Page 163 - Introduction to chemical reaction engineering and kinetics

6.5 Transition State Theory (TST) 145

transition state because the important Cl-C bond is stretched. The increased C-C-Cl bond
angle also increases the value of the smallest moment of inertia. Thus, the ratio Q&,/Q,,,
is greater than 1. An exact calculation requires a quantitative estimate of the bond lengths
and angles. The transition state has the same number of vibrational modes, but several
of the vibrational frequencies in the transition state are expected to be somewhat lower,
particularly those involving both the weakened C-Cl bond stretch and the affected C-H
bond. It is also possible to form the transition state with any of the three hydrogen atoms
on the CH, group, and so a symmetry number of 3 accrues to the transition state. The
internal rotation around the C-C bond is inhibited in the transition state, which decreases
the contribution of this model to Qz , but the rest of the considerations increase it, and the
net effect is that (es/Q,) > 1. From the value of the A factor in Example 6-4, Al(kTlh) =
(Q$/Q,) = 38.4. As with many theories, the information flows two ways: (1) measured
rate constants can be used to study the properties of transition states, and (2) information
about transition states gained in such studies, as well as in calculations, can be used to
estimate rate constants.

Consider a bimolecular reaction, A + B -+ products. Confining two molecules A and B
to be together in the transition state in a bimolecular reaction always produces a loss of
entropy. This is dominated by the ratio of the translational partition functions:

<Q~~lV>l<Q,,lV>(Q,,,lV> = W-m A+B kgTlh2)3’2/[2~mAkgTlh2)3’2(2n-mg kBTlh2)3’2]
= (2r,u kBTlh2)-3i2
where p is the reduced mass, equation 6.4-6. This ratio introduces the volume units to the
rate constant, and is always less than 1 for a bimolecular (and termolecular) reaction. At
500 K, and for a reduced mass of 30 g mol-l, this factor is 1.7 X 1O-6 L mol-’ s-l, and
corresponds to an entropy change of - 110 J mol-’ K-l. The number of internal modes
(rotation and vibration) is increased by 3, which partly compensates for this loss of entropy.
If A and B are atoms, the two rotational modes in the transition state add 70 J mol-’
K-’ to the entropy of the transition state. The total AS”* is therefore approximately -40
J mol-’ K-l, a value in agreement with the typical value given in Table 6.1. For each of
the two rotational modes, the moment of inertia cited in Table 6.2 is I = pdi,; the value
above is calculated using dAB = 3 X lo-lo m.

6.54 Comparison of TST with SCT

Qualitatively, both the TST and the SCT are in accord with observed features of kinet-
ics:

(1) Both theories yield laws for elementary reactions in which order, molecularity,
and stoichiometry are the same (Section 6.1.2).
(2) The temperature dependence of the reaction rate constant closely (but not
exactly) obeys the Arrhenius equation. Both theories, however, predict non-
Arrhenius behavior. The deviation from Arrhenius behavior can usually be
ignored over a small temperature range. However, non-Arrhenius behavior is
common (Steinfeld et al., 1989, p. 321). As a consequence, rate constants are
often fitted to the more general expression k = BPexp( -EIRT), where B , IZ,
and E are empirical constants.
The activation energy in both theories arises from the energy barrier at the transition
state, and is treated similarly in both. The relationship between the pre-exponential fac-
tors in the two theories is not immediately obvious, since many of the terms which arise