Page 127 - Adsorption by Powders and Porous Solids

112 ADSORPTION BY POWDERS AND POROUS SOW

Equation (4.47) and obtain an isotherm equation in which the distribution function,
f(B) was expressed in an analytical form (Huber et al., 1978; Bansal et al., 1988). I,,
principle, f(B) provides an elegant basis for relating the micropore size distribution to
the adsorption data. However, it must be kept in mind that the validity of the approaa
rests on the assumption that the DR equation is applicable to each pore group and ~lat
there are no other complicating factors such as differences in sugace heterogeneity.

4.4.5. Empirical isotherm equations
Many different equations have been applied to physisorption isotherms on mi-
porous adsorbents. The first and best known empirical equation was proposed by
Freundlich (1926) in the form
= kp'lm (4.49)
where k and m are constants (m > 1).
According to Equation (4.49), the plot of In In] against In [p] should be linear. In
general, activated carbons give isotherms which obey the Freundlich equation in the
middle range of pressure (Brunauer, 1945). but the agreement is usually poor at high
pressures and low temperatures. These limitations are partly due to the fact that the
Freundlich isotherm does not give a limiting value of n as p -+ =.
It is possible to achieve an improved fit at higher pressures by a combination of the
Freundlich and Langmuir equations (Sips, 1948),
n/nL = (kp) 'lm/[l + (kp)'lm] (4.50)
where n, is the limiting adsorption capacity. Equation (4.50) has been applied as a
'generalized Freundlich' isotherm to multisite occupancy by long-chain hydrocar-
bons (Rudzinski and Everett, 1992, p. 491). However, as in the case of the Freundlich
isotherm itself, the Sips equation does not reduce to Henry's law as y -+ 0.
Another empirical variant is Toth equation,
n/n, = p/(b +pm)'lm (4.5 1)
which also contains three adjustable parameters (n,, b and m), but has the advantage
that it appears to give the correct limits for both p -+ 0 and p -+ m. Thus. although it
was originally proposed for monolayer adsorption (Toth, 1962), the Toth equation
actually gives a more extensive range of fit when applied to Type I isotherms
(Rudzinski and Everett, 1992).
In spite of the availability of modem computer-aided techniques for curve fitting,
these and other relatively simple, empirical equations are still found to be useful for
the analysis of chemical engineering data (e.g. in the context of pressure swing
adsorption or other separation processes).
There is a growing interest in the presentation of physisorption isotherms in a gen-
eralized integral form. This approach was first applied to physisorption in the sub-
monolayer region (Adamson et al., 1961), but much of the current interest is centred
on the analysis of micropore filling isotherms. An apparent advantage is that it
provides a means of constructing a series of model isotherms by mi sterna tic all^