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Jason R. Green added Irreversible kinetics.tex
over 9 years ago
Commit id: 6bde5c0bef536560f44509729f6179270019b856
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We consider the irreversible reaction types
\begin{equation}
A^i \to \mathrm{products}\quad\quad\textrm{for}\quad i=1,2,3,\ldots,n
\end{equation}
with the differential rate laws
\begin{equation}
\frac{dC_i(t)}{dt} = k_i(t)\left[C_i(t)\right]^i.
\end{equation}
Experimental data corresponds to the integrated form of the rate law, a concentration profile. 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) = \frac{C_i(t)}{C_i(0)} = \sqrt[i-1]{\frac{1}{1+(i-1)\omega tC_i(0)^{i-1}}},
\end{equation}
which we will use as the input to our theory.
