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the porous medium. A similar assumption was made by van Genuchten and
Wierenga (1976) for equilibrium linear and Freundlich-type reactions and by
Selim, Schulin, and Flühler (1987) for selectivity coefficients for homovalent
ion-exchange reactions. Specifically, as
t
→ ∞, that is, when both the dynamic
(or active) sites and the sites in the stagnant region achieve local equilibrium,
Equations 8.23 and 8.24 yield the following expressions. For the active sites
associated with the mobile region:
m
m
m
−ρ
m
=
0
as
t
→ ∞
Θ
k
C
k
S
(8.28)
1
2
or
m
m
S
=
Θ
ρ
k
1
=
ω
as
t
→∞
(8.29)
1
m
m
C
k
2
and for the sites associated with the immobile region we have:
im
im
im
−ρ
im
=
0
as
t
→ ∞
Θ
k
C
k
S
(8.30)
1
2
or
im
im
im
S
=
Θ
ρ
k
1
=
ω
as
t
→∞
(8. 31)
2
im
C
k
2
Here ω
1
and ω
2
represent equilibrium constants for the retention reactions
associated with the mobile and immobile regions, respectively. The for-
mulation of Equations 8.30 and 8.32 are analogous to expressions derived
for the second-order two-site model discussed previously. In this sense,
the equilibrium constants ω
1
and ω
2
resemble the Langmuir coefficients,
with
S
max
as the maximum sorption capacity (Selim and Amacher, 1988).
These equilibrium constants are also analogous to the selectivity coef-
ficients associated with ion exchange reactions (see Selim, Schulin, and
Flühler, 1987).
The dimensionless forms of Equations 8.3, 8.4, 8.23, and 8.24 are
2
Ω
∂
∂
m
+µ
∂
∂
m
=
µ
∂
∂
m
−
∂
∂
m
s
T
c
TP
c
X
c
X
f
−α − µ
(
mim
)
m
cc
k c
(8.32)
s
2
∂
∂
im
+Ω
−
−µ
1
1
f
∂
∂
im
c
T
s
T
=α −−
(
mim
)
im
cc k c
(8.33)
s
∂
∂
im
s
T
=µ
Φ
m
m
−
m
k
c
k
s
(8.34)
1
2
∂
∂
=−µ
Φ
im
s
T
(1
)
1
im
im
−
im
k
c
k
s
(8.35)
2
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