Competitive Sorption and Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 195

matrix. In a standard mass action formulation, the exchange reaction for two
competing ions i and j , having valencies v i and v j , respectively, may be writ-
ten as:

ν
ν

j

()
()

* a

a

= ( 7. 2)

T

K ij i

*

where T K ij denotes the thermodynamic equilibrium constant and α * (omit-
ting the subscripts) are the ion activity in soil solution and on the exchanger
surfaces, respectively. Based on Equation 7.2, one can denote the parameter
v K ij as:

T

K

ij

v

=

K

( 7. 3)

ij

ν
ν

j

()
()

ζ
ζ

j

i

j

where v K ij is the Vanselow selectivity coefficient and ζ the activity coefficient
on the soil surface. It is recognized that in soils, ion exchange involves a wide
range of thermodynamically different sites. As a result, a common practice
is to ignore the activity coefficients of the adsorbed phase (ζ) in general. In
addition, the much simpler Gaines and Thomas (1953) selectivity coefficient
G K ij may be used, where:

vi

ν i ν j

( γ

γ

)

s
C

s
C

j j

i

G

=

K ( 7. 4) ij

ν j

( )

i j

i

This formulation is more conveniently incorporated into the dispersion-
convection transport equation (7.1). In Equation 7.4, γ i and γ j are dimension-
less solution-phase activity coefficients where a = i i γ . In addition, the

terms s i and s j are dimensionless, representing the solid-phase concentra-
tions expressed in terms of equivalent fraction, s i i =Ω . Here the term Ω
is the cation exchange (or adsorption) capacity of the soil (mmol c kg -1 soil)
and S i is the concentration (mmol c kg -1 ) of adsorbed-phase soil. Although Ω
is often assumed as invariant, it is recognized that Ω has been observed to
be dependent on soil pH and the counterions present in the soil. Moreover,
there are several other ways to express the adsorbed-phase concentration on
a fractional basis, including as a molar rather than as an equivalent value.
For the simple case of binary homovalent ions, that is, v i = v j = v , and assum-
ing similar ion activities in the solution phase (γ i = γ j = 1), Equation 7.3 can be
rewritten as:

C

=

s
C

s
C

j

i

/

K

( 7. 5 )

ij

i j