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4.1 Basic Principles of Adsorption and Ion Exchange 265

For the needs of the present book, in the analysis of adsorption and ion-e xchange kinet-
ferich’
ics, Helfs approach is followed—the particle, consisting of pore fluid and a solid
matrix, is treated as a single quasi-homogeneous phase, reardless of its inhomogeneities g
in molecular or colloidal dimensions and its particular geometrical structure (Helf ferich,
v
1962). The model solutions gien by Helfferich are equialent to the solutions of a solid v
diffusion–controlled process. Eq. (4.1) is valid and the corresponding diffusion coefficient
fecti
v
is an “ef kinetic parameter characterizing the ease of moement of the solutes in v
e”
the particle in macropores and micropores. Furthermore, the diffusion coeficient is con- f
sidered to be constant, or to be more realistic, an aalue as it can be dependent on v erage v
the solute solid-phase concentration.
Following this simplified approach, the time dependence of the concentration is related
s second la to the flux by Fick’w (material balance or condition of continuity):
C
i div J
t i (4.2)

Combining eqs. (4.1) and (4.2), and for spherical particles, the following diffusion equa-
tion, written in spherical coordinates ( r ), describes the mass transfer process:

C i  C 2 i 2 C i 
t D   r 2 r   (4.3)
r

where D is the solid dif icient and fusion coef f C i the solid-phase concentration of the solute
(in the following chapters, this concentration is denoted by q ). Eq. (4.3) must be solv ed
under the appropriate initial and boundary conditions. Here, it should be noted that for
most nonspherical particles, their representation as an equialent sphere is an acceptable v
approximation, and thus the aboe-mentioned equation can be used in this case as well. v
Apart from the diffusion step in the particle, when the uptake process occurs from a
binary or multicomponent fluid mixture, there maybe an additional resistance to mass
transfer associated with the transport of solutes through the fluid layer surrounding the par-
ticle. The driving force in this case is the concentration difference across the boundary
layer, and the flux at the particle surface is

(4.4)
J k C ( f i f, C i s , )
i

where k is the mass transfer coefficient and C i, f and C i ,s are the concentration of solute in
f
the bulk fluid and at the particle surface phase, respecti . v ely
Thus, the analysis of the rate-determining step, as analyzed for heterogeneous processes
in Section 3.1.2, is equally applied in adsorption and ion eThe only dif xchange. ference is
that the diffusion processes in the fluid film and in the particle are followed by physical
adsorption or ion exchange and not by a reaction step as in catalysis.
The analysis of adsorption and ion-exchange kinetics is presented in detail in Section
4.2.1, and is based on the diffusion processes and equations rather than on some kind of