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

8.3 Generalized Lindemann-Hinshelwood mechanism 184
8.3.2 Equilibrium state populations
As seen from the master equation analysis, the unimolecular reaction rate constant
in the high and in the low pressure regimes depends on the equilibrium (Boltzmann)
state distributions (for discrete states) or () (continuous state distributions). In
a polyatomic molecule, due to the large number of vibrational degrees of freedom, the
number of vibrational states increases very rapidly with increasing vibrational excitation
energy. The quanta of vibrational excitation can be distributed over the oscillators. In
order to calculate the number of combinations, we start with a single oscillator and
expand our scope first to two oscillators and then to =3 − 6 oscillators.
a) Equilibrium state population for a single oscillator:
I
Boltzmann distribution:
with the energy quanta
the degeneracy factor
and the partition function
= −
(8.31)
= × ; =0 1 2 (8.32)
=
(8.33)
1
1 − −
(8.34)
I Limiting case for high T (transition to classical mechanics): ¿ y
classical
=
(8.35)
since
− ≈ 1 − for → 0 (8.36)
Note that this classical description of vibrations is not so bad for unimolecular reactions,
where we are interested in the kinetic behavior at high temperatures.
b) Equilibrium state population for s oscillators:
y
() → () (8.37)
() = () −
(8.38)
with () being the density of vibrational states at the energy and
=
Y
=1
1
1 − −
(8.39)
and
=3 − 6 (resp. =3 − 5) (8.40)