Environmental Engineering Reference
In-Depth Information
ifthereactionsarecomplexandoccurinseveralsteps.Figure5.3representsthechange
in concentrations of reactantA and product B for a first-order reaction such asA
→
B.
The derivative
d[A]/d
t
is the rate of disappearance ofA with time (this is also equal
to d[B]/d
t
at any time
t
). Since the reaction is first order in A, we have
−
[
]
d
t
=−
[
]
d
t
=
1
ν
A
d
A
d
A
r
A
=
[
]
k
A
,
(5.28)
since
ν
A
=−
1. The rate is directly proportional to [A]. Since at
t
=
0,
[
A
]=[
A
]
0
,a
constant, one can integrate the above expression to get
ln
[
A
]
=−
kt
.
(5.29)
[
A
]
0
→
If the reaction is
n
th order in A, that is,
n
A
B, the integrated rate law is
1
[
1
n
1
·
n
−
1
−
=
kt
.
(5.30)
−
1
A
]
n
−
1
0
[
A
]
Thusforafirst-orderreactionaplotofln
]
0
versus
t
willgive
k
,therateconstant.
Similar plots can be made for other values of
n
. By finding the most appropriate
integrated rate expression to fit a given data, both
n
and
k
can be obtained.
If the reaction involves two or more components, the expression will be different.
For example, if a reaction is second order (first order in A and first order in B),
A
[
A
]
/
[
A
+
B
→
products, then
−
d
[
A
]
/
d
t
=
k
[
A
][
B
]
.If
x
is the concentration of A that has
reacted, then
[
A
]=[
A
]
0
−
x
and
[
B
]=[
B
]
0
−
x
. Hence
d
[
]
d
t
=
A
d
x
d
t
=
−
k(
[
A
]
0
−
x)(
[
B
]
0
−
x)
.
(5.31)
[A
0
]
[B]
[I]
[A]
=
[A
0
]
-
[B]
t
/
min
FIGURE 5.3
Change in concentrations of A and B for the first-order reaction A
→
B.
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