View source for Solving the Rate Equation

Consider the following reaction:

:<math>
\mathrm{B} \rightarrow \mathrm{X}
</math>

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

:<math>
\frac{\mathrm{d}\left[\mathrm{B}\right]}{\mathrm{d}t} = -k\left[\mathrm{B}\right]\ ,
</math>

where <math>k</math> is the rate constant.

Equations of this type occur frequently in science. We obtain the solution by the '''[http://en.wikipedia.org/wiki/Separation_of_variables separation of variables]''', firstly dividing by <math>\left[\mathrm{B}\right]</math>.

:<math>
\frac{1}{\left[\mathrm{B}\right]}\frac{\mathrm{d}\left[\mathrm{B}\right]}{\mathrm{d}t} = -k\ ,
</math>

:<math>
\frac{1}{\left[\mathrm{B}\right]}\mathrm{d}\left[\mathrm{B}\right] = -k\mathrm{d}t\ ,
</math>

:<math>
\int\frac{1}{\left[\mathrm{B}\right]}\mathrm{d}\left[\mathrm{B}\right] = -\int k\mathrm{d}t\ ,
</math>

:<math>
\ln{\left[\mathrm{B}\right]} = -kt + C\ .
</math>

Finally, we apply the '''[http://en.wikipedia.org/wiki/Boundary_conditions boundary conditions]''',

:<math>
\left[\mathrm{B}\right] \bigg|_{t=0} = \left[\mathrm{B}\right]_0\ ,
</math>

and obtain the solution,

:<math>
\left[\mathrm{B}\right] = \left[\mathrm{B}\right]_0 e^{-kt}\ .
</math>