Rate Law and Integrated Rate Law for a Third-Order Reaction: A + 2B \u2192 P<\/h3>\n\n\n\n
Rate Law (Differential Form):<\/strong><\/p>\n\n\n\n For the reaction:A+2B\u2192PA + 2B \\rightarrow PA+2B\u2192P<\/p>\n\n\n\n The overall reaction is third-order<\/strong> (first-order in A and second-order in B). The rate law is expressed as:Rate=\u2212d[A]dt=k[A][B]2\\text{Rate} = -\\frac{d[A]}{dt} = k[A][B]^2Rate=\u2212dtd[A]\u200b=k[A][B]2<\/p>\n\n\n\n To derive the integrated rate law, we must account for the stoichiometry. Let:<\/p>\n\n\n\n Substituting into the rate law:dxdt=k([A]0\u2212x)([B]0\u22122x)2\\frac{dx}{dt} = k([A]_0 – x)([B]_0 – 2x)^2dtdx\u200b=k([A]0\u200b\u2212x)([B]0\u200b\u22122x)2<\/p>\n\n\n\n This is a complex differential equation and typically cannot be integrated analytically in a simple form unless specific assumptions are made. However, if [B]_0 \u226b [A]_0<\/strong>, then [B] remains approximately constant during the reaction. This is known as the pseudo-first-order approximation<\/strong>.<\/p>\n\n\n\n Under this approximation:Rate=k\u2032[A],where k\u2032=k[B]2\\text{Rate} = k'[A], \\quad \\text{where } k’ = k[B]^2Rate=k\u2032[A],where k\u2032=k[B]2<\/p>\n\n\n\n Integrating:d[A][A]=\u2212k\u2032[B]2dt\u21d2ln\u2061[A]=\u2212k[B]2t+ln\u2061[A]0\u21d2[A]=[A]0e\u2212k[B]2t\\frac{d[A]}{[A]} = -k'[B]^2 dt \\Rightarrow \\ln[A] = -k[B]^2t + \\ln[A]_0 \\Rightarrow [A] = [A]_0 e^{-k[B]^2t}[A]d[A]\u200b=\u2212k\u2032[B]2dt\u21d2ln[A]=\u2212k[B]2t+ln[A]0\u200b\u21d2[A]=[A]0\u200be\u2212k[B]2t<\/p>\n\n\n\n This is valid under pseudo-first-order conditions only.<\/p>\n\n\n\n In chemical kinetics, the rate law<\/strong> expresses the speed of a reaction as a function of the concentrations of reactants. For a reaction A+2B\u2192PA + 2B \\rightarrow PA+2B\u2192P, it is third-order overall: first-order in A and second-order in B. The rate law is written as:Rate=\u2212d[A]dt=k[A][B]2\\text{Rate} = -\\frac{d[A]}{dt} = k[A][B]^2Rate=\u2212dtd[A]\u200b=k[A][B]2<\/p>\n\n\n\n Here, kkk is the rate constant, and the exponents correspond to the reaction order with respect to each reactant. The negative sign indicates that the concentration of A decreases over time.<\/p>\n\n\n\n To determine the integrated rate law<\/strong>, which relates concentrations to time, we consider the stoichiometric relationship between the reactants. If xxx is the amount of A reacted at time ttt, then [A]=[A]0\u2212x[A] = [A]_0 – x[A]=[A]0\u200b\u2212x, and [B]=[B]0\u22122x[B] = [B]_0 – 2x[B]=[B]0\u200b\u22122x, because two moles of B react per mole of A. Substituting these into the rate law yields:dxdt=k([A]0\u2212x)([B]0\u22122x)2\\frac{dx}{dt} = k([A]_0 – x)([B]_0 – 2x)^2dtdx\u200b=k([A]0\u200b\u2212x)([B]0\u200b\u22122x)2<\/p>\n\n\n\n This equation is not easily integrable in closed form and often requires numerical methods or simplifying assumptions. A common simplification is the pseudo-first-order approximation<\/strong>, applied when [B]_0 is much larger than [A]_0. In this case, [B] remains nearly constant during the reaction, and the rate law simplifies to:Rate=k\u2032[A],with k\u2032=k[B]2\\text{Rate} = k’ [A], \\quad \\text{with } k’ = k[B]^2Rate=k\u2032[A],with k\u2032=k[B]2<\/p>\n\n\n\n This results in a first-order integrated rate law:[A]=[A]0e\u2212k[B]2t[A] = [A]_0 e^{-k[B]^2 t}[A]=[A]0\u200be\u2212k[B]2t<\/p>\n\n\n\n This exponential decay equation describes how [A] decreases over time in such conditions.<\/p>\n\n\n\n Derive the rate law equation for a third-order reaction A+2B\u2192P. Express the integrated rate law in concentration terms. The Correct Answer and Explanation is: Rate Law and Integrated Rate Law for a Third-Order Reaction: A + 2B \u2192 P Rate Law (Differential Form): For the reaction:A+2B\u2192PA + 2B \\rightarrow PA+2B\u2192P The overall reaction is third-order […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-231652","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231652","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=231652"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231652\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Integrated Rate Law (in terms of concentration):<\/h3>\n\n\n\n
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Full Explanation<\/h3>\n\n\n\n
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