Guggenheim model

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

The problem of standard states as well as a comparison of AG° values obtained from different isotherms was discussed by Torrent et Using the Guggenheim model of the interphase,they derived a relation between the chemical potentials of all components in the interphase and in the bulk that are required to describe the equilibrium state. 

In the Gibbs model all volume terms disappear. In the Guggenheim model one must also keep P constant during the integration. 

In order to use these equations it is necessary to devise a model which enables the surface excess concentration of species i in the interphase to be defined. The interphase itself is defined by enclosing it in two arbitrary planes positioned in such a way that the bulk phases extend homogeneously up to these planes. Any changes in the thermodynamic properties must occur between these planes (Fig. 5.3). If the thickness of the phase is known, then the concentration of species i can be mathematically expressed this is the Guggenheim model. An alternative approach due to Gibbs introduces a third arbitrary plane called the dividing surface which acts as a reference surface. The concentration of species i can then be expressed in terms of an excess or deficiency of component i at the reference surface with respect to its concentration in the bulk phase. 

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. 

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). 

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... 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

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). 

Thermodynamics of the ITIES was developed by several authors [2-6] on the basis of the interfacial phase model of Gibbs or Guggenheim. General treatments were outlined by Kakiuchi and Senda [5] and by Girault and Schiffrin [6]. At a constant temperature T and pressure p the change in the surface tension y can be related to the relative surface excess concentrations Tf " of the species i with respect to both solvents [6],... 

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

The regular solution model, originally introduced by Hildebrand [2] and further developed by Guggenheim [3], is the most used physical model beside the ideal... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

The method preferred in the NBA Thermochemical Data Base review is the specific ion interaction model in the form of the Bronsted-Guggenheim-Scatchard approach. 

The quasi-chemical model was derived by Guggenheim for application to organic fluid mixtures. Applying it to crystalline solids is not immediate, because it necessitates conceptual modifications of operative parameters, such as the above-mentioned contact factor. Empirical methods of derivation of the above parameters, based on structural data, are available in the literature (Green, 1970 Sax-ena, 1972). We will not treat this model, because it is of scanty application in geochemistry. More exhaustive treatment can be found in Guggenheim (1952) and Ganguly and Saxena (1987). 

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