Fowler-Guggenheim equation

In the case of the patchwise adsorbent, the mean field approximation leads to the set of equations (Fowler-Guggenheim type) representing adsorption equilibrium on the individual patches... 

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... 

Even with the simpler Frumkin or Fowler-Guggenheim approach (Eqs. 6.50 and 6.52), treating the coadsorption and surface reaction of different adsorbates leads immediately to mathematically intractable expressions and to the introduction of new parameters, whereas equation... 

This isotherm is consistent with the modified electrochemical Langmuir isotherm, the Nemst equation and the potential-work function equivalence. For intermediate 0j and Pj values the isotherm of Eq. (6.58) is well approximated both by the Fowler-Guggenheim and by the Temkin isotherms. 

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). 

The Frumkin Equation (also referred to as the Frumkin-Fowler-Guggenheim, FFG, equation) has been specifically developped to take lateral interactions at the surface into account. In the FFG equation, the term 6 / (1-6) in (4.10b) is multiplied by the factor exp(-2 a0) which reflects the extent of lateral interactions... 

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. 

The adsorption of surfactants on solid substrates may be described by the Frumkin-Fowler-Guggenheim equation,... 

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

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. 

Finally, it is necessary to introduce the Fowler-Guggenheim isotherm equation... 

Computations were carried through for values of 0.05 < 0 < 0.95 in increments of 0.05 unit, with C — 2, 3, 4, and 5. It was assumed that lateral interactions were due to attractive van der Waals-London dispersion forces, where the leading term in the energy expansion varies with distance as r-1/6 with R = V2 one finds C = C1/8. Calculations were also carried out in the Fowler-Guggenheim approximation this simply requires the determination of the zero-order inputs Po(a 0), Pj(b °K and P/P°. The results are exhibited in Figures 2 and 3 the broken curves refer to isotherms calculated according to Equations 22 and 23. 

The second type of isotherm which commonly occurs arises when the lateral interactions between adsorbed molecules are taken into account or the Van der Waals type of equation is used to describe the state of the adsorbate (J7). The equations for these isotherms usually have the form u = f(v), for example, the Fowler-Guggenheim isotherm studied with some care in the earlier work (2. ... 

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 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]. 

At high surface coverage (0>O.l) the lateral interaction between the chains must be taken into account, by introducing a constant A, for example using the Frumkin-Fowler-Guggenheim (FFG) equation [17] ... 

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... 

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 fundamental equations, Langmuir, Volmer, Fowler-Guggenheim and Hill de Boer, will form a basis for the study of heterogeneous adsorbents as we shall discuss briefly in Chapter 3 and in further detail in Chapter 6. 

This equation is known in the literature as the Fowler-Guggenheim equation, or the quasi approximation isotherm. This equation can also be derived from the statistical thermodynamics (Rudzinski and Everett, 1992). Due to the lateral interaction term exp(-c0), the Fowler-Guggenheim equation and the Hill-de Boer... 

Fowler-Guggenheim equation (2.3-29) is one of the simplest equations allowing for the lateral interaction. Before discussing the two dimensional condensation phenomenon, we first investigate the isosteric heat behaviour. 

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

Similarly for the Hill-de Boer equation, we obtain the same isosteric heat of adsorption as that for the case of Fowler-Guggenheim equation. This is so as we have discussed in the section 2.3.3 for the case of Volmer equation that the mobility of adsorbed molecule does not influence the way in which solid interacts with adsorbate. 

Then the Fowler-Guggenheim equation will take the form ... 

The following algorithm describes the behaviour of the Fowler-Guggenheim equation. 

Similar analysis of the Hill-deBoer equation (2.3-27) shows that the two dimensional condensation occurs when the attraction between adsorbed molecules is strong and this critical value of c is 27/4. The fractional loading at the phase transition point is 1/3, compared to 1/2 in the case of Fowler-Guggenheim equation. A computer code Hill.m is provided with this book for the calculation of the fractional loading versus pressure for the case of Hill-de Boer equation. Figure 2.3-4 shows plots of the fractional loading versus nondimensional pressure (bP) for various values of c= 5, 7, 10, 15. ... 

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 ... 

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