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Jason R. Green deleted file Pseudo-First Order Reactions.tex
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\section{Pseudo-first-order reactions}
Second order kinetics can easily be simplified using the isolation method.[cite] The isolation method involves an using excess of one reactant and reacting it with a much smaller amount of a second reactant. While the excess reactant's concentration essentially remains constant over time, the other reactant's concentration varies at a measurable rate over time, allowing the order of that reactant to be determined, giving details about the reaction's mechanism. In essence, the isolation method simply turns a second order reaction into a first order reaction, or vice versa. The rate law of a second order irreversible decay reaction between A and B, where the reaction is first order in both A and B, decaying into products is
\begin{equation}
\frac{-d[A]}{dt}=\omega{[A][B]}
\end{equation}
With the concentration of B is in excess with respect to A and staying constant, a pseudo first order rate coefficient can be defined. The pseudo first order rate constant, $\omega'$ is
\begin{equation}
\omega'=\omega[B]
\end{equation}
Now the rate law can be written as
\begin{equation}
\frac{-d[A]}{dt}=\omega'[A]
\end{equation}
The inequality between the statistical length and divergence measures the amount of disorder a rate coefficient has over a period of time, $\Delta t$, for a first order irreversible decay reaction.
The statistical length can be calculated as the integral of k'(t).
\begin{equation}
\mathcal{L}(\Delta t)^2=\left[\int_0^{\Delta t} \omega'dt\right]^2=(\omega'\Delta t)^2
\end{equation}
The divergence,$\mathcal{J}(\Delta{t})$, can also be calculated from k'(t)
\begin{equation}
\Delta t \int_0^{\Delta t}k'(t)^2dt=\omega'^2\Delta t^2
\end{equation}
The inequality between the statistical length and divergence works in a pseudo-first order irreversible reaction. The difference between $\mathcal{J}(\Delta{t})$ and $\mathcal{L}(\Delta{t})^2$ not only measures the amount of static and dynamic disorder in a pseudo-first order decay, but also how psuedo-first order the reaction really is. A large difference between $\mathcal{J}(\Delta{t})$ and $\mathcal{L}(\Delta{t})^2$ may also suggest that there is not enough excess of one reactant to consider the reaction to be completely pseudo-first order. The only time a single pseudo-first order rate coefficient is sufficient to describe the population decaying over time is when $\omega'^2\Delta{t}^2=(\omega\Delta{t})^2$. This is also the only time where the reaction is truly pseudo-first order.
