Solving the Rate Equation

Consider the following reaction:

$\mathrm{B}\to \mathrm{X}$

The reaction is first order in B, and has the rate equation,

$$\begin{gathered} \frac{\mathrm{d}\left[\mathrm{B}\right]}{\mathrm{d}t}=-k\left[\mathrm{B}\right] \\ , \end{gathered}$$

where $k$ is the rate constant.

Equations of this type occur frequently in science. We obtain the solution by the separation of variables, firstly dividing by $\left[\mathrm{B}\right]$.

$$\begin{gathered} \frac{1}{\left[\mathrm{B}\right]}\frac{\mathrm{d}\left[\mathrm{B}\right]}{\mathrm{d}t}=-k \\ , \end{gathered}$$

$$\begin{gathered} \frac{1}{\left[\mathrm{B}\right]}\mathrm{d}\left[\mathrm{B}\right]=-k\mathrm{d}t \\ , \end{gathered}$$

$$\begin{gathered} \int \frac{1}{\left[\mathrm{B}\right]}\mathrm{d}\left[\mathrm{B}\right]=-\int k\mathrm{d}t \\ , \end{gathered}$$

$$\begin{gathered} \ln \left[\mathrm{B}\right]=-kt+C \\ . \end{gathered}$$

Finally, we apply the boundary conditions,

$$\begin{gathered} \left[\mathrm{B}\right]{|}_{t=0}={\left[\mathrm{B}\right]}_0 \\ , \end{gathered}$$

and obtain the solution,

$$\begin{gathered} \left[\mathrm{B}\right]={\left[\mathrm{B}\right]}_0{e}^{-kt} \\ . \end{gathered}$$