Fowler-Guggenheim adsorption isotherm equation

Using the van t Hoff equation (2.2-11), we can obtain the following heat of adsorption for the Fowler-Guggenheim adsorption isotherm ... 

This equation is sometimes called the Frumkin-Fowler-Guggenheim (FFG) isotherm [374— 376], For j3 = nEP/RT < 4 lateral interactions cause a steeper increase of the adsorption isotherm in the intermediate pressure range. Characteristic of all Langmuir isotherms is a saturation at high partial pressures P/Po —> 1. 

Langmuir-Type and Fowler-Guggenheim-Type Adsorption Isotherm Equations... 

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

The local adsorption isotherm equations of the form Langmuir, Volmer, Fowler-Guggenheim and Hill-de Boer have been popularly used in the literature and are shown in the following Table 6.3-1. The first column shows the local adsorption equation in the case of patchwise topography, and the second column shows the corresponding equations in the case of random topography. Other form of the local isotherm can also be used, such as the Nitta equation presented in Chapter 2 allowing for the multisite adsorption. 

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... 

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). 

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. 

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). 

The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermod)mamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic fimctions are then derived from these partition fimctions, using the usual relationships of statistical thermodynamics. 

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]. 

We start this book with a chapter (Chapter 2) on the fundamentals of pure component equilibria. Results of this chapter are mainly applicable to ideal solids or surfaces, and rarely applied to real solids. Langmuir equation is the most celebrated equation, and therefore is the cornerstone of all theories of adsorption and is dealt with first. To generalise the fundamental theory for ideal solids, the Gibbs approach is introduced, and from which many fundamental isotherm equations, such as Volmer, Fowler-Guggenheim, Hill-de Boer, Jura-Harkins can be derived. A recent equation introduced by Nitta and co-workers is presented to allow for the multi-site adsorption. We finally close this chapter by presenting the vacancy solution theory of Danner and co-workers. The results of Chapter 2 are used as a basis for the... 

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. 

At first the BET equation was derived from the kinetic considerations analogous to those proposed by Langmuir while deriving the monomolecular adsorption isotherm. First, statistical thermodynamic derivation was carried out by Cassie [123]. Lately, a slightly modified derivation has been proposed by HiU [124-126], Fowler and Guggenheim [127]. 

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... 

For the derivation of Volmer s and Fowler Guggenheim isotherms from the gas phase equations of state and Gibbs adsorption isotherms, we will refer the reader to Ruthven (1984). 

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... 

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