Transition State Theory - Derivation of The Eyring Equation

Derivation of The Eyring Equation

One of the most important feature introduced by Eyring, Polanyi and Evans was the notion that activated complexes are in quasi-equilibrium with the reactants. The rate is then directly proportional to the concentration of these complexes multiplied by the frequency (k_BT/h) with which they are converted into products.

Quasi-Equilibrium Assumption

It should be noted that quasi-equilibrium is different from classical chemical equilibrium, but can be described using the same thermodynamic treatment. Consider the reaction below

where complete equilibrium is achieved between all the species in the system including activated complexes, ‡ . Using statistical mechanics, concentration of ‡ can be calculated in terms of the concentration of A and B.

TST assumes that even when the reactants and products are not in equilibrium with each other, the activated complexes are in quasi-equilibrium with the reactants. As illustrated in Figure 2, at any instant of time, there will be a few activated complexes, and some were reactant molecules in the immediate past, which are designated ‡ (since they are moving from left to right). The remainder of them were product molecules in the immediate past (‡). Since the system is in complete equilibrium, the concentrations of ‡ and ‡ are equal, so that each concentration is equal to one-half of the total concentration of activated complexes:

In TST, it is assumed that the flux of activated complexes in the two directions are independent of each other. That is, if all the product molecules were suddenly removed from the reaction system, the flow of ‡ will stop, but there will still be a flow from left to right. Hence, to be technically correct, the reactants are in equilibrium only with ‡, the activated complexes that were reactants in the immediate past.

The activated complexes do not follow a Boltzmann distribution of energies, but an "equilibrium constant" can still be derived from the distribution they do follow. The equilibrium constant K‡ɵ for the quasi-equilibrium can be written as

So, the concentration of the transition state AB‡ is

Therefore the rate equation for the production of product is

Where the rate constant k is given by

k‡ is directly proportional to the frequency of the vibrational mode responsible for converting the activated complex to the product; the frequency of this vibrational mode is ν. Every vibration does not necessarily lead to the formation of product, so a proportionality constant κ, referred to as the transmission coefficient, is introduced to account for this effect. So k‡ can be rewritten as

For the equilibrium constant K‡, statistical mechanics leads to a temperature dependent expression given as

where

Combining the new expressions for k‡ and K‡, a new rate constant expression can be written, which is given as

Since ΔG = ΔH –TΔS, the rate constant expression can be expanded, giving the Eyring equation

TST's rate constant expression can be used to calculate the Δ‡Gɵ, Δ‡Hɵ, Δ‡Sɵ, and even Δ‡V (the volume of activation) using experimental rate data.