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\section{Theory}  We consider the irreversible elementary reaction types  \begin{equation}  A^i \to \mathrm{products}\quad\quad\textrm{for}\quad i=1,2,3,\ldots,n  \end{equation}  with the mass-action rate laws  \begin{equation}  \frac{dC_{A^i}(t)}{dt} = k_iC_{A^i}(t).  \end{equation}  Experimentalists deduce rate laws from data that is the integrated rate law. For the $i^{th}$-order reaction the integrated rate law is  \begin{equation}  \frac{1}{[C_A(t)]^{i-1}} = \frac{1}{[C_A(0)]^{i-1}}+(i-1)\omega t.  \end{equation}  Survival functions are the input to our theory. They are a measure of the concentration of species at a time $t$ compared to the initial concentration  \begin{equation}  S(t) = \frac{C_A(t)}{C_A(0)},  \end{equation}  which come from the integrated rate law. The survival function for this class of reactions is  \begin{equation}  S(t) = \sqrt[i-1]{\frac{1}{1+(i-1)\omega tC_A(0)^{i-1}}}  \end{equation}  From the survival function, the time dependent rate coefficient is determined by taking various time derivatives of the survival function, depending on the total order of reaction. For first order irreversible decay reactions, $A\to B$, the rate law defines the time dependent rate coefficient  \begin{equation}  \frac{-d\ln S(t)}{dt} = k(t)