Second-Order Transport Modeling - Transport and Fate of Chemicals in Soils: Principles and Applications - page 160

terms of activities rather than concentrations. However, we use the implicit
assumption that solution-phase ion activity coefficients are constant in a
constant ionic strength medium. Moreover, the solid-phase ion activity coef-
ficients are assumed to be incorporated in the selectivity coefficients (ω 1 and
ω 2 ) as in ion-exchange formulations.
Incorporation of Equations 6.1 to 6.4 into Equations 6.9 and 6.10 and further
rearrangement yields the following expressions for the amounts retained by
type 1 and type 2 sites at t → ∞,

C C

S ω

+ ω

S ω

+ ω

1 1 2 1

= ,and =

(6.11)

( S ) 1 C ( S ) 1 C

max 1 1 max 2 2

Therefore, the total amount sorbed in the soil S (= S 1 + S 2 ), is

S C C

ω
+ ω

ω
+ ω

1 2

= F + (1 − F ) (6.12)

S 1 C 1 C

max 1 2

Equation 6.12 is analogous to the two-site Langmuir formulation where
the amount sorbed in each region is clearly expressed. Such Langmuir for-
mulations are commonly used to obtain independent parameter estimates
for S max and the affinity constants ω 1 and ω 2 .
Let us now consider the case where only one type of active sites is domi-
nant in the soil system. In a similar fashion to the formulations of Equations
6.9 and 6.10, the kinetics of the reaction can be described by the following
equation (Murali and Aylmore, 1983):

ρ ∂

S
t ∂ =Θφ−ρ

C S (6.13)

k k f b

Here k f and k b (h -1 ) are the forward and backward retention rate coeffi-
cients and S is the total amount of solute retained by the soil matrix surfaces.
This formulation is often referred to as the kinetic Langmuir equation. In
fact, Equation 6.13 at equilibrium obeys the widely recognized Langmuir
isotherm equation:

S ω

+ω

C

=

(6.14)

S 1 C

max

where ω = (Θ k f /ρ k b ) is equivalent to that of Equations 6.9 and 6.10. For a dis-
cussion of the formulation of the kinetic Langmuir equation see Rubin (1983).
It should be recognized that the unfilled or vacant sites (ϕ) in Equations 6.7,
6.8, and 6.13 are not strictly vacant. They are occupied by hydrogen, hydroxyl,