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

8.3 Generalized Lindemann-Hinshelwood mechanism 188
(5) With (empirical) Whitten-Rabinovitch corrections:
() = ( + () ) −1
( − 1)! Q
=1
(8.63)
where () is a correction factor that is of the order of 1 except at very low
energies.
(6) Exact values of () for harmonic oscillators can be obtained by direct state
counting algorithms.
(7) Corrections for anharmonicity can be applied using different means.
8.3.4 Unimolecular reaction rate constant in the low pressure regime
From the master equation analysis above, we obtained the thermal unimolecular reaction
rate constant in the low pressure regime as
0 = X X
−1() [M]
(8.64)
The reduction of the excited state populations compared to the equilibrium (Boltzmann)
distribution is shown in Fig. 8.13. Two cases can be distinguished:
(1) With the strong collision assumption (h∆ iÀ), we can apply the so-called
equilibrium theories which assume that = for ≤ 0 . Thus, the state
population below 0 remains as in the high pressure regime, whereas it is reduced
above 0 .Then,
y
0 = X
X
−1() [M]
= X
Z ∞
0 = [M]
X
−1() [M] = [M] X
(8.65)
0
() (8.66)
Using the expressions for () () and in the classical limit ( ¿ )
derived above and doing the integration by parts, we obain the Hinshelwood
equation for 0 :
µ −1 0 1
0 = [M]
( − 1)! − 0
(8.67)
This equation is usually used to describe experimental data with as a fit parameter
(usually about half the real ).
Since ∝ 05 and = 2 ln , the low pressure activation energy
becomes
0 = 0 − ( − 15) (8.68)
Thus, may be significantly smaller than 0 . This can be understood as
illustrated in Fig. 8.14.