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\section{Second-order decay, $A+A\to P$}  Taking our definitions of the integrated rate law, survival function, and time dependent rate coefficient, we are able to apply them to to second order irreversible decay. The integrated second-order rate law gives the survival function  %\begin{equation}  % C_A(t) = \frac{C_A(0)}{1+\omega tC_A(0)}  %\end{equation}  \begin{equation}  S(t) = \frac{C_A(t)}{C_A(0)} = \frac{1}{1+\omega tC_A(0)}  \end{equation}  In second order, we define the time dependent rate coefficient as the time derivative of the inverse of the survival function[insert citation]  \begin{equation}  k_2(t) \equiv \frac{d}{dt}S(t)^{-1} = \omega C_A(0)