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272 4. Adsorption and Ion Exchange

Furthermore, the selecti f vity coef icient K may be defined as follo ws:
A B

q Z B C Z A
K A B (4.31)
AB Z A Z B
q B C A

where Z is the absolute value of the ions charge. Here, concentration units are in eq/L. If
i
the ion A is preferred, then the selecti whereas if ion B vity coef icient is lar ger than unity f ,
is preferred, it is smaller than unityvious that due to the e dif . It is ob xponents, ferent units
ferent v result in difalues of the selecti vity coef icient. f
Ion-exchange equilibrium can be considered to be analogous to chemical equilibrium.
From that point of vie the mass-action law can be used to express the state of equilib-
,
w
xclusi
v
rium despite the fact that this law is defined eely for homogeneous systems.
ay
,
Derived this w the so-called pseudo-equilibrium constant K is not really a constant,
e
since it depends on the total concentration:
Y Z B X Z A q  o  Z A Z B
K A B K AB  (4.32)
e Z A Z B C   
Y B X A o
However, if the valences of the exchanging cations are equal, the selecti f icient or vity coef
pseudo-equilibrium constant is not affected by concentration. As already mentioned, one
isotherm corresponds to a specific temperature in the case of adsorption or ion e xchange
,
of equal valence ions, whereas additionally the same normality is required for the e xis-
tence of only one isotherm in the case of ion exchange of different valence ions, due to the
concentration–valence effect (Helfferich, 1962). The determination of the true equilibrium
constant should be based on the thermodynamic activities (activity coeficients) of the f
species rather than concentrations. It is clear that the dificulties in the determination of f
activity coeficients also complicate the determination of the true equilibrium constant
f
(Culfaz and Yagiz, 2004).

4.2 DESIGN OF ADSORPTION AND ION-EXCHANGE PROCESSES
4.2.1 Adsorption and ion-exchange kinetics in batch operations

In this section, the basic theory required for the analysis and interpretation of adsorption
and ion-exchange kinetics in batch systems is presented. For this analysis, we consider the
transient adsorption of a single solute from a dilute solution in a constant v well- olume,
,
mixed batch system, or equi adsorption of a pure gas. Moreo uniform spheri- , er v
alently
v
cal particles and isothermal conditions are assumed. Finally diffusion coeficients are f
,
considered to be constant. Heat transfer has not been taken into account in the follo wing
analysis, since adsorption and ion exchange are not chemical reactions and occur princi-
pally with little eolution or uptake of heat. Furthermore, in environmental applications,
v