Environmental Engineering Reference
In-Depth Information
For species A we obtain
k
2
k
f1
k
b2
+
d
[
]
d
t
=−
A
B
∗
]=−
k
f1
[
]+
k
b1
[
[
]
A
A
(5.50)
k
2
and for species C we have
k
2
k
f1
k
b1
+
[
d
]
d
t
=
[
C
B
∗
]=
k
2
[
A
]
.
(5.51)
k
2
These expressions can be readily integrated with the appropriate boundary conditions
to obtain the concentrations of species A and C. If
k
b1
k
2
,
d
[
]
d
t
=
k
2
K
eq
[
C
A
]
,
(5.52)
where
K
eq
=
k
f1
/k
b1
is the equilibrium constant for the first step in the reaction. This
occurs if the intermediate B formed is converted to A much more rapidly than to C.
The rate of the reaction is then controlled by the value of
k
2
. Thus B
→
C is said to be
the
rate-determiningstep
.Another situation is encountered if
k
2
k
b1
, in which case
d[C]/d
t
k
f1
[A]. This happens if the intermediate is rapidly converted to C. Then
k
f1
determines the rate. The
rate-determining step
is then said to be the equilibrium
reaction A
=
B.
The following are illustrations of how the concepts of integrated rate laws can be
used to analyze particular environmental reaction schemes. An example from water
chemistry and another one from air chemistry are chosen.
E
XAMPLE
5.6 S
OLUTION OF
I
NORGANIC
G
ASES IN
W
ATER
A reaction of environmental relevance is the dissolution of an inorganic gas (e.g., CO
2
)
in water. The reaction proceeds in steps. The important step is the hydration of CO
2
followed by dissolution into HCO
3
species in water. Stumm and Morgan (1996) and
Butler (1982) have analyzed this reaction, and we shall adopt their approach here. In
analyzing these reactions, we shall consider water to be in excess such that its concen-
tration does not make any contribution toward the overall rate. The overall hydration
reaction can be written as
k
f
k
b
HCO
3
(
aq
)
+
H
+
(
aq
)
,
CO
2
(
aq
)
+
H
2
O
(5.53)
with
k
f
=
0.03 s
−
1
and
k
b
=
7
×
10
4
mol/L/s. Since H
2
O concentration is constant, we
shall consider the functional reaction to be of the form A
k
f
k
b
B
+
C, where A represents
CO
2
, B represents HCO
3
, and C represents H
+
. Laidler (1965) provides the integrated
rate law for the above reaction:
ln
[
A
]
0
[
B
]
eq
+[
B
]
(
[
A
]
0
−[
B
]
eq
)
[
A
]
0
(
[
B
]
eq
−[
B
]
)
[
B
]
eq
2
[
A
]
0
−[
B
]
eq
=
k
f
t
,
(5.54)
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