Page 161 - Introduction to chemical reaction engineering and kinetics

6.5 Transition State Theory (TST) 143
AGot = A@ - TAS”$ (6.5-8)

= 250,000 - 500(22)
= 239,000 J mol-’

(Comment: the normally expected value of AS’S for a unimolecular reaction, based on
A = 1013 to 1014, is = 0 (Table 6.1); the result here is greater than this.)

A method for the estimation of thermodynamic properties of the transition state and
other unstable species involves analyzing parts of the molecule and assigning separate
properties to functional groups (Benson, 1976). Another approach stemming from sta-
tistical mechanics is outlined in the next section.

6.5.3 Quantitative Estimates of Rate Constants Using TST with Statistical Mechanics

Quantitative estimates of Ed are obtained the same way as for the collision theory, from
measurements, or from quantum mechanical calculations, or by comparison with known
systems. Quantitative estimates of the A factor require the use of statistical mechanics,
the subject that provides the link between thermodynamic properties, such as heat ca-
pacities and entropy, and molecular properties (bond lengths, vibrational frequencies,
etc.). The transition state theory was originally formulated using statistical mechanics.
The following treatment of this advanced subject indicates how such estimates of rate
constants are made. For more detailed discussion, see Steinfeld et al. (1989).
Statistical mechanics yields the following expression for the equilibrium constant, Kj ,

Kz = (Qs/Q,)exp( -EzIRT) (6.5-16)

The function Qs is the partition function for the transition state, and Q, is the product
of the partition functions for the reactant molecules. The partition function essentially
counts the number of ways that thermal energy can be “stored” in the various modes
(translation, rotation, vibration, etc.) of a system of molecules, and is directly related to
the number of quantum states available at each energy. This is related to the freedom
of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy
change is related to the ratio of the partition functions:
AS”” = Rln(QslQ,) (6.5-17)

An increase in the number of ways to store energy increases the entropy of a system.
Thus, an estimate of the pre-exponential factor A in TST requires an estimate of the
ratio Q$/Q,. A common approximation in evaluating a partition function is to separate
it into contributions from the various modes of energy storage, translational (tr), rota-
tional (rot), and vibrational (vib):

Q = Q,,Q,,,Q,,Q(electronic, symmetry) (6.5-18)

This approximation is valid if the modes of motion are completely independent-an
assumption that is often made. The ratio in equation 6.5-17 can therefore be written as
a product of ratios:

(Qs/Q,> = (Qj,/Q,,)(Q$,,/Q,,,)(Q,s,b/Q,,> . . .
Furthermore, each Q factor in equation 6.5-18 can be further factored for each individ-
ual mode, if the motions are independent; for example,