The Fowler-Guggenheim Equation

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 Fowler-Guggenheim equation exhibits phase-transition loops when Z(o/RT 4. The values of the coverage limits of the isotherm step and the corresponding step position, p/K, have been tabulated. 

Fowler and E. A. Guggenheim, Statistical Thermodynamics , Cambridge University Press, 

Broekhoff and R. H. van Dongen, Physical and Chemical Aspects of Adsorbents and 

Catalysts , ed. B. G. Linsen, Academic Press, London and New York, 1970, Ch. 2. 

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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). 

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. 

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... 

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

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

When n = 1, the Nitta s equation (2.4-6) is reduced to the Fowler-Guggenheim equation (2.3-29). In the case of no adsorbate-adsorbate interaction, that is u = 0, we have the following isotherm ... 

The Langmuirian equation does not allow for the interaction between adsorbed molecules. To allow for this, we need to consider an equation that does so such as the quasi-chemical treatment of Fowler-Guggenheim. The Fowler-Guggenheim equation has the form ... 

For pressures lower than the condensation pressure P., the fractional loading is very low and taking the limit of the Fowler-Guggenheim equation at low pressure we get... 

Other forms of the local isotherm can be used to allow for the adsorbate-adsorbate interaction, such as the Fowler-Guggenheim equation ... 

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 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]... 

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... 

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). 

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. ... 

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. 

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. 

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. ... 

Since there are many fundamental equations which can be derived from various equations of state, we will limit ourselves to a few basic equations such as the Henry law equation, the Volmer, the Fowler-Guggenheim, and the Hill-de Boer equation. Usage of more complex fundamental equations other than those just mentioned needs justification for doing so. 

To calcnlate the adsorption energy distribution functions, the Fowler-Guggenheim (FG) equation was nsed to describe localized monolayer adsorption with lateral interactions ... 

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