Page 284 - Adsorption, Ion Exchange & Catalysis- 2007, Elsevier - Copy

Else_AIEC-INGLE_cH004.qxd 7/1/2006 6:53 PM Page 280
280 4. Adsorption and Ion Exchange

ions of different charge, size, and nature. In the follo the equations co wing, v ering this more
general case are presented.

Solid diffusion control and infinite solution v olume In this case, the diffusion coef f i-
cient is not a constant but depends on the concentration of the ions in the solid phase. The
basic diffusion equation to be solved is the follo 1962): ferich, wing (Helf

q 1  q B 
B r D 2
t r 2   AB  
r
r
(4.68)
where

DD z q 2 z q 2
D AB A 2 B A A 2 B B
zq D z q D
A A A B B B (4.69)

is the coupled interdiffusion coefficient, q the solid-phase concentration of ion species, D i
the self-diffusion coefficient of the ion i , and z its char i ge.
These equations, for the case of solid dif are solved by arith- fusion–controlled kinetics,
metic methods resulting in some analytical approximate expressions. One common and
useful solution is the so-called Nernst–Plank approximation. This equation holds for the
case of complete conersion of the solid phase to A-form. The complete conersion of v
v
solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion,
,
olume,
requires an excess of liquid v and thus w 1. Consequently in practice, the
v restriction of complete conersion is equialent to the infinite solution volume condition. v
The solution of the diffusion equation is

Ut ()  p x m ( )   0.5 (4.70)
1e


m 1B c T 2 B c T 2 3B 3 (4.71)
2
c T
where

D
B
D (4.72)
A
Dt B
T B 2 (4.73)
r
o