D15.3 Second-Order Reaction

For the generic reaction “A ⟶ products”, the integrated rate law is this:

\displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]^m} = -kt}

When the reaction is second order with respect to [A], that is, when m = 2, this becomes:

\begin{array}{rcl} \displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]^2} &=& -kt \\[1.5em] -\dfrac{1}{[\text{A}]_t}-\left(-\dfrac{1}{[\text{A}]_0}\right) &=& -kt} \end{array}

This integrated rate law for a second-order reaction can be alternatively expressed as:

\dfrac{1}{[\text{A}]_t} = kt\;+\;\dfrac{1}{[\text{A}]_0}

 

Exercise: Calculate Concentration from Time

The integrated rate law for a second-order reaction also has a standard linear equation format:

\begin{array}{rcl} \dfrac{1}{[\text{A}]_t} &=& \; kt \; + \dfrac{1}{[\text{A}]_0} \\[1em] y &=& mx + \;\; b \end{array}

Hence, if a reaction is second order in [A], a plot of \frac{1}{[\text{A}]_t} vs t should yield a straight line, where the slope equals k and the y-intercept is \frac{1}{[\text{A}]_0} . If the plot is not a straight line, then the reaction is not second order with respect to [A].

Activity: Order from Integrated Rate Law

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Chem 104 Summer 2024 Copyright © by Jia Zhou; John Moore; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.