Environmental Engineering Reference
In-Depth Information
The integrated rate law is
1
[
ln
[
B
]
0
(
[
A
]
0
−
x)
=
kt
.
(5.32)
A
]
0
−[
B
]
0
[
A
]
0
(
[
B
]
0
−
x)
A plot of the term on the left-hand side versus
t
would give
k
as the slope.
Physical chemists have investigated a large variety of possible kinetic rate expres-
sions over the years and the integrated rate laws have been tabulated in the literature
(Laidler, 1965; Moore and Pearson, 1981). Table 5.1 lists some of the rate laws most
frequently encountered in environmental engineering.
An important parameter that is useful in analyzing rate data is the
half-life of a
reactant, t
1
/
2
. This is defined as the time required for the conversion of one half of
the reactant to products. For a first-order reaction this is
(
ln 2
)/k
and is independent
of [A]
0
. For a second-order reaction the half-life is 1/(
k
]
0
)
and is inversely propor-
tional to [A]
0
. Similarly, for all higher-order reactions appropriate half-lives can be
determined.
[
A
TABLE 5.1
Integrated Rate Laws Encountered in Environmental Systems
Reaction Type
Order Rate Law
t
1
/
2
A
→
B
+···
0
kt
=
x
[
A
]
0
/
2
k
1
kt
=
ln
(
[
A
]
0
/
[
A
]
0
−
x)
(
ln 2
)/k
x)
n
−
1
n
−
1
0
(
2
n
−
1
n
−
1
0
≥
2
kt
={
1
/n
−
1
}{
1
/(
[
A
]
0
−
−
1
/(
[
A
]
}
−
1
)/(n
−
1
)k
[
A
]
[
B
]
0
−[
A
]
0
·
ln
[
A
]
0
(
[
B
]
0
−
x)
1
A
+
B
→
C
+
D
+···
kt
=
1
/k
[B]
0
[
B
]
0
(
[
A
]
0
−
x)
[
A
]
0
·
ln
;
k
f
[
A
]
eq
[
B
]
eq
[
A
]−[
A
]
eq
A
B
k
b
t
=
=
k
b
K
eq
]
eq
)
·
ln
[
B
]
(
[
A
]
0
−
2
[
B
]
eq
)
+[
A
]
0
[
B
]
eq
[
B
]
eq
A
+
B
C
+
D
k
f
t
=
2
[
A
]
0
(
[
A
]
0
−[
B
[
A
]
0
(
[
B
]
eq
−[
B
]
)
k
1
−−→
B
k
2
]
0
e
−
k
1
t
A
−→
X
[
A
]=[
A
e
−
k
1
t
−
e
−
k
2
t
[
B
]=
[
A
]
0
k
1
k
2
−
k
1
k
2
(
1
−
e
−
k
1
t
)
−
k
1
(
1
−
e
−
k
2
t
)
]
0
k
2
−
k
1
[
A
[
C
]=
k
1
−−→
C
[
A
]=[
A
]
0
e
−
(k
1
+
k
2
)t
A
1
−
e
−
(k
1
+
k
2
)t
]
0
k
1
+
k
2
k
1
[
A
k
2
−−→
D
A
[
C
]=
1
−
e
−
(k
1
+
k
2
)t
k
2
[
A
]
0
k
1
+
k
2
[
D
]=
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