Page 132 - Mechanism and Theory in Organic Chemistry

The Transition State Theory of Isotope Effects 121

The transition state sum omits the reaction coordinate degree of freedom since
it is not a bound vibration and does not contribute to the zero-point energy in the
transition state. E, and E, are respectively the energy of the potential energy
surface at transition state and reactants. Then,

and

where ui = hvi/kT, and AE is the energy difference along the potential surface
from reactants to transition state. The expression for k is now given by Equation
A2.8 :

The isotope effect is now found by taking the ratio of rate constants for the
two isotopic systems (Equation A2.9).

QADQ~H 3N.i-7 1 3N, -6
- - (uin - ID) I (~2.9)
k~ -
n exp [- ?cUfH - .ID)] n exp
kD QAHQ~D t
The energy difference AE is independent of isotopic substitution and cancels. We
have assumed that the isotopic substitution is in A, so Q, cancels also.
We now refer to Appendix 1 to write the partition functions in terms of
their translational, rotational, and vibrational components. Of the quantities
appearing in the expressions for these components, only the molecular mass My
the moments of inertia I, the vibrational frequencies u,, and the symmetry num-
bers o are different for the isotopic molecules; all other factors cancel, leaving
Equation A2.10.

This expression can fortunately be simplified by use of a theorem known as
the Teller-Redlich rule, which expresses the molecular mass and moment of
inertia ratios in terms of a ratio of a product of all the atomic masses mj and the
vibrational frequencies :a

" (a) K. B, Wiberg, Physical Organic Chemistry, Wiley, New York, 1964, p. 275; (b) J. Bigeleisen and
M. Wolfsberg, Advan. Chem. Phys., 1, 15 (1958).