GitHub, the theme both integrate with KaTeX and MathJax for math supports.
@preview
Under the styles directory of the repository, several line of scss styles came across my eyes,
KaTeX tweak fix styles in theme
◎ KaTeX tweak fix styles in theme
this scss lines only add a horizontal trackbar at the bottom of each overflowing equation while maximizing the vertical height of the equation, when user is on a small screened device as follows,
Horizontal scrollbar of equations on small screens
◎ Horizontal scrollbar of equations on small screen
Consequently, the fix is easy.
1.4: Solution ​
If you are running a documentation site like this version controlled via Node, especially VitePress without native scss supports, your fix would be installing scss support globally across the project, then add the scss styles and finally import the stylesheets to take effect globally.
Install global scss support using your favored package manager.
sh
$ npm install -D scss
$ npm install -D scss
sh
$ yarn add --dev scss
$ yarn add --dev scss
sh
$ pnpm add -D scss
$ pnpm add -D scss
Then in your scss stylesheets, add the following and link or import them to your global styles, you should be off to go with a complete fix-up for KaTeX.
scss
...
 
 // katex responsiveness fix
 .katex {
@@ -132,8 +132,8 @@
 
 .katex-error {
     color: #f00;
-}
To be continued.
Reference

-    
+}

To be continued.

Reference

+ \ No newline at end of file diff --git a/application/vitepress-plugin-shiki-twoslash/api/annotations.html b/application/vitepress-plugin-shiki-twoslash/api/annotations.html index d20f4445..9745695a 100644 --- a/application/vitepress-plugin-shiki-twoslash/api/annotations.html +++ b/application/vitepress-plugin-shiki-twoslash/api/annotations.html @@ -13,7 +13,7 @@ - + @@ -36,7 +36,7 @@ -
Toshiki's Note

Appearance

Queries ​

Author:Anda Toshiki
Updated:a minute ago
Words:541
Reading:3 min

Sometimes the thing you want to say is about the code, annotations provide a way to provide outside commentary on your code.

@annotate: [left|right] [overrides] - [text] ​

Annotate has a lot more controls than most of the other Twoslash commands, because each use of it probably needs to feel a bit different. Here's an example based on the TypeScript home page, click it to get it running so we can talk about what it does:

ts
ts
function compact(arr) {
    if (.length > 10) return arr.trim(0, 10)
Cannot find name 'orr'.2304Cannot find name 'orr'.
    return arr
}

+    
Toshiki's Note
Appearance
Queries ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:541
 Reading:3 min
Sometimes the thing you want to say is about the code, annotations provide a way to provide outside commentary on your code.
@annotate: [left|right] [overrides] - [text] ​
Annotate has a lot more controls than most of the other Twoslash commands, because each use of it probably needs to feel a bit different. Here's an example based on the TypeScript home page, click it to get it running so we can talk about what it does:
ts
ts
function compact(arr) {
    if (.length > 10) return arr.trim(0, 10)
Cannot find name 'orr'.2304Cannot find name 'orr'.
    return arr
}

 

   
     
@@ -67,7 +67,7 @@
     return arr
 }
 // @annotate: left { "arrowRot": "90deg 8px 27px", "textDegree": "3deg", "top": "0rem" } - Discovered a typo, the param is arr, not orr!
-```
First up, cool — it adds some text to the left hand side of the code. It features quite a few different options, so lets go through them one by one:
  • left or right: It's currently left. It's worth noting the arrow flips also, and 90deg isn't a great option. Let's look at that next.
  • { "arrrowRot": "90deg 8px 27px" } - This JSON object is used to manipulate the annotation, you have 3 controls for arrow positioning and rotation: degrees x y. I recommend keeping those in degrees and px, but it's your life. These are overrides from defaults which are okay, but not really something you ever want to ship.
  • { "textDegree": "3deg" } - Rotates the text, you probably want something between -3deg and 3deg. Optional, defaults to 0.
  • { "top": "0rem" } - Sets the y coordinates for the annotation relative to the code sample, if it's not included then it becomes [lineNum]rem.
What's not included in this sample is flipped, which can be used to flip the arrow's orientation. Here's some examples:
A horizontal right example:
ts
ts
function compact(arr) {
    if (.length > 10) return arr.trim(0, 10)
Cannot find name 'orr'.2304Cannot find name 'orr'.
    return arr
}

+```
First up, cool — it adds some text to the left hand side of the code. It features quite a few different options, so lets go through them one by one:
  • left or right: It's currently left. It's worth noting the arrow flips also, and 90deg isn't a great option. Let's look at that next.
  • { "arrrowRot": "90deg 8px 27px" } - This JSON object is used to manipulate the annotation, you have 3 controls for arrow positioning and rotation: degrees x y. I recommend keeping those in degrees and px, but it's your life. These are overrides from defaults which are okay, but not really something you ever want to ship.
  • { "textDegree": "3deg" } - Rotates the text, you probably want something between -3deg and 3deg. Optional, defaults to 0.
  • { "top": "0rem" } - Sets the y coordinates for the annotation relative to the code sample, if it's not included then it becomes [lineNum]rem.
What's not included in this sample is flipped, which can be used to flip the arrow's orientation. Here's some examples:
A horizontal right example:
ts
ts
function compact(arr) {
    if (.length > 10) return arr.trim(0, 10)
Cannot find name 'orr'.2304Cannot find name 'orr'.
    return arr
}

 

   
     
@@ -98,7 +98,7 @@
     return arr
 }
 // @annotate: left { "arrowRot": "90deg 8px 27px", "textDegree": "3deg", "top": "0rem" } - Discovered a typo, the param is arr, not orr!
-```
Upside down arrow pointing at the error, using flipped to re-flip the arrow:
ts
ts
function compact(arr) {
    if (.length > 10) return arr.trim(0, 10)
Cannot find name 'orr'.2304Cannot find name 'orr'.
    return arr
}

+```
Upside down arrow pointing at the error, using flipped to re-flip the arrow:
ts
ts
function compact(arr) {
    if (.length > 10) return arr.trim(0, 10)
Cannot find name 'orr'.2304Cannot find name 'orr'.
    return arr
}

 

   
     
@@ -129,8 +129,8 @@
     return arr
 }
 // @annotate: left { "arrowRot": "90deg 8px 27px", "textDegree": "3deg", "top": "0rem" } - Discovered a typo, the param is arr, not orr!
-```

-    
+```
+ \ No newline at end of file diff --git a/application/vitepress-plugin-shiki-twoslash/api/cutting.html b/application/vitepress-plugin-shiki-twoslash/api/cutting.html index 3b6c01e3..1ad07c34 100644 --- a/application/vitepress-plugin-shiki-twoslash/api/cutting.html +++ b/application/vitepress-plugin-shiki-twoslash/api/cutting.html @@ -13,7 +13,7 @@ - + @@ -36,7 +36,7 @@ -
Toshiki's Note

Appearance

Cutting ​

Author:Anda Toshiki
Updated:a minute ago
Words:242
Reading:1 min

Every Twoslash code example needs to be a complete TypeScript program so it can compile. Quite often to make it compile, there is a bunch of code which isn't relevant to the user. This can be extracted out of the code examples via cut comments.

---cut--- ​

Cut works after TypeScript has generated the project and pulled out all the editor information (like identifiers, queries, highlights etc) and then amends all of their offsets and lines to re-fit the smaller output. What your user sees is everything below the ---cut---.

ts
ts
console.log(level)
ts
console.log(level)
md
```ts twoslash
+    
Toshiki's Note
Appearance
Cutting ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:242
 Reading:1 min
Every Twoslash code example needs to be a complete TypeScript program so it can compile. Quite often to make it compile, there is a bunch of code which isn't relevant to the user. This can be extracted out of the code examples via cut comments.
---cut--- ​
Cut works after TypeScript has generated the project and pulled out all the editor information (like identifiers, queries, highlights etc) and then amends all of their offsets and lines to re-fit the smaller output. What your user sees is everything below the ---cut---.
ts
ts
console.log(level)
ts
console.log(level)
md
```ts twoslash
 const level: string = 'Danger'
 // ---cut---
 console.log(level)
@@ -44,7 +44,7 @@
 const level: string = 'Danger'
 // ---cut---
 console.log(level)
-```
Cutting even works across multiple files. This is why // @filename: [file] is specifically the only Twoslash command which is not removed, because if it's not relevant it can be ---cut--- away.
ts
ts
import { ```
Cutting even works across multiple files. This is why // @filename: [file] is specifically the only Twoslash command which is not removed, because if it's not relevant it can be ---cut--- away.
ts
ts
import { helloWorld } from './a'
console.log(helloWorld)
ts
import { helloWorld } from './a'
console.log(helloWorld)
// ---cut---
 import { helloWorld } from './a'
 console.log(helloWorld)
-```

---cut-after--- ​

The sibling to ---cut---, which trims anything after the sigil:

tsx
tsx
    <Container>
        <ImportantComponent />
    </Container>
tsx
    <Container>
        <ImportantComponent />
    </Container>
md
```tsx twoslash
+```

---cut-after--- ​

The sibling to ---cut---, which trims anything after the sigil:

tsx
tsx
    <Container>
        <ImportantComponent />
    </Container>
tsx
    <Container>
        <ImportantComponent />
    </Container>
md
```tsx twoslash
 const Page = () => (
     // ---cut---
     <Container>
@@ -80,8 +80,8 @@ import helloWorld">helloWorld)
</Container> // ---cut-after--- ) -```
- +```
+ \ No newline at end of file diff --git a/application/vitepress-plugin-shiki-twoslash/api/emit.html b/application/vitepress-plugin-shiki-twoslash/api/emit.html index e24129dd..1690cd61 100644 --- a/application/vitepress-plugin-shiki-twoslash/api/emit.html +++ b/application/vitepress-plugin-shiki-twoslash/api/emit.html @@ -13,7 +13,7 @@ - + @@ -36,13 +36,13 @@ -
Toshiki's Note

Appearance

Emit ​

Author:Anda Toshiki
Updated:a minute ago
Words:264
Reading:1 min

Running a Twoslash code example is a full TypeScript compiler run that will create files inside the virtual file system. You can replace the contents of your code examples with the results of running TypeScript over the project.

@showEmit ​

// @showEmit is the main command to tell Shiki Twoslash that you want to replace the output of your code example with the equivalent .js file.

ts
ts
"use strict";
const level = 'Danger';
 
ts
"use strict";
const level = 'Danger';
 
md
```ts twoslash
+    
Toshiki's Note
Appearance
Emit ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:264
 Reading:1 min
Running a Twoslash code example is a full TypeScript compiler run that will create files inside the virtual file system. You can replace the contents of your code examples with the results of running TypeScript over the project.
@showEmit ​
// @showEmit is the main command to tell Shiki Twoslash that you want to replace the output of your code example with the equivalent .js file.
ts
ts
"use strict";
const level = 'Danger';
 
ts
"use strict";
const level = 'Danger';
 
md
```ts twoslash
 // @showEmit
 const level: string = 'Danger'
 ```
```ts twoslash
 // @showEmit
 const level: string = 'Danger'
-```
@showEmittedFile: [file] ​
While the .js file is probably the most useful file out of the box, TypeScript does emit other files if you have the right flags enabled (.d.ts and .map) but also when you have a multi-file code sample — you might need to tell Twoslash which file to show. For all these cases you can also add @showEmittedFile: [file] to tell Twoslash which file you want to show.
Shows emitted .d.ts for a TypeScript code example:
md
```ts twoslash
+```
@showEmittedFile: [file] ​
While the .js file is probably the most useful file out of the box, TypeScript does emit other files if you have the right flags enabled (.d.ts and .map) but also when you have a multi-file code sample — you might need to tell Twoslash which file to show. For all these cases you can also add @showEmittedFile: [file] to tell Twoslash which file you want to show.
Shows emitted .d.ts for a TypeScript code example:
md
```ts twoslash
 // @declaration
 // @showEmit
 // @showEmittedFile: index.d.ts
@@ -52,7 +52,7 @@
 // @showEmit
 // @showEmittedFile: index.d.ts
 export const hello = 'world'
-```
ts
ts
export declare const hello = "world";
 
ts
export declare const hello = "world";
 
Shows emitted .map files:
md
```ts twoslash
+```
ts
ts
export declare const hello = "world";
 
ts
export declare const hello = "world";
 
Shows emitted .map files:
md
```ts twoslash
 // @sourceMap
 // @showEmit
 // @showEmittedFile: index.js.map
@@ -62,7 +62,7 @@
 // @showEmit
 // @showEmittedFile: index.js.map
 export const hello = 'world'
-```
ts
ts
{"version":3,"file":"index.js","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,MAAM,CAAC,MAAM,KAAK,GAAG,OAAO,CAAA"}
ts
{"version":3,"file":"index.js","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,MAAM,CAAC,MAAM,KAAK,GAAG,OAAO,CAAA"}
md
```ts twoslash
+```
ts
ts
{"version":3,"file":"index.js","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,MAAM,CAAC,MAAM,KAAK,GAAG,OAAO,CAAA"}
ts
{"version":3,"file":"index.js","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,MAAM,CAAC,MAAM,KAAK,GAAG,OAAO,CAAA"}
md
```ts twoslash
 // @declaration
 // @declarationMap
 // @showEmit
@@ -74,8 +74,8 @@
 // @showEmit
 // @showEmittedFile: index.d.ts.map
 export const hello = 'world'
-```
ts
ts
{"version":3,"file":"index.d.ts","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,eAAO,MAAM,KAAK,UAAU,CAAA"}
ts
{"version":3,"file":"index.d.ts","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,eAAO,MAAM,KAAK,UAAU,CAAA"}

-    
+```
ts
ts
{"version":3,"file":"index.d.ts","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,eAAO,MAAM,KAAK,UAAU,CAAA"}
ts
{"version":3,"file":"index.d.ts","sourceRoot":"","sources":["index.ts"],"names":[],"mappings":"AAAA,eAAO,MAAM,KAAK,UAAU,CAAA"}
+ \ No newline at end of file diff --git a/application/vitepress-plugin-shiki-twoslash/api/errors.html b/application/vitepress-plugin-shiki-twoslash/api/errors.html index d7474c7d..1bf46ccb 100644 --- a/application/vitepress-plugin-shiki-twoslash/api/errors.html +++ b/application/vitepress-plugin-shiki-twoslash/api/errors.html @@ -13,7 +13,7 @@ - + @@ -36,7 +36,7 @@ -
Toshiki's Note

Appearance

Errors ​

Author:Anda Toshiki
Updated:a minute ago
Words:227
Reading:1 min

Most of the time, you want to avoid errors in your code examples. Strictly speaking, this usually means setting the right compiler flags and environment in each code example.

Sometimes however, you do want to raise a compiler error — to show incorrect states. In those cases, twoslash has a way to mark the compiler errors you expect.

@errors: [num] ​

All TypeScript compiler errors have a number, this number is relatively arbitrary and can change between TypeScript versions. For our case these numbers are useul in declaring what we expect to see.

You can use // @errors: [num] to tell Twoslash that you expect this error to occur. This moves the compiler error message into the code example.

ts
ts
const a = '123'
 = 132
Cannot assign to 'a' because it is a constant.2588Cannot assign to 'a' because it is a constant.
ts
const a = '123'
 = 132
Cannot assign to 'a' because it is a constant.2588Cannot assign to 'a' because it is a constant.
md
```ts twoslash
+    
Toshiki's Note
Appearance
Errors ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:227
 Reading:1 min
Most of the time, you want to avoid errors in your code examples. Strictly speaking, this usually means setting the right compiler flags and environment in each code example.
Sometimes however, you do want to raise a compiler error — to show incorrect states. In those cases, twoslash has a way to mark the compiler errors you expect.
@errors: [num] ​
All TypeScript compiler errors have a number, this number is relatively arbitrary and can change between TypeScript versions. For our case these numbers are useul in declaring what we expect to see.
You can use // @errors: [num] to tell Twoslash that you expect this error to occur. This moves the compiler error message into the code example.
ts
ts
const a = '123'
 = 132
Cannot assign to 'a' because it is a constant.2588Cannot assign to 'a' because it is a constant.
ts
const a = '123'
 = 132
Cannot assign to 'a' because it is a constant.2588Cannot assign to 'a' because it is a constant.
md
```ts twoslash
 // @errors: 2588
 const a = '123'
 a = 132
@@ -44,7 +44,7 @@
 // @errors: 2588
 const a = '123'
 a = 132
-```
@noErrors ​
Sometimes you have needs in which a broken TypeScript build is okay. A good example of this is using a completion query, which requires a broken TypeScript project to work. You can use // @noErrors to supress all errors in a code sample, and not have them show inline.
ts
ts
const a = '123'
a = 132
ts
const a = '123'
a = 132
md
```ts twoslash
+```
@noErrors ​
Sometimes you have needs in which a broken TypeScript build is okay. A good example of this is using a completion query, which requires a broken TypeScript project to work. You can use // @noErrors to supress all errors in a code sample, and not have them show inline.
ts
ts
const a = '123'
a = 132
ts
const a = '123'
a = 132
md
```ts twoslash
 // @noErrors
 const a = '123'
 a = 132
@@ -52,8 +52,8 @@
 // @noErrors
 const a = '123'
 a = 132
-```

-    
+```
+ \ No newline at end of file diff --git a/application/vitepress-plugin-shiki-twoslash/api/includes.html b/application/vitepress-plugin-shiki-twoslash/api/includes.html index df73f192..2c5c4f4b 100644 --- a/application/vitepress-plugin-shiki-twoslash/api/includes.html +++ b/application/vitepress-plugin-shiki-twoslash/api/includes.html @@ -13,7 +13,7 @@ - + @@ -56,7 +56,7 @@ // - afterUserDefinitions type SomeGroup = { name: string; members: SomeUser[] } // - afterGroupDefinitions -```

Including a whole block ​

To include a re-usable block, add // @include: [block name] in your code block.

twoslash
ts
ts
type SomeString = string
const a: SomeString = 'string'
ts
type SomeString = string
const a: SomeString = 'string'
md
```twoslash include myBlock
+```
Including a whole block ​
To include a re-usable block, add // @include: [block name] in your code block.
twoslash
ts
ts
type SomeString = string
const a: SomeString = 'string'
ts
type SomeString = string
const a: SomeString = 'string'
md
```twoslash include myBlock
 type SomeString = string
 ```
 
@@ -70,7 +70,7 @@
 ```ts twoslash
 // @include: myBlock
 const a: SomeString = 'string'
-```
Including a block step ​
To include a re-usable block at a specific step, add // @include: [block name]-[step name] in your code block.
twoslash
ts
ts
type SomeString = string
type ```
Including a block step ​
To include a re-usable block at a specific step, add // @include: [block name]-[step name] in your code block.
twoslash
ts
ts
type SomeString = string
type SomeUser = { name: string; mail?: ```ts twoslash
 // @include: myBlockWithSteps-afterUserDefinitions
 const mail: SomeUserMail = { content: 'some-email', verified: true }
-```
Hiding re-used code ​
Re-using a lot of TypeScript code can easily bloat your documentation and obstruct the main point of your code block. You can hide re-used code to keep your code blocks clean and concise by cutting right after the @include statement.
ts
ts
const mail: ```
Hiding re-used code ​
Re-using a lot of TypeScript code can easily bloat your documentation and obstruct the main point of your code block. You can hide re-used code to keep your code blocks clean and concise by cutting right after the @include statement.
ts
ts
const mail: SomeUserMail = { content: 'some-email', verified: true }
ts
const mail: // @include: myBlockWithSteps-afterUserDefinitions
 // ---cut---
 const mail: SomeUserMail = { content: 'some-email', verified: true }
-```

-    
+```

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/api/logging.html b/application/vitepress-plugin-shiki-twoslash/api/logging.html
index b91e1089..6321a813 100644
--- a/application/vitepress-plugin-shiki-twoslash/api/logging.html
+++ b/application/vitepress-plugin-shiki-twoslash/api/logging.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -36,7 +36,7 @@
     
   
   
-    
Toshiki's Note
Appearance
Logging ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:152
 Reading:1 min
When you first see a Twoslash code example with an inline compiler error, you instinctively trust that the compiler error is correct because the design shows that it is not a part of the code sample. The logging tools lets you do that, but abuses the systemic trust because your code is not being evaluated to generate the logs.
This feature is effectively a facade, people will trust your output and it will look better.
@log:, @warn:, @error: ​
The names are based on the functions on the console object:
ts
ts
console.log('Hello log')
Hello log
 
console.warn('Hello warn')
Hello warn
 
console.error('Hello error')
Hello error

+    
Toshiki's Note
Appearance
Logging ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:152
 Reading:1 min
When you first see a Twoslash code example with an inline compiler error, you instinctively trust that the compiler error is correct because the design shows that it is not a part of the code sample. The logging tools lets you do that, but abuses the systemic trust because your code is not being evaluated to generate the logs.
This feature is effectively a facade, people will trust your output and it will look better.
@log:, @warn:, @error: ​
The names are based on the functions on the console object:
ts
ts
console.log('Hello log')
Hello log
 
console.warn('Hello warn')
Hello warn
 
console.error('Hello error')
Hello error

 
ts
console.log('Hello log')
Hello log
 
console.warn('Hello warn')
Hello warn
 
console.error('Hello error')
Hello error
md
```ts twoslash
 console.log('Hello log')
 // @log: Hello log
@@ -55,8 +55,8 @@
 
 console.error('Hello error')
 // @error: Hello error
-```

-    
+```

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/api/multi-file.html b/application/vitepress-plugin-shiki-twoslash/api/multi-file.html
index 7bef0937..4e9d1903 100644
--- a/application/vitepress-plugin-shiki-twoslash/api/multi-file.html
+++ b/application/vitepress-plugin-shiki-twoslash/api/multi-file.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -36,19 +36,19 @@
     
   
   
-    
Toshiki's Note
Appearance
Multi-file ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:477
 Reading:2 min
Twoslash code examples aren't limited to creating a single file, by using // @filename: [file] you can write any file to the virtual file system used by TypeScript to power your code examples.
@filename: [file] ​
Most of the time, you don't need to think about the underlaying virtual file system in a code example, but when you have imports between them it becomes important to know. Twoslash will default to creating an index.[type] based on the langauge passed to the code example:
ts
ts
// I'm index.ts
ts
// I'm index.ts
md
```ts twoslash
+    
Toshiki's Note
Appearance
Multi-file ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:477
 Reading:2 min
Twoslash code examples aren't limited to creating a single file, by using // @filename: [file] you can write any file to the virtual file system used by TypeScript to power your code examples.
@filename: [file] ​
Most of the time, you don't need to think about the underlaying virtual file system in a code example, but when you have imports between them it becomes important to know. Twoslash will default to creating an index.[type] based on the langauge passed to the code example:
ts
ts
// I'm index.ts
ts
// I'm index.ts
md
```ts twoslash
 // I'm index.ts
 ```
```ts twoslash
 // I'm index.ts
-```
tsx
tsx
// I'm index.tsx
tsx
// I'm index.tsx
md
```tsx twoslash
+```
tsx
tsx
// I'm index.tsx
tsx
// I'm index.tsx
md
```tsx twoslash
 // I'm index.tsx
 ```
```tsx twoslash
 // I'm index.tsx
-```
js
js
// I'm index.tjs
js
// I'm index.tjs
md
```js twoslash
+```
js
js
// I'm index.tjs
js
// I'm index.tjs
md
```js twoslash
 // I'm index.tjs
 ```
```js twoslash
 // I'm index.tjs
-```
Then until Twoslash hits another // @filename: [file], the parser keeps adding new lines into the same file. After seeing @filename Twoslash creates a new virtual file-system file and adds the new lines to that. You can't edit a file after it was created, but you can overwrite it.
It can be any file. For example, if you want to quickly fake a node module:
ts
ts
// @filename: node_modules/@types/mylib/index.d.ts
export function doit(): string
 
// @filename: index.ts
import { ```
Then until Twoslash hits another // @filename: [file], the parser keeps adding new lines into the same file. After seeing @filename Twoslash creates a new virtual file-system file and adds the new lines to that. You can't edit a file after it was created, but you can overwrite it.
It can be any file. For example, if you want to quickly fake a node module:
ts
ts
// @filename: node_modules/@types/mylib/index.d.ts
export function doit(): string
 
// @filename: index.ts
import { doit } from 'mylib'
console.log(doit)
ts
// @filename: node_modules/@types/mylib/index.d.ts
export function doit(): string
 
// @filename: index.ts
import { doit } from 'mylib'
console.log(doit)
// @filename: index.ts
 import { doit } from 'mylib'
 console.log(doit)
-```
You can also set up a JSON object which can be imported in a TypeScript file:
ts
ts
// @filename: app.json
{ "version": "23.2.3" }
 
// @filename: index.ts
import appSettings from "./app.json"
appSettings.version
              
(property) "version": string
ts
// @filename: app.json
{ "version": "23.2.3" }
 
// @filename: index.ts
import appSettings from "./app.json"
appSettings.version
              
(property) "version": string
md
```ts twoslash
+```
You can also set up a JSON object which can be imported in a TypeScript file:
ts
ts
// @filename: app.json
{ "version": "23.2.3" }
 
// @filename: index.ts
import appSettings from "./app.json"
appSettings.version
              
(property) "version": string
ts
// @filename: app.json
{ "version": "23.2.3" }
 
// @filename: index.ts
import appSettings from "./app.json"
appSettings.version
              
(property) "version": string
md
```ts twoslash
 // @resolveJsonModule
 // @filename: app.json
 { "version": "23.2.3" }
@@ -84,7 +84,7 @@ import doit">doit)
import appSettings from "./app.json"
 appSettings.version
 //           ^?
-```
Finally, the following code allows importing non-TypeScript content. There is a .d.ts file which globally says 'md files are OK to import' and 'the module "react" exists, but don't worry about the details'.
Then for a user, they only see the imports and exports inside index.tsx.
ts
ts
import ```
Finally, the following code allows importing non-TypeScript content. There is a .d.ts file which globally says 'md files are OK to import' and 'the module "react" exists, but don't worry about the details'.
Then for a user, they only see the imports and exports inside index.tsx.
ts
ts
import React from "react"
import MultiFileDocs from "./MultiFileDocs.mdx"
 
export default () => <MultiFileDocs/>
ts
import React from "react"
import MultiFileDocs from "./MultiFileDocs.mdx"
 
export default () => <MultiFileDocs/>
md
```ts twoslash
 // @filename: ambient.d.ts
@@ -118,8 +118,8 @@ import React">React fromimport MultiFileDocs from "./MultiFileDocs.mdx"
 
 export default () => <MultiFileDocs/>
-```

-    
+```

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/api/queries.html b/application/vitepress-plugin-shiki-twoslash/api/queries.html
index afde13fe..62d98204 100644
--- a/application/vitepress-plugin-shiki-twoslash/api/queries.html
+++ b/application/vitepress-plugin-shiki-twoslash/api/queries.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -36,7 +36,7 @@
     
   
   
-    
Toshiki's Note
Appearance
Queries ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:167
 Reading:1 min
One of the key features of Twoslash is the ability to use the TypeScript compiler to pull out type information about your code. Twoslash comes with two different ways to query your code: ?^ and ?|.
Extract Type ^? ​
Using ^? you can pull out type information about a particular identifier in the line of code above it.
ts
ts
const hi = 'Hello'
const msg = hi + ', world'
      
const msg: string
ts
const hi = 'Hello'
const msg = hi + ', world'
      
const msg: string
md
```ts twoslash
+    
Toshiki's Note
Appearance
Queries ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:167
 Reading:1 min
One of the key features of Twoslash is the ability to use the TypeScript compiler to pull out type information about your code. Twoslash comes with two different ways to query your code: ?^ and ?|.
Extract Type ^? ​
Using ^? you can pull out type information about a particular identifier in the line of code above it.
ts
ts
const hi = 'Hello'
const msg = hi + ', world'
      
const msg: string
ts
const hi = 'Hello'
const msg = hi + ', world'
      
const msg: string
md
```ts twoslash
 const hi = 'Hello'
 const msg = hi + ', world'
 //    ^?
@@ -44,7 +44,7 @@
 const hi = 'Hello'
 const msg = hi + ', world'
 //    ^?
-```
Completions ^| ​
Using ^| you can pull out information about a what the auto-complete looks like at a particular location.
ts
ts
console.e
         
  • error
ts
console.e
         
  • error
md
```ts twoslash
+```
Completions ^| ​
Using ^| you can pull out information about a what the auto-complete looks like at a particular location.
ts
ts
console.e
         
  • error
ts
console.e
         
  • error
md
```ts twoslash
 // @noErrors
 console.e
 //       ^|
@@ -52,8 +52,8 @@
 // @noErrors
 console.e
 //       ^|
-```
INFO
Note that the compiler flag for // @noErrors is set, because console.e is a failing TypeScript code sample but we don't really care about that.

-    
+```
INFO
Note that the compiler flag for // @noErrors is set, because console.e is a failing TypeScript code sample but we don't really care about that.

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/api/types.html b/application/vitepress-plugin-shiki-twoslash/api/types.html
index 6048d002..217d0bda 100644
--- a/application/vitepress-plugin-shiki-twoslash/api/types.html
+++ b/application/vitepress-plugin-shiki-twoslash/api/types.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -36,7 +36,7 @@
     
   
   
-    
Toshiki's Note
Appearance
@types ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:230
 Reading:1 min
For most examples, you probably need to import external libraries into your code examples.
Twoslash works by faking a virtual file system over your existing file system. This means any @types or libraries with TypeScript definitions should work out of the box with no config.
Local Sources ​
Simply import locally installed libraries and Twoslash can pick up types:
ts
ts
import { 
Toshiki's Note
Appearance
@types ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:230
 Reading:1 min
For most examples, you probably need to import external libraries into your code examples.
Twoslash works by faking a virtual file system over your existing file system. This means any @types or libraries with TypeScript definitions should work out of the box with no config.
Local Sources ​
Simply import locally installed libraries and Twoslash can pick up types:
ts
ts
import { defineConfig } from 'vitepress'
const config = defineConfig({})
        
const config: UserConfig<DefaultTheme.Config>
export default config
ts
import { defineConfig } from 'vitepress'
const config = defineConfig
 const config = defineConfig({})
 //    ^?
 export default config
-```
Globals ​
Setting up globals is a little bit more complex, but not drastically. You need to use the triple slash reference which adds a particular library to the global scope.
For example, setting up Node imports and globals etc.
ts
ts
import { ```
Globals ​
Setting up globals is a little bit more complex, but not drastically. You need to use the triple slash reference which adds a particular library to the global scope.
For example, setting up Node imports and globals etc.
ts
ts
import { writeFileSync } from 'fs'
writeFileSync('myfile.txt', '// TODO')
ts
import { writeFileSync } from 'fs'
writeFileSync// ---cut---
 import { writeFileSync } from 'fs'
 writeFileSync('myfile.txt', '// TODO')
-```
APIs like Vitest are similar cases where you would use a triple slash reference.
ts
ts
test('my tests', () => {
    expect('hello').toEqual('hello')
      
const expect: ExpectStatic
})
ts
test('my tests', () => {
    expect('hello').toEqual('hello')
      
const expect: ExpectStatic
})
md
```ts twoslash
+```
APIs like Vitest are similar cases where you would use a triple slash reference.
ts
ts
test('my tests', () => {
    expect('hello').toEqual('hello')
      
const expect: ExpectStatic
})
ts
test('my tests', () => {
    expect('hello').toEqual('hello')
      
const expect: ExpectStatic
})
md
```ts twoslash
 /// <reference types="vitest/globals" />
 // ---cut---
 
@@ -80,8 +80,8 @@ import writeFileSync">writeFileSync    expect('hello').toEqual('hello')
     // ^?
 })
-```

-    
+```

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/config/flags.html b/application/vitepress-plugin-shiki-twoslash/config/flags.html
index d670a4fe..d901f3e3 100644
--- a/application/vitepress-plugin-shiki-twoslash/config/flags.html
+++ b/application/vitepress-plugin-shiki-twoslash/config/flags.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -626,8 +626,8 @@
 Emit ECMAScript-standard-compliant class fields..
 
 // @useUnknownInCatchVariables
-Type catch clause variables as 'unknown' instead of 'any'..

-    
+Type catch clause variables as 'unknown' instead of 'any'..

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/config/reference.html b/application/vitepress-plugin-shiki-twoslash/config/reference.html
index 212f3475..3fad73a9 100644
--- a/application/vitepress-plugin-shiki-twoslash/config/reference.html
+++ b/application/vitepress-plugin-shiki-twoslash/config/reference.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -118,8 +118,8 @@ import defineConfig">defineConfig
             wrapFragments: true, 
         },
     })
-)

-    
+)

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/guide/custom-theme.html b/application/vitepress-plugin-shiki-twoslash/guide/custom-theme.html
index 75128467..f47f6346 100644
--- a/application/vitepress-plugin-shiki-twoslash/guide/custom-theme.html
+++ b/application/vitepress-plugin-shiki-twoslash/guide/custom-theme.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -120,8 +120,8 @@ import defineConfig">defineConfig
 }
 html.dark pre.shiki[class*='-light'] {
     display: none;
-}

-    
+}

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/guide/markdown-extensions.html b/application/vitepress-plugin-shiki-twoslash/guide/markdown-extensions.html
index 018c0ce9..531388fd 100644
--- a/application/vitepress-plugin-shiki-twoslash/guide/markdown-extensions.html
+++ b/application/vitepress-plugin-shiki-twoslash/guide/markdown-extensions.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -36,14 +36,14 @@
     
   
   
-    
Toshiki's Note
Appearance
Markdown Extensions ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:181
 Reading:1 min
Code Groups ​
Code Groups and Twoslash multi-file support.
ts
ts
import { 
Toshiki's Note
Appearance
Markdown Extensions ​
 Author:Anda Toshiki
 Updated:a minute ago
 Words:181
 Reading:1 min
Code Groups ​
Code Groups and Twoslash multi-file support.
ts
ts
import { name } from './name'
export function hello(name: string) {
    console.log(`Hello, ${name}!`)
}
hello(name)
       
(alias) const name: "twoslash"
 import name
ts
import { name } from './name'
export function hello(name: string) {
    console.log(`Hello, ${name}!`)
}
hello(name)
       
(alias) const name: "twoslash"
-import name
ts
ts
export const name = 'twoslash'
ts
export const name = 'twoslash'
Unsupported Extensions ​
Since VitePress Twoslash uses it's own Shiki highlighter, the following syntax highlighting extensions are not currently compatible.
If you are interested in adding support, please start a new GitHub Discussion.

-    
+import name
ts
ts
export const name = 'twoslash'
ts
export const name = 'twoslash'
Unsupported Extensions ​
Since VitePress Twoslash uses it's own Shiki highlighter, the following syntax highlighting extensions are not currently compatible.
If you are interested in adding support, please start a new GitHub Discussion.

+    
     
   
 
\ No newline at end of file
diff --git a/application/vitepress-plugin-shiki-twoslash/index.html b/application/vitepress-plugin-shiki-twoslash/index.html
index df68b154..c2a1e667 100644
--- a/application/vitepress-plugin-shiki-twoslash/index.html
+++ b/application/vitepress-plugin-shiki-twoslash/index.html
@@ -13,7 +13,7 @@
     
     
     
-    
+    
     
     
     
@@ -58,7 +58,7 @@
 import defineConfig">defineConfig } from 'vitepress'
 
export default defineConfig({
    ti,
      
  • title
  • titleTemplate
})
ts
import { defineConfig } from 'vitepress'
 
export default defineConfig({
    ti,
      
  • title
  • titleTemplate
})
The name Twoslash refers to specially formatted comments (e.g. // ^?) which can be used to set up your environment, like compiler flags or separate input files. It couldn't be easier to set up and start creating incredible code examples!
Install ​
Install @andatoshiki/vitepress-plugin-shiki-twoslash (requires vitepress@>=1.0.0-alpha.61).
bash
pnpm add @andatoshiki/vitepress-plugin-shiki-twoslash
pnpm add @andatoshiki/vitepress-plugin-shiki-twoslash
bash
npm i @andatoshiki/vitepress-plugin-shiki-twoslash
npm i @andatoshiki/vitepress-plugin-shiki-twoslash
bash
yarn add @andatoshiki/vitepress-plugin-shiki-twoslash
yarn add @andatoshiki/vitepress-plugin-shiki-twoslash
WARNING
Until shiki-twoslash uses the same version of shiki as VitePress, you must override the following packages' shiki versions for syntax highlighting to look the same.
json
{
+import defineConfig">defineConfig({
    ti,
      
  • title
  • titleTemplate
})
The name Twoslash refers to specially formatted comments (e.g. // ^?) which can be used to set up your environment, like compiler flags or separate input files. It couldn't be easier to set up and start creating incredible code examples!
Install ​
Install @andatoshiki/vitepress-plugin-shiki-twoslash (requires vitepress@>=1.0.0-alpha.61).
bash
pnpm add @andatoshiki/vitepress-plugin-shiki-twoslash
pnpm add @andatoshiki/vitepress-plugin-shiki-twoslash
bash
npm i @andatoshiki/vitepress-plugin-shiki-twoslash
npm i @andatoshiki/vitepress-plugin-shiki-twoslash
bash
yarn add @andatoshiki/vitepress-plugin-shiki-twoslash
yarn add @andatoshiki/vitepress-plugin-shiki-twoslash
WARNING
Until shiki-twoslash uses the same version of shiki as VitePress, you must override the following packages' shiki versions for syntax highlighting to look the same.
json
{
     "pnpm": {
         "overrides": {
             "remark-shiki-twoslash>shiki": "^0.14.1",
@@ -146,8 +146,8 @@ import defaultTheme">defaultTheme
LockedAccount>
           
type UnlockedAccount = {
     id: string;
     name: string;
-}

-    
+}

+    
     
   
 
\ No newline at end of file
diff --git a/assets/academic_chemistry_index.md.ea7d7e52.js b/assets/academic_chemistry_index.md.acc4bb6b.js
similarity index 92%
rename from assets/academic_chemistry_index.md.ea7d7e52.js
rename to assets/academic_chemistry_index.md.acc4bb6b.js
index 08335087..c73efda4 100644
--- a/assets/academic_chemistry_index.md.ea7d7e52.js
+++ b/assets/academic_chemistry_index.md.acc4bb6b.js
@@ -1 +1 @@
-import{_ as a}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as o,o as c,c as r,H as s,k as e,a as i}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const k=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1712687763000}'),m={name:"academic/chemistry/index.md"},n=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[i("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1);function d(l,_,h,p,f,x){const t=a;return c(),r("div",null,[n,s(t,{readTime:"1",words:"3"})])}const N=o(m,[["render",d]]);export{k as __pageData,N as default};
+import{_ as a}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as o,o as c,c as r,H as s,k as e,a as i}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const k=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1712870707000}'),m={name:"academic/chemistry/index.md"},n=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[i("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1);function d(l,_,h,p,f,x){const t=a;return c(),r("div",null,[n,s(t,{readTime:"1",words:"3"})])}const N=o(m,[["render",d]]);export{k as __pageData,N as default};
diff --git a/assets/academic_chemistry_index.md.ea7d7e52.lean.js b/assets/academic_chemistry_index.md.acc4bb6b.lean.js
similarity index 92%
rename from assets/academic_chemistry_index.md.ea7d7e52.lean.js
rename to assets/academic_chemistry_index.md.acc4bb6b.lean.js
index 08335087..c73efda4 100644
--- a/assets/academic_chemistry_index.md.ea7d7e52.lean.js
+++ b/assets/academic_chemistry_index.md.acc4bb6b.lean.js
@@ -1 +1 @@
-import{_ as a}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as o,o as c,c as r,H as s,k as e,a as i}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const k=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1712687763000}'),m={name:"academic/chemistry/index.md"},n=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[i("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1);function d(l,_,h,p,f,x){const t=a;return c(),r("div",null,[n,s(t,{readTime:"1",words:"3"})])}const N=o(m,[["render",d]]);export{k as __pageData,N as default};
+import{_ as a}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as o,o as c,c as r,H as s,k as e,a as i}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const k=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1712870707000}'),m={name:"academic/chemistry/index.md"},n=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[i("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1);function d(l,_,h,p,f,x){const t=a;return c(),r("div",null,[n,s(t,{readTime:"1",words:"3"})])}const N=o(m,[["render",d]]);export{k as __pageData,N as default};
diff --git a/assets/academic_chemistry_notes_12-5.md.80389f1d.js b/assets/academic_chemistry_notes_12-5.md.452880b7.js
similarity index 99%
rename from assets/academic_chemistry_notes_12-5.md.80389f1d.js
rename to assets/academic_chemistry_notes_12-5.md.452880b7.js
index 5d1aa544..56b0c166 100644
--- a/assets/academic_chemistry_notes_12-5.md.80389f1d.js
+++ b/assets/academic_chemistry_notes_12-5.md.452880b7.js
@@ -1 +1 @@
-import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as e,o as n,c as m,H as i,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const cs=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"keywords":["chemistry","reaction","mechanism","reaction-mechanism","tutorial","explanation","textbook","reference"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1712687763000}'),r={name:"academic/chemistry/notes/12-5.md"},c=s("h1",{id:"_12-5-reaction-mechanism",tabindex:"-1"},[a("12-5: Reaction Mechanism "),s("a",{class:"header-anchor",href:"#_12-5-reaction-mechanism","aria-label":'Permalink to "12-5: Reaction Mechanism"'},"​")],-1),p=s("h2",{id:"_12-5-1-learning-objectives",tabindex:"-1"},[a("12-5-1: Learning Objectives "),s("a",{class:"header-anchor",href:"#_12-5-1-learning-objectives","aria-label":'Permalink to "12-5-1: Learning Objectives"'},"​")],-1),h=s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"Learning Objectives"),s("p",null,"One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.")],-1),o=s("p",null,"In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),u=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),d=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),v=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),y=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),k=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),f=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),z=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),O=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),_=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),M=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),N=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),C=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),L=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),A=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),B=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),T=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),R=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),I=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),q=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),S=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),P=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),D=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),H=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),E=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),F=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),j=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),U=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),$=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),V=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),W=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),G=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),J=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),Y=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),K=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),Q=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),X=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1);function Z(ss,as,ts,ls,es,ns){const t=l;return n(),m("div",null,[c,i(t,{readTime:"11",words:"1.9k"}),p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,B,T,R,I,q,S,P,D,H,E,F,j,U,$,V,W,G,J,Y,K,Q,X])}const ps=e(r,[["render",Z]]);export{cs as __pageData,ps as default};
+import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as e,o as n,c as m,H as i,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const cs=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"keywords":["chemistry","reaction","mechanism","reaction-mechanism","tutorial","explanation","textbook","reference"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1712870707000}'),r={name:"academic/chemistry/notes/12-5.md"},c=s("h1",{id:"_12-5-reaction-mechanism",tabindex:"-1"},[a("12-5: Reaction Mechanism "),s("a",{class:"header-anchor",href:"#_12-5-reaction-mechanism","aria-label":'Permalink to "12-5: Reaction Mechanism"'},"​")],-1),p=s("h2",{id:"_12-5-1-learning-objectives",tabindex:"-1"},[a("12-5-1: Learning Objectives "),s("a",{class:"header-anchor",href:"#_12-5-1-learning-objectives","aria-label":'Permalink to "12-5-1: Learning Objectives"'},"​")],-1),h=s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"Learning Objectives"),s("p",null,"One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.")],-1),o=s("p",null,"In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),u=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),d=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),v=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),y=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),k=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),f=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),z=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),O=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),_=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),M=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),N=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),C=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),L=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),A=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),B=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),T=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),R=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),I=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),q=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),S=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),P=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),D=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),H=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),E=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),F=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),j=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),U=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),$=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),V=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),W=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),G=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),J=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),Y=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),K=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),Q=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),X=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1);function Z(ss,as,ts,ls,es,ns){const t=l;return n(),m("div",null,[c,i(t,{readTime:"11",words:"1.9k"}),p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,B,T,R,I,q,S,P,D,H,E,F,j,U,$,V,W,G,J,Y,K,Q,X])}const ps=e(r,[["render",Z]]);export{cs as __pageData,ps as default};
diff --git a/assets/academic_chemistry_notes_12-5.md.80389f1d.lean.js b/assets/academic_chemistry_notes_12-5.md.452880b7.lean.js
similarity index 99%
rename from assets/academic_chemistry_notes_12-5.md.80389f1d.lean.js
rename to assets/academic_chemistry_notes_12-5.md.452880b7.lean.js
index 5d1aa544..56b0c166 100644
--- a/assets/academic_chemistry_notes_12-5.md.80389f1d.lean.js
+++ b/assets/academic_chemistry_notes_12-5.md.452880b7.lean.js
@@ -1 +1 @@
-import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as e,o as n,c as m,H as i,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const cs=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"keywords":["chemistry","reaction","mechanism","reaction-mechanism","tutorial","explanation","textbook","reference"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1712687763000}'),r={name:"academic/chemistry/notes/12-5.md"},c=s("h1",{id:"_12-5-reaction-mechanism",tabindex:"-1"},[a("12-5: Reaction Mechanism "),s("a",{class:"header-anchor",href:"#_12-5-reaction-mechanism","aria-label":'Permalink to "12-5: Reaction Mechanism"'},"​")],-1),p=s("h2",{id:"_12-5-1-learning-objectives",tabindex:"-1"},[a("12-5-1: Learning Objectives "),s("a",{class:"header-anchor",href:"#_12-5-1-learning-objectives","aria-label":'Permalink to "12-5-1: Learning Objectives"'},"​")],-1),h=s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"Learning Objectives"),s("p",null,"One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.")],-1),o=s("p",null,"In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),u=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),d=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),v=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),y=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),k=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),f=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),z=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),O=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),_=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),M=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),N=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),C=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),L=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),A=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),B=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),T=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),R=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),I=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),q=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),S=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),P=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),D=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),H=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),E=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),F=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),j=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),U=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),$=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),V=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),W=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),G=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),J=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),Y=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),K=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),Q=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),X=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1);function Z(ss,as,ts,ls,es,ns){const t=l;return n(),m("div",null,[c,i(t,{readTime:"11",words:"1.9k"}),p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,B,T,R,I,q,S,P,D,H,E,F,j,U,$,V,W,G,J,Y,K,Q,X])}const ps=e(r,[["render",Z]]);export{cs as __pageData,ps as default};
+import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as e,o as n,c as m,H as i,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const cs=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"keywords":["chemistry","reaction","mechanism","reaction-mechanism","tutorial","explanation","textbook","reference"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1712870707000}'),r={name:"academic/chemistry/notes/12-5.md"},c=s("h1",{id:"_12-5-reaction-mechanism",tabindex:"-1"},[a("12-5: Reaction Mechanism "),s("a",{class:"header-anchor",href:"#_12-5-reaction-mechanism","aria-label":'Permalink to "12-5: Reaction Mechanism"'},"​")],-1),p=s("h2",{id:"_12-5-1-learning-objectives",tabindex:"-1"},[a("12-5-1: Learning Objectives "),s("a",{class:"header-anchor",href:"#_12-5-1-learning-objectives","aria-label":'Permalink to "12-5-1: Learning Objectives"'},"​")],-1),h=s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"Learning Objectives"),s("p",null,"One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.")],-1),o=s("p",null,"In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),u=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),d=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),v=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),y=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),k=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),f=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),z=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),O=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),_=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),M=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),N=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),C=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),L=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),A=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),B=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),T=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),R=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),I=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),q=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),S=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),P=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),D=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),H=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),E=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),F=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),j=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),U=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),$=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),V=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),W=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),G=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),J=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),Y=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),K=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),Q=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),X=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1);function Z(ss,as,ts,ls,es,ns){const t=l;return n(),m("div",null,[c,i(t,{readTime:"11",words:"1.9k"}),p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,B,T,R,I,q,S,P,D,H,E,F,j,U,$,V,W,G,J,Y,K,Q,X])}const ps=e(r,[["render",Z]]);export{cs as __pageData,ps as default};
diff --git a/assets/academic_chemistry_problems_02-20.md.7d9a135a.js b/assets/academic_chemistry_problems_02-20.md.b4bac8f3.js
similarity index 99%
rename from assets/academic_chemistry_problems_02-20.md.7d9a135a.js
rename to assets/academic_chemistry_problems_02-20.md.b4bac8f3.js
index f86343f9..28d57a19 100644
--- a/assets/academic_chemistry_problems_02-20.md.7d9a135a.js
+++ b/assets/academic_chemistry_problems_02-20.md.b4bac8f3.js
@@ -1 +1 @@
-import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const L=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/02-20.md"},p=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),h=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),g=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),u=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),d=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),v=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),y=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),w=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1);function z(f,x,H,k,_,O){const t=l;return n(),e("div",null,[p,r(t,{readTime:"2",words:"412"}),c,h,o,g,u,d,v,y,w,b])}const T=m(i,[["render",z]]);export{L as __pageData,T as default};
+import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const L=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/02-20.md"},p=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),h=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),g=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),u=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),d=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),v=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),y=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),w=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1);function z(f,x,H,k,_,O){const t=l;return n(),e("div",null,[p,r(t,{readTime:"2",words:"412"}),c,h,o,g,u,d,v,y,w,b])}const T=m(i,[["render",z]]);export{L as __pageData,T as default};
diff --git a/assets/academic_chemistry_problems_02-20.md.7d9a135a.lean.js b/assets/academic_chemistry_problems_02-20.md.b4bac8f3.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_02-20.md.7d9a135a.lean.js
rename to assets/academic_chemistry_problems_02-20.md.b4bac8f3.lean.js
index f86343f9..28d57a19 100644
--- a/assets/academic_chemistry_problems_02-20.md.7d9a135a.lean.js
+++ b/assets/academic_chemistry_problems_02-20.md.b4bac8f3.lean.js
@@ -1 +1 @@
-import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const L=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/02-20.md"},p=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),h=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),g=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),u=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),d=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),v=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),y=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),w=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1);function z(f,x,H,k,_,O){const t=l;return n(),e("div",null,[p,r(t,{readTime:"2",words:"412"}),c,h,o,g,u,d,v,y,w,b])}const T=m(i,[["render",z]]);export{L as __pageData,T as default};
+import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const L=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/02-20.md"},p=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),h=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),g=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),u=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),d=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),v=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),y=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),w=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1);function z(f,x,H,k,_,O){const t=l;return n(),e("div",null,[p,r(t,{readTime:"2",words:"412"}),c,h,o,g,u,d,v,y,w,b])}const T=m(i,[["render",z]]);export{L as __pageData,T as default};
diff --git a/assets/academic_chemistry_problems_03-02-1.md.e45de42c.js b/assets/academic_chemistry_problems_03-02-1.md.28aa3eea.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-1.md.e45de42c.js
rename to assets/academic_chemistry_problems_03-02-1.md.28aa3eea.js
index 1110a3e1..c5235325 100644
--- a/assets/academic_chemistry_problems_03-02-1.md.e45de42c.js
+++ b/assets/academic_chemistry_problems_03-02-1.md.28aa3eea.js
@@ -1 +1 @@
-import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as n,o as e,c as m,H as c,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const q=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/03-02-1.md"},p=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),h=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),g=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),u=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),y=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),v=s("p",null,"the rate of reaction is defined as",-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),x=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),f=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),k=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),z=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),M=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1);function _(N,O,L,D,B,P){const l=t;return e(),m("div",null,[p,c(l,{readTime:"2",words:"404"}),r,o,h,g,u,y,d,v,b,x,w,f,k,z,M])}const S=n(i,[["render",_]]);export{q as __pageData,S as default};
+import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as n,o as e,c as m,H as c,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const q=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/03-02-1.md"},p=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),h=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),g=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),u=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),y=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),v=s("p",null,"the rate of reaction is defined as",-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),x=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),f=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),k=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),z=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),M=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1);function _(N,O,L,D,B,P){const l=t;return e(),m("div",null,[p,c(l,{readTime:"2",words:"404"}),r,o,h,g,u,y,d,v,b,x,w,f,k,z,M])}const S=n(i,[["render",_]]);export{q as __pageData,S as default};
diff --git a/assets/academic_chemistry_problems_03-02-1.md.e45de42c.lean.js b/assets/academic_chemistry_problems_03-02-1.md.28aa3eea.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-1.md.e45de42c.lean.js
rename to assets/academic_chemistry_problems_03-02-1.md.28aa3eea.lean.js
index 1110a3e1..c5235325 100644
--- a/assets/academic_chemistry_problems_03-02-1.md.e45de42c.lean.js
+++ b/assets/academic_chemistry_problems_03-02-1.md.28aa3eea.lean.js
@@ -1 +1 @@
-import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as n,o as e,c as m,H as c,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const q=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/03-02-1.md"},p=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),h=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),g=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),u=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),y=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),v=s("p",null,"the rate of reaction is defined as",-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),x=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),f=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),k=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),z=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),M=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1);function _(N,O,L,D,B,P){const l=t;return e(),m("div",null,[p,c(l,{readTime:"2",words:"404"}),r,o,h,g,u,y,d,v,b,x,w,f,k,z,M])}const S=n(i,[["render",_]]);export{q as __pageData,S as default};
+import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as n,o as e,c as m,H as c,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const q=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/03-02-1.md"},p=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),h=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),g=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),u=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),y=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),v=s("p",null,"the rate of reaction is defined as",-1),b=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),x=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),f=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),k=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),z=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),M=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1);function _(N,O,L,D,B,P){const l=t;return e(),m("div",null,[p,c(l,{readTime:"2",words:"404"}),r,o,h,g,u,y,d,v,b,x,w,f,k,z,M])}const S=n(i,[["render",_]]);export{q as __pageData,S as default};
diff --git a/assets/academic_chemistry_problems_03-02-2.md.b6d0b4ce.js b/assets/academic_chemistry_problems_03-02-2.md.b8e7fb41.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-2.md.b6d0b4ce.js
rename to assets/academic_chemistry_problems_03-02-2.md.b8e7fb41.js
index 5a2bc8f5..db14d91f 100644
--- a/assets/academic_chemistry_problems_03-02-2.md.b6d0b4ce.js
+++ b/assets/academic_chemistry_problems_03-02-2.md.b8e7fb41.js
@@ -1 +1 @@
-import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const Y=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/03-02-2.md"},c=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),p=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),h=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),d=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),y=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1);function w(k,b,f,L,M,_){const l=t;return n(),e("div",null,[c,r(l,{readTime:"1",words:"205"}),p,o,h,d,u,g,y,v,x])}const N=m(i,[["render",w]]);export{Y as __pageData,N as default};
+import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const Y=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/03-02-2.md"},c=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),p=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),h=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),d=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),y=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1);function w(k,b,f,L,M,_){const l=t;return n(),e("div",null,[c,r(l,{readTime:"1",words:"205"}),p,o,h,d,u,g,y,v,x])}const N=m(i,[["render",w]]);export{Y as __pageData,N as default};
diff --git a/assets/academic_chemistry_problems_03-02-2.md.b6d0b4ce.lean.js b/assets/academic_chemistry_problems_03-02-2.md.b8e7fb41.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-2.md.b6d0b4ce.lean.js
rename to assets/academic_chemistry_problems_03-02-2.md.b8e7fb41.lean.js
index 5a2bc8f5..db14d91f 100644
--- a/assets/academic_chemistry_problems_03-02-2.md.b6d0b4ce.lean.js
+++ b/assets/academic_chemistry_problems_03-02-2.md.b8e7fb41.lean.js
@@ -1 +1 @@
-import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const Y=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/03-02-2.md"},c=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),p=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),h=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),d=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),y=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1);function w(k,b,f,L,M,_){const l=t;return n(),e("div",null,[c,r(l,{readTime:"1",words:"205"}),p,o,h,d,u,g,y,v,x])}const N=m(i,[["render",w]]);export{Y as __pageData,N as default};
+import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,o as n,c as e,H as r,k as s,a}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const Y=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/03-02-2.md"},c=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),p=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),o=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),h=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),d=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),y=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),x=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1);function w(k,b,f,L,M,_){const l=t;return n(),e("div",null,[c,r(l,{readTime:"1",words:"205"}),p,o,h,d,u,g,y,v,x])}const N=m(i,[["render",w]]);export{Y as __pageData,N as default};
diff --git a/assets/academic_chemistry_problems_03-02-3.md.4d9ff6c1.js b/assets/academic_chemistry_problems_03-02-3.md.04a75589.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-3.md.4d9ff6c1.js
rename to assets/academic_chemistry_problems_03-02-3.md.04a75589.js
index 5dc9942e..e4691340 100644
--- a/assets/academic_chemistry_problems_03-02-3.md.4d9ff6c1.js
+++ b/assets/academic_chemistry_problems_03-02-3.md.04a75589.js
@@ -1 +1 @@
-import{_ as e}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as l,o as n,c as m,H as r,k as a,a as s,Q as c}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const P=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/03-02-3.md"},o=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),h=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),p=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),d=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),g=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),u=c('
Solution ​
',2),x=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),w=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1);function y(k,f,b,M,v,_){const t=e;return n(),m("div",null,[o,r(t,{readTime:"1",words:"246"}),h,p,d,g,u,x,w])}const S=l(i,[["render",y]]);export{P as __pageData,S as default};
+import{_ as e}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as l,o as n,c as m,H as r,k as a,a as s,Q as c}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const P=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/03-02-3.md"},o=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),h=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),p=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),d=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),g=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),u=c('
Solution ​
',2),x=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),w=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1);function y(k,f,b,M,v,_){const t=e;return n(),m("div",null,[o,r(t,{readTime:"1",words:"246"}),h,p,d,g,u,x,w])}const S=l(i,[["render",y]]);export{P as __pageData,S as default};
diff --git a/assets/academic_chemistry_problems_03-02-3.md.4d9ff6c1.lean.js b/assets/academic_chemistry_problems_03-02-3.md.04a75589.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-3.md.4d9ff6c1.lean.js
rename to assets/academic_chemistry_problems_03-02-3.md.04a75589.lean.js
index 89d017eb..d39ef14c 100644
--- a/assets/academic_chemistry_problems_03-02-3.md.4d9ff6c1.lean.js
+++ b/assets/academic_chemistry_problems_03-02-3.md.04a75589.lean.js
@@ -1 +1 @@
-import{_ as e}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as l,o as n,c as m,H as r,k as a,a as s,Q as c}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const P=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1712687763000}'),i={name:"academic/chemistry/problems/03-02-3.md"},o=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),h=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),p=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),d=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),g=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),u=c("",2),x=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),w=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1);function y(k,f,b,M,v,_){const t=e;return n(),m("div",null,[o,r(t,{readTime:"1",words:"246"}),h,p,d,g,u,x,w])}const S=l(i,[["render",y]]);export{P as __pageData,S as default};
+import{_ as e}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as l,o as n,c as m,H as r,k as a,a as s,Q as c}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const P=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1712870707000}'),i={name:"academic/chemistry/problems/03-02-3.md"},o=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),h=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),p=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),d=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),g=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),u=c("",2),x=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),w=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1);function y(k,f,b,M,v,_){const t=e;return n(),m("div",null,[o,r(t,{readTime:"1",words:"246"}),h,p,d,g,u,x,w])}const S=l(i,[["render",y]]);export{P as __pageData,S as default};
diff --git a/assets/academic_cis105_cis105-l1-lecture-note.md.9a8ef371.js b/assets/academic_cis105_cis105-l1-lecture-note.md.64262986.js
similarity index 99%
rename from assets/academic_cis105_cis105-l1-lecture-note.md.9a8ef371.js
rename to assets/academic_cis105_cis105-l1-lecture-note.md.64262986.js
index fbf64738..07f40377 100644
--- a/assets/academic_cis105_cis105-l1-lecture-note.md.9a8ef371.js
+++ b/assets/academic_cis105_cis105-l1-lecture-note.md.64262986.js
@@ -1,4 +1,4 @@
-import{_ as t}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as n,o,c as l,H as r,k as e,a,Q as i}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const D=JSON.parse('{"title":"CIS105: Computer Applications & Information Systems Lec. 1.","description":"","frontmatter":{},"headers":[],"relativePath":"academic/cis105/cis105-l1-lecture-note.md","filePath":"academic/cis105/cis105-l1-lecture-note.md","lastUpdated":1712687763000}'),p={name:"academic/cis105/cis105-l1-lecture-note.md"},c=e("h1",{id:"cis105-computer-applications-information-systems-lec-1",tabindex:"-1"},[a("CIS105: Computer Applications & Information Systems Lec. 1. "),e("a",{class:"header-anchor",href:"#cis105-computer-applications-information-systems-lec-1","aria-label":'Permalink to "CIS105: Computer Applications & Information Systems Lec. 1."'},"​")],-1),m=e("h2",{id:"chapter-1-everything-changes",tabindex:"-1"},[a("Chapter 1: Everything Changes "),e("a",{class:"header-anchor",href:"#chapter-1-everything-changes","aria-label":'Permalink to "Chapter 1: Everything Changes"'},"​")],-1),h=e("h3",{id:"_1-1-the-historic-timelines",tabindex:"-1"},[a("1.1: The Historic Timelines "),e("a",{class:"header-anchor",href:"#_1-1-the-historic-timelines","aria-label":'Permalink to "1.1: The Historic Timelines"'},"​")],-1),d=e("ul",null,[e("li",null,[e("p",null,"1903: Wright bros, invented the airplane.")]),e("li",null,[e("p",null,"1969: Armstrong walks on the moon.")]),e("li",null,[e("p",null,"2004: SpaceX Falcon 9 lands upright.")]),e("li",null,[e("p",null,"2015: Electreic plane crosses English Channel")]),e("li",null,[e("p",null,"20xx: Google Live goes online.")]),e("li",null,[e("p",null,[a("Boeing 777, commercial airline airplanes normally cost approximately "),e("span",{class:"katex"},[e("span",{class:"katex-mathml"},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("semantics",null,[e("mrow",null,[e("mi",{mathvariant:"normal"},"$"),e("mn",null,"200"),e("mo",{separator:"true"},","),e("mn",null,"000"),e("mo",{separator:"true"},","),e("mn",null,"000"),e("mo",{separator:"true"},","),e("mn",null,"000")]),e("annotation",{encoding:"application/x-tex"},"\\$ 200,000,000,000")])])]),e("span",{class:"katex-html","aria-hidden":"true"},[e("span",{class:"base"},[e("span",{class:"strut",style:{height:"0.9444em","vertical-align":"-0.1944em"}}),e("span",{class:"mord"},"$200"),e("span",{class:"mpunct"},","),e("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),e("span",{class:"mord"},"000"),e("span",{class:"mpunct"},","),e("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),e("span",{class:"mord"},"000"),e("span",{class:"mpunct"},","),e("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),e("span",{class:"mord"},"000")])])]),a(" for a singular aircraft.")])])],-1),u=i(`
1.2: Analytical Engne ​
A mechanical computing device, was a special-purpose machine designed to tabulate logarithms and trigonometric functions by evaluating finite differences to create approximating polynomials.
1.3: Ada Lovelace ​
It tok the powerful insights of a mathematicial named Ada Lovelace to realize the true potentilal of the analytical engine. She was the first person to recognize that the machine could be used for more than pure calculations. She developed the first algorithm for the engine. It was the very first example of computer programming.
1.4: Information Technology is for People ​
1.5: Moore's Law ​
1.6: The Cuff Smartwatch? ​
US rapper/producer/entrepreneur Will.i.Am announced his foray into the world of wearable tech in 2014, proclaiming to have created a device so life-changing and futuristic it'd blow our archaic mind.
1.7: Types of Computer ​
1.8: What is System Software? ​