Page 130 - Mechanism and Theory in Organic Chemistry

Derivation of Transition State Theory Expression for a Rate Constant 119
where the limits of integration in the numerator are set at 0 to + oo because we are
interested only in positive v. Evaluation of the integrals yields Equation A1.37:

The rate at which transition states pass over the barrier from left to right is
just their velocity, 6, divided by the distance they must go, which we called 6
above in the partition function. Therefore

and, from Equation A1.34,

Equation A1.39 reduces to Equation A1.40 :

Another way to express this result is to define a new equilibrium constant, K*,
which includes all features of the transition state except the reaction coordinate,
and write

We then define free energy of activation as the free energy of the transition state
excluding the reaction coordinate mode, so that Equations A1.42 and A1.43 hold.

AG* = -RTln K* (A1.42)
- AG*
K* = exp
A factor K, called the transmission coe@cient, is sometimes included in the expression
for k, to allow for the possibility that some transition states may be reflected back
at the barrier, or that some may tunnel through it even though classically they
do not have the requisite energy. These corrections are usually considered to be
small, and we shall simply disregard them. We leave it as an exercise to the reader
to extend the transition state treatment to bimolecular reactions.