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where
K
ij
represents the affinity of ion
i
over
j
or a separation factor for the
affinity of ions on exchange sites. Rearrangement of Equation 7.5 yields the
following isotherm relation for equivalent fraction of ion 1 as a function of
c
1
:
K
c
12
1
s
=
( 7. 6 )
1
1(
+
K
-1)
c
12
1
and
c
1
relative concentration (dimensionless) and
C
T
(mmol
c
L
-1
) represent
the total solution concentration:
C
C
∑
i
c
=
and
=
C
C
i
T
i
( 7. 7 )
T
i
The respective isotherm equation for ion 2 (i.e.,
s
2
versus
c
2
) can be easily
deduced. This dimensionless isotherm relation is represented in Figure 7.1
for different values of
K
12
. For
K
12
= 1, a linear isotherm relation is produced,
represented by the solid line in Figure 7.1. This clearly illustrates a 1:1 rela-
tionship between relative concentration in solution and that on the adsorbed
phase (i.e.,
s
1
=
c
1
). This also implies that the two ions 1 and 2 each have equal
affinity for the exchange sites. In contrast, for
K
12
different than 1, we have
nonlinear sorption isotherms. Specifically, for
K
12
> 1, sorption of ion 1 is
preferred and the isotherms are convex. For
K
12
< 1, sorption affinity is appo-
site and the isotherms are concave. Examples of homovalent ion exchange
isotherms are illustrated in Figure 7.2 for Ca-Mg in soils and clay mineral
1.0
K
12
= 10
0.8
2
1
0.6
1/2
0.4
1/10
0.2
0
0
0.2
0.4
0.6
0.8
1.0
Relative Concentration
FIGURE 7.1
Exchange isotherms as affected by different values of selectivity coefficients (
K
12
). (From H. M.
Selim, and M. C. Amacher. 1997.
Reactivity and Transport of Heavy Metals in Soils
. Boca Raton, FL:
CRC Press. With permission.)
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