D32.2 First-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}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-223fa25d0949cbede50ff4a6d6282ad9_l3.png "Rendered by QuickLaTeX.com")
If the reaction is first order with respect to [A], that is, when m = 1, this becomes:
![\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-01d97b4b723b5c581ad152a7e5e904d7_l3.png "Rendered by QuickLaTeX.com")
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](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-54489c326e95cf37a5850ed0a716069b_l3.png "Rendered by QuickLaTeX.com")
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}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-b19eee34c76d774b007ed3977a427448_l3.png "Rendered by QuickLaTeX.com")
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}](https://wisc.pb.unizin.org/app/uploads/quicklatex/quicklatex.com-dd0d9c704092c676cda289eb7d8d363c_l3.png "Rendered by QuickLaTeX.com")
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 linear plot is −k and the y-intercept is 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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