54
4 The Smoluchowski Model
Fig. 4.2 Model of a chemical reaction indicating that the activated complex is a very shallow
energy well ∼
1
2 k B T at the top of the barrier
where G
‡
f is the Gibbs free-energy difference between the activated complex and
the reactants. To obtain the rate of crossing we divide the mean velocity of crossing
by the distance to be traversed. We assume a crossing will involve a single oscillation
so that the distance involved is the order of 2δ, the length transversed in the time of
a single vibrational period T ,
Rate of barrier crossing =
v
2δ
(4.22)
The period of a classical harmonic oscillator is given by
T = 2π
m
κ
(4.23)
The ground state energy of an oscillator from quantum theory is
0 =
h
4π
κ
m
=
h
2T
(4.24)
where h is the Planck’s constant. The right-hand equality comes from introducing
Eq. 4.23 into Eq. 4.24. Noting our assumption that the depth of the well is ∼
1
2 k B T
we get
4 The Smoluchowski Model
Fig. 4.2 Model of a chemical reaction indicating that the activated complex is a very shallow
energy well ∼
1
2 k B T at the top of the barrier
where G
‡
f is the Gibbs free-energy difference between the activated complex and
the reactants. To obtain the rate of crossing we divide the mean velocity of crossing
by the distance to be traversed. We assume a crossing will involve a single oscillation
so that the distance involved is the order of 2δ, the length transversed in the time of
a single vibrational period T ,
Rate of barrier crossing =
v
2δ
(4.22)
The period of a classical harmonic oscillator is given by
T = 2π
m
κ
(4.23)
The ground state energy of an oscillator from quantum theory is
0 =
h
4π
κ
m
=
h
2T
(4.24)
where h is the Planck’s constant. The right-hand equality comes from introducing
Eq. 4.23 into Eq. 4.24. Noting our assumption that the depth of the well is ∼
1
2 k B T
we get