Environmental Engineering Reference
In-Depth Information
The overall equilibrium constant for the formation of complex Z is
[
][
]
k
f
k
b
=
Z
Y
K
eq
=
.
(5.116)
[
X
][
A
]
If we start with initial concentrations [X]
0
and [A]
0
, and [Z] is the concentration of
the intermediate complex, then
[
X
]=[
X
]
0
−[
Z
]
and
[
A
]=[
A
]
0
−[
Z
]
. We can now
obtain an expression that is only dependent on [Z] and [Y]:
[
Z
][
Y
]
K
eq
=
]
)
.
(5.117)
(
[
X
]
0
−[
Z
]
)(
[
A
]
0
−[
Z
If, for example,
[
A
]
0
[
X
]
0
, then
K
eq
[
]
0
[
]
0
X
A
[
Z
]=
(5.118)
K
eq
[
A
]
0
+[
Y
]
and the rate of product formation is
K
eq
[
X
]
0
[
A
]
0
k
[
k
=
][
]=
]
[
]
r
Z
W
W
.
(5.119)
K
eq
[
A
]
0
+[
Y
AtmosphericchemicalreactionsfollowtheaboverateexpressionwhenYisabsentand
W is either O
2
or N
2
(see Example 5.7). In those cases where
K
eq
[
,
r
=
k
[X]
0
[W] and the rate is linear in [X]
0
and independent of [A]
0
. In many environmental
systems where acid or base catalysis prevails, the condition of interest is
K
eq
[A]
0
[
A
]
0
[
Y
]
. The rate is then proportional to the first power in [A]
0
.
A different reaction rate will ensue if we start with a high catalyst concentration,
[X]
0
[
Y
]
A
]
0
. We then have
[
X
]=[
X
]
0
,
[
A
]=[
A
]
0
−[
Z
]
, and hence the following
rate:
K
eq
[
X
]
0
[
A
]
0
k
r
=
.
(5.120)
K
eq
[
X
]
0
+[
Y
]
The catalyst concentration enters the rate expression in a distinctly nonlinear fashion.
If the second reaction (dissipation of the complex Z) is extremely fast, then the
rate of dissipation can be handled using a pseudo-steady-state approximation. Thus,
d
[
]
d
t
=
Z
k
[
0
=
k
f
[
X
][
A
]−
k
b
[
Z
][
Y
]−
Z
][
W
]
.
(5.121)
, [Z] is small, and [Z]
2
is even smaller,
Since
[
X
]=[
X
]
0
−[
Z
]
,
[
A
]=[
A
]
0
−[
Z
]
k
f
[
X
]
0
[
A
]
0
[
Z
]=
(5.122)
k
f
(
[
X
]
0
+[
A
]
0
)
+
k
b
[
Y
]+
k
[
W
]
and the rate is
]
0
k
f
(
[
X
]
0
+[
A
]
0
)
+
k
b
[
Y
]+
k
[
W
]
k
f
[
X
]
0
[
A
k
[
r
=
W
]
.
(5.123)
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