Potential model, average

By a statistical model of a solution we mean a model which does not attempt to describe explicitly the nature of the interaction between solvent and solute species, but simply assumes some general characteristic for the interaction, and presents expressions for the thermodynamic functions of the solution in terms of an assumed interaction parameter. The quasi-chemical theory is of this type, and we have noted that a serious deficiency is its failure to consider the vibrational effects in the solution. It is of interest, therefore, to consider briefly the average-potential model which does include the effect of vibrations. 

It is difficult to point to the basic reason why the average-potential model is not better applicable to metallic solutions. Shimoji60 believes that a Lennard-Jones 6-12 potential is not adequate for metals and that a Morse potential would give better results when incorporated in the same kind of model. On the other hand, it is possible that the main trouble is that metal solutions do not obey a theorem of corresponding states. More specifically, the interaction eAB(r) may not be expressible by the same function as for the pure components because the solute is so strongly modified by the solvent. This point of view is supported by considerations of the electronic models of metal solutions.46 The idea that the solvent strongly modifies the solute metal is reached also through a consideration of the quasi-chemical theory applied to dilute solutions. This is the topic that we consider next. 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

Attractive intermolecular forces, 60 Average potential model, 134... 

STATISTICAL MECHANICS OF MIXTURES— THE AVERAGE POTENTIAL MODEL... 

On account of its underlying assumptions this model has alternatively been called the Average Potential Model (APM)11 or the Corresponding States Theory of Mixtures.12 Various slightly different versions of it have been developed, depending on the precise definition of the average potential acting on a molecule j11 12 extensions and comments on this model have been published by several authors.13-16... 

Mixtures, Statistical Mechanics of—The Average Potential Model (Bellemans, Mathot, and Simon)... 

Statistical Mechanics of Mixtures—The Average Potential Model... 

In general the two-fluid average potential model gives better results than the one-fluid model. The three-fluid model counts all binary inter actions and is, therefore, properly used at low densities. The van der Waals one-fluid... 

A. Bellemens, V. Mathot, and M. Simon, Statistical Mechanics of Mixtures - The Average Potential Model , Advances in Chemical Physics, vol. XI, ed. I. Prigogine, Interscience, 1967. 

Throughout this chapter we confine as to the particular case taa = AB = Tbb - The general case which has been handled by Salsburg and Kirkwood will be treated later by the average potential method, which is more satisfactory than the cell method (Ch. IX, X, XI). In Ch. XI we will compare Salsburg and Kirkwood s calculated excess functions with those of the average potential model. 

Ch. X and first study the consequeaces of the present model which may be called the "average potential model. 

In the crude approximation of the average potential model we consider in this chapter, the mixture is treated as a pure substance (apart from the ideal entropy of mixing) characterized by the reduced variables (cf. 2.4.11)... 

Discussion of the Excess Functions-Efiect of Lattice Deformations 197. 9. Limitations of the Average Potential Model 200-... 

In the present more refined version of the average potential model we shall introduce separate average potential constants ei >, < >... 

Let us recall the fundamental assumptions of the average potential model. 

In the present situation of statistical mechanics it seems impossible to give a quantitative estimate of the approximation introduced by the average potential model. We wish however to emphasize a few points which indicate clearly the limitations of this approach. 

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