Second-Order Transport Modeling - Transport and Fate of Chemicals in Soils: Principles and Applications - page 158

where ( S max ) 1 and ( S max ) 2 are the total number of type 1 sites and type 2 sites,
respectively. If F represents the fraction of type 1 sites to the total number of
sites or the adsorption capacity for an individual soil, we thus have:

( ) = F S and( ) = (1 − F S )

S S max

(6.2)

max max max

1 2

We now denote ϕ as the number of unfilled or vacant sites in the soil, such that:

φ = ( ) − = F SS −

S S

(6.3)

max 1 max 1

1 1

φ = ( ) − =− − (1 F SS )

S S (6.4) max 2 max 2

2 2

where ϕ and ϕ 2 are the number of vacant sites and S 1 and S 2 are the amounts
of solute retained (or the number of filled sites) on type 1 and type 2 sites,
respectively. As the sites become filled or occupied by the reactive solute, the
number of vacant sites approaches zero, that is, (ϕ 1 + ϕ 2 ) → 0. In the meantime,
the amount of solute retained by the soil matrix approaches that of the total
capacity (or maximum number) of sites, ( S 1 + S 2 ) → S max .
We commonly express the amount of reactive chemical retained, such as
S 1 and S 2 in Equations 6.3 and 6.4 as the mass of solute per unit mass of soil
(mg kg -1 soil). Based on the above formulations, the total number of sites S max ,
( S max ) 1 , and ( S max ) 2 and vacant or unfilled sites ϕ 1 and ϕ 2 must also have similar
dimensions. Here the units used for S and ϕ will be in terms of milligrams of
solute per kilogram soil mass (mg kg -1 soil).
Based on this approach, reactive chemical retention mechanisms are
assumed to follow a second-order kinetic reaction where the forward pro-
cess is controlled by the product of the solution concentration C (mg L -1 ) and
the number of unoccupied or unfilled sites (ϕ) (Selim and Amacher, 1988).
Specifically, the reactions for type 1 and type 2 sites may be expressed by the
reversible processes; t → ∞:

k
k

+ φ →

←

C

1

S (6.5)

1

1

2

and

k
k

+ φ →

←

C

3

S (6.6) 2

2

4

Therefore, the differential form of the kinetic rate equations for reactive
chemical retention may be expressed as:

ρ ∂

S
t

1

∂ =Θφ −ρ fortype1sites

C

k k S

(6.7)

1 2 1

1