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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