Page 159 - Introduction to chemical reaction engineering and kinetics

6.5 Transition State Theory (TST) 141

In the TST, molecularity (m) is the number of reactant molecules forming one
molecule of the transition state. In reaction 6.5-2, m = 2 (AB and C); that is, the
formation is bimolecular. Other possibilities are m = 1 (unimolecular) and m = 3
(termolecular). The molecularity of formation of the transition state affects the form
of Kj, and the order of the reaction equals m.

6.5.2 Thermodynamic Formulation
The reaction isotherm of classical thermodynamics applied to the formation of the tran-
sition state relates K: to AGO’, the standard Gibbs energy of formation of the activated
complex:

AGoS = -RT In K’ c (6.5-7)

Also

AG”~ = AH’S - T&q’* (6.543)

where AHot and ASot are, respectively, the (standard) enthalpy of activation and (stan-
dard) entropy of activation. Combining equations 6.5-6 to -8, we obtain

k = (kBTlh)e AS”*IRe-AH”IRT (6.5-9)
for the rate constant according to the TST. As with the SCT, we may compare this
expression with observed behavior

k o b s = A~-EAIRT

to obtain interpretations of the Arrhenius parameters A and EA in terms of the TST
quantities.
We first relate EA to AHot. From equation 6.5-6,

AU”*
dlnk
-= dlnK,S _ 1 I (6.510)
dT dT T RT2
where AU” is the internal energy of activation, and we have used the analogue of
the van? Hoff equation (3.1-5) for the temperature-dependence of K: (Denbigh, 1981,
p.147). For the activation step as a gas-phase reaction of molecularity m involving ideal
gases, from the definition H = U + PV,

AHoS = AU”t + (1 - m)RT. (6.5-11)

From equations 3.1-8 (i.e., from 3.1-6), and 6.5-10 and -11,

E A = AHoS + mRT (6.542) j

We next relate the pre-exponential factor A to ASOz. From equations 6.5-9 and 6.5-12,

k = (k,T/h)eASoi/Reme-Ea/RT (6.5-13) '