{"id":7952,"date":"2021-06-08T21:58:17","date_gmt":"2021-06-08T21:58:17","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/activation-energy-and-the-arrhenius-equation\/"},"modified":"2021-10-26T18:45:56","modified_gmt":"2021-10-26T18:45:56","slug":"activation-energy-and-the-arrhenius-equation","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/activation-energy-and-the-arrhenius-equation\/","title":{"raw":"Activation Energy and the Arrhenius Equation","rendered":"Activation Energy and the Arrhenius Equation"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>To gain an understanding of activation energy.<\/li>\r\n \t<li>To determine activation energy graphically or algebraically.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_1051\" align=\"aligncenter\" width=\"239\"]<img class=\"wp-image-1051 size-medium\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2021\/06\/Arrhenius2-239x300-1.jpg\" alt=\"Middle-aged man with a moustache and wearing a bowtie.\" width=\"239\" height=\"300\" \/> Figure 17.11 \"Svante Arrhenius.\" Swedish scientist Svante Arrhenius (1859\u20131927).[\/caption]\r\n\r\nEarlier in the chapter, reactions were discussed in terms of effective collision frequency and molecule energy levels. In 1889, a Swedish scientist named Svante Arrhenius proposed an equation that\u00a0relates these concepts with the rate constant:\r\n<p style=\"text-align: center;\">[latex]k=Ae^{\\frac{-E_a}{RT}}[\/latex]<\/p>\r\nwhere <em>k<\/em> represents the rate constant, <em>E<\/em><sub>a<\/sub> is the activation energy, <em>R<\/em> is the gas constant [latex]\\left(\\dfrac{8.3145\\text{ J}}{\\text{K mol}}\\right)[\/latex], and <em>T<\/em> is the temperature expressed in Kelvin. <em>A<\/em> is known as the <b>frequency factor<\/b>, having units of L mol<sup>\u22121<\/sup> s<sup>\u22121<\/sup>, and takes into account the frequency of reactions and likelihood of correct molecular orientation.\r\n\r\nThe Arrhenius equation allows us to calculate activation energies if the rate constant is known, or vice versa. As well, it mathematically expresses the relationships we established earlier: as activation energy term <em>E<\/em><sub>a<\/sub> increases, the rate constant <i>k<\/i> decreases and therefore the rate of reaction decreases.\r\n<h1>Determining the Activation Energy<\/h1>\r\n<h2>Graphically<\/h2>\r\nWe can graphically determine the activation energy by manipulating the Arrhenius equation to put it into the form of a straight line. Taking the natural logarithm of both sides gives us:\r\n<p style=\"text-align: center;\">[latex]\\ln k=-\\dfrac{E_a}{RT}+\\ln A[\/latex]<\/p>\r\nA slight rearrangement of this equation then gives us a straight line plot (<em>y<\/em> = <em>mx<\/em> + <em>b<\/em>) for ln <i>k<\/i> versus [latex]\\dfrac{1}{T}[\/latex], where the slope is [latex]-\\dfrac{E_a}{R}[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\ln k=-\\dfrac{E_a}{R}\\left(\\dfrac{1}{t}\\right)+\\ln A[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing the data from the following table, determine the activation energy of the reaction:\r\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 60%; height: 80px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<th style=\"width: 50%; height: 16px;\" scope=\"col\">Temperature (K)<\/th>\r\n<th style=\"width: 50%; height: 16px;\" scope=\"col\">Rate Constant, <em>k<\/em> (s<sup>\u22121<\/sup>)<\/th>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 50%; height: 16px;\">375<\/td>\r\n<td style=\"width: 50%; height: 16px;\">1.68 \u00d7 10<sup>\u22125<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 50%; height: 16px;\">400<\/td>\r\n<td style=\"width: 50%; height: 16px;\">3.5 \u00d7 10<sup>\u22125<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 50%; height: 16px;\">500<\/td>\r\n<td style=\"width: 50%; height: 16px;\">4.2 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 50%; height: 16px;\">600<\/td>\r\n<td style=\"width: 50%; height: 16px;\">2.11 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<em>Solution<\/em>\r\n\r\nWe can obtain the activation energy by plotting ln <em>k<\/em> versus [latex]\\dfrac{1}{T}[\/latex], knowing that the slope will be equal to [latex]-\\dfrac{E_a}{R}[\/latex].\r\n\r\nFirst determine the values of ln <em>k<\/em> and [latex]\\dfrac{1}{T}[\/latex], and plot them in a graph:\r\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 60%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\" scope=\"col\">[latex]\\frac{1}{T}[\/latex]<\/th>\r\n<th style=\"width: 50%; text-align: center;\" scope=\"col\">ln <em>k<\/em><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0.002667<\/td>\r\n<td style=\"width: 50%; text-align: center;\">\u221210.9941<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0.0025<\/td>\r\n<td style=\"width: 50%; text-align: center;\">\u221210.2602<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0.002<\/td>\r\n<td style=\"width: 50%; text-align: center;\">\u22127.77526<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0.001667<\/td>\r\n<td style=\"width: 50%; text-align: center;\">\u22126.16107<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_1268\" align=\"aligncenter\" width=\"481\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/05\/17.5-graphical-determination-of-Ea-example-plot.jpg\"><img class=\"wp-image-1052 size-full\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2021\/06\/17.5-graphical-determination-of-Ea-example-plot-1.jpg\" alt=\"Graphical determination of Ea example plot\" width=\"481\" height=\"289\" \/><\/a> Graphical determination of <em>E<\/em><sub>a <\/sub>example plot[\/caption]\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}\r\n\\text{Slope}&amp;=-\\dfrac{E_a}{R} \\\\ \\\\\r\n-4865\\text{ K}&amp;=-\\dfrac{E_a}{8.3145\\text{ J K}^{-1}\\text{ mol}^{-1}} \\\\ \\\\\r\nE_a&amp;=4.0\\times 10^4 \\text{ J\/mol}\r\n\\end{align*}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Algebraically<\/h2>\r\nThe activation energy can also be calculated algebraically if <i>k<\/i> is known at two different temperatures:\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}\r\n\\text{At temperature 1: }\\ln k_1&amp;=-\\dfrac{E_a}{RT_1}+\\ln A \\\\ \\\\\r\n\\text{At temperature 2: }\\ln k_2&amp;=-\\dfrac{E_a}{RT_2}+\\ln A\r\n\\end{align*}[\/latex]<\/p>\r\nWe can subtract one of these equations from the other:\r\n<p style=\"text-align: center;\">[latex]\\ln k_1-\\ln k_2=\\left(-\\dfrac{E_a}{RT_1}+\\ln A\\right)-\\left(-\\dfrac{E_a}{RT_2}+\\ln A\\right)[\/latex]<\/p>\r\nThis equation can then be further simplified to:\r\n<p style=\"text-align: center;\">[latex]\\ln \\dfrac{k_1}{k_2}=\\dfrac{E_a}{R}\\left(\\dfrac{1}{T_2}-\\dfrac{1}{T_1}\\right)[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine the value of <i>E<\/i><sub>a<\/sub> given the following values of<i> k<\/i> at the temperatures indicated:\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}\r\n600\\text{ K: }k&amp;=2.75\\times 10^{-8}\\text{ L mol}^{-1}\\text{s}^{-1} \\\\\r\n800\\text{ K: }k&amp;=1.95\\times 10^{-7}\\text{ L mol}^{-1}\\text{s}^{-1}\r\n\\end{align*}[\/latex]<\/p>\r\n<em>Solution<\/em>\r\n\r\nSubstitute the values stated into the algebraic method equation:\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}\r\n\\ln \\dfrac{k_1}{k_2}&amp;=\\dfrac{E_a}{R}\\left(\\dfrac{1}{T_2}-\\dfrac{1}{T_1}\\right) \\\\ \\\\\r\n\\ln \\dfrac{2.75\\times 10^{-8}\\text{ L mol}^{-1}\\text{s}^{-1}}{1.95\\times10^{-7}\\text{ L mol}^{-1}\\text{s}^{-1}}&amp;=\\dfrac{E_a}{8.3145\\text{ J K}^{-1}\\text{mol}^{-1}}\\left(\\dfrac{1}{800\\text{ K}}-\\dfrac{1}{600\\text{ K}}\\right) \\\\ \\\\\r\n1.96&amp;=\\dfrac{E_a}{8.3145\\text{ J K}^{-1}\\text{mol}^{-1}}\\left(-4.16\\times10^{-4}\\text{ K}^{-1}\\right) \\\\ \\\\\r\n4.704\\times 10^{-3}\\text{ K}^{-1}&amp;=\\dfrac{E_a}{8.3145\\text{ J K}^{-1}\\text{mol}^{-1}} \\\\ \\\\\r\nE_a&amp;=3.92\\times 10^4\\text{ J\/mol}\r\n\\end{align*}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>The activation energy can be graphically determined by manipulating the Arrhenius equation.<\/li>\r\n \t<li>The activation energy can also be calculated algebraically if <em>k<\/em> is known at two different temperatures.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h3>Media Attributions<\/h3>\r\n<ul>\r\n \t<li><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Arrhenius2.jpg\">\"Arrhenius2\"<\/a> \u00a9 1909 by Meisenbach, Riffarth, &amp; Co. Leipzig is licensed under a <a href=\"https:\/\/creativecommons.org\/publicdomain\/mark\/1.0\/\">Public Domain<\/a> license<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>To gain an understanding of activation energy.<\/li>\n<li>To determine activation energy graphically or algebraically.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure id=\"attachment_1051\" aria-describedby=\"caption-attachment-1051\" style=\"width: 239px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1051 size-medium\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2021\/06\/Arrhenius2-239x300-1.jpg\" alt=\"Middle-aged man with a moustache and wearing a bowtie.\" width=\"239\" height=\"300\" \/><figcaption id=\"caption-attachment-1051\" class=\"wp-caption-text\">Figure 17.11 &#8220;Svante Arrhenius.&#8221; Swedish scientist Svante Arrhenius (1859\u20131927).<\/figcaption><\/figure>\n<p>Earlier in the chapter, reactions were discussed in terms of effective collision frequency and molecule energy levels. In 1889, a Swedish scientist named Svante Arrhenius proposed an equation that\u00a0relates these concepts with the rate constant:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-b05dd13acc316f7d9a7dd9ac7d508ed3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#65;&#101;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#69;&#95;&#97;&#125;&#123;&#82;&#84;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"83\" style=\"vertical-align: 0px;\" \/><\/p>\n<p>where <em>k<\/em> represents the rate constant, <em>E<\/em><sub>a<\/sub> is the activation energy, <em>R<\/em> is the gas constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-a017375312478403b940d649e293e5a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#56;&#46;&#51;&#49;&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#32;&#109;&#111;&#108;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"88\" style=\"vertical-align: -17px;\" \/>, and <em>T<\/em> is the temperature expressed in Kelvin. <em>A<\/em> is known as the <b>frequency factor<\/b>, having units of L mol<sup>\u22121<\/sup> s<sup>\u22121<\/sup>, and takes into account the frequency of reactions and likelihood of correct molecular orientation.<\/p>\n<p>The Arrhenius equation allows us to calculate activation energies if the rate constant is known, or vice versa. As well, it mathematically expresses the relationships we established earlier: as activation energy term <em>E<\/em><sub>a<\/sub> increases, the rate constant <i>k<\/i> decreases and therefore the rate of reaction decreases.<\/p>\n<h1>Determining the Activation Energy<\/h1>\n<h2>Graphically<\/h2>\n<p>We can graphically determine the activation energy by manipulating the Arrhenius equation to put it into the form of a straight line. Taking the natural logarithm of both sides gives us:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-621be8088ebecdab7207682d9c5aa1ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#32;&#107;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#84;&#125;&#43;&#92;&#108;&#110;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"148\" style=\"vertical-align: -12px;\" \/><\/p>\n<p>A slight rearrangement of this equation then gives us a straight line plot (<em>y<\/em> = <em>mx<\/em> + <em>b<\/em>) for ln <i>k<\/i> versus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-f3a7c345fc7f74b763faaca50ec4007c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"13\" style=\"vertical-align: -12px;\" \/>, where the slope is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-e2bbf1dbd2d83545bea3afc8f6f0406c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -12px;\" \/>:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-932905c482c9ba08f7510f3a5873b470_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#32;&#107;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#110;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"184\" style=\"vertical-align: -17px;\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using the data from the following table, determine the activation energy of the reaction:<\/p>\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 60%; height: 80px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<th style=\"width: 50%; height: 16px;\" scope=\"col\">Temperature (K)<\/th>\n<th style=\"width: 50%; height: 16px;\" scope=\"col\">Rate Constant, <em>k<\/em> (s<sup>\u22121<\/sup>)<\/th>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">375<\/td>\n<td style=\"width: 50%; height: 16px;\">1.68 \u00d7 10<sup>\u22125<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">400<\/td>\n<td style=\"width: 50%; height: 16px;\">3.5 \u00d7 10<sup>\u22125<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">500<\/td>\n<td style=\"width: 50%; height: 16px;\">4.2 \u00d7 10<sup>\u22124<\/sup><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">600<\/td>\n<td style=\"width: 50%; height: 16px;\">2.11 \u00d7 10<sup>\u22123<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><em>Solution<\/em><\/p>\n<p>We can obtain the activation energy by plotting ln <em>k<\/em> versus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-f3a7c345fc7f74b763faaca50ec4007c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"13\" style=\"vertical-align: -12px;\" \/>, knowing that the slope will be equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-e2bbf1dbd2d83545bea3afc8f6f0406c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<p>First determine the values of ln <em>k<\/em> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-f3a7c345fc7f74b763faaca50ec4007c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"13\" style=\"vertical-align: -12px;\" \/>, and plot them in a graph:<\/p>\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 60%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\" scope=\"col\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-ee7a811b4f8bd892126cd6ff8e7c899b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"10\" style=\"vertical-align: -6px;\" \/><\/th>\n<th style=\"width: 50%; text-align: center;\" scope=\"col\">ln <em>k<\/em><\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0.002667<\/td>\n<td style=\"width: 50%; text-align: center;\">\u221210.9941<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0.0025<\/td>\n<td style=\"width: 50%; text-align: center;\">\u221210.2602<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0.002<\/td>\n<td style=\"width: 50%; text-align: center;\">\u22127.77526<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0.001667<\/td>\n<td style=\"width: 50%; text-align: center;\">\u22126.16107<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"attachment_1268\" aria-describedby=\"caption-attachment-1268\" style=\"width: 481px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/05\/17.5-graphical-determination-of-Ea-example-plot.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1052 size-full\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2021\/06\/17.5-graphical-determination-of-Ea-example-plot-1.jpg\" alt=\"Graphical determination of Ea example plot\" width=\"481\" height=\"289\" \/><\/a><figcaption id=\"caption-attachment-1268\" class=\"wp-caption-text\">Graphical determination of <em>E<\/em><sub>a <\/sub>example plot<\/figcaption><\/figure>\n<p style=\"text-align: center;\">\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 165px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-dd00c588b6b86129191ba2e637e88806_l3.png\" height=\"165\" width=\"263\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#116;&#101;&#120;&#116;&#123;&#83;&#108;&#111;&#112;&#101;&#125;&#38;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#125;&#32;&#92;&#92;&#32;&#92;&#92; &#45;&#52;&#56;&#54;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#125;&#38;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#56;&#46;&#51;&#49;&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#32;&#75;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#32;&#92;&#92;&#32;&#92;&#92; &#69;&#95;&#97;&#38;&#61;&#52;&#46;&#48;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#49;&#48;&#94;&#52;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#47;&#109;&#111;&#108;&#125; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<\/div>\n<\/div>\n<h2>Algebraically<\/h2>\n<p>The activation energy can also be calculated algebraically if <i>k<\/i> is known at two different temperatures:<\/p>\n<p style=\"text-align: center;\">\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 113px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-575e2749e629858bf6789be119ab58f5_l3.png\" height=\"113\" width=\"311\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#116;&#101;&#120;&#116;&#123;&#65;&#116;&#32;&#116;&#101;&#109;&#112;&#101;&#114;&#97;&#116;&#117;&#114;&#101;&#32;&#49;&#58;&#32;&#125;&#92;&#108;&#110;&#32;&#107;&#95;&#49;&#38;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#84;&#95;&#49;&#125;&#43;&#92;&#108;&#110;&#32;&#65;&#32;&#92;&#92;&#32;&#92;&#92; &#92;&#116;&#101;&#120;&#116;&#123;&#65;&#116;&#32;&#116;&#101;&#109;&#112;&#101;&#114;&#97;&#116;&#117;&#114;&#101;&#32;&#50;&#58;&#32;&#125;&#92;&#108;&#110;&#32;&#107;&#95;&#50;&#38;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#84;&#95;&#50;&#125;&#43;&#92;&#108;&#110;&#32;&#65; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<p>We can subtract one of these equations from the other:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-4325f3851ca3d4f2e3f77b25d330971e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#32;&#107;&#95;&#49;&#45;&#92;&#108;&#110;&#32;&#107;&#95;&#50;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#84;&#95;&#49;&#125;&#43;&#92;&#108;&#110;&#32;&#65;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#84;&#95;&#50;&#125;&#43;&#92;&#108;&#110;&#32;&#65;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"389\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>This equation can then be further simplified to:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-a13fe40889da77724dadeb165f0c24a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#110;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#107;&#95;&#49;&#125;&#123;&#107;&#95;&#50;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#95;&#50;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#95;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"179\" style=\"vertical-align: -17px;\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine the value of <i>E<\/i><sub>a<\/sub> given the following values of<i> k<\/i> at the temperatures indicated:<\/p>\n<p style=\"text-align: center;\">\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 45px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-068df527d93c17d893cde6a0bc04c4c5_l3.png\" height=\"45\" width=\"274\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#54;&#48;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#58;&#32;&#125;&#107;&#38;&#61;&#50;&#46;&#55;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#49;&#48;&#94;&#123;&#45;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#76;&#32;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#94;&#123;&#45;&#49;&#125;&#32;&#92;&#92; &#56;&#48;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#58;&#32;&#125;&#107;&#38;&#61;&#49;&#46;&#57;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#49;&#48;&#94;&#123;&#45;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#76;&#32;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#94;&#123;&#45;&#49;&#125; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<p><em>Solution<\/em><\/p>\n<p>Substitute the values stated into the algebraic method equation:<\/p>\n<p style=\"text-align: center;\">\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 323px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/ql-cache\/quicklatex.com-c79a03697d8a12006e3d72889b3b2fc4_l3.png\" height=\"323\" width=\"541\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#108;&#110;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#107;&#95;&#49;&#125;&#123;&#107;&#95;&#50;&#125;&#38;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#82;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#95;&#50;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#95;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#32;&#92;&#92; &#92;&#108;&#110;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#46;&#55;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#49;&#48;&#94;&#123;&#45;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#76;&#32;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#123;&#49;&#46;&#57;&#53;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#76;&#32;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#38;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#56;&#46;&#51;&#49;&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#32;&#75;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#48;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#125;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#48;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#32;&#92;&#92; &#49;&#46;&#57;&#54;&#38;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#56;&#46;&#51;&#49;&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#32;&#75;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#46;&#49;&#54;&#92;&#116;&#105;&#109;&#101;&#115;&#49;&#48;&#94;&#123;&#45;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#32;&#92;&#92; &#52;&#46;&#55;&#48;&#52;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#49;&#48;&#94;&#123;&#45;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#75;&#125;&#94;&#123;&#45;&#49;&#125;&#38;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#69;&#95;&#97;&#125;&#123;&#56;&#46;&#51;&#49;&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#32;&#75;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#32;&#92;&#92;&#32;&#92;&#92; &#69;&#95;&#97;&#38;&#61;&#51;&#46;&#57;&#50;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#49;&#48;&#94;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#74;&#47;&#109;&#111;&#108;&#125; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>The activation energy can be graphically determined by manipulating the Arrhenius equation.<\/li>\n<li>The activation energy can also be calculated algebraically if <em>k<\/em> is known at two different temperatures.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h3>Media Attributions<\/h3>\n<ul>\n<li><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Arrhenius2.jpg\">&#8220;Arrhenius2&#8221;<\/a> \u00a9 1909 by Meisenbach, Riffarth, &amp; Co. Leipzig is licensed under a <a href=\"https:\/\/creativecommons.org\/publicdomain\/mark\/1.0\/\">Public Domain<\/a> license<\/li>\n<\/ul>\n","protected":false},"author":90,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["jessie-a-key"],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[49],"license":[54],"class_list":["post-7952","chapter","type-chapter","status-publish","hentry","contributor-jessie-a-key","license-cc-by"],"part":7933,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7952","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":5,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7952\/revisions"}],"predecessor-version":[{"id":9021,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7952\/revisions\/9021"}],"part":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/parts\/7933"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7952\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/media?parent=7952"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapter-type?post=7952"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/contributor?post=7952"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/license?post=7952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}