Geology Reference
In-Depth Information
Fig. 3.2 Gibbs energy
change during a reaction
D
G
m
G
m
G
m
+
D
G
m
REACTION COORDINATE
where the fc
ðÞ
are now taken to represent an amount or concentration of potential
reaction species subject to fluctuations of energy, m is the frequency of the fluc-
tuations (the suffixes +,- distinguish the forward and reverse cases, respectively),
and exp
G
m
=
R
ð Þ
is the probability that a given fluctuation will reach G
m
:
Under
the assumptions that the ''attempt'' rates f
þ
c
ðÞ
m
þ
and f
c
ðÞ
m
in the two direc-
tions are the same (they must be so at equilibrium, when DG
m
¼
0
;
) the net rate of
reaction
n
¼
n
þ
n
will be given by
n
¼
n
þ
1
e
DG
m
ð
3
:
10a
Þ
RT
(See Lasaga
1981
for a more satisfying derivation of
3.10
using transition state
theory). In the case DG
RT
;
we then have
n
n
þ
DG
m
RT
¼
fc
ðÞ
DG
m
me
G
RT
ð
3
:
10b
Þ
RT
or substituting (
3.8
) and dropping the suffixes, the rate coefficient, now also the
specific rate, is given by
n
fc
ðÞ
DG
m
me
G
RT
k
¼
ð
3
:
11
Þ
RT
This case thus corresponds to the ''near to equilibrium'' linear case discussed
earlier in connection with (
3.4
).
The above argument is put on a somewhat sounder basis in transition state
theory, although still under the assumption that there is a thermodynamically
definable intermediate state, known as the activated complex, which is in equi-
librium with the reactants in a steady-state reaction. On statistical mechanical
grounds (Atkins
1978
, Chap. 27; Christian
1975
, Chap. 3; Flynn
1972
, Chap. 7;
