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276 4. Adsorption and Ion Exchange

where

CD
o f
QD M s (4.45)
p

mr n 2 o 2 r o 1 2
A (4.46)
n 2 2
mr r r 1
n o o o
Finally, the quantities m n are the roots of the equation

r
mr n o (1 )t o an( mr ) n o (4.47)
Note that for adsorption, the parameter q e is used in the place of Q M .

Finite fluid volume and solid diffusion contr ol The solution is the following (Perry
and Green, 1999):

 
exp  pD 2 t 2 
 n s r 
Ut () ∑ o 2 (4.48)
1 6
n 1 w 9 (1 w) (1 w p ) n

where p n are the positie roots of the equation (Figure 4.14): v

tan( p ) 3
fp ) n 0 (4.49)
(
n
p n 3+ p ( 2 n w 1)
Helfferich (1962) gives a similar equation in a different form for isotopic e xchange:

∞ 2
2 exp ST
Ut () ∑ n
1
w 3 n 1 S 2 n
1+ (4.50)
ww 9( 1)
where S n are the roots of the equation

S 2
S co t S () n (4.51)
1
n n
w 3