Solving the Rate Equation

Consider the following reaction:

$\mathrm{B} \rightarrow \mathrm{X}$

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

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

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 $\left[\mathrm{B}\right]$.

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

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

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

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

Finally, we apply the boundary conditions,

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

and obtain the solution,

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