Page 111 - Mechanism and Theory in Organic Chemistry

at the transition state going in each direction over the barrier, and we shall con-
centrate on one direction only, A+ B. We are therefore dealing with rate
constant k, ; exactly the same arguments will apply to the A + B reaction and k- ,.
We suppose that the transition state molecules moving from left to right, At, are
at equilibrium with the bulk of the A molecules in the restricted sense specified
above. The concentration [Ax] can therefore be written in terms of an equilibrium
constant, Kr (Equation (2.54)). The rate of the reaction from left to right is

kx[Ax], the concentration of molecules at the transition state multiplied by a rate
constant characterizing their rate of passage over the barrier. Then since k,[A]
is the conventional reaction rate,

and, substituting for [A:] from Equation 2.54, and cancelling [A] from both sides,
we find Equation 2.56 relating the first-order rate constant to properties of the
transition state.

The equilibrium constant K* is then analyzed by the methods of statistical
thermodynamics to separate out the contribution of the reaction coordinate from
other contributions. The rate constant kx is also calculated by statistical thermo-
dynamic methods. These calculations are given in Appendix 1 to this chapter.
The results of the analysis are expressed by Equation 2.57, where k is the Boltz-

mann constant, h is Planck's constant, T is the Kelvin temperature, and Kx
is a new equilibrium constant that excludes the contributions from the reaction
coordinate. The new equilibrium constant K: can be written in terms of a free
energy of activation, AG: (Equation 2.58), and AG: can in turn be divided into

kT AH% AS$
kl = - exp (- =) (R)
h
exp
contributions from enthdpy of activation, AH:, and entropy of activation, AS:
(Equation 2.59). Equations 2.60 and 2.61 then follow. Equation 2.60 is called
the Eyring equation, after Henry Eyring, who was instrumental in the develop-
ment of the transition state theory.