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Jonathan Nichols edited $n^{th}$ Order Irreversible Decay.tex
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We now shift our attention to $n^{th}$ order reactions. These $n^{th}$ order reactions are of the form of one reactant turning into product. The inequality between the statistical distance and Fisher divergence can also be derived for these irreversible decay reactions.
The integrated rate law of an $n^{th}$ order irreversible decay reaction is [cite]
\begin{equation}
\frac{1}{[A_t]^{n-1}}=\frac{1}{[A_0]^{n-1}}+(n-1)\omega t
\end{equation}
From the integrated rate law, we obtain the survival function
\begin{equation}
S(t)=\frac{[A_t]}{[A_0]}\sqrt[n-1]{\frac{1}{1+(n-1)\omega t[A_0]^{n-1}}}
\end{equation}
The time dependent rate coefficient is
\begin{equation}
k(t)\equiv(\frac{d\frac{1}{(S(t)^{n-1)}}}{dt})\equiv(n-1)\omega[A_0]^{n-1}