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

8.4 The specific unimolecular reaction rate constants () 197
(2) Number of states () and ‡ ( − 0 ): Since () = () , the
number of states from =0to can be obtained by integration,
a) RRK expression:
y
() =
Z
0
() =
() =
Z
0
Z
0
() (8.98)
−1
( − 1)! () =
() =
Z
1 1
( − 1)! () −1
0
(8.99)
!() (8.100)
b) Allowing for different frequencies, we modify this expression to
() =
! Q
=1
(8.101)
c) As above, a correction for zero-point energy gives
() = ( + )
! Q
=1
(8.102)
d) With the Whitten-Rabinovitch correction, this becomes
() = ( + () )
! Q
=1
(8.103)
e) Correction for → 0: At =0,wemusthaveatleastonestate,i.e.,
‡ () =1+ ( + () )
! Q
=1
− ( (0) )
! Q
=1
(8.104)
(3) Application to unimolecular reactions:
a) At the TS, we have only − 1 oscillators since the RC has to be excluded.
Furthermore, the states range from the zero-point energy 0 of the TS to
, while we count from the zero-point of the reactants. y
‡ ( − 0 )=1+
³
− 0 + ‡ ( − 0 ) ‡ ´−1
−
³
‡ (0) ‡ ´−1
( − 1)! Q −1
=1 ‡ ( − 1)! Q −1
=1 ‡
(8.105)
This expression gives the correct value of ‡ ( − 0 )=1at = 0 .