Page 71 - Adsorbents fundamentals and applications

56 PORE SIZE DISTRIBUTION

the analysis adsorbate is smaller than the adsorbent channels. The mathematical
expression relating relative pressure of the adsorbing gas to the pore size of the
adsorbent has come to be known as the Horv´ ath-Kawazoe (HK) model.
Over the past two decades, different types of HK models have been developed
depending upon the pore geometry. The original HK model discussed slit-shaped
pores (Horv´ ath and Kawazoe, 1983), whereas models for cylindrical pores (Saito
and Foley, 1991), and spherical pores (Cheng and Yang, 1994) have also been
proposed. The basic framework for all the different HK models is the same:

P
RT ln = U 0 + P a (4.2)
P 0
where U 0 and P a denote the sorbate–sorbent and sorbate–sorbate interaction
energies, respectively. Thus the R.H.S. is a function of the pore geometry and
dimension, which is related to the relative pressure of the adsorbate. The calcu-
lation of the PSD is now relatively simple. By using different values of a pore
dimension (pore width in case of a slit-pore, pore diameter in case of pores with
curvature), the threshold sorption relative pressure (P/P 0 ) at which the pore fill-
ing will occur (or “filling” pressure) can be obtained over the expected pore size
range. From the adsorption measurements using a suitable sorbate, the fractional
adsorbed amount (q = w/w ∞ ) is obtained as a function P/P 0 .Fromacombi-
nation of the above two functional relationships, the adsorbed amount w/w ∞
can be plotted as a one-to-one function of pore dimension L, thus giving the
cumulative PSD. A differential PSD can be further obtained by calculating the
derivative d(w/w ∞ )/dL as a function of L.
The original HK equation included an implicit assumption that the adsorption
isotherm follows Henry’s law. In order to correct for the deviation of isotherm
data from Henry’s law at higher relative pressures, Cheng and Yang (1994)
proposed a correction factor, which incorporated a Langmuir isotherm fit to the
data. The HK model with the Cheng–Yang correction (henceforth referred to as
the HK–CY equation) is given as:

P RT 1

RT ln + RT − ln = U 0 + P a (4.3)
P 0 θ 1 − θ
Despite the immense utility of the HK model, there exist certain conceptual
defects in the original model. The derivation of the original HK model proceeds
by calculating the energy potential of a single adsorbate molecule with a layer
of sorbent molecules along the pore periphery of a particular geometry by using
the Lennard–Jones 6–12 potential. This potential is calculated by incorporating
the adsorbate–adsorbent dispersion term as calculated by the Kirkwood–M¨ uller
formalism in the potential energy minimum (Horv´ ath and Kawazoe, 1983). In
order to include the adsorbate–adsorbate interactions, the adsorbate–adsorbate
dispersion constant is appended to the adsorbate–adsorbent dispersion term in
the potential energy minimum, as will be shown in detail later. However, no