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274 4. Adsorption and Ion Exchange

e Adsorption and isotopic e xc hang
In the following analysis, adsorption models for solid diffusion control are applicable for
xchange,
isotopic e i.e. exchange of isotopes, while in the case of liquid diffusion control
and the intermediate case, only adsorption models for linear equilibrium can be used for
isotopic e xchange.
Furthermore, as will be analyzed in practical applications, the adsorption models can
also be used as a first approximation for ion-exchange systems, i.e. in the exchange of ions
of different valences.
Infinite fluid volume and solid diffusion contr ol Practically, infinite solution v olume
condition ( w << 1) amounts to constant liquid-phase concentration. For a constant dif fu-
olume,
sivity and an infinite fluid v the solution of the diffusion equations is (Helf ferich,
en, 1962; Ruthv 1984)
∞
6 1 22
Ut () 2 ∑ exp n T
1
n 1 n 2 (4.35)
where U ( t ) is the fractional approach to equilibrium at time t and is defined as

(4.36)
Ut () q q) / ( q )
q(
t o ∞ o
where:
q o solute initial concentration in the solid phase
q equilibrium concentration in the solid phase
q t the a v erage concentration of the solute in solid phase at time t , def ined as
(Ruthven, 1984)

3 r o 2
q 2 ∫ qr r d
t (4.37)
r o 0
where r is the particle radius. For a solid phase initially free from solute, o q o is zero and thus

q C C
Ut () t o t (4.38)
q ∞ C o C ∞

where C ’s are the corresponding fluid-phase concentrations. i
The dimensionless time T is defined as

Dt
T s (4.39)
r 2
o
where D s icient. is the solid dif f fusion coef