Page 119 - Mechanism and Theory in Organic Chemistry

Isotope effects in linear transition states Let us now analyze the
kinetic isotope effect in a simple system, a transfer of hydrogen from AH to B
through a linear transition state (Equation 2.74).53 We assume that A and B are
polyatomic fragments. In the reactants we have to consider the A-H stretching

and A-H bending modes. In the transition state, the A-H stretch has become
the reaction coordinate (28),

and contributes nothing to the transition state term in Equation 2.72, leaving
exp[+t(u,, - ui,)] for this mode to contribute to the reactant term. It is this
factor that we evaluated earlier as being about 6.4. But there are also in the
transition state other vibrations to be considered. There will be two degenerate
bends, 29 and 30, which are identical but occur in mutually perpendicular
d d
7 A H B
@@a3

planes. These motions are not present in the reactants, and it is difficult to know
how to deal with them. They are, however, roughly comparable to the reactant
A-H bending; and since bending frequencies are lower than stretching and
therefore contribute less to the isotope effect in any event, bending frequencies
are usually considered to cancel approximately between reactant and transition
state when a primary isotope effect is being evaluated.54
We are then left with one final transition state vibration, a symmetric
stretch (31), which has no counterpart in the reactants. If the transition state is

-0 0 w
A H B
3 1
highly symmetric, so that the A .... H and the H-B force constants are equal, this
stretch will involve only A and B moving in and out together, with no motion of
the H (or D). The frequency will then be the same for H and D, and its contribu-
tion to the transition state term in Equation 2.72 will cancel. We shall then be left
with only the reaction coordinate mode, and an isotope effect around 6.4. If the
transition state is not symmetric, the H (D) will be closer to A or to B; then the
H (D) will move in the symmetric stretch and since v, > v,, exp[-t(u, - u,)]

63 See (a) note 48(b, c), p. 105.
L~~ See Wiberg, Physical Organic Chemistry, pp. 332-361, for calculations that roughly justify this
assumption for a specific example.