# Presentation problem: 02-20

## Question

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

$$
\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{g}) \longrightarrow \mathrm{CH}_3 \mathrm{CHO}(g)+\mathrm{H}_2(g)
$$

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

| Time (s) | $P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }$ |
| -------- | ------------------------------------------------------------ |
| 0        | 250.                                                         |
| 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 $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}$ after $900 . \mathrm{s}$ from the start of the reaction. (Hint: To determine the order of the reaction with respect to $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}$, compare how the pressure of $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}$ decreases with each time listing.)

## Solution

![solution graph](https://r2.toshiki.dev/image/chemistry/solution1.png){data-zoomable}

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

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