Sorption Kinetics - Transport and Fate of Chemicals in Soils: Principles and Applications - page 41

(i.e., as t → ∞ ), when the rate of retention approaches zero, Equation 2.2
yields:

θ
ρ

k
k

b f

SKCwhere = K =

(2.3)

f f

b

Equation 2.3 is analogous to the Freundlich equilibrium equation where K f
is the solute partitioning coefficient (cm 3 /g). Therefore, one may regard the
parameter K f as the ratio of the rate coefficients for sorption (forward reac-
tion) to that for desorption or release (backward reaction).
The parameter b is a measure of the extent of the heterogeneity of sorption
sites of the soil matrix. In other words, sorption sites have different affinities
for heavy metal retention by matrix surfaces, where sorption by the highest-
energy sites takes place preferentially at the lowest solution concentrations.
For the simple case where b = 1, we have the linear form:

θ
ρ

k
k

f

SKCwhere = K =

(2.4)

d d

b

The parameter K d is the solute distribution coefficient (cm 3 /g) and of similar
form to the Freundlich parameter K f . There are numerous examples of cation
and heavy metal retention, which were described successfully using the lin-
ear or the Freundlich equation (Sparks, 1989; Buchter et al., 1989). The lack of
nonlinear or concentration-dependent behavior of sorption patterns as indi-
cated by the linear case of Equation 2.1 is indicative of the lack of heterogene-
ity of sorption-site energies. For this special case, sorption-site energies for
linear sorption processes of heavy metals may be best regarded as relatively
homogeneous. A partial list of kinetic models is presented in Table 1.1 of
Chapter 1.

2.1.2 Second-Order and Langmuir Kinetics

An alternative to the above first- and nth-order models is that of the second-
order kinetic approach. Such an approach is commonly referred to as
Langmuir kinetics and has been used for predictions of phosphorus reten-
tion and heavy metals (Selim and Amacher, 1997). Based on second-order
formulation, it is assumed that the retention mechanisms are site specific
where the rate of reaction is a function of the solute concentration present in
the soil solution phase ( C ) and the number of available or unoccupied sites ϕ
(μg/g soil), by the reversible process:

k
k

+φ →

←

f
b

C S

(2.5)