D34.2 First-Order Reaction
For the generic reaction “A ⟶ products”, the integrated rate law:
![Rendered by QuickLaTeX.com \displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]^m} = -kt}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-da1c90c392b7ac1ecbd981f957e5ef46_l3.png)
for when the reaction is first order with respect to [A], that is, when m = 1, is:
![Rendered by QuickLaTeX.com \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}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-08fe89437d237cfcfa3c4039ac1bbdf1_l3.png)
This integrated rate law for a first-order reaction can be alternatively expressed as:
![Rendered by QuickLaTeX.com \text{ln}\left(\dfrac{[\text{A}]_t}{[\text{A}]_0}\right) = -kt](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-0a9f0d231e3412a9e42482c99cc0ea16_l3.png)
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:
![Rendered by QuickLaTeX.com \dfrac{[\text{A}]_t}{[\text{A}]_0} = e^{-kt} \;\;\;\;\;\text{or}\;\;\;\;\; [\text{A}]_t = [\text{A}]_0e^{-kt}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-558d194735d5ac5b619f35e013d72c69_l3.png)
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:
![Rendered by QuickLaTeX.com \begin{array}{rcl} \text{ln}[\text{A}]_t &=& -kt + \text{ln}[\text{A}]_0 \\[0.5em] y &=& \; mx +\; b \end{array}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-a28da174104f3a3986847fc063f7e1b1_l3.png)
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
Please use this form to report any inconsistencies, errors, or other things you would like to change about this page. We appreciate your comments. 🙂