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

8.4 The specific unimolecular reaction rate constants () 196
I RRKM expression with angular momentum conservation: Eq. 8.90 can be generalized
to include other conserved quantities, e.g., total angular momentum,
() = ‡ ( − 0 )
()
(8.91)
I Relation between RRKM and RRK theory: To elucidate the relation between Eqs.
8.86 and 8.90, we look at () and ( − 0 ) more closely.
(1) Density of vibrational states ():
() =
number of states in energy interval
energy interval
=
()
(8.92)
a) RRK result for the number of indistinguishable permutations of quanta
over oscillators; = À :
y
() = 1 ( + − 1)!
!( − 1)!
() =
≈ 1 −1
( − 1)! = 1 () −1
( − 1)!
(8.93)
−1
( − 1)! () (8.94)
b) It is relatively easy to show that for oscillators with different frequencies, the
expression for () becomes
() =
−1
( − 1)! Q
=1
(8.95)
c) With corrections for zero-point energy:
() =
( + ) −1
( − 1)! Q
=1
(8.96)
d) With (empirical) Whitten-Rabinovitch correction:
() = ( + () ) −1
( − 1)! Q
=1
(8.97)
where () is a correction factor that is of the order of 1 except at very
low energies.
e) Exact values of () for harmonic oscillators can be obtained by direct
state counting algorithms. Corrections for anharmonicity can be applied
using different means.