{"id":1318,"date":"2020-06-23T16:21:00","date_gmt":"2020-06-23T20:21:00","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/chbe220\/?post_type=chapter&p=1318"},"modified":"2020-08-11T15:39:54","modified_gmt":"2020-08-11T19:39:54","slug":"reaction-rate-law","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/chbe220\/chapter\/reaction-rate-law\/","title":{"raw":"Reaction Rate Law","rendered":"Reaction Rate Law"},"content":{"raw":"
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nBy the end of this section, you should be able to:\r\n\r\nDefine<\/strong> reaction rate law and r<\/span>eaction rate constant (k)<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n\r\nReaction rate law Definition:<\/strong> The relationship between the rate of reaction and the concentration of reactants.\r\n\r\nThe rate law is usually proportional to the concentrations of reactants raised to a certain power:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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\r\n\r\nTake the reaction we used as an example before: [latex]A + 2B \u2192 3C + D[\/latex]\r\n

The general form for reaction rate law is<\/p>\r\n\r\n\r\n\r\n\r\n
[latex]r=k_{r}[A]^a[B]^b[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

For gas cases, we can use partial pressure<\/p>\r\n

[latex]r=k_{r}p_{A}^a p_{B}^b[\/latex]<\/p>\r\nThe rate constant [latex]k_{r}[\/latex] is independent of species concentration but generally dependent on temperature.\r\n\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n

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\r\n\r\nFor example, let's look at the rate of the gas-phase decomposition of dinitrogen pentoxide,\r\n

[latex] 2 N_{2}O_{5} \u21cc 4 NO_{2} + O_{2}[\/latex]<\/p>\r\nSay the rate law is found to be directly proportional to the concentration of [latex]N_{2}O_{5}[\/latex], we can express the rate law by[latex]^{[1]}[\/latex]:\r\n

[latex] r = k_{r} [N_{2}O_{5}][\/latex]<\/p>\r\nReaction rate laws can be complicated and may tell us about the mechanism of the reactions. For example, consider the reaction between hydrogen and bromine:\r\n\r\nSimple stoichiometry:\r\n

[latex] H_{2(g)} + Br_{2(g)} \u2192 2 HBr_{(g)}[\/latex]<\/p>\r\nComplicated rate law:\r\n

[latex]r = \\frac{k_{a}[H_{2}][Br_{2}]^{3\/2}}{[Br_{2}]+k_{b}[HBr]}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\nRate Law vs. Equilibrium Constant<\/strong>\r\n\r\nBe careful not to confuse equilibrium constant expressions with rate law expressions. The expression for [latex]K_{eq}[\/latex] can always be written by inspecting the balanced<\/strong> reaction equation, and often contains a term for each species of the reaction (raised to the power of its coefficient) whose concentration changes during the reaction. The equilibrium constant for the reaction [latex] 2 N_{2}O_{5} \u21cc 4 NO_{2} + O_{2}[\/latex] is given below:\r\n\r\n[latex]K_{eq}=\\frac{[NO_{2}]^4[O_{2}]}{[N_{2}O_{5}]^2}[\/latex]<\/span>\r\n\r\nIn contrast, the expression for the rate law generally bears no relation to the reaction equation and must be determined experimentally. [latex]^{[1]}[\/latex]<\/span>\r\n\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Reaction Rate Law Units<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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\r\n\r\nReaction rate (r) is generally expressed in units of concentration over time (e.g. [latex]\\frac{mol}{L\u00b7s}[\/latex], [latex]\\frac{kPa}{min}[\/latex], [latex]\\frac{mol}{m^3\u00b7h}[\/latex] ).\r\n\r\nThis means the rate constant [latex]k_{r}[\/latex] needs to be such that r is expressed in units of concentration over time.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Exercise: Rate Constant Units<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFor the following example, what are the units for the reaction rate constant ([latex]k_{r}[\/latex])?\r\n

[latex]r=k_{r}*p_{A}*p_{B}^2[\/latex]<\/p>\r\nwith p in Pa and time in seconds\r\n\r\n<\/div>\r\n<\/div>\r\n

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Solution<\/h3>\r\nSince r is expressed in concentration over time, the units of r are [latex]\\frac{Pa}{s}[\/latex].\r\n\r\n\\begin{align*}\r\n\\frac{Pa}{s}& = k_{r}*Pa*Pa^2 \\\\\r\nk_{r}& =\\frac{1}{Pa^2s}\r\n\\end{align*}\r\n\r\n<\/div>\r\n<\/div>\r\n
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References<\/h2>\r\n
\r\n\r\n[1] Chemistry LibreTexts. 2020. The Rate Law.<\/i> [online] Available at: <https:\/\/chem.libretexts.org\/Bookshelves\/Physical_and_Theoretical_Chemistry_Textbook_Maps\/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)\/Kinetics\/Rate_Laws\/The_Rate_Law<\/a>> [Accessed 23 April 2020].\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>","rendered":"
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Learning Objectives<\/p>\n<\/header>\n

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By the end of this section, you should be able to:<\/p>\n

Define<\/strong> reaction rate law and r<\/span>eaction rate constant (k)<\/span><\/p>\n<\/div>\n<\/div>\n

 <\/p>\n

Reaction rate law Definition:<\/strong> The relationship between the rate of reaction and the concentration of reactants.<\/p>\n

The rate law is usually proportional to the concentrations of reactants raised to a certain power:<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n
\n

Take the reaction we used as an example before: [latex]A + 2B \u2192 3C + D[\/latex]<\/p>\n

The general form for reaction rate law is<\/p>\n\n\n\n
[latex]r=k_{r}[A]^a[B]^b[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

For gas cases, we can use partial pressure<\/p>\n

[latex]r=k_{r}p_{A}^a p_{B}^b[\/latex]<\/p>\n

The rate constant [latex]k_{r}[\/latex] is independent of species concentration but generally dependent on temperature.<\/p>\n

 <\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n

<\/div>\n
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For example, let’s look at the rate of the gas-phase decomposition of dinitrogen pentoxide,<\/p>\n

[latex]2 N_{2}O_{5} \u21cc 4 NO_{2} + O_{2}[\/latex]<\/p>\n

Say the rate law is found to be directly proportional to the concentration of [latex]N_{2}O_{5}[\/latex], we can express the rate law by[latex]^{[1]}[\/latex]:<\/p>\n

[latex]r = k_{r} [N_{2}O_{5}][\/latex]<\/p>\n

Reaction rate laws can be complicated and may tell us about the mechanism of the reactions. For example, consider the reaction between hydrogen and bromine:<\/p>\n

Simple stoichiometry:<\/p>\n

[latex]H_{2(g)} + Br_{2(g)} \u2192 2 HBr_{(g)}[\/latex]<\/p>\n

Complicated rate law:<\/p>\n

[latex]r = \\frac{k_{a}[H_{2}][Br_{2}]^{3\/2}}{[Br_{2}]+k_{b}[HBr]}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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 <\/p>\n

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Rate Law vs. Equilibrium Constant<\/strong><\/p>\n

Be careful not to confuse equilibrium constant expressions with rate law expressions. The expression for [latex]K_{eq}[\/latex] can always be written by inspecting the balanced<\/strong> reaction equation, and often contains a term for each species of the reaction (raised to the power of its coefficient) whose concentration changes during the reaction. The equilibrium constant for the reaction [latex]2 N_{2}O_{5} \u21cc 4 NO_{2} + O_{2}[\/latex] is given below:<\/p>\n

[latex]K_{eq}=\\frac{[NO_{2}]^4[O_{2}]}{[N_{2}O_{5}]^2}[\/latex]<\/span><\/p>\n

In contrast, the expression for the rate law generally bears no relation to the reaction equation and must be determined experimentally. [latex]^{[1]}[\/latex]<\/span><\/p>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Reaction Rate Law Units<\/h2>\n<\/div>\n<\/div>\n<\/div>\n
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Reaction rate (r) is generally expressed in units of concentration over time (e.g. [latex]\\frac{mol}{L\u00b7s}[\/latex], [latex]\\frac{kPa}{min}[\/latex], [latex]\\frac{mol}{m^3\u00b7h}[\/latex] ).<\/p>\n

This means the rate constant [latex]k_{r}[\/latex] needs to be such that r is expressed in units of concentration over time.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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 <\/p>\n

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Exercise: Rate Constant Units<\/p>\n<\/header>\n

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For the following example, what are the units for the reaction rate constant ([latex]k_{r}[\/latex])?<\/p>\n

[latex]r=k_{r}*p_{A}*p_{B}^2[\/latex]<\/p>\n

with p in Pa and time in seconds<\/p>\n<\/div>\n<\/div>\n

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Solution<\/h3>\n

Since r is expressed in concentration over time, the units of r are [latex]\\frac{Pa}{s}[\/latex].<\/p>\n

\\begin{align*}
\n\\frac{Pa}{s}& = k_{r}*Pa*Pa^2 \\\\
\nk_{r}& =\\frac{1}{Pa^2s}
\n\\end{align*}<\/p>\n<\/div>\n<\/div>\n

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References<\/h2>\n
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[1] Chemistry LibreTexts. 2020. The Rate Law.<\/i> [online] Available at: <https:\/\/chem.libretexts.org\/Bookshelves\/Physical_and_Theoretical_Chemistry_Textbook_Maps\/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)\/Kinetics\/Rate_Laws\/The_Rate_Law<\/a>> [Accessed 23 April 2020].<\/p>\n<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":948,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1318","chapter","type-chapter","status-publish","hentry"],"part":1286,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/1318","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/wp\/v2\/users\/948"}],"replies":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/wp\/v2\/comments?post=1318"}],"version-history":[{"count":10,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/1318\/revisions"}],"predecessor-version":[{"id":2603,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/1318\/revisions\/2603"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/parts\/1286"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/1318\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/wp\/v2\/media?parent=1318"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/pressbooks\/v2\/chapter-type?post=1318"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/wp\/v2\/contributor?post=1318"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chbe220\/wp-json\/wp\/v2\/license?post=1318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}