Page 133 - Mechanism and Theory in Organic Chemistry

For the transition state, of course, one of the 3N - 6 vibrations is really a transla-
tion; for the moment we single it out and write for its frequency ratio vLH/vLD.
When Equation A2.11 is substituted into Equation A2.10, the products of atomic
masses will cancel, leaving Equation A2.12:

This expression gives the isotope effect in terms of vibrational frequencies
only; if the molecules are simple enough, a complete vibrational analysis and
direct calculation of the isotope effect will be possible. But for most purposes we
want an expression that will be easier to apply. Some simplification can be
achieved by noting that for all those vibrational modes that involve no substantial
motion at the isotopically substituted position, v,, = v,, (and therefore also
u,, = u,,) in both reactant and transition state. These modes will therefore
cancel and need not be considered further. Moreover, any mode that does involve
motion at the isotopically substituted position but that has the same force constant
in reactant and transition state will have v, in the reactant equal to v, in the
transition state and likewise for v,, and will also cancel. We therefore need con-
sider only those modes for which force constants of vibrations involving the
isotopically substituted position change on going from reactant to transition
state. For vibrations involving hydrogen, most of which have frequencies above
1000 cm-l, the factor 1 - ecU is approximately unity. Furthermore, since all the
ratios vH/vD should be about fly they will approximately ~ancel.~ If we ignore
for the moment the symmetry number ratio, which can always be put in later if
needed, we then have

where the products are over only those vibrations that involve force constant
changes at isotopically substituted positions.
It is frequently also necessary to assess isotope effects on equilibria. For an
equilibrium
KHID
AH + BD AD + BH (A2.14)
the appropriate expression isC

b See (a) L. Melander, Isotope Effects on Reaction Rates, Ronald Press, New York, 1960, p. 38; (b) J.
Bigeleisen, Pure Afifil. Chem., 8, 2 17 (1964), for further discussion.
Wiberg, Physical Organic Chemistry, p. 275.