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278 4. Adsorption and Ion Exchange
1
0.9
0.8
0.12
0.7
0.1
0.6
0.08
erf(x) 0.5 0.06
0.4
0.04
0.3 0.02
0.2 0
0 0.02 0.04 0.06 0.08 0.1
0.1
0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75
x
Figure 4.15 The erf(x) function. Note that erf(– x ) erf( x ).
responsible for the movement of ions. The error of Patterson’s approximation is below 2%
for the whole range of U ( t ), compared to the exact solution.
Finite solution volume and liquid diffusion contr ol General case : The general solu-
tion for the Langmuir isotherm is gien by Perry and Green (1999): v
b(1 ) La ∞ 1 ( 2 cU t ( ) b d )( )
d
b
1 ln
d
2 c d ( 2 cU t ( ) b d )( )
b
1 La
ln 1 ( ) bU t cU ( ) t 2
2c (4.59)
3 kt C
f o
r o pe q
where
1 La
b w
1 La (4.60)
c 1 La (4.61)
w
d 2 c 4 0.5 (4.62)
b