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

8.3 Generalized Lindemann-Hinshelwood mechanism 187
(2) Using the approximation
( + − 1)!
!( − 1)!
≈
−1
( − 1)!
(Eq. 8.49) for À gives
() = 1 −1
( − 1)!
(8.55)
(3) Inserting = into (), weobtain
() = 1 () −1
( − 1)!
(8.56)
(4) Result:
() =
−1
( − 1)! () (8.57)
d) Notes:*
(1) The above derivation may seem artificial because in a real molecule not all oscillators
are identical. However, the derivation can be easily generalized.
(2) For instance, we can start from the number of states of the first oscillator 1 ( 1 )
and convolute this with the density of states 2 () of the second oscillator
2 ( 2 )= 2 ( − 1 )= 1
(8.58)
and compute the total (combined) density of states for two oscillators by convolution
according to
() =
Z
etc. up to oscillators. The result is
0
1 ( 1 ) 2 ( − 1 ) 1 (8.59)
() =
−1
( − 1)! Q
=1
(8.60)
(3) We can also use inverse Laplace transforms: Since is the Laplace transform
of () according to
=
Z ∞
0
() − = L [ ()] (8.61)
we can determine () from (which we know) by inverse Laplace transformation.
(4) With corrections for zero-point energy:
() =
( + ) −1
( − 1)! Q
=1
(8.62)