Fowler and Guggenheim

The following derivation is modified from that of Fowler and Guggenheim [10,11]. The adsorbed molecules are considered to differ from gaseous ones in that their potential energy and local partition function (see Section XVI-4A) have been modified and that, instead of possessing normal translational motion, they are confined to localized sites without any interactions between adjacent molecules but with an adsorption energy Q. 

In the Lewis and Gibson statement of the third law, the notion of a perfect crystalline substance , while understandable, strays far from the macroscopic logic of classical thennodynamics and some scientists have been reluctant to place this statement in the same category as the first and second laws of thennodynamics. Fowler and Guggenheim (1939), noting drat the first and second laws both state universal limitations on processes that are experunentally possible, have pointed out that the principle of the unattainability of absolute zero, first enunciated by Nemst (1912) expresses a similar universal limitation ... 

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). 

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

R. H. Fowler and E. A. Guggenheim [Statistical Thermodynamics (Cambridge University Press, Cambridge, 1939)] criticized this statement as well as similar statements (to be quoted below) which imply that the entropy of perfect crystalline substances is zero. According to Fowler and Guggenheim, the only valid third-law inference is the unattainability of absolute zero, as expressed in the following statement ... 

Fowler and Guggenheim (1939) It is impossible to reduce the temperature of any system to absolute zero in a finite number of steps. 

The Lennard-Jones-Devonshire theory (as summarized by Fowler and Guggenheim, 1952, pp. 336ff) averaged the pair potentials of Equation 5.24a and b between the solute and each water, for Zi molecules in the surface of the spherical cavity to obtain a cell potential r) of... 

This is Fowler and Guggenheim s crude approximation (40). As has been shown before (38), the equation for equilibrium then becomes... 

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] ... 

Solutions that satisfy conditions 3 and 4 and that show an ideal entropy effect and no volume effect on mixing but do show a thermal effect are called regular solutions (Hildebrand, Fowler and Guggenheim). In ideal solutions the thermal effect is also absent. 

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 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 function may be calculated from molecular data by tlie methods of quantum statistical mechanics. It may also be obtained from experimentally determined heat capacity data, from which the required entropies are deduced using the third law of thermodynamics. The calculation may be put into several equivalent forms, and is discussed in the text-books. An account of the theory is given by Fowler and Guggenheim 7 its application by... 

Fowler and Guggenheim (1939) have shown that the chemical potential or partial free energy y of an uncharged substance distributed at random among a limited number of similar sites is given by... 

For a definition of long-range and short-range order see Fowler and Guggenheim [20], chap. XIII. 

---