D34.1 Integrated Rate Law

In addition to the method of initial rates, integrated rate law is another method for determining the rate law and rate constant from experimental data. An integrated rate law relates the concentration of a reactant or product to the elapsed time of the reaction. Thus, it can be used to determine the concentration of a reactant or a product present after a certain period of time, or to estimate the time required for a reaction to proceed to a certain extent.

For a given rate equation, calculus can be used to derive an appropriate integrated rate law. If you are unfamiliar with calculus and integration, don’t worry—you can still use the integrated rate law for a reaction without deriving it yourself.

For a generic reaction:

A ⟶ products

The rate of the reaction can be expressed as:

 \text{rate} = - \dfrac{\Delta[\text{A}]}{{\Delta}t} = k[\text{A}]^m

In calculus, the definition of a derivative is:

 \dfrac{d[\text{A}]}{dt} = \lim_{{\Delta}t \rightarrow 0} \dfrac{\Delta [\text{A}]}{{\Delta}t}

Therefore, the instantaneous rate for this reaction can also be written as:

 \text{rate} = -\dfrac{d[\text{A}]}{dt} = k[\text{A}]^m

Rearranging the equation gives:

 -\dfrac{d[\text{A}]}{[\text{A}]^m} = k\;dt

Integrating both sides of this equation gives:

 \displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} -\dfrac{d[\text{A}]}{[\text{A}]^m} = \int^t_0 k\;dt}

Assuming that the reaction starts at t = 0, the right side integration simply becomes kt. Then multiplying both sides by -1 gives:

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

[A]0 is the initial reactant concentration and [A]t is the reactant concentration at time t. The solution to the left side integration has a different mathematical form depending on the order of the reaction with respect to [A] (that is, it depends on the value of m). We will explore cases where m = 1, 2, and 0 in the next three sections.

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Chemistry 109 Fall 2021 Copyright © by John Moore; Jia Zhou; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.