Page 39 - Adsorbents - fundamentals and applications

24 SORBENT SELECTION: CRITERIA

by Wei (1994). Wei’s analysis included concentration dependence of single-file
diffusion as well as some unsolved problems of zeolite diffusion. Here we will
only briefly discuss the concentration dependence of diffusivity and prediction
of multicomponent diffusivities from pure-component diffusivities.
The Fickian diffusivities for diffusion in zeolites and microporous materi-
als are generally concentration dependent. Although a variety of concentration
dependence has been reported (Yang, 1987), an increase in the diffusivity with
concentration is generally the rule. The observed concentration dependence is
similar to that seen for surface diffusion, that is,
1
D s,θ
= (3.23)
D s,θ=0 1 − θ
where θ is the fractional surface coverage and D s is the surface diffusivity. This
dependence can be explained by the HIO model (Higashi et al., 1963), based on
the random walk (or hop) of molecules. It assumes that the transit time between
sites is negligibly relative to the residence time, t, at each site, given by:

1 − E/RT
= ve (3.24)
t

where v is the vibrational frequency of the bond that holds the molecule to
the site, and E is the effective bond energy, that is, the difference in energy
between the states corresponding to adsorption at the ground vibrational level
of the bond and the free mobility on the surface. The surface diffusivity is thus
obtained by the Einstein equation, Eq. 3.21. It is further assumed that when a
molecule encounters a site already occupied by another molecule, it immediately
bounces off and continues without stopping until it finds an unoccupied site at
which to rest. The average number of jumps a molecule takes to find an empty
site at surface coverage θ is

∞
k−1 1
η θ = k(1 − θ)θ = (3.25)
1 − θ
k=1
which takes the same length of time, t. Thus the relation in Eq. 3.23 is obtained.
The HIO model predicts values in agreement with experimental data reason-
ably well up to θ = 0.6 or 0.7. At higher values of θ, the values predicted
become greater than those measured. When θ = 1, the model would give a value
of infinity. This discrepancy has been circumvented by a modified model in
which multilayer adsorption is allowed and a finite residence time is assigned
to the second and higher-number layers (Yang et al., 1973). To account for the
second-layer adsorption, the result from the modified model gives:

D s 1
= (3.26)
D s,θ=0 1 − θ + θ(v 1 /v 2 ) exp[−( E 1 − E 2 )/RT ]