Agriculture Reference
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also be mentioned that expressions similar to the above model have been
used to describe mass transfer between mobile and immobile water in
porous media (Coats and Smith, 1964), as well as chemical kinetics (Parker
and Jardine, 1986).
Other kinetic expressions have also been employed to describe the kinetic
exchange of ions in mineral and soils. The parabolic diffusion model is based
on the assumption of diffusion controlled rate-limited processes in media
with homogeneous particle size. The parabolic diffusion equation was
derived from Fick's second law of diffusion in a radial coordinate system.
S
S
4
Dt
r
Dt
r
m
m
=
( 7.11)
2
2
()
π
eq
i
where r is the average radius of soil or mineral particle, and D m is the molecu-
lar diffusion constant. The Elovich model is anther empirical kinetic reten-
tion model, which may be expressed as:
S
t
i
BS i
=
ae
( 7.1 2)
where a is the initial adsorption rate and B is an empirical constant. Jardine
and Sparks (1984) have compared the first-order, parabolic diffusion, and
Elovich approaches described above to describe the kinetic exchange of K-Ca
in clay minerals and soils. They found that the pseudo first-order model pro-
vided the best overall goodness-of-fit of the experimental results. Recently,
a fractional power approach was introduced by Serrano et al. (2005) having
the form:
β
St
( 7.1 3)
where κ and β are empirical constants. They compared the overall sorption
kinetics of Pb and Cd for single and binary systems. Their results showed
that the simultaneous presence of the competing metal did not affect the
estimated apparent sorption rate, which indicated that the rate-limiting pro-
cesses of the sorption of heavy metal ions were not impacted by the compet-
ing ions.
7.4 E x a mples
Several examples are given here to illustrate the capability of the competitive
model to describe heavy metals transport in soils. Figure 7.7 is an example of
Cd miscible displacement results for Eustis fine sandy soil from Selim et al.
(1992). The sequence of input solutions was 10 mmol c L -1 of Ca(NO 3 ) 2 followed
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