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Jason R. Green renamed Theory.tex to Definition of time-dependent rate coefficient.tex
over 9 years ago
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\section{Definition of time-dependent rate coefficient}
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}
k_1(t) \equiv \frac{-d\ln S(t)}{dt}
\end{equation}
In traditional kinetics, irreversible decay is only dependent on one rate coefficient, $k(t)\to\omega$.
The time-dependent rate coefficient, $k(t)$, is determined by integrating the rate law of the reaction and forming a survival function from the integrated rate law.
In order to determine $k(t)$ in higher order reactions, we use the survival function, but it is not necessary. From the survival function, the time dependent rate coefficient is
\begin{equation}
k_i(t) \equiv \frac{d}{dt}\frac{1}{S(t)^{i-1}}
\end{equation}
