Page 126 - Mechanism and Theory in Organic Chemistry

Derivation of Transition State Theory Expression for a Rate Constant 115

the number in the lowest energy state. Then, using Equation Al.l, we obtain

The ratio NA/noA is defined as the partition function for A, Q,,

If we want to know the equilibrium constant for the isomeriziition A + B,
we need again K = NB/NA. But we have NA and NB summed up over all the
energy states in the partition functions,
NA = noAQA (~1.11)
N, = ~~BQB (A1.12)
and the equilibrium constant is given by Equation A1.13:

The ratio noB/noA, the numbers in the lowest states, is, as we have already seen,
just exp[ - (E:, - EEA)/RT], so our equilibrium constant is

QB [- (EiB - GA)]
K = - exp
Q A RT

RTlnK = -AE,"+ RTln (A1.15)

For an isomerization, in which there is no change in the number of molecules,
the expression A1.15 is equal to -AG& When the number of molecules changes,
Equation A1.15 must be modified to Equation A1.l 6.C

The Components of the Partition Function
For thinking about transition states, it is useful to divide the energy levels into
categories, and to associate a fraction of the partition function with each category.

'The other thermodynamic functions are readily derived from Equation A1.16. See, for example,
Wiberg, Physical Organic Chemistry, p. 216.