toshiki-notebook/assets/academic_chemistry_problems_03-02-3.md.177c75c1.lean.js

import{_ as l}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.65c6b98c.js";import{_ as n,o as m,c as r,k as a,a as s,E as i,O as c}from"./chunks/framework.c989bd33.js";import"./chunks/commonjsHelpers.725317a4.js";const f=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":1726418622000}'),o={name:"academic/chemistry/problems/03-02-3.md"};function h(p,t,d,g,u,x){const e=l;return m(),r("div",null,[t[0]||(t[0]=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)),i(e,{readTime:"1",words:"246"}),t[1]||(t[1]=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to 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")])])]),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 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"),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)),t[5]||(t[5]=c('<ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It increases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> It decreases by a factor of 2.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It decreases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It does not change.</li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-label="Permalink to &quot;Solution&quot;">​</a></h2>',2)),t[6]||(t[6]=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)),t[7]||(t[7]=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))])}const b=n(o,[["render",h]]);export{f as __pageData,b as default};