Reduction of Contaminants in Soil and Water By Direct Electric Current - Electrokinetics for Petroleum and Environmental Engineers - page 47

with the experimental data reported by Wittle and Pamukcu (1993) and
Acar and Alshawabkeh (1996), among many others.
When charged ions transport under the influence of an externally applied
electrical field, their concentration distributions change with time which
lead to a change in local electrical conductivity, as discussed previously.
The change of local electric conductivity directly alters the value of a poten-
tial gradient at that specific point. Hence, the changing electric conductiv-
ity and electrical field strength describe the transport process of the species
implicitly. Furthermore, the mobility of the charged species decreases as
they congregate in the direction of their migration. The increasing concen-
tration of the ions leads to a decrease in equivalent conductance in their
migration direction (Kortüm and Bockris, 1951). According to Kohlrausch's
law of the independent migration of ions, the equivalent conductance of an
electrolyte is additively composed of the ionic conductivities of the con-
stituent ions. As ions move under an applied electric field at their terminal
ionic velocity ( u i ) of infinite dilution, two other phenomena occur, which
retard ion velocity and effectively reduce the bulk conductivity. These are
called the “retardation” and “electrophoretic” effects. The retardation effect
comes about when the ionic atmosphere becomes unsymmetrical about an
ion in motion, as the charge density decrease in front of the ion and increase
behind it. For a moving positive ion, a net negative charge trail behind it
exerting an electrostatic force in the opposite direction. The electropho-
retic effect occurs as the negative ions in the ionic atmosphere around the
positive ions migrate in the opposite direction taking their solvent sheaths
with them. The positive ion therefore travels against a medium moving in
the opposite direction, hence a viscous drag. A quantitative formulation of
ion mobility, relating the magnitudes of the electrostatic retardation force
and the viscous drag force to the radius of the ionic atmosphere directly
provides a theoretical basis for the well-known empirical relation known
as the “Kohlrausch square root law” (1900), in terms of equivalent conduc-
tance, Λ v [ l 2 Ohm -1 ] (Kortüm and Bockris, 1951). The Λ v is also the sum of

individual ionic conductivities ( F. u + for positive ions; and F. u - for negative
ions), hence the ionic mobility of a given ion, u i can be expressed in terms
of the equivalent conductance (Bard and Faulkner, 1980):

'

Λ t

F

(2.12) u =

vi

i

where t' i = u i /(u + +u - ) is the transference number of the of species i .
Equation 2.12 effectively incorporates the retardation effects into the
mobility determination for high concentration solutions. As an example,