The Mathematics of Transition State Theory

Our analysis is not going to use statistical mechanics. Instead, we focus immediately upon the use of the common thermodynamic parameters (AG, AH, and AS). We start our analysis by considering a bimolecular reaction of A and B giving C in a single step (Eq. 7.7), where the activated complex is represented by AB (a unimolecular reaction is also perfectly amenable to this analysis). The rate of a bimolecular reaction can be expressed as the rate of change of the concentration of the product, d[C] j dt, as given by Eq. 7.8. 

In TST analysis, the rate by which C would be produced is directly proportional to the concentration of the activated complex AB and the intrinsic rate constant k for its decom- 

Given the thermodynamic relationships introduced in Section 2.1, we can write an equilibrium expression for the formation of AB, where K represents the equilibrium constant (Eq. 7.10). This is the primary postulate of TST—that the reactants are in equilibrium with the activated complex. 

Now we need to solve for K. This is the point at which TST turns into a statistical mechanics analysis. We invoke statistical mechanics because a transition state does not have a Boltzmann distribution of states (see the next section), because its lifetime is so fleeting. Using statistical mechanics, it is found that K is proportional to a new equilibrium constant K ) that can be viewed in the same manner as the simple equilibrium constants given in Chapter 2. Thus, this K is equal to exp(-AG /RT). The exact expression found for K is Eq. 7.13, where k, h, v, and T are the Boltzmann constant, Planck s constant, vibrational frequency, and absolute temperature, respectively (see Appendix 1 for values). 

Substituting Eqs. 7.12 and 7.13 into Eq. 7.11 gives the Eyring equation, named after Henry Eyring (Eq. 7.14). The equation consists of the term/c bT/which is near 10 s and is similar in magnitude to the frequency of many bond vibrations, as well as the K term. 

---