Page 113 - Adsorption by Powders and Porous Solids

98 ADSORPTION BY POWDERS AND POROUS SOLIDS

Here the pre-exponential factor, K, is equal to the ratio of the adsorption and desorp
tion coefficients, alp. Alternatively, b may be regarded as a function of the enthalpy
and entropy of adsorption (Everett, 1950; Barrer, 1978, p. 117).
In his early treatment, Langmuir assumed that the energy of adsorption for the first
layer is generally considerably larger than for the second and higher layers, and
therefore multilayer formation is possible only at much greater pressures than the
pressure required for monolayer completion. Thus, the formation of second or higher
layers would be indicated by the appearance of a discontinuous isotherm. In fact, this
situation does arise in the case of a stepwise, Type VI, isotherm. However, the lateral
adsorbate-adsorbate interactions, which are associated with all known stepwise
isotherms, are not compatible with the Langmuir model.
It is evident that Equation (4.1 1) is of a very general mathematical form (i.e. a
hyperbolic function). At low 0 it reduces to Henry's law; at high surface coverage, a
plateau is reached as 0+ 1. Other equations of the same mathematical form as
Equation (4.1 1) have been derived from a classical thermodynamic standpoint
(Brunauer, 1945) and by application of the principles of statistical mechanics
(Fowler, 1935).
Equation (4.1 1) is usually applied in the linear form

pin = 11% b +pin, (4.13)
where n is the specific amount of gas adsorbed at the equilibrium pressure p and n,
is the monolayer capacity (as before, 8 = n/n,).
Many systems give linear plots of pin against p over a limited ranges of pressure,
but such linearity does not by itself imply conformity with the Langmuir model. As
already indicated, a second condition is that the energy of adsorption should be inde-
pendent of surface coverage. Thirdly, the differential entropy of adsorption should
vary in accordance with the ideal localized model (Everett, 1950). That no real
system has been found to satisfy all these requirements is not surprising in view of
the complexities noted here and in subsequent chapters.
Various attempts have been made to modify the Langmuir model. One of the best
known is that of Fowler and Guggenheim (1939), which allowed for adsorbate-
adsorbate interactions in a localized monolayer on a uniform surface. However, on an
empirical basis the Fowler-Guggenheim equation turns out to be no more successful
than the original Langmuir isotherm. The highly complex problem of localized
adsorption on heterogeneous surfaces has been discussed by Rudzinski and Everea
(1992).

4.2.4. The Brunauer-Emmett-Teller (BET) theory
By introducing a number of simplifying assumptions, Brunauer, Emmett and Teller
(1938) were able to extend the Langmuir mechanism to multilayer adsorption and
obtain an isotherm equation (the BET equation), which has Type I1 character. The
original BET treatment involved an extension of the Langmuir kinetic theory of
monomolecular adsorption to the formation of an infinite number of adsorbed layers.
According to the BET model, the adsorbed molecules in one layer can act as