Sorption Isotherms - Transport and Fate of Chemicals in Soils: Principles and Applications - page 8

By choosing the accurate density function g(ζ), any type of isotherm
can be described by the general isotherm equation (1.6; Hinz, Gaston, and
Selim, 1994; Hinz, 2001). Consequently, various density functions that were
capable of describing observed adsorption isotherms have been derived.
Examples include Langmuir, two-site Langmuir, general Freundlich, general
Langmuir-Freundlich, Freundlich, and Rothmund-Kkornfeld. For example,
to describe Freundlich isotherms, Sposito (1980) suggested an affinity dis-
tribution function whose curve closely resembles a log-normal distribution.
As outlined by Kinniburgh et al. (1983) and later by Hinz, Gaston, and
Selim (1994), a Langmuir isotherm can be derived by use of the Dirac delta
function δ for g(ζ) as

g (ζ ) = δ (ζ - k) (1.8)

Incorporation of Equation 1.8 in Equation 1.7 and integration yields the
Langmuir isotherm,

S kC

kC

=

(1.9)

1 +

S

max

where k is an “overall” affinity coefficient that is equivalent to ω of Equation
1.3. Selecting w(ζ) in the form and proceeding as above yields the two-
surface Langmuir isotherm equation,

F k C
k C

+

k C
k C

F

(1.10)

1 1 2 2

S = .

S max

1 + 1 +

1 2

Consequently, a complete isotherm may be regarded as the sum of all indi-
vidual Langmuir isotherms.
A major drawback of the above approach is that the distribution functions
g(ζ) for the various isotherms are not very verifiable. In other words, there is
no evidence or independent measure that verifies that such a specific distri-
bution g(ζ) exists. In fact, distribution functions of the normal and log of nor-
mal type do not represent the constituents that make up the soil matrix. This
is illustrated in Figure 1.2 for several commonly used probability distribution
functions (PDFs). For most if not all soils, the dominant fractions are those
associated with low-affinity sites such as sand and silt. In contrast, high-affin-
ity sites, which are mainly associated with clay minerals, organic matter, and
oxides, often are associated with much smaller fractions. Therefore, normal
as well as log-normal distribution functions are unrealistic representations
of the actual affinity distribution of a soil since it grossly underestimates the
low-affinity fractions. Another drawback is that such affinity distribution
functions g(ζ) are not considered unique. Therefore, distribution functions
cannot be precluded from describing site affinity distributions.