Page 72 - Adsorbents fundamentals and applications

HORV ´ ATH–KAWAZOE APPROACH 57

clear explanation is provided for doing so. Moreover, the energy potential due
to adsorbate–adsorbate interactions in a filled pore can be interpreted as the
energy profile due to the adsorbate molecules placed, rather unrealistically, at
the same position as the adsorbent molecules. This is because, according to
the original model, the distance z of a sorbate molecule from another sorbate
molecule is the same as that between a sorbate molecule and a sorbent molecule.
Furthermore, the average interaction energy is calculated by integration of this
energy profile over the entire pore characteristic length (pore width for slit-pores
or radius in case of curved geometries). This implies that in a filled pore there is
a continuous distribution of molecules along the characteristic length. However,
micropores have dimensions comparable with molecular widths, and hence the
molecules in a filled pore can occupy only discrete positions relative to each
other, if minor thermal vibrations are neglected. The original HK model has
also been criticized for not taking the distribution of energy into account while
calculating the average interaction energy (Webb and Orr, 1997). As a result of
these and other shortcomings, it has been observed that pore sizes estimated by
the original HK method for materials in the higher micropore size range (8–20 ˚ A)
are unrealistically low in some cases (e.g., Carrott et al., 1998).
The original HK model will be given first, followed by the corrected model
by Rege and Yang (2000).

4.2.1. The Original HK Slit-Shaped Pore Model
The basis for obtaining the energy profile in a pore is the Lennard–Jones 6–12
potential:

12 6
σ σ
∗
ε 12 (z) = 4ε 12 − (4.4)
z z
Halsey and co-workers gave the interaction energy of one adsorbate molecule
with a single infinite-layer plane of adsorbent molecules to be (Sams et al., 1960)

4 10
N S A S σ σ
ε(z) = − + (4.5)
2σ 4 z z
Everett and Powl (1976) extended the above result to two parallel infinite lattice
planes whose nuclei are spaced at distance L apart:

4 10 4 10
N S A S σ σ σ σ
ε(z) = − + − + (4.6)
2σ 4 z z L − z L − z
where the internuclear distance at zero-interaction energy (in Angstrom):
σ = (2/5) 1/6 d 0 ; d 0 is the average of the adsorbate and adsorbent molecule
diameters, i.e., (d 0 + d A )/2, N S is the number of sorbent molecules per unit
area, and z is the internuclear distance between the adsorbate and adsorbent