Page 80 - Adsorbents - fundamentals and applications

HORV ´ ATH–KAWAZOE APPROACH 65

will no longer be collinear, the density of the molecules adjacent to the middle
layer will be decreased and its interaction energy will be lower. This factor is
taken into account by using the interpolating Eq. 4.22, which will consider only
a fraction of the adsorbate–adsorbate potential ε 3 (z) for this molecule in the final
average. Equation 4.22 will thus be exact when M is a whole number and will
be an interpolated approximation when M lies between two whole numbers.
The PSD can be now determined following the methodology outlined by
Horv´ ath and Kawazoe (1983). As was explained in the previous section regarding
the original HK model, the free energy change upon adsorption, RT ln(P/P 0 ),is
equated to the molar average interaction energy, N Av · ε. Thus the modified-HK
equation can be written as:

P

RT ln = N Av · ε (4.23)
P 0
On applying the correction for isotherm non-linearity given by Cheng and Yang
(1994), the modified HK–CY equation is obtained as:

P 1

RT ln + RT 1 + ln(1 − θ) = N Av · ε (4.24)
P 0 θ
In Eqs. 4.23 and 4.24 above:

when 1 ≤ M< 2
ε = ε 1
(4.25)
2ε 2 + (M − 2)ε 3
= when M ≥ 2
M
The above model is simple and can be easily implemented by using a program-
ming code or a spreadsheet. A slight discontinuity in ε may occur at M = 2as
a result of the above two interpolating functions, but the difference in the values
was found to be negligible for the examples studied in this work. The algorithm
for determining pore-size, by using the corrected HK equation is summarized in
Table 4.1.
We can now compare predictions of PSD by using different models from
isotherm data from the literature. The physical parameters used in the calcu-
lation are given in Table 4.2 (Rege and Yang, 2000). The PSD of HGS-638
molecular sieve carbon predicted by the original HK Eq. 4.12, original HK–CY
Eq. 4.13, modified-HK Eq. 4.23 and the modified HK–CY Eq. 4.24 are shown
in Figure 4.5. It can be seen that there is virtually no shift in the peak pore size
when the interaction energy model was changed. This condition is so because
the peak pore size of HGS-638 lies in the 0.4–0.6 nm range, which corresponds
to roughly twice the adsorbate molecule (N 2 ) diameter. As shown by Rege and
Yang (2000), the interaction energy predicted by the two models is nearly the
same until the slit width is about 2d A ; hence the predicted pore widths are also
similar in this size range. Also, it was observed that the points on the figure corre-
sponding to the higher extreme of the pore width in the original model appeared