Environmental Engineering Reference
In-Depth Information
Box 7.6.1
(
Continued
)
will reach another cavity and if it has a negative velocity it will fall back to the cavity
from which it came. Combining the probability to be on top of the barrier with the
positive average velocity,
|ν
a
|
, of the Maxwell distribution gives us a hopping rate:
1/ 2
()
()
exp
−
Fz
* /
kT
1
8
kT
(
)
B
B
k
cavity
→
cavity
=
1
2
2
m
π
∫
exp
Fz kTdz
/
−
B
cavity
Combining all these approximations, we get the hopping rate for our model
membrane:
L
exp
−
U
/
k T
1
(
)
wy
w
B
k
cavity
→
cavity
=
ν
1
2
a
2
2
L
exp
−
U
/
k T
pz
C
B
From the hopping rate we can get the self-diffusion coeffi cient, using our random
walk model:
L
exp
−
U
/
k T
1
4
wy
w
B
s
2
D
=ν
L
a
z
2
L
exp
−
U
/
k T
pz
C
B
From this hopping rate, we can get the self-diffusion coeffi cient using
our random walk model, derived in Section 7.4:
L
exp
−
U
/
k T
1
4
wy
w
B
s
2
D
=ν
L
a
z
2
L
exp
U
/
k T
−
pz
C
B
Permeability
To get the permeance of our membrane we combine the resulting
expression for the diffusion coeffi cient with the expression for the Henry
coeffi cient:
2
1
V
V
L
c
−
UkT
/
w
−
UkT
/
cz
−
UkT
/
H
=
e
CB
+
e
≈
e
cB
wB
kT V
V
LLkT
B
y
z
B
This gives for the permeance of our membrane:
2
L
exp
−
U
/
k T
1
L
wy
w
B
−
UkT
/
s
2
cz
P=DH
′
≈
ν
L
e
CB
a
z
4
2
LLk T
L
exp
U
/
k T
−
yzB
cz
C
B
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