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

7.1 Foundations of transition state theory 150
(1) We start by assuming equilibrium between the reactants and products:
A+BC 1 À
2
ABC ‡ 3
À
4
P (7.11)
(2) We consider two dividing surfaces at the TS separated by a distance of ≤ .
is the “length” of a “phase space cell” in semiclassical theory which is determined
by Heisenberg’s uncertainty principle:
∆ ‡ × ∆ ‡ = × ∆ ‡ = (7.12)
(3) We now consider the number densities of molecules passing through these surfaces
from left to right ( −→ ‡ )andfromrighttoleft( ←−
‡ ). The total number density ‡
at the TS in equilibrium is their sum, i.e.,
‡ = −→ ‡ + ←−
‡ = ‡ A BC (7.13)
In equilibrium, we must have
Thus,
−→
‡ = ←−
‡ (7.14)
−→
‡ = ←−
‡ = 1 2 ‡ = ‡
2 A BC (7.15)
(4) We now suddenly remove all the products. Then ←−
‡ =0 However, there is no
reason for −→ ‡ to change, because the molecules on the reactant side remain in
internal equilibrium (this is the quasi-equilibrium assumption). 43 y
−→
‡ = ‡
2 = ‡
2 A BC (7.16)
Thus, we can still express −→ ‡ using the equilibrium constant ‡ even if there is
no equilibrium between reactants and products.
(5) We now turn to the reaction rate, which is given by the number density of molecules
−→ ‡ in the time interval that are crossing the DS in the forward direction.
We can write this as
−→ ‡
= −→ −→
‡ ‡ ‡
=
‡ (7.17)
where
• −→ ‡ is the decay rate of the TS in the forward direction (or the frequency
factor for the molecules crossing the DS in the forward direction),
43 The quasi-equilibrium assumption for −→ ‡ is made in canonical TST. In a microcanonical derivation
· ·
−→ ←−
of the TST expression for (), we consider the fluxes ‡ and ‡ through the dividing surface
at the TS. Then, the individual molecules “dont’t know” that the products have been removed and
·
·
←−
−→
‡ =0. Since they don’t know about this, the flux ‡ has to remain unchanged.