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

8.3 Generalized Lindemann-Hinshelwood mechanism 186
I
Answer: We are asking for the number of distinguishable possibilities to distribute
quanta between oscillators, (). This is identical to the problem of distributing
balls between boxes, for example for =11and =8:
••• • • • •• • •• (8.45)
That number is identical to the number of distinguishable permutations of balls ( • )
and − 1 walls ( | ),
( + − 1)!
() = (8.46)
!( − 1)!
where ( + − 1)! is the total number of permutations, which is corrected by dividing
through !( − 1)! to account for the number of indistinguishable permutations among
the balls (!) andwalls(( − 1)!) among each other (which give indistinguishable results).
I Answer: We make the following approximation for À : 49
( + − 1)!
!( − 1)!
≈
−1
( − 1)!
(8.49)
I
Question: How can we compute ()?
I
Answer:
() =
()
(8.50)
(1) We consider the energy interval
∆ = (8.51)
at the excitation energy
= × (8.52)
In this energy interval, we have the number of states
∆() = () (8.53)
y
() =
()
∆ ()
≈
∆ = 1 ( + − 1)!
!( − 1)!
(8.54)
49 The approximation
( + − 1)!
≈ −1 (8.47)
!
can be derived using Stirling’s formula (ln ! = ln − ) and the power series expansion of
for || ¿1 (see homework assignment).
ln (1 − ) ≈ (8.48)