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

7.1 Foundations of transition state theory 151
• −→ ‡ = −→ ‡ is the mean speed of the molecules crossing the DS in the forward
direction.
(6) We now use the equilibrium constant to determine ‡ via
‡ = ‡ ABC
A BC
(7.18)
y ‡ = ‡ ABC = ‡ A BC (7.19)
y
−→ −→
‡
‡
=
‡ A BC = A BC (7.20)
with
−→
‡
=
‡ (7.21)
(7) From statistical thermodynamics (see Appendix G), we have the following expression
for the equilibrium constant ‡ ( ) in terms of the molecular partition
functions:
‡ ( )=
∗
=
µ
exp − ∆
0
(7.22)
• The ’s are the respective molecular partition functions (per unit volume).
• ∗ is written with respect to the zero-point level of the reactants.
• The exp (−∆ 0 ) factor arises subsequently, because we now express
the value of partition function for the TS relative to the zero-point level of
the TS ( ∗ = exp (−∆ 0 )).
(8) Separating the motion along the reaction coordinate ‡ from the other degrees of
freedom, we write
= ‡ ‡
(7.23)
where ‡ is the 1-D (translational) partition function describing the translational
motion of the molecules along ‡ across the TS and ‡
is the partition function
for all 3 − 7 other (rotational-vibrational-electronic) degrees of freedom.
Here,
• the 1-D translational partition function is
‡ =
µ 2
2 12
× (7.24)
• the mean speed along ‡ for crossing the DS in the forward direction is given
by
R
−→ ∞
‡ −2 2 µ
12
0
= R ∞
−∞ −2 2 =
(7.25)
2