Fowler-Guggenheim model

The two-component Fowler-Guggenheim model can be given as follows [3] ... 

Over 40 years ago, Kiselev [30] presented an interesting concept of the associating adsorbate. He assumed that all interactions in the monolayer might be described as a series of reversible quasichemical reactions between admolecules and adsorption sites and between adsorbate molecules in the monolayer. These interactions were characterized by means of suitable reaction constants. This theory was extended by Berezin and Kiselev [31] their final isotherm involves dispersive interactions according to the Fowler-Guggenheim model and specific interactions which cause formation different associates in the surface phase. 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

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 ... 

The Fowler-Guggenheim-Jovanovic model [3] assumes (as it was the earlier case also) the occurrence of intermolecular interactions among the molecules adsorbed as a monolayer but is based on the Jovanovic isotherm. The single-component isotherm is represented by the equation ... 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

FIGURE 2.23 Peak profiles calculated for the Fowler-Guggenheim isotherm model for % = 3 and the concentrations of 2, 1.5, 1, and 0.5 mol L (peaks from the largest to the smallest, respectively). 

Bumble and Honig (I) have discussed the HB procedure as applied to gas adsorption and the approximations inherent in the model used below. Honig (6) gives additional background information, where the Fowler-Guggenheim isotherm equation is derived in an elementary fashion. 

The statistical thermodynamic approach, along lines already indicated, has been more tractable and suggestive. Models have been based on the Fowler-Guggenheim treatment of localized monolayers, in which account is taken of energy terms arising from interaction between point defects in nearest neighbor... 

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 Everett (1992). 

In practice, most models are based on the assumption of localization, the reasoning being that many substances adhere rather strongly to surface sites. As a consequence, frequently used Individual isotherms Include those of Langmuir or Frumkin-Fowler-Guggenheim (FFG) but not those of Volmer or Hlll-De Boer (see app. 1). 

Another model, which takes into account lateral interaction and surface heterogeneity, is the Fowler-Guggenheim-Jovanovic isotherm. 

Figure 4.17 Competitive experimental isotherm data for 2-phenylethanol and 3-phenylpropanol fitted to the competitive Fowler-Guggenheim/Langmuir-Freundlich model. Reproduced from I. Quinones, G. Guiochon, J. Chromatogr. A, 796 (1998) 15 (Figs. 3 and 4).

The Langmuir and Volmer equations are special cases of the Fowler-Guggenheim and Hill-de Boer equations, respectively, in which lateral interactions are allowed to vanish the Brunauer-Emmett-Teller equation is a special case of the Broekhoff-van Dongen equation with n = oo and null lateral interactions the model in which all layers are mobile is a special case of Broekhoff-van Dongen model with n = 0. 

Many different equations have been used to interpret monolayer—multilayer isotherms [7, 11, 18, 21, 22] (e.g., the equations associated with the names Langmuir, Vohner, HiU-de Boer, Fowler-Guggenheim, Brunauer-Emmett-Teller, and Frenkel-Halsey-Hill). Although these relations were originally based on adsorption models, they are generally applied to the experimental data in an empirical manner and they all have Hmitations of one sort or another [7, 10, 11]. 

The greatest barrier in the application of the Multicomponent Fowler-Guggenheim or Bragg-Williams Lattice gas model to, a practical situation like Pet-reforming, is the absence of experimental interaction parameters. In the simulations of the earlier sections, representative values were used. In general, for an n component system, we need to fix n(n+l) / 2 interaction parameters of the symmetric W matrix (91 for a 13 component Model ). Mobil has used successfully a 13 lump KINPTR model(5), which essentially uses a Hougen-Watson Langmuir-Hinshelwood approach. This results in a psuedo-monomolecular set of reactions, which is amenable to matrix analysis. 

Next, we will discuss one of the recent equations introduced by Nitta and his co-workers. This theory based on statistical thermodynamics has some features similar to the Langmuir theory, and it encompasses the Langmuir equation as a special case. Basically it assumes a localised monolayer adsorption with the allowance that one adsorbate molecule can occupy more than one adsorption site. Interaction among adsorbed molecules is also allowed for in their theory. As a special case, when the number of adsorption sites occupied by one adsorbate molecule is one, their theory is reduced to the Fowler-Guggenheim equation, and further if there is no adsorbate-adsorbate interaction this will reduce to the Langmuir equation. Another model of Nitta and co-workers allowing for the mobility of adsorbed molecules is also presented in this chapter. 

The first such solutions were carried out by Ross and Olivier [1, p. 129 6,7]. Using Gaussian distributions of adsorptive potential of varying width, they computed tables of model isotherms using kernel functions based on the Hill-de Boer equation for a mobile, nonideal two-dimensional gas and on the Fowler-Guggenheim equation [Eq. (14)] for localized adsorption with lateral interaction. The fact that these functions are implicit for quantity adsorbed was no longer a problem since they could be solved iteratively in the numerical integration. 

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... 

The Langmuir approach was a starting point for developing the more realistic formalism in the framework of the lattice gas theories based on the Ising model [24]. It seems intuitively obvious that the lattice gas model is well suited for representing localized adsorption. The adsorbed phase is considered a two-dimensional lattice gas. The most popular isotherm involving molecular interaction effects is the Fowler-Guggenheim equation [25]... 

Apart from the Fowler-Guggenheim local adsorption isotherm, the Berezin-Kiselev equation has been extended to adsorption on heterogeneous surfaces by Jaroniec and Borowko [89]. Their results have an instructive character the method allow us to investigate the influence of various geometrical distributions of active sites on adsorption in the fi amework of simple and clearly constructed model. They have considered localized monolayer adsorption on the siuface consisting of two types of adsorption site. The lateral interactions caused the formation of double associates. The total adsorption was the sum of the surface coverage on adsorption sites of both kinds, which may be calculated fi om... 

Model of Fowler Guggenheim s quasi-chemical solution... 

Statistical Thermodynamic Isotherm Models. These approaches were pioneered by Fowler and Guggenheim (21) and Hill (22). Examples of the appHcation of this approach to modeling of adsorption in microporous adsorbents are given in references 3, 23—27. Excellent reviews have been written (4,28). 

The Langmuir model was extended to include interaction between the adsorbed atoms/molecules by Fowler and Guggenheim [31], The model now becomes... 

The nature of approximations involved in the derivation of the LPBE remains obscure (2), and after the analyses of Fowler (12), Onsager (13), and Kirkwood (14), it appears that no more can be learned about them from a statistical-mechanical argument. Following the early pronouncement of Guggenheim and Fowler (15), supported by other analysis (4), we consider the LPBE the only logical choice for a model in which the ions obey the laws of electrostatics. 

For an evaluation of the local model isotherm 6(p,T,Q) with constant interaction energy Q, the effects of multi-layer adsorption and lateral interactions between neighboring adsorbed molecules are considered by applying two modifications to the Langmuir isotherm (i) a multi-layer correction according to the well known BET-concept and (ii) a correction due to lateral interactions with neighboring gas molecules introduced by Fowler and Guggenheim (FG) [105] ... 

The model used (independently) by Floiy and Huggins (following an earlier suggestion by Fowler and Guggenheim) assumes a large molecule such as a polymer can be treated as a set of linked segments. Each of these segments is not necessarily equal in size to the chemical repeat unit, but defined to have a molar volume equal to that of the solvent. 

This is a model of a strictly regular solution as used by R. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, New York, 1949. 

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