Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

7.1 Foundations of transition state theory 149
7.1.2 Formal kinetic model
In order to derive the TST expression for ( ), wefirst look at a simple formal kinetic
model. This formal kinetic model has its roots in the original idea that the TS can be
considered as some sort of an “activated complex” with a local energy minimum on
the PES. It is not a good derivation of the TST expression for ( ), but leads to the
correct expression.
I
Activated complex model of TST:
y
A+BC 1 À
2
ABC ‡ 3 → P (7.3)
£
ABC
‡ ¤ = 1 [A] [BC]
2 + 3
(7.4)
I Assumption: 3 ¿ 2 y
£ ¤ ABC
‡ = 1
[A] [BC] =
12 [A] [BC] (7.5)
2
y Equilibrium between the reactants (A+BC)andtheTS(ABC ‡ ):
y
[P]
= 3
£
ABC
‡ ¤
(7.6)
= 3 | {z 12 [A] [BC] (7.7)
}
=
I
ACT expression for rate constant:
= 3 12 (7.8)
Below, we will find that
and
3 = ‡ =
12 ∝ exp
µ− ∆‡
(7.9)
(7.10)
7.1.3 The fundamental equation for k (T )
The above derivation of the TST rate constant was based on an oversimplified picture,
since the TS is not at a local minimum but at a saddle point on the PES. In this section,
we derive the TST fundamental equation more carefully by bringing in some dynamical
aspects. In (1) - (4), we explore the implications of the quasi-equilibrium assumption,
and in (5) - (9) we develop a dynamical semiclassical model: 42
42 Semiclassical because the translation is handled classically while all other degrees of freedom are
described quantum mechanically.