D32.3 Second-Order Reaction

Jia Zhou; John Moore

D32.3 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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