## N

Nm where O(x3) denotes the rest of the order of x3. Equation (6.21) indicates that, at relative pressures where the contributions of order x2 become important, the isotherms on fractal and on planar surfaces begin to differ. This regime is important for the determination of the monolayer capacity Nm.

For any value of the relative pressure x, the amount of adsorbed molecules per surface site, 0 = N/Nm, decreases with increasing Ds. This occurs because, for a given number of surface sites, increasing Ds imposes increasing spatial restrictions on the growth of the film thickness. When the pressure approaches the saturated vapor pressure, i.e. for p ^ p0 or for x ^ 1, the asymptotic behavior of the adsorption isotherms is described by the power law 

— -- for 2 < Ds < 3 and — = ln[1/(1 - x)] for Ds = 3

where T is the gamma function. The divergence of the isotherm for p ^ p0 means that the fractal surface, similarly to a geometrically flat surface adsorbs infinitely thick layers. For Ds = 3, the divergence is logarithmic.

Equation (6.17) suffers from the deficiency that during its derivation, 'multiple-surface' effects were neglected. Nonetheless, for the filling of a slit-like pore, the film grows from two opposite adsorbing surfaces and stops growing when the two films meet. A BET-like theory that takes into account multiple-surface effects has been developed by Cole et al. . Such an approach replaces the Ds-dimensional surface by an equivalent system of pores, each of which can be treated as independent. In each pore, coverage can be evaluated from standard BET theory for adsorption between two plates by assuming that the maximum number of adsorbed layers «max and the pore width H are related via 2ffnmax = H. In other words, adsorption in a single pore is given by

The total number of adsorbed molecules is calculated by summing up the contributions from individual pores:

0 0