Environmental Engineering Reference
In-Depth Information
is in keeping with the
principle of microscopic reversibility
enunciated by Tolman
(1927), which states that
at equilibrium the rate of the forward reaction is the same
as that of the backward reaction
. As an example, let us choose the reaction
A
k
f
k
b
B,
(5.33)
where the forward and backward reactions are both first order. The rate of the forward
reaction is
r
f
=
k
b
[B]. The net rate of
change in [A] is that due to the decrease inA by the forward reaction and the increase
in the same by the reverse reaction. Thus
k
f
[A] and that of the backward reaction is
r
b
=
[
]
d
t
=
d
A
r
A
=−
k
f
[
]−
k
b
[
]
A
B
.
(5.34)
If [A]
0
is the initial concentration of A, then by the mass conservation principle
[
A
]
0
=[
A
]+[
B
]
at all times
(t >
0
)
. Therefore, we have
d
[
]
d
t
=−
A
(k
f
+
k
b
)
[
A
]+
k
b
[
A
]
0
.
(5.35)
This is a first-order ordinary differential equation, which can be easily solved to obtain
k
b
+
.
k
f
e
−
(k
f
+
k
b
)t
k
b
+
[
A
]=[
A
]
0
·
(5.36)
k
f
]
eq
.
As
t
→∞
,
[
A
]→[
A
]
eq
and [B]
→[
B
k
b
k
f
+
[
A
]
eq
=[
A
]
0
,
k
b
(5.37)
k
f
k
f
+
[
B
]
eq
=[
A
]−[
A
]
eq
=[
A
]
0
.
k
b
The ratio [B]
eq
/[A]
eq
is the equilibrium constant of the reaction,
K
eq
. It is important
to note that
[
]
eq
B
k
f
k
b
K
eq
=
]
eq
=
.
(5.38)
[
A
The connection between thermodynamics and kinetics becomes apparent. In practice,
for most environmental processes, if one of the rate constants is known, then the other
can be inferred from the equilibrium constant. It should be noted that whereas the
ratio
k
f
/k
b
describes the final equilibrium position, the sum (
k
f
+
k
b
)
determines how
fast equilibrium is established.
An example of a reversible reaction is the exchange of compounds between soil
and water. Previously, we showed that this equilibrium is characterized by a partition
coefficient
K
sw
. Consider the transfer as a reversible reaction
k
f
k
b
A
water
A
soil
.
(5.39)
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