Biomedical Engineering Reference
In-Depth Information
H
$s þ
NH
$s
%
NH
2
$s þ s
(9.127d)
H
$s þ
NH
2
$s
%
NH
3
$s þ s
(9.127e)
NH
3
$s
%
NH
3
þ s
(9.127f)
þÞ ¼
overall
N
þ
3
H
2
%
2
NH
3
(9.127g)
The adsorption of Nitrogen was found to be the rate-limiting step and all the other steps
are rapid equilibrium steps. As such, the overall rate of reaction is determined by the adsorp-
tion of nitrogen or reaction
(9.127a)
. The rate is given by
r ¼r
N
2
¼ k
N
2
C
2
2
p
N
2
k
N
2
C
2
2
N
s
q
s
q
(9.128)
Noting that
E
0
ad
þEaq
N
RT
E
0
ad
RT
e
E
ad
RT
E
aq
N
RT
k
N
2
¼ k
0
N
2
e
¼ k
0
N
2
e
¼ k
0
N
2
e
(9.129)
E
0
ad
þE
max
DH
0
ad
E
des
RT
e
ð
Emax
E
aÞq
N
k
N
2
¼ k
0
N
2
e
¼ k
0
(9.130)
N
2
e
RT
RT
Combining the constants together,
Eqn (9.130)
is reduced to
E
a
q
N
RT
N
e
ð
E
max
E
a
Þq
N
r ¼ k
N
2
q
k
N
2
q
2
p
N
2
e
2
(9.131)
RT
Temkin isotherm, which was derived from the linear distribution of adsorption heat, is
given by
RT
E
max
ðK
N
2
p
N
2
e
E
max
q
N
z
ln
Þ
(9.132)
RT
Since the nitrogen adsorption is the rate-limiting step and the surface coverage of Nitrogen is
in equilibrium with hydrogen and ammonia via other surface reaction steps. The pressure in
the adsorption isotherm,
p
N
2
, is thus not the actual partial pressure of nitrogen. This virtual
partial pressure of N
2
is evaluated by the partial pressures of hydrogen and ammonia by
p
2
NH
3
p
N
2
p
3
K
P
¼
(9.133)
H
2
Substituting
Eqn (9.133)
into
Eqn (9.132)
, we obtain
ln
!
K
N
2
p
2
NH
3
RT
E
max
e
E
max
q
N
¼
(9.134)
RT
K
P
p
3
H
2
Since the dependence of fractional coverage is dominated by the exponential terms, one can
neglect the change in the power-law terms of the fractional coverage in
Eqn (9.131)
while
applying the Temkin approximation of
q
N
. Thus,
Eqn (9.131)
is approximated by
!
!
E
RT
E
max
E
a
RT
RT
E
max
RT
E
max
K
N
2
p
2
K
N
2
p
2
r ¼ k
N
2
p
N
2
e
E
max
k
N
2
e
E
max
NH
3
K
P
p
3
NH
3
K
P
p
3
(9.135)
RT
RT
H
2
H
2
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