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282 4. Adsorption and Ion Exchange

Obviously, in the case of isotopic e xchange, = 1. Furthermore, in isotopic e the xchange,
f diffusion coeficients are equal to the self-diffusion coeficient of each ion. For dif f ferent
ions (not isotopes), the self-dif fusion coeficient is substituted by an empirical constant dif- f
fusion coefficient. In the general case of ion exchange, the dif f icient is not con- fusion coef
erage”
v
stant and for practical reasons, an appropriate “a value is used. For the case of
complete conversion into A-form (infinite solution volume), this “a dif erage” v fusion coef-
ficient is (Helf 1962) ferich,

DD A z B A ( z )
B
D
avr (4.86)
zD A A z D B B
The following equation is more general (Ruthv 1984): en,

q
1 2
D ∫ Dq ()d q
avr (4.87)
q
q
2 1 q
1
This diffusion coeficient is the mean diffusion coeficient for a solid phase loading rang- f f
ing from q to q .
1 2
The use of simplified “diffusion” models in ion-exchange systems It is well-kno wn
that in practice, seeral simplified models for the representation and design of ion-
v
exchange systems are used. In the follo such models are analyzed. wing sections,
The use of isotopic models in the literactical limits of usa e—pr atur e g : As mentioned
above, simplified solutions are employed in ion exchange for the estimation of dif fusion
coefficients. For example, the equations of Vermeulen and Patterson, deried from isotopic v
v
exchange systems, hae been successfully used, een in processes that are not isotopic.
v
Inglezakis and Grigoropoulou (2001) conducted an extended review of the literature on the
use of isotopic models for ion-exchange systems.
The same authors (2001) studied the common case of bialent (liquid phase)–mono v a- v
lent (solid phase) exchange. In this study two isotopic models, i.e. Vs and ermeulen’
,
Patterson’s and the Nernst–Plank model for the exchange of ions of different valence, were
compared in terms of applicability (Table 4.16). Specifically, the authors studied the range

Table 4.16
Applicability range for isotopic models

U ( t ) limits ( w ) limits ( ) limits Model

0.3–0.8 0.1 1–20 Vermeulen
0–0.8 Any finite value 1–20 Patterson