Page 73 - Adsorbents fundamentals and applications

58 PORE SIZE DISTRIBUTION

molecules (in Angstrom). The dispersion constants A S and A A are calculated by
6
the Kirkwood–M¨ uller formulae (erg × cm ) as follows:
2
6mc α S α A
A S = (4.7)
α S α A
+
χ S χ A
3 2
A A = mc α S α A (4.8)
2
Further, Horv´ ath and Kawazoe (1983) proposed that the potential is increased
by the interaction of adsorbate molecules within the pore. They included this
additional interaction by adding an adsorbate dispersion term (N A A A )inthe
4
numerator of the depth of potential energy minimum (N S /2σ ) in Eq. 4.6
as follows:

4 10 4 10
N S A S + N A A A σ σ σ σ
ε(z) = − + − +
2σ 4 z z L − z L − z
(4.9)
However, no clear justification could be found in the literature for incorporating
the adsorbate–adsorbate–adsorbent interaction in this manner. Further, ε(z) in
Eq. 4.9 may be split as follows:

ε(z) = ε A−S (z) + ε A−A (z)

4 10 4 10
N S A S σ σ σ σ
= − + − +
2σ 4 z z L − z L − z

4 10 4 10
N A A A σ σ σ σ
+ − + − + (4.10)
2σ 4 z z L − z L − z
The first term ε A–S (z) gives the adsorbate-surface interaction and the second term
ε A–A (z) gives the adsorbate–adsorbate interaction. Because the distance z of the
gas molecule is the same in both ε A–S (z) as well as ε A–A (z), Eq. 4.10 gives
an erroneous interpretation that the adsorbate–adsorbate interaction is caused
between a gas molecule and two parallel infinite sheets of gas molecules, imprac-
tically placed at the same position as the sorbent molecules. Besides, the internu-
clear distance at zero interaction energy (σ)usedin ε A–A (z) is expected to differ
fromthatusedinthe ε A–S (z) expression; however, the original model does not
take this fact into account.
The next step in the derivation involved obtaining the average interaction
energy by integrating the above profile over free space in the slit-pore:

L−d 0
ε(z) dz
d 0
ε(z) = (4.11)

L−d 0
dz
d 0