Adsorption isotherms random distribution

There are several isotherm models for which the isotherm shapes and peak prohles are very similar to that for the anti-Langmuir case. One of these models was devised by Fowler and Guggenheim [2], and it assumes ideal adsorption on a set of localized active sites with weak interactions among the molecules adsorbed on the neighboring active sites. It also assumes that the energy of interactions between the two adsorbed molecules is so small that the principle of random distribution of the adsorbed molecules on the adsorbent surface is not significandy affected. For the liquid-solid equilibria, the Fowler-Guggenheim isotherm has been empirically extended, and it is written as ... 

For a random distribution of adsorption sites, statistical thermodynamics gives the following equation for the local isotherm [77,78] ... 

Q)A on random surfaces or g = (1 - 20 <)/4 on patchwise surfaces. At a random surface the sites of different adsorption energy are randomly distributed over the lattice, whereas at a patchwise surface the sites of equal energy are present in groups. The total adsorption isotherm may be calculated from... 

In conclusion, the lifetime analysis of the experimental data of Drazer and Zanette provided important clues regarding the kinetics and mechanism of the desorption process. In particular, the knowledge of the fractal exponents of the adsorption isotherm and of the tail of the lifetime distribution was enough to elucidate the shape and structure of the activation energy barrier. We have also shown that the experimental data indicate that the adsorption rate and the adsorption activation energies are constant and only the desorption rate and desorption activation energy are random. The application of the method may involve detailed theoretical developments for different systems nevertheless we expect that the results are worth the effort. 

As a first-order deviation from the Langmuir model one may consider ideal adsorption on a set of localized sites with weak interaction between adsorbed molecules on neighboring sites. Such a model has been investigated theoretically by Lacher and by Fowler and Guggenheim. If the interaction is sufficiently weak that the random distribution of the adsorbed molecules is not significantly affected the resulting expression for the isotherm is... 

The Fowler-Guggenheim Equation. This local isotherm is based on a localized model of adsorption but includes average nearest-neighbour interactions. This is handled on the basis of a random distribution of atoms among the... 

It is well known that in the hterature there are more than 100 isotherm equations derived based on various physical, mathematical, and experimental considerations. These variances are justified by the fact that the different types of adsorption, solid/gas (S/G), solid/liquid (S/L), and liquid/gas (L/G), have, apparently, various properties and, therefore, these different phenomena should be discussed and explained with different physical pictures and mathematical treatments. For example, the gas/solid adsorption on heterogeneous surfaces have been discussed with different surface topographies such are arbitrary, patchwise, and random ones. These models are very useful and important for the calculation of the energy distribution functions (Gaussian, multi-Gaussian, quasi-Gaussian, exponential) and so we are able to characterize the solid adsorbents. Evidently, for these calculations, one must apply different isotherm equations based on various theoretical and mathematical treatments. However, as far as we know, nobody had taken into account that aU of these different isotherm equations have a common thermodynamical base which makes possible a common mathematical treatment of physical adsorption. Thus, the main aim of the following parts of this chapter is to prove these common features of adsorption isotherms. 

The molecular interactions in the surface films formed on heterogeneous adsorbents have been taken into account in the studies of munerous scientists [5]. The Fowler-Guggenheim local adsorption isotherm has been used for the patchwise and random topographies for different energy distributions. The systematic analysis of the results can be found in Refs. 6-8. 

Using Eqs. (75) [or Eq. (76)] and (74), we can easily obtain the adsorption isotherm assumed for the adsorption energy distribution. In the framework of the mean field approximation expressions for any thermodynamic quantity (e.g. internal energy, heat capacity) can be readily derived [234]. Adsorption on randomly heterogeneous surfaces has been studied in terms of the above-described approach. It has been demonstrated that this mean-field-type theory was valid only at very high temperatures. Below the critical two-dimensional temperature, the predictions of theory seriously underestimate the heterogeneity effects on phase transitions in adsorbed monolayers [12,234],... 

The individual variants of the lattice model differ fi om each other in the way the spatial distribution of the molecules of the individual components is taken into account. The simplest solution is the Bragg-Williams (B-W) approach which assumes a random distribution of molecules within the bulk phase. The thermodynamical meaning of this assumption is that the mixture is regular. In the adsorption layer, however, it is only in two dimensions (i.e., within the individual sublayers that a statistical distribution of molecules is assumed). Pioneering work in this field was published by Ono [92-94] and Ono and Kondo [95,96]. The method was later applied to the description of L/G interfaces by Lane and Johnson [97] and later taken up by Altenberger and Stecki [98]. Analytic isotherm equations have also been derived from the above... 

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