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%\end{equation}  \begin{equation}  S(t) = \frac{C_A(t)}{C_A(0)} = \frac{1}{1+\omega tC_A(0)}  \end{equation}From this survival function, taking the inverse of the survival function and then the time derivative yields the time dependent rate coefficient.  \begin{equation}  k_2(t) = \frac{d}{dt}S(t)^{-1}  \end{equation}