Page 76 - Adsorbents fundamentals and applications

HORV ´ ATH–KAWAZOE APPROACH 61

observations that show that the energy of adsorption of a monolayer of adsorbate
molecules on a clean sorbent surface is the highest and that on subsequently
adsorbed layers of gas molecules is a nearly constant lower value. Any interac-
tion with molecules not in the immediate proximity of the molecule or with those
lying above or below the molecule in the same layer, is assumed to be negligible.
The average interaction energy is then calculated by averaging the energy poten-
tials of the individual gas molecule layers weighted by an approximate molecular
population of each layer.
Consider a slit pore with the nuclei of the sorbent molecules in the lattice
planes forming the pore-wall spaced at a distance L apart. If the diameter of the
sorbent molecules is denoted as d S and that of the adsorbate molecules as d A ,
then the number of molecule layers M that can be accommodated laterally as the
pore gets filled up can be estimated as:

L − d S
M = (4.16)
d A
It should be noted that M can be a whole number or it may also be a frac-
tion. The physical interpretation of a fractional number of layers is that the
molecules in the layers are not arranged with their centers oriented in the same
straight line but are slightly skewed with respect to each other so as to afford a
greater packing of molecules. Such a packing arrangement will of course result
in a decrease in molecular density in the neighboring layers, but the resulting
decrease in interaction energy will be accounted for when taking the average of
the energy potentials.
When 1 ≤ M< 2, only one layer of molecules can be accommodated within
the slit pore as shown in Figure 4.1. In this case, each adsorbate molecule will
interact with the two lattice planes forming the pore wall of the sorbent. The
interaction energy of this monolayer, denoted as ε 1 (z), is given by an expression
similar to Eq. 4.7. An examination of Eq. 4.7 reveals that the minimum energy
potential exists at a distance of d 0 from either sorbent lattice planes. Because the
Boltzmann law of energy distribution law suggests that the molecule will most
probably rest at a position at which the energy potential is the minimum, each
molecule is assumed to exist at a distance of d 0 from one of the two sorbent
lattice planes. The interaction energy would then be

4 10 4 10
N S A S σ S σ S σ S σ S
ε 1 (z) = − + − + (4.17)
2σ 4 d 0 d 0 L − d 0 L − d 0
S
Note that the internuclear distance at zero interaction energy for an
adsorbate–sorbent system is now denoted as σ S to differentiate it from σ A ,which
is the zero interaction energy distance for an adsorbate–adsorbate system. The
expressions for σ S and σ A are as follows:

σ S = (2/5) 1/6 d 0 (4.18)
σ A = (2/5) 1/6 d A (4.19)