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266 4. Adsorption and Ion Exchange

adsorption or ion-exchange kinetics. However, the shrinking core model and some kinetic-
type models are presented in brief.

4.1.4 Equilibrium

A common way to represent the equilibrium in adsorption and ion-exchange systems is the
equilibrium isotherm. The equilibrium isotherm represents the distribution of the adsorbed
material between the adsorbed phase and the solution phase at equilibrium. This isotherm
is characteristic for a specific system at a particular temperature.
The basic difference between adsorption and ion exchange is that while there is only one
isotherm at a specified temperature for adsorption, more than one isotherm can exist at a
specified temperature for different normalities of the solution in the exchange of ions of
different valences due to the concentration–valence effect (Helfferich, 1962). a spe- Thus,
cific ion-exchange system presents one equilibrium curve (isotherm) only under constant
,
. temperature and normalityThis is why while the term “isotherms” is used for the equi-
librium curves in the case of adsorption, the term “isotherm–isonormal” should be used for
xchange. ion e
In the follo the most important isotherm types are presented. wing sections,
Adsorption

Langmuir isotherm Adsorbents that exhibit the Langmuir isotherm behavior are sup-
vidual sites,
ix posed to contain f each of which equally adsorbs only one molecule,
ed indi
forming thus a monolayer namely a layer with the thickness of a molecule (Perry and
,
,
Green, 1999):
q KC
e e
Q M 1 KC e (4.5)

where q e is the solid-phase concentration in equilibrium with the liquid-phase concentra-
tion C , Q M is the final sorptive capacity (most commonly in mg/mg), and K is an equilib-
e
rium constant (most commonly in L/mg). The units of K are L/mol provided that C e is
expressed in (mol/L). Applying the same equation for C e = C , o

q KC
max o
Q 1 KC (4.6)
M o

where q max is the solid-phase concentration in equilibrium with C . Dividing the abo e v
o
equations:

q C 1 KC
e e o (4.7)
q max C o 1 KC e