D34.2 First-Order Reaction

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

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

for when the reaction is first order with respect to [A], that is, when m = 1, is:

 \begin{array}{rcl} \displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]} &=& -kt \\[2em] \text{ln}[\text{A}]_t - \text{ln}[\text{A}]_0 &=& -kt} \end{array}

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

 \text{ln}\left(\dfrac{[\text{A}]_t}{[\text{A}]_0}\right) = -kt

It is easier to use this form of the equation when trying to calculate the time required for a reaction to proceed to a certain extent.

On the other hand, if you raise e (the base of the natural logarithm system) to the power of each side of the equation, it gives:

 \dfrac{[\text{A}]_t}{[\text{A}]_0} = e^{-kt} \;\;\;\;\;\text{or}\;\;\;\;\; [\text{A}]_t = [\text{A}]_0e^{-kt}

It is easier to use this form of the equation when trying to determine the concentration of reactant remaining after a certain period of time.

Exercise: Integrated Rate Law for First Order Reaction

The integrated rate law for a first-order reaction can be rearranged to have a standard linear equation format:

 \begin{array}{rcl} \text{ln}[\text{A}]_t &=& -kt + \text{ln}[\text{A}]_0 \\[0.5em] y &=& \; mx +\; b \end{array}

Hence, if a reaction is first order in [A], a plot of “ln[A]t vs. t” must give a straight line. The slope of such a plot would be −k and the y-intercept would correspond to ln[A]0. If the plot is not a straight line, the reaction is not first order with respect to [A].

Activity: First-order Rate Constant from Graph

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Chemistry 109 Fall 2021 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.