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