Page 90 - Adsorbents fundamentals and applications

THE INTEGRAL EQUATION APPROACH 75

function. The limits for the integration can be replaced by the upper and lower
bounds for the pore sizes.
Equation 4.33 is the Fredholm integral equation of the first kind. Except for
a few special cases, no solution for G(x) exists (Tricomi, 1985). Numerical
solutions can be used. To solve for G(x) from a given q(P ), one needs to have
an individual pore isotherm, which must be related to the pore size. Moreover,
the integral equation is ill-defined, that is, the solution for G(x) is not unique,
unless a functional form for G(x) is assumed. It is clear then that there are as
many solutions for the PSD as the number of assumed functional forms.
Earlier work focused on analytical solutions for the PSD function (Stoeckli,
1977; Jaroniec and Madey, 1988; Stoeckli, 1990; Rudzinski and Everett, 1992).
They remain useful because of their simplicity. An outline of the solution by
Jaroniec and Choma (Jaroniec et al., 1991) is given next.
For a homogeneous or nearly homogeneous microporous material, the Dubinin-
Radushkevich isotherm (Eq. 3.9) (Dubinin and Radushkevich, 1947) is applicable:

RT P 0
2
V = V 0 exp −B ln = f(B, P ) (4.34)
β P
where V is the amount adsorbed at relative pressure P/P 0 , β is the affinity
coefficient, and B is a structural parameter, which increases with an increase in
pore size. A semi-empirical relationship was found between B and the half-width
x of the slit-like micropores (Dubinin and Stoeckli, 1980; Jaroniec and Madey,
1988; Stoeckli, 1990; Chen and Yang, 1996):

B = kx 2 (4.35)

where the constant k is characteristic of the sorbent material. The value of k can
be obtained by using a limiting probe molecule (Baksh et al., 1992). A theoretical
basis has been given for this relationship by Chen and Yang (1996). Jaroniec and
co-workers used a Gamma-type pore size distribution function:

2(ku) n+1 2n+1 2
G(x) = x exp(−kux ) (4.36)
(n + 1)
where u is a constant. Upon substitution of this distribution function into Eq. 4.33,
Jaroniec obtained the following solution, referred to as Jaroniec–Choma isotherm:

  n+1
V  u 
  (4.37)
2
=  

V 0  RT P 0 
u + ln
β P