Page 118 - Mechanism and Theory in Organic Chemistry

Isotope Effects 107
difference is therefores1

The resulting isotope effect would be appr~ximately~~

Since the C-H stretching vibration appears in the infrared spectrum around
3000 cm-l, the isotope effect at T = 300°K would be

This model is, however, too crude to account for the observed range of isotope
effects. There are other changes occurring in the vibrations, and a more careful
treatment must take them into account.
Appendix 2 to this chapter gives a derivation that shows that the isotope
effect is more closely approximated by Equation 2.72. The II symbols signify a

product of terms; the first is a product over nsal modes of vibration of the
transition state, and the second over normal modes of the reactants. The quantity
ui is defined as hvi/kT, where vi is the frequency of normal mode i; each of the
exponential terms thus contains a difference in vibrational frequency between
the hydrogen and the deuterium compound. The products are over bound vibra-
tions only. In other words, the reaction coordinate itself, which is a vibration in
the reactants but not in the transition state, contributes only to the reactant part
of Equation 2.72. It is necessary to include in Equation 2.72 only those vibrations
that involve changes of force constants at isotopically substituted positions. An
expression for the isotope effect on an equilibrium is given in Appendix 2.
The following qualitative statement of the direction of an isotope effect is
sometimes useful. The heavy isotope will concentrate at that site where it is bound
more strongly, that is, has the larger force constant and frequency. For a kinetic
effect, this statement means that deuterium will prefer the reactant, where the
force constant is higher, and the hydrogen will prefer the transition state, where
the force constant is lower; the hydrogen compound will react faster. For an
equilibrium,

if the force constant is higher in AH(D) than in BH(D), the deuterium will prefer
to be in A and the hydrogen will prefer to be in B; K,,, will be greater than 1.

51 Multiplication by the speed of light, c, converts frequency expressed in cm-I to sec-l.
5a The units of hcv are erg molecule-l. To convert to cal mole-l, multiply by 1.439 x 10la; then
the gas constant R = 1.987 cal mole-l OK-l m ust be used in place of k.