Page 124 - Mechanism and Theory in Organic Chemistry

Appendix 1

DERIVATION OF THE

TRANSITION STATE THEORY

EXPRESSION FOR A

RATE CONSTANTa

In order to analyze the transition state equilibrium, we need to know how a
collection of molecules divides up the available energy.

THE BOLTZMANN DISTRIBUTION
Molecules distribute their total energy among translational, rotational, vibra-
tional, and electronic motions. These motions are all quantized, with energy-level
separations very small for translation, larger for rotation, still larger for vibration,
and very large for electronic motion. There are therefore many discrete energy
states available. At very_ low temperature, approaching absolute zero, nearly all
the_~o!ecules are in their lowest energy state; but as the temperature is raised,
the molecules acquire more energy and begin to populate higher states. The ratio
of numbers of molecules in any two states depends on the energy difference be-
tween the states and on the temperature, and is given by the Boltzmann distribu-
tion law,

Here n, is the number of molecules in state 1, energy E,, n, is the number of
molecules in state 2, energy E,, k is the Boltzmann constant, 1.3806 x 10-l6
erg OK-l, and T is the absolute temperature in degrees Kel~in.~

' Derivations may be found in the sources cited in note 36 of Chapter 2.
For a derivation see K. B. Wiberg, Physical Organic Chemistry, Wiley, New York, 1964, p. 21 1, or
W. J. Moore, Physical Chemistry, 3rd ed., Prentice-Hall, Englewood Cliffs, N.J,, 1962, p. 619.