Model Vacancy Solution

Not all of the isotherm models discussed in the following are rigorous in the sense of being thermodynamically consistent. For example, specific deficiencies in the Freundhch, Sips, Dubinin-Radushkevich, Toth, and vacancy solution models have been identified (14). 

Isotherm Models for Adsorption of Matures. Of the following models, all but the ideal adsorbed solution dieory (IAST) and the related heterogeneous ideal adsorbed solution theory (H1AST) have been shown to contain some thermodynamic inconsistencies. They include Markham and Benton, die Leavitt loading ratio correlation (LRC) method, lire ideal adsorbed solution (IAS) model, the heterogeneous ideal adsorbed solution theory (HIAST), and the vacancy solution model (VSM). 

Derylo-Marczewska and Jaroniec [28] have reviewed the adsorption of organic solutes from dilute solutions and have provided a useful compilation of published experimental data for both single- and multisolute adsorption isotherms on carbonaceous adsorbents. They also presented a survey of theoretical approaches used to describe the solute adsorption equilibria, including the Polanyi adsorption model, the solvophobic interaction model, the Langmuir adsorption theory, the vacancy solution model, as well as considerations based on the energetic heterogeneity of the adsorbent. In particular, these authors emphasize the... 

The fifth approach is more a field than a concise method, since it embodies so many theoretical concepts and associated methods. All are grouped together as adsorbed mixture models. Basically, this involves treating the adsorbed mixture in the same manner that the liquid is treated when doing VLE calculations. The major distinction is that the adsorbed phase composition cannot be directly measnred (i.e., it can only be inferred) hence, it is difficult to pursue experimentally. A mixture model is nsed to account for interactions, which may be as simple as Raoult s law or as involved as Wilson s equation. These correspond roughly to the Ideal Adsorbed Solution theory and Vacancy Solution model, respectively. Pure component and mixture equilibrium data are required. The unfortunate aspect is that they require iterative root-finding procedures and integration, which complicates adsorber simnlation. They may be the only route to acceptably accurate answers, however. It would be nice if adsorbents could be selected to avoid both aspects, but adsorbate-adsorbate interactions may be nearly as important and as complicated as adsorbate-adsorbent interactions. 

Vacancy Solution Model. The initial model (37) considered the adsorbed phase to be a mixture of adsorbed molecules and vacancies (a vacancy solution) and assumed that nonidealities of the solution can be described by the two-parameter Wilson activity coefficient equation. Subsequendy, it was found that the use of the three-parameter Flory-Huggins activity coefficient equation provided improved prediction of binary isotherms (38). 

The vacancy solution model is applied with good success in many systems, and it is readily extended to multicomponent systems because the inherent feature of this model is the interaction between gas molecules and the vacancies. 

The two VSM isotherm equations are given in eqs. (2.7-5) and (2.7-7) depending on whether the Wilson equation or the Flory-Huggin equation is used to calculate the activity coefficient. Observing the form of these equations, the vacancy solution model equation can be written in general form as follows ... 

The vacancy solution theory or vacancy solution model (VSM) was proposed by Suwanayuen and Danner [6,7] and later modified by Cochran et al. [8], The isosteric heat of adsorption of this model was obtained by Talu and Kabel [9]. In this theory. 

TABLE 4.6. Systems to which Vacancy Solution Model Has Been Successfully Applied... 

For these systems the deviations of the binaries from the predictions of the vacancy solution model are significant and the deviations of the ternaries arc even more pronounced. ... 

Talu, 0., and Kabel, R.L., Isosteric heat of adsorption and vacancy solution model, AlChE J., 33(3), 510-514... 

The vacancy solution model was extended to describe mixture adsorption isotherm (Suwanayuen and Danner, 1980). 

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