9.4 Application to Chemical Kinetics
501
K
‡
eq =
k f
k b
,
in which k f and k b are the rate constants for the forward and backward reactions.
Finally, we know from statistical mechanical considerations [see Eq. (9.2.12)] that
we may also write K
‡
eq in terms of appropriate canonical partition functions as
K
‡
eq =
Z
(AB) ‡
e −βE ‡
Z(A)Z(B)
,
(9.4.2)
in which E ‡ is the energy required to form the activated complex (AB) ‡ .
The application of basic chemical kinetics arguments to step (2) in the above
reaction sequence gives rise to an expression for the net rate, namely,
Net rate =
d[products]
dt
= k
(AB)
‡
= kK
‡
eq [A][B] ,
while from the net reaction, the net rate may be expressed as
d[products]
dt
= k 2 [A][B] .
The equivalence of these two expressions then requires that the effective bimolecular
rate constant k 2 be given by
k 2 = kK
‡
eq .
(9.4.3)
Now let us consider the breakup of (AB) ‡ to give the products of the reaction as
occurring when one of the vibrational modes of what is referred to as the ‘activated
complex’ (AB) ‡ receives a large surge of energy that causes fragmentation of the
complex into products. If we can consider the complex as being made up of two
stable fragments weakly bound together, then one of the vibrational modes of the
complex will approximate diatom stretching, and we can ascribe the force constant
for the vibrational motion with the bound energy for that mode.
It is really only at this point that statistical mechanics enters. Let us designate the
canonical partition function for chemical species α by Z(α). We have seen that there
are two basic principles governing the distribution of molecules amongst various
states, these being
(i) tendency to maximize entropy: all states of a system having equal energy have
equal probability of being achieved;
(ii) tendency to minimize energy: two states having different energies E 1 and E 2
will have a population ratio of
p 1
p 2
=
ω 1 e −βE 1
ω 2 e −βE 2
=
ω 1
ω 2
e
−β(E 1 −E 2 ) .
501
K
‡
eq =
k f
k b
,
in which k f and k b are the rate constants for the forward and backward reactions.
Finally, we know from statistical mechanical considerations [see Eq. (9.2.12)] that
we may also write K
‡
eq in terms of appropriate canonical partition functions as
K
‡
eq =
Z
(AB) ‡
e −βE ‡
Z(A)Z(B)
,
(9.4.2)
in which E ‡ is the energy required to form the activated complex (AB) ‡ .
The application of basic chemical kinetics arguments to step (2) in the above
reaction sequence gives rise to an expression for the net rate, namely,
Net rate =
d[products]
dt
= k
(AB)
‡
= kK
‡
eq [A][B] ,
while from the net reaction, the net rate may be expressed as
d[products]
dt
= k 2 [A][B] .
The equivalence of these two expressions then requires that the effective bimolecular
rate constant k 2 be given by
k 2 = kK
‡
eq .
(9.4.3)
Now let us consider the breakup of (AB) ‡ to give the products of the reaction as
occurring when one of the vibrational modes of what is referred to as the ‘activated
complex’ (AB) ‡ receives a large surge of energy that causes fragmentation of the
complex into products. If we can consider the complex as being made up of two
stable fragments weakly bound together, then one of the vibrational modes of the
complex will approximate diatom stretching, and we can ascribe the force constant
for the vibrational motion with the bound energy for that mode.
It is really only at this point that statistical mechanics enters. Let us designate the
canonical partition function for chemical species α by Z(α). We have seen that there
are two basic principles governing the distribution of molecules amongst various
states, these being
(i) tendency to maximize entropy: all states of a system having equal energy have
equal probability of being achieved;
(ii) tendency to minimize energy: two states having different energies E 1 and E 2
will have a population ratio of
p 1
p 2
=
ω 1 e −βE 1
ω 2 e −βE 2
=
ω 1
ω 2
e
−β(E 1 −E 2 ) .