kinetics - Rate-limiting in transition state theory

Suppose an initial state $A$ can transition to either state $C$ or state $D$. Suppose further that both of these two processes are rate-limited by a transition to the same intermediate state $B$, as follows:

Note that $\Delta G_{AB}> \Delta G_{BC}>\Delta G_{BD}$.

According to transition state theory, rates are determined by the rate-limiting step. So the rate of each process will be the same,

\begin{align} r_{A\to C}&=r_{A\to B} \\ r_{A\to D}&=r_{A\to B} \end{align}

from which it follows that the number of transitions made during some large $\Delta t$ will also be the same,

\begin{align} N_{A\to C}&=r_{A\to B}\Delta t \\ N_{A\to D}&=r_{A\to B}\Delta t \end{align}

However, if we were to consider the two pathways collectively then we would predict more transitions to be made to state $D$ than $B$ since $\Delta G_{BD}<\Delta G_{BC}$.

How do we resolve this apparent discrepancy?

Answer

In transition state theory (TST, goldbook) one of the necessary assumptions is that reactants and products are in equilibrium. In principle this gives us \begin{align}\ce{ A &<=> [AB]^{$\ddagger$} -> B\\ B &<=> [AB]^{$\ddagger$} -> A\\ A &<=> B, }\end{align} and in addition to this, also \begin{align}\ce{ B &<=> [BC]^{$\ddagger$} -> C & B &<=> [BD]^{$\ddagger$} -> D\\ C &<=> [BC]^{$\ddagger$} -> B & D &<=> [BD]^{$\ddagger$} -> B\\ B &<=> C & B &<=> D, }\end{align} as well as \begin{align}\ce{ C <=> [BC]^{$\ddagger$} -> & B <=> [BD]^{$\ddagger$} -> D\\ D <=> [BD]^{$\ddagger$} -> & B <=> [BC]^{$\ddagger$} -> C\\ C <=> & B <=> D, }\end{align} and $$\ce{ C <=> A <=> D\\ [AB]^{$\ddagger$} <=> [BC]^{$\ddagger$} <=> [BD]^{$\ddagger$} }.$$

You can use TST to estimate rate constants. $$k = \frac{\mathcal{k}_\mathrm{B} T}{\mathrm{h}} \exp\left\{ −\frac{\Delta^\ddagger{}G^\circ}{\mathcal{R} T}\right\}$$

Since all components have to be in equilibrium, you can predict a ratio between them via Boltzmann statistics, hence: $$\frac{N([BC]^{\ddagger})}{N([BD]^{\ddagger})} = \exp\left\{ −\frac{\Delta^\ddagger{}G^\circ([BC]^{\ddagger})-\Delta^\ddagger{}G^\circ([BD]^{\ddagger})}{\mathcal{R} T}\right\}=\frac{N(C)}{N(D)} $$

The preceding equilibrium does not need to be considered to determine the ratio of the products. However, it is important to note, that there are many assumptions necessary for TST to work. For example, at higher temperatures, anharmonic corrections have to be considered. More crucial is the assumption, that the transition can be described as a translatory movement, i.e. it is treated with classical mechanics.

For a system with a preceding equilibrium, it is well possible, that the TST approximation breaks down completely.