Second-Order Reactions
We can derive the equation for calculating the half-life of a second order as follows: $$\frac{1}{[A]_t}=kt+\frac{1}{[A]_0}$$
For a half-life, $ t=t_{\frac{1}{2}}$ and $[A]_t=\frac{1}{2}[A]_0$
Substituting into the integrated rate law: $\frac{1}{\frac{1}{2}[A]_0}=k t_{\frac{1}{2}} +\frac{1}{[A]_0}$
And rearranging:
$$ \frac{1}{ \frac{1}{2}[A]_0 }- \frac{1}{[A]_0} =k t_{\frac{1}{2}} \\
\frac{2}{[A_0]}- \frac{1}{[A]_0} =k t_{\frac{1}{2}} \\
\frac{1}{[A]_0} =k t_{\frac{1}{2}} $$
$$ t_{\frac{1}{2}} =\frac{1}{k[A]_0} \label{eq1}\tag{1}$$
For a second-order reaction, $ t_{\frac{1}{2}} $ is inversely proportional to the concentration of the reactant, and the half-life increases as the reaction proceeds because the concentration of reactant decreases. Consequently, we find the use of the half-life concept to be more complex for second-order reactions than for first-order reactions. Unlike with first-order reactions, the rate constant of a second-order reaction cannot be calculated directly from the half-life unless the initial concentration is known.