deletions | additions       diff --git a/Kinetic model with dynamic disorder.tex b/Kinetic model with dynamic disorder.tex  index 5053a59..c047243 100644  --- a/Kinetic model with dynamic disorder.tex  +++ b/Kinetic model with dynamic disorder.tex 
...
\frac{d}{dt}[S(t)]=\frac{1}{n-1}(\frac{1}{1+z^\beta t^\beta})^{\frac{2-n}{n-1}}(\frac{-z^\beta\beta t^{\beta -1}}{(1+z^\beta t^\beta)^2})  \end{equation}  Such that z contains all the time independent variables $\omega C_a(o)^{n-1}$  Assuming an overall second-order process with a time depedent rate coefficient the survival function is $S_2(t) = 1/\left(1+(\omega tC_2(0))^{\beta}\right)$. We can then simplify the rate law expression.  \begin{equation}  \frac{d}{dt}[S(t)]=\frac{-\beta (\omega C_A(0))^\beta t^{\beta -1}}{(1+(\omega C_A(0))^\beta t^\beta)^2}=-k(t)S^2(t)  \end{equation}  The time-depedent rate coefficient characterizing the decay is $k_{2,KWW}(t) = \beta(C_A(0)\omega t)^{\beta}/t$, from the time-derivative of the inverse of the survival function. This definition of the rate coefficient depends on the initial concentration of the reactants which is consistent with units of rate constants in traditional kinetics. Integrating the time-depedent rate coefficient gives the statistical length \begin{equation}  \mathcal{L}_{KWW}(\Delta{t}) = \left(C_A(0)\omega t\right)^{\beta}\big|_{t_i}^{t_f}  \end{equation}