deletions | additions       diff --git a/Nonlinear irreversible kinetics.tex b/Nonlinear irreversible kinetics.tex  index 1b1b6bc..c772046 100644  --- a/Nonlinear irreversible kinetics.tex  +++ b/Nonlinear irreversible kinetics.tex 
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\begin{equation}  \frac{dC_i(t)}{dt} = k_i(t)\left[C_i(t)\right]^i.  \end{equation}  Experimental data is typically a concentration profile corresponding to the integrated form of the rate law. For example, in Normalizing  the case of the $i^{th}$-order reaction, concentration profile, by comparing  the traditional integrated rate law and concentration at  a rate ``constant'', $k_i(t)\to\omega$, is time $t$ to the initial concentration, leads to the survival function  \begin{equation}  \frac{1}{C_i(t)^{i-1}} = \frac{1}{C_i(0)^{i-1}}+(i-1)\omega t.  \end{equation}  Normalizing the concentration profile, by comparing the concentration at a time $t$ to the initial concentration, leads to the survival function  \begin{equation}  S_i(t) = \frac{C_i(t)}{C_i(0)} = \sqrt[i-1]{\frac{1}{1+(i-1)\omega tC_i(0)^{i-1}}}, \frac{C_i(t)}{C_i(0)},  \end{equation}  which we will use as the input to our theory. Namely, we define the effective rate coefficient, $k_i(t)$, through an appropriate time derivative of the survival function that depends on the order $i$ of reaction  \begin{equation} 
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\displaystyle +\frac{d}{dt}\frac{1}{S_i(t)^{i-1}} & \text{if } i \geq 2.  \end{cases}  \end{equation}  For example, in the case of the $i^{th}$-order reaction, the traditional integrated rate law and a rate ``constant'', $k_i(t)\to\omega$, is  \begin{equation}  \frac{1}{C_i(t)^{i-1}} = \frac{1}{C_i(0)^{i-1}}+(i-1)\omega t.  \end{equation}  Normalizing the concentration profile, by comparing the concentration at a time $t$ to the initial concentration, leads to the survival function  \begin{equation}  S_i(t) = \sqrt[i-1]{\frac{1}{1+(i-1)\omega tC_i(0)^{i-1}}},  \end{equation}  In traditional kinetics, the rate coefficient of irreversible decay is assumed constant, in which case $k(t)\to\omega$. However, this will not be the case when the kinetics are statically or dynamically disordered. In these cases, we will use the above definitions of $k(t)$.