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\end{equation}  The second order survival function is  \begin{equation}  S(t)=\frac{C_A(t)}{C_A(0)}=\frac{1}{1+\omega tC_A(0)]} 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}