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range of concentrations found in the miscible displacement experiment. The
only exception is for the irreversible parameter k s , which increased as the
input concentration ( C o ) increased.
For Olivier soil, predicted BTCs using the SOM indicate that the use of
batch rate coefficients from either C o of 10 or 25 mg L -1 provided surpris-
ingly good overall descriptions of the experimental results (Figure 6.15). Less
than adequate predictions were obtained for the highly kinetic Cecil and
Windsor soils, however. In fact, no one set of batch rate coefficients success-
fully described both the adsorption and the desorption sides of Windsor or
Cecil BTCs. For both soils, closest predictions were realized using batch rate
coefficients from C o of 10 or 25 mg L -1 . This is a similar finding to that based
on the predictions for Olivier soil.
6.4.1 Modified Second-Order Approach
A major modification of the second-order model (SOM) approach was intro-
duced by Ma and Selim (1994, 1998) where few model parameters are required.
The primary difference of the modified approach from that of the original
second-order formulation presented earlier is the assumption that vacant sites
are equally accessible and can thus be occupied by either S 1 or S 2 . That is, S 1
and S 2 can compete for the unoccupied adsorption sites regardless of whether
they are type 1 or type 2 sites. Therefore, it is assumed that adsorption sites
are related and affected by each other and adsorption of one species may
block the adsorption sites of the other type. In other words, the total adsorp-
tion sites ( S max ) were not partitioned between S 1 and S 2 phases based on a frac-
tion of sites F . Instead, it is now assumed that the vacant sites are available
to both types of sites and F is no longer required and the amount of solute
adsorbed on each type of sites is only determined by the rate coefficients asso-
ciated with each type of sites. As a result, sites associated with fast reactions
will compete for available sites before slow sites are filled. Such mechanisms
are in line with observations where rapid (equilibrium type) sorption is first
encountered followed by slow types of retention reactions. Therefore, based
on the above, we defined ϕ (μg g -1 ) as the number of vacant sites, which is
dependent on the number of sites occupied by S 2 and S 2 such that:
φ= −−
SSS
max
(6.33)
1
2
and governing kinetic expressions for the rate of reactions for solutes present
in the soil solution phase (C) for the type 1 and type 2 sites may be written as:
ρ
S
t
1
=Θφ−ρ
C
fortype
1
sites
k
k
S
(6.34)
1
2
1
ρ
S
t
2
=Θφ−ρ
C
fortype
2
sites
k
k
S
(6.35)
3
4
2
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