Environmental Engineering Reference
In-Depth Information
Marcus also derived the following relationship between individual ionic interactions
relating
k
AB
to
k
AA
and
k
BB
(k
AB
k
BB
k
AB
f )
1
/
2
k
AB
=
(5.180)
(
1
/
4
)(
ln
K
AB
)
2
/
ln
(k
AA
k
BB
/Z
2
)
.
Z
is the collision frequency between
uncharged A and B (
≈
10
12
L/mol s). From the expression for
k
AB
, we can obtain
where ln
f
=
G
AB
1
2
1
2
RT
ln
f
.
G
AB
=
G
AA
+ Δ
G
BB
+ Δ
Δ
Δ
−
(5.181)
G
AB
∼
Formostreactions
0.Notethat
k
AA
and
k
BB
aresodefinedthattheyrepresent
the following redox reactions
Δ
A
red
A
red
+
A
ox
+
A
ox
,
B
ox
+
B
red
B
red
+
B
ox
.
(5.182)
The above equations represent
self-exchangereactions
. The overall redox process has
K
AB
as the equilibrium constant
K
AB
=
[
A
red
]+[
B
ox
]
.
(5.183)
[
A
ox
][
B
red
]
Since for one-electron exchange reactions we have already seen that
(E
HA
−
E
HB
)
0.059
pe
A
−
pe
B
,
ln
K
AB
=
=
(5.184)
we can compare the equilibrium constant calculated using Marcus theory to that deter-
mined experimentally. Good agreement is generally observed between the observed
and predicted values, lending validity to the Marcus relationship.
The limiting case of the Marcus relationship leads to a simple LFER between
k
AB
and
K
AB
for OS electron transfer reactions of the type A
ox
L
+
B
ox
. If a plot of ln
k
AB
versus ln
K
AB
is made, a linear relationship with slope of
0.5 is observed, provided
f
+
B
red
→
A
red
L
G
AB
is small representing near equilibrium
conditions. For very endergonic reactions, the slope is
≈
Δ
1 and
1. Several examples of such
relationshipsinenvironmentalreactionshavebeenestablished(Wehrli,1990).Several
auto-oxidation reactions of interest in environmental science were considered by
Wehrli (1990) and data tabulated for both
k
AB
and
K
AB
for reactions of the type
A
red
+
≈
O
2
. Figure 5.17 is a plot showing the unit slope for the LFER
involving different redox couples.
It is appropriate at this stage to summarize the various LFERs that we have dis-
cussed so far for the prediction of rate constants and equilibrium constants in a variety
of contexts. These are compiled in Table 5.7. Armed with a knowledge of these
categories of LFERs, one should be able to predict the rates of many common environ-
mental reactions and/or the equilibrium constants for various partitioning processes.
This can be especially useful if only few values are available within a group.
O
2
→
A
ox
+
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