Page 128 - Mechanism and Theory in Organic Chemistry

Derivation of Transition State Theory Expression for a Rate Constant 117

axes and a is fhe symmetry number, the number of equivalent ways of orienting
the molecule.
The vibrational partition function, which is the one of most concern for our
purposes, is found by summing over the vibrational energy levels for each vibra-
tional mode and multiplying together the results for all the modes. Assuming
simple harmonic motion, the lowest energy level for a normal mode, the zero-
point level, has energy €, = +hv, measured from the minimum of the potential
energy curve. Here v is the excitation frequency for the vibration (equal to the
frequency observed for that mode in the infrared or Raman spectrum). The other
levels are spaced upwards from this one at intervals of hv. The levels thus fall at
integral multiples of hv above the lowest and, since there are no degeneracies, the
vibrational partition function for each normal mode is

aj nhv,
fpde i = 2 exp --
kT
n=o
Since an infinite sum of terms of the form e-ax converges to 1/(1 - e-ax), the
partition function A1.24 is more simply written

fyode ' = [l - exp (-u,)]-l (A1.25)
where u, = hvi/kT. The total vibrational partition function is then a product of
terms for the 3 N - 6 modes,

The electronic partition function can be evaluated by summing over
spectroscopically determined electronic states, but as the electronic energy-level
separations are large, the number of molecules in excited electronic states is
negligibly small at ordinary temperatures and the electronic partition function is
unity and will be ignored henceforth.

THE TRANSITION STATE EQvILIBRIUM
Now consider Reaction A1.27 in the k, direction.

We have from Section 2.6 the following relations:

We express the equilibrium constant KT in terms of the partition function ratio
Qt/QA to yield Equation A1.31, where AE; is the difference between the lowest

energy level of A and the lowest energy level of the transition state.