Shane Flynn edited Kinetic model with dynamic disorder.tex  over 9 years ago

Commit id: 03647b653bf2308315d5d94f05a8a3a80bb79a3e

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The effects of dynamic disorder on higher ordered kinetics can be investigated through the Kohlrausch-Williams-Watts (KWW) model. This expression takes the expotential decay predicted by classical kinetics and stretches the curve through a time depedent rate coefficient $\omega$ and the coopertavity of decay $\beta$, making it a conventient expression for interpreting dynamic disorder. The use of this model is intended to be a proof of principle, however any system with a time depedent rate coefficient may be analyzed.   Assuming an overall second order process with a time depedent rate coefficient the survival function is written as $S(t)=\frac{1}{1+(\omega t[A_0])^{\beta}}$.Thorugh the time derivitive of the inverse of the survival function the time depedent rate coefficient characterising the deacy is expressed as $k(t)=\frac{\beta([A_0]\omega t)^{\beta}}{t}$. We see this definition of the rate coefficient maintains a depedence on the initial concentration of the reactants which is a unique result when compared to first order kinetics. Through integrating over the time depedent rate coefficient we find the statistical length.   \begin{equation}  \mathcal{L}_{KWW}(\Delta{t})=[A_0]^{\beta}\omega^{\beta}t^{\beta}\bigg|_{t_i}^{t_f} \mathcal{L}_{KWW}(\Delta{t})=[A_0\omega t]^{\beta}\bigg|_{t_i}^{t_f}  \end{equation}