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Jonathan Nichols added Pseudo-First Order Reactions.tex
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In practice, second order reactions can be difficult to analyze because it can be difficult to precisely measure the concentration of two reactants at the same time. Pseudo first order kinetics can be used to accommodate this issue by using the isolation method.[cite] The isolation method involves an excess of one reactant and reacting it with a much smaller amount of a second reactant. While the excess reactant's concentration 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, which gives details about the reaction's mechanism. 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 great 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}
L(\Delta t)=\int_0^{\Delta t} \omega'dt=\omega'\Delta t
\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 \int_0^{\Delta t}dt
\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 $\omega'(\Delta{t})^2=(\omega\Delta{t})^2$. This is also the only time where the reaction is truly pseudo-first order.
