Environmental Engineering Reference
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a pore size distribution function, the distribution of pore volumes as a function of
pore width H. Therefore, all of the heterogeneities of less crystalline porous solids
are approximated by the distribution of pore sizes. If (P,H) can be obtained from
the molecular statistics, f(H) can be determined by the best lit to the observed
experimental isotherm. The width H in f(H) is not the effective pore width that
as mentioned in above models. In order to derive the molecular density in a pore,
statistical approaches to fluids have been used. Seaton et al. applied the mean
field theory to calculate (P,H). The mean field theory is an approximate theory
of inhomogeneous fluids in which the interactions between the fluid molecules
are divided into a short-range, repulsive part and a long-range, attractive part.
Each is treated separately for faster calculation than full molecular simulation.
The contribution of the long-range forces to the fluid properties is treated in the
mean field approximation, while the effect of the short-range forces is modeled
by an equivalent array of hard spheres. There are two approaches to get the short-
range forces-the Local mean field theory and the non-loca1 one, where the former
neglects the short-range correlation, bat the latter takes it into account. Seaton et
al. adopted the local density approach for their calculation. They calculated (P,H)
by the above method. How can we determine f(H) from the calculated (P, H) and
the experimental adsorption isotherm N(P)? It has a mathematical difficulty. They
used the following bimodal log-normaldistribution, which is flexible to represent
the various pore size distributions and is zero for all negative pore widths:
{
}
{
}
{
}
{
}
(
[
]
[
]
(48)
)
(
)
1/ 2
2
2
2
1/ 2
2
fH
=
V
/
sπ
H
2
exp - log
H
-
ms sπ
/ 2
+
V
/
H
(2
)
×
exp - log
H
-
ms
/ 2
1
1
1
2
2
2
2
where Vi is the pore volume of the distribution i, and a, and, u, are the parameters
defining the distribution shape. These six parameters in Eq. (48) are determined
from the best fit to the experimental adsorption isotherm. The limit of H
mi
cor-
responds to the smallest pore into which the N
2
molecule can enter. On the other
hand, the upper limit of H, is determined by the width of the mesopore, which
condenses at the highest experimental pressure. This calculation can determine
the pore size distribution from micropore to mesopore. In that work the applica-
bility for the pores of less than 1.3 nm was not shown. Lastoskie et al. extended
the above method to the non-local mean field theory. The non-local mean field
theory gives a quantitative accurate description of even ultra micropore structures.
They compared the pore size distributions from the local and non-local mean field
theories as to real adsorption isotherms by activated carbons; the local theory un-
derestimates the pore size distribution compared with the non-local theory. As the
calculation with the mean field density theory often gives a qualitative agreement
rather than quantitative one. The grand canonical ensemble Monte Carlo simula-
tion is also necessary for such an approach. They also got good results. The mo-
lecular simulation studies on the pore size distribution have shown a new picture
on the adsorption in the wide range of pores from ultramicropores to mesopores.
Understanding of micropore filling and capillary condensation proceeds rapidly.
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