Statistical Thermodynamic Approach

The parameters A, A, A, and A are known from the fit of the single-component isotherms but additional assumptions are required in order to estimate the cross coefficients A,.  

For many systems it has been found that the cross coefficients are related by 

In order to show the relationship with the Myers and Prausnitz treatment we ignore the vacancies and consider the adsorbed phase simply as a mixture of the adsorbed species. Equation. (3.69) becomes 

If we choose the standard state such that it = ir°, j/, = s/° we recover Eq. (3.49). The difference between the vacancy solution and the Myers and Prausnitz formulations is seen to lie in the choice of standard state and the way in which adsorbed phase concentrations are expressed. 

The approach to the modeling of adsorption equilibria outlined in Section 3.3 depends on the implicit assumption of a mobile adsorbed phase, the thermodynamic behavior of which can be represented by an equation of state. An alternative view is to consider the adsorbed molecules as localized at distinct surface sites, with only relatively infrequent movement of the molecules between sites. Each site may then be regarded as a separate subsystem and the equilibrium behavior may be deduced by the methods of statistical 

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The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

Statistical Thermodynamic Approach. Helgeson (y) has described the dissociation for complexes (such as for NH.,OH ". NH + OH ) in terms of two functions - an electrostatic temperature function and a non-electrostatic (dielectric) temperature function. The following equation has been suggested to obtain AGT for the dissociation of complexes in solution ... 

Delarue, M. and Qrland, H. (2000). General formalism for phase combination and phase refinement a statistical thermodynamics approach in reciprocal space. Actu Crystallogr. A 56, 562-574. 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

With the determination of hydrate structure, more rigorous predictive methods were formulated for hydrate thermodynamic property predictions. Barrer and Stuart (1957) initially suggested a statistical thermodynamic approach to determining gas hydrate properties. In a similar yet more successful approach,... 

Hydrate Nonstoichiometry. The cause of the nonstoichiometric properties of hydrates has been considered. Evidence for the view that only a fraction of the cavities need to be occupied is obtained from both the experimental observations of variation in composition, and the theoretical success of the statistical thermodynamic approach of van der Waals and Platteeuw (1959) in Chapter 5. Typical occupancies of large cavities are greater than 95%, while occupancy of small cavities vary widely depending on the guest composition, temperature, and pressure. 

C A Statistical Thermodynamic Approach to Hydrate Phase Equilibria... 

Section 5.1 presents the fundamental method as the heart of the chapter— the statistical thermodynamics approach to hydrate phase equilibria. The basic statistical thermodynamic equations are developed, and relationships to measurable, macroscopic hydrate properties are given. The parameters for the method are determined from both macroscopic (e.g., temperature and pressure) and microscopic (spectroscopic, diffraction) measurements. A Gibbs free energy calculation algorithm is given for multicomponent, multiphase systems for comparison with the methods described in Chapter 4. Finally, Section 5.1 concludes with ab initio modifications to the method, along with an assessment of method accuracy. 

A Statistical Thermodynamic Approach to Hydrate Phase Equilibria and the ideal gas chemical potential fx is calculated by... 

A general thermodynamic theory and a statistical thermodynamic approach are presented, which describe the phase transitions in insoluble monolayers, particularly the inclined transition from a liquid-expanded to a liquid-condensed phase. 

E. Treatment of Experimental Data, Based on the Nonequilibrium Statistical Thermodynamics Approach... 

We note that such problem does not appear in the nonequilibrium statistical thermodynamic approach (Sec. Ill), according to which micropores are considered together with their solid environment (micropore walls). Therefore, unlike the case of pyrolytic carbons, micropores in polymeric materials cannot be described in their own energy terms (chemical potential, etc.). 

The nonequilibrium statistical thermodynamic approach in the description of microporous materials prepared by pyrolysis of organic materials (active carbons) comprises two principal methods these for systems with regular and random fluxes. These methods comprise steady-state and nonsteady-state models which can be formulated for homo- and heterogeneous systems. 

The statistical thermodynamic approach, along lines already indicated, has been more tractable and suggestive. Models have been based on the Fowler-Guggenheim treatment of localized monolayers, in which account is taken of energy terms arising from interaction between point defects in nearest neighbor... 

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