Lattice solution sites

NH3 is similar to H2O in that they both possess large dipole moments and are both small molecules. The presence of NH3 in a zeolite is chemically similar to the presence of H2O in a zeolite. Therefore, the hydrated cation distribution in zeolites is probably more typical of NH3 adsorption in zeolites than the dehydrated cation distribution. According to Breck (18), for hydrated zeolite X, cations are found in sites SI, SI, SII, and SIV. Of these sites, SI, SII, and SIV would all be adsorption lattice solution sites. The cationic and anionic lattice solution sites (in the supercavity of NaX) are illustrated in Figure 8. For NH3, the subscript J1 will refer to SII sites, the subscript J2 will refer to SI sites, and J3 will refer to SIV sites. The anionic sites are two and are (l) in the center U-membered ring of the connecting frame and (2) near the center of the 0(2)—0(1)—0(l) triad of oxygen atoms. For NH3, the subscript il will refer to the first anionic site the subscript i2 will refer to the second anionic site. 

Flory ) and Huggins ) derived a now classical mean-field expression for the configurational entropy and energy of mixing, using a lattice model. The solution, containing moles of solvent and n moles of polymer, is described as a lattice of sites, of which are occupied by solvent and N

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Figure 35.5 Lattice model (schematic, in two dimensions here) for polymer molecule in a solution. Sites not occupied by polymer segments are occupied by solvent molecules (one per site). (From T. L. Hill, Introduction to Statistical Mechanics. Reading, Mass. Addison-Wesley, 1960.)...

In order to describe the collapse of a long-chain polymer in a poor solvent, Flory developed a nice and simple theory in terms of entropy and enthalpy of a solution of the polymer in water [14]. In order to obtain these two competing thermodynamic functions, he employed a lattice model which can be justified by the much larger size of the polymer than the solvent molecules. The polymer chains are represented as random walks on a lattice, each site being occupied either by one chain monomer or by a solvent molecule, as shown in Figure 15.8. The fraction of sites occupied by monomers of the polymer can be denoted as 0, which is related to the concentration c, i.e., the number of monomers per cm by 0 = ca, where is the volume of the unit cell in the cubic lattice. Though the lattice model is rather abstract, the essential features of the problem are largely preserved here. This theory provides a convenient framework to describe solutions of all concentrations. 

The pH dependence of the rate constants proves that chemical bonds of Fe atoms with solution species contribute to a determining extent to the generation of active sites of dissolution, even in atomistic models where surfece lattice features (kinks, steps, terraces, etc.) are generally put forward. Large similarities of glassy metals (amorphous) with crystalline ones [83,61] must also be regarded as arguing in fevor of chemical bonds as predominant entities with respect to lattice related sites. 

Wilhoul solute (t) OJ, Mole Iraclion of solvetil in lattice surface sites 1 so, Mole fraction 1 of solid surface 1 molecules... 

With solute fe) Sm, Mole traction of solvent in lattice surface sites 1 Mole fraction 1 of solute—solid [ surface complexes... 

Acrfv coefficient quotients Without solute fl, Activity coetticient of solvent molecule in lattice surface sites [ /so. Activity 1 coefficient of solid [ surface molecules... 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

There is nothing unique about the placement of this isolated segment to distinguish it from the placement of a small molecule on a lattice filled to the same extent. The polymeric nature of the solute shows up in the placement of the second segment This must be positioned in a site adjacent to the first, since the units are covalently bonded together. No such limitation exists for independent small molecules. To handle this development we assume that each site on the lattice has z neighboring sites and we call z the coordination number of the lattice. It might appear that the need for this parameter introduces into the model a quantity which would be difficult to evaluate in any eventual test of the model. It turns out, however, that the z s cancel out of the final result for, so we need not worry about this eventuality. 

The logic that leads us to this last result also limits the applicability of the ensuing derivation. Applying the fraction of total lattice sites vacant to the immediate vicinity of the first segment makes the model descriptive of a relatively concentrated solution. This is somewhat novel in itself, since theories of solutions more commonly assume dilute conditions. More to the point, the model is unrealistic for dilute solutions where the site occupancy within the domain of a dissolved polymer coil is greater than that for the solution as a whole. We shall return to a model more appropriate for dilute solutions below. For now we continue with the case of the more concentrated solution, realizing... 

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