Page 75 - Adsorbents fundamentals and applications

60 PORE SIZE DISTRIBUTION

Taking a limit of Eq. 4.14 as P approaches the saturation pressure P 0 ,we
have θ = K/(1 + K). Now, applying the above limit of P/P 0 approaching 1
to Eq. 4.13, we obtain the following:

K + 1

lim N Av · ε(z) = RT 1 − ln(K + 1) (4.15)
P→P 0 K
Thus, at high pressures, the average interaction energy should approach the value
given by Eq. 4.15. Near the saturation pressure, the size of the largest pore,
which can be filled with adsorbate, tends towards infinity. As can be seen from
Eq. 4.12, as L →∇ the average interaction energy approaches zero. Intuitively,
it is apparent that there will always be some adsorbate–adsorbate interaction
and that the limiting interaction energy cannot go to zero even at large pore size.
Thus, the original model for average interaction energy in the pore is inconsistent
with the HK equation by using the Cheng–Yang correction. The proposed model
is found to have a better agreement in this regard.

4.2.2. Modified HK Model for Slit-Shaped Pores
The shortcomings in the original HK model provide the motivation for devel-
oping a new model based on an improved energy profile for a micropore filled
with adsorbate molecules (Rege and Yang, 2000). The new model developed
by Rege and Yang (2000) proposes that molecules occupy discrete positions in
the adsorbate-filled micropore. Furthermore, the interaction energy of a molecule
is calculated by using an intermolecular spacing corresponding to a minimum
energy potential between that molecule and the immediate neighboring molecules.
The average interaction energy is obtained by a population-weighted average of
energy potentials rather than by integration. New models for PSD, based on the
Horv´ ath-Kawazoe principle, have been developed for slit-, cylindrical-, as well as
spherical-shaped micropores (Rege and Yang, 2000). Comparison of the average
interaction energy versus pore-size profiles for the original and modified models
showed a good agreement for small pore widths, but a considerable deviation
was observed at pore widths measuring more than two adsorbate molecule diam-
eters. Pore-size distribution using the modified energy profile has been obtained
by using isotherm data in the literature for all three pore geometries, and the
results are comparable with that used in the original model. Clear and signifi-
cant improvements have been obtained by using the modified models (Rege and
Yang, 2000).
The general strategy for obtaining the PSD for the three pore geometries is
basically the same. The modified model for calculating the average interaction
energy in a filled micropore proceeds by first estimating the number of gas adsor-
bate molecule layers within the filled micropore. Each gas molecule is assumed
to rest preferentially at a position at which its energy potential within the pore
would be minimum, in accordance with the Boltzmann law of energy distribution.
Furthermore, a gas molecule is assumed to interact most effectively only with its
laterally immediate molecular layer. This fact is in agreement with experimental