Second-Order Transport Modeling - Transport and Fate of Chemicals in Soils: Principles and Applications - page 159

and

ρ ∂

S
t

2

∂ =Θφ −ρ fortype2sites

C

k k S (6.8)

3 4 2

2

where k 1 and k 2 (h -1 ) are forward and backward rate coefficients for type 1
sites, whereas k 3 and k 4 are rate coefficients for type 2 reaction sites. In addi-
tion, Θ is the soil water content (cm 3 cm -3 ), ρ is the soil bulk density (g cm -3 ),
and t is time (h). If ϕ 1 and ϕ 2 are omitted from Equations 6.7 and 6.8, the
above equations yield two first-order kinetic retention reactions (Lapidus
and Amundson, 1952). However, a major disadvantage of first-order kinetic
reactions is that as the concentration in solution increases, a maximum solute
sorption is not attained, which implies that there is an infinite solute reten-
tion capacity of the soil or infinite number of exchange sites on the matrix
surfaces. In contrast, the approach proposed here achieves maximum sorp-
tion when all unfilled sites become occupied (i.e., ϕ and ϕ 2 → 0).
In a fashion similar to the nonequilibrium two-site concept proposed by
Selim, Davidson, and Mansell (1976), it is possible to regard type 1 sites as
those where equilibrium is rapidly reached (i.e., in a few minutes or hours).
In contrast, type 2 sites are highly kinetic and may require several days or
months for apparent local equilibrium to be achieved. Therefore, for type 1
sites the rate coefficients k 1 and k 2 are expected to be several orders of magni-
tude larger than k 3 and k 4 of the type 2 sites. As t → ∞, that is, when both sites
achieve local equilibrium, Equations 6.7 and 6.8 yield the following expres-
sions. For type 1 sites:

= Θ
ρ

S k

k

1 1

Θ−ρ= C 0, or = ω

k k S 1

(6.9)

1 2 1

1

C

2

1

and for type 2 sites:

= Θ
ρ

S k

k

2 3

Θ− ρ= C 0, or = ω

k k S

2

(6.10)

3 4 2

2

C

4

2

Here ω 1 and ω 2 represent equilibrium constants for the retention reactions
associated with type 1 and type 2 sites, respectively. The formulations of
Equations 6.9 and 6.10 are analogous to expressions for homovalent ion-
exchange equilibrium reactions. In this sense, the equilibrium constants ω 1
and ω 2 resemble the selectivity coefficients for exchange reactions and S max
resembles the exchange capacity (CEC) of soil matrix surfaces. However,
a major difference between ion exchange and the proposed second-order
approach is that no consideration of other competing ions in solution or on
matrix surfaces is incorporated into the second-order rate equations. In a
strict thermodynamic sense, Equations 6.9 and 6.10 should be expressed in