Page 38 - Adsorbents - fundamentals and applications

EQUILIBRIUM ISOTHERMS AND DIFFUSION 23

for all components. This can be shown by substituting the Langmuir isotherm
into Eq. 3.14:

P 0 P 0
1 B 1 2 B 2
q m1 dP = q m2 dP (3.19)
0 1 + B 1 P 0 1 + B 2 P
0
0
If q m1 = q m2 ,then B 1 P = B 2 P . Substituting this result into Eq. 3.15, the fol-
1 2
lowing is obtained:
B 1 P 1
X 1 = (3.20)
B 1 P 1 + B 2 P 2
This equation can be readily obtained from the extended Langmuir model,
Eq. 3.4, for the binary system. Thus, when the saturated amounts for the pure
gases are the same, the IAS is identical to the extended Langmuir model.
It is interesting to note that similarity also exists between the IAS theory and
the potential model of Grant and Manes (1966). The similarity has been shown
by Belfort (1981) and summarized in Yang (1987).

3.1.4. Diffusion in Micropores: Concentration Dependence
and Predicting Mixed Diffusivities
For diffusion in gases and colloidal systems, concentration is the origin (or “driv-
ing force”) for diffusion. Einstein first showed in 1905 that from the concentration
gradient, the diffusivity is (Kauzmann, 1966)

δ 2
D = (3.21)
2 t
where δ is the mean distance between collisions, or Brownian motion in time
t in the x direction, and D is the diffusivity that relates the flux with the
concentration gradient by Fick’s law:
dc
j =−D (3.22)
dx

For diffusion in liquid mixtures, it has been argued that chemical potential is the
“driving force.” (Haase and Siry, 1968).
The mechanisms of diffusion in these two systems (gas and liquid) are differ-
ent and unrelated; diffusion in gases is the result of the collision process, whereas
that in liquids is an activated process (Bird et al., 1960). Diffusion in microp-
orous materials is neither gaseous nor liquid diffusion. The closest case for such
diffusion is surface diffusion, where molecules “hop” within the surface force
field (see review by Kapoor et al., 1989b). Fick’s law is used for both appli-
cation (in modeling of adsorption processes) and experimental measurement of
diffusion. Extensive reviews are available on diffusion in microporous materials
and zeolites (Karger and Ruthven, 1992; Do, 1998). A lucid discussion on the
nonlinear, and in some cases peculiar, phenomena in zeolite diffusion was given