Page 85 - Adsorbents - fundamentals and applications

70 PORE SIZE DISTRIBUTION

will preferentially occupy positions at which the energy potential is minimum,
the first layer of molecules is supposed to exist at a radius of (L − d 0 ) from the
center. Thus making the required substitution, the energy potential of the first
layer (e 1 ) is obtained as:

∞ ∞
3 N S A S 21 10 2k 4 2k
ε 1 = π a 1 α k (b ) − a 1 β k (b ) (4.27)
1
1
4 d 0 4 32 k=0 k=0
where a 1 and b 1 are given by
a 1 = d 0 /L and b 1 = (L − d 0 )/L (4.28)

As can be seen from Eq. 4.27, the constant a 1 (and in general, a i ) corresponds to
a ratio of internuclear spacing d 0 to the radius of the enclosing layer of molecules,
whereas b 1 (and in general, b i ) is the ratio of distance of the layer from the central
axis of the pore to the radius of the enclosing layer of molecules. Note that in the
following discussion the layers are numbered in increasing order starting from
the outermost layer to the innermost.
One can write the equation for the energy potential for the ith layer (i> 1)
as follows:

∞ ∞
3 N A A A 21 10 2k 4 2k
ε i = π a i α k (b ) − a i β k (b ) (4.29)
i
i
4
4 d A 32 k=0 k=0
and the constants a i and b i are given by:
d A
a i =
L − d 0 − (i − 2)d A
L − d 0 − (i − 1)d A
b i = (4.30)
L − d 0 − (i − 2)d A
The maximum number of molecules of diameter d that can be accommodated
with their centers along the circumference of a circle of diameter D is given by
N = int[p/sin −1 (d/D)]for d ≤ D and N = 1for d> D.For the i th layer of
adsorbate molecules within the pore, the molecules lie along a circle of diameter
2[L − d 0 − (i − 1)d A ], and hence the maximum number of molecules of diameter
d A that are present in a horizontal cross section of the i th layer (when d A ≤
2[L − d 0 − (i − 1)d A ] can be written as:

π
N i = (4.31)
−1 d A
sin
2(L − d 0 − (i − 1)d A )
However, if d A > 2[L − d 0 − (i − 1)d A ], N i = 1.