Biomedical Engineering Reference
In-Depth Information
Total site balance leads to
1
¼ q þ q
A
þ q
B
¼ q þ
ðK
C
k
1
þ K
A
k
1
ÞC
A
þ K
C
k
A
C
B
K
A
q þ
ðk
1
þ K
A
k
1
ÞC
B
þ k
B
C
A
(E9-4.36)
B
S
A
S
K
B
q
K
C
k
1
þ K
A
k
1
þ k
1
A
K
C
k
1
þ K
A
k
1
þ k
1
A
B
S
B
S
Thus,
1
q ¼
(E9-4.37)
þ
ðK
A
K
C
k
1
þ k
1
S
þ k
B
K
B
ÞC
A
þðK
C
k
A
K
A
þ k
A
K
B
þ K
A
k
1
S
K
B
ÞC
B
B
1
K
C
k
1
þ K
A
k
1
þ k
1
A
B
S
Substituting
Eqn (E9-4.37)
into
Eqns (E9-4.34) and (E9-4.35)
, we obtain the fractional cover-
ages on the catalyst surface.
The overall rate of reaction:
r ¼r
A
¼ r
1
¼ k
A
C
A
qC
s
k
A
q
A
C
s
(E9-4.38)
That is,
r ¼ k
A
ðC
A
q q
A
=K
A
ÞC
s
(E9-4.39)
Substituting the fractional coverage of A and the fraction of vacant sites,
Eqn (E9-4.40)
renders
"
C
A
ðK
C
k
1
#
þ K
A
k
1
ÞC
A
þ K
C
k
A
C
B
C
A
K
C
C
B
K
C
k
1
B
S
r ¼ k
A
qC
s
¼
þ k
A
qC
s
(E9-4.40)
K
C
k
1
þ K
A
k
1
þ k
1
A
þ K
A
k
1
B
S
B
S
or
C
A
K
C
C
B
K
C
k
1
C
s
þ K
A
k
1
þ k
1
A
B
S
r ¼
(E9-4.41)
þ
ðK
A
K
C
k
1
þ k
1
S
þ k
B
K
B
ÞC
A
þðK
C
k
A
K
A
þ k
A
K
B
þ K
A
k
1
S
K
B
ÞC
B
B
1
K
C
k
1
þ K
A
k
1
þ k
1
A
B
S
which is the rate expression for the first case shown in
Table 9.3
. However, the rate expression
is derived from PSSH, not from a rate-limiting step assumption. The expression
(E9-4.41)
is
quite similar to the three rate expressions based on different rate limiting assumptions,
although the rate constants are different from any one of them. In fact, one can reduce
Eqn
(E9-4.41)
to the three expressions in
Table 9.3
case 1) by assuming
k
1
S
0 and
k
1
B
¼
¼
0 for
k
1
A
0and
k
1
B
adsorption of A being rate limiting;
¼
¼
0 for surface reaction being rate
limiting; and
k
A
¼
0 and
k
1
S
0 for desorption of B being rate limiting. Therefore, PSSH
is more general and carries less degree of approximation/simplifications than the LHHW
approximations.
¼
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