Environmental Engineering Reference
In-Depth Information
are needed to describe the ion-exchange process. Equilibrium can be useful to predict
the maximum amount of separation that can be achieved for a particular solution and
resin.
8.6.1
Applying the law of mass action
The generalized form of Equation (8.2) for cation exchange between two ions (one (
A
)in
solution, and one (
B
) originally in the resin) is:
z
A
B
z
B
z
B
A
z
A
z
B
A
z
A
z
A
B
z
B
+
↔
+
,
(8.6)
where
z
A
and
z
B
are the valence and charge of ions
A
and
B
. Since this reaction is reversible,
we can write an expression for the thermodynamic equilibrium constant:
(
a
A
)
z
B
(
a
B
)
z
A
(
a
B
)
z
A
(
a
A
)
z
B
.
K
a
=
(8.7)
It is often difficult to determine the activity coefficients for the resin phase while the
activity coefficients are approximately 1 for dilute solutions. So, a selectivity coefficient
K
c
is often used:
(
C
A
)
z
B
(
C
B
)
z
A
(
C
B
)
z
A
(
C
A
)
z
B
.
(
K
c
)
B
=
(8.8)
For a binary system, this coefficient can be written in terms of ion fractions
C
C
z
A
−
z
B
(
y
A
)
z
B
(
x
B
)
z
A
(
y
B
)
z
A
(
x
A
)
z
B
.
(
K
c
)
B
=
(8.9)
A distribution factor can be defined as
C
A
C
A
=
y
A
C
x
A
C
.
m
A
=
(8.10)
The separation factor
α
is:
y
A
/
x
A
y
A
x
B
y
B
x
A
,
α
AB
=
x
B
=
(8.11)
y
B
/
where
y
represents the resin phase and
x
represents the solution (fluid) phase.
For univalent exchange, Equation (8.2),
z
A
=
z
B
=
1. Equation (8.8) becomes
y
A
x
B
y
B
x
A
=
α
AB
.
(
K
c
)
B
=
(8.12)
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