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 for the quasi-equilibrium can be written as
So, the chemical activity of the transition state AB is
Therefore, the rate equation for the production of product is
where the rate constant k is given by
Here, 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 . Every vibration does not necessarily lead to the formation of product, so a proportionality constant , 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
Combining the new expressions for k and K, a new rate constant expression can be written, which is given as
Since, by definition, ΔG = ΔH –TΔS, the rate constant expression can be expanded, to give an alternative form of the Eyring equation:
For correct dimensionality, the equation needs to have an extra factor of (c) for reactions that are not unimolecular:
where c is the standard concentration 1 mol⋅L and m is the molecularity.