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

8.4 The specific unimolecular reaction rate constants () 191
8.4 The specific unimolecular reaction rate constants k (E)
We may intuitively expect that the specific uinimolecular reaction rate constants ()
depend on the energy content (i.e., the excitation energy ) of the energetically activated
molecules molecules A ∗ . More precisely, we will also have to take into account
the dependence of the specific rate constants on other conserved degrees of freedom,
in particular total angular momentum (⇒ ()), and perhaps other quantities
(symmetry, components of if -mixing is weak, (⇒ (; ; ))).
8.4.1 Rice-Ramsperger-Kassel (RRK) theory
I RRK model: In order to derive an expression for the specific rateconstants (),
werealizethatitisnotenoughthatthemoleculesareenergizedtoabove 0 . Indeed,
for the reaction to take place, the energy also has to be in the right oscillator, namely
in the RC. In particular, for the reaction to take place, of the total excitation energy ,
the amount † has to be concentrated in the RC such that † ≥ 0 . Thus, we can
write the reaction scheme as
∗ †
→ † → (8.74)
where † denotes those excited molecules that have enough energy in the RC to overcome
0 . We assume that once this critical configuration ( † ) is reached, the molecules
will immediately cross the TS and go to the product side.
I Probabilities of A † and A ∗ : The probability of † compared to ∗ can be derived
using statistical arguments by considering the ratio of the density of states. We consider
amoleculewith
• oscillators,
• = quanta which have to be distributed over the oscillators,
• a threshold energy of 0 such that a minimum of = 0 quanta have to be
concentrated in the RC and only − quanta can be distributed freely,
• À and − À .
The probability of † relative to ∗ is given by
¡ †¢
( ∗ ) = ¡ †¢
( ∗ )
(8.75)
where
( ∗ ) ∝ ∗ () = 1 ( + − 1)!
(8.76)
!( − 1)!
and, since in the critical configuration only − quanta can be distributed freely,
¡ †¢ ∝ † () = 1 ( − + − 1)!
( − )! ( − 1)!
(8.77)