Statistical thermodynamics stoichiometry

The concentrations of point defects in thermal equilibrium can be calculated for many types of defect structures by statistical thermodynamics. Stoichiometry is determined by the interior of crystals the surfaces of crystals often do not follow the stoichiometry of the bulk. 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

Let us consider the compounds which show a small deviation from the stoichiometric composition and whose non-stoichiometry is derived from metal vacancies. The free energy of these compounds, which take the composition MX in the ideal or non-defect state, can be calculated by the method proposed by Libowitz. To readers who are well acquainted with the Fowler-Guggenheim style of statistical thermodynamics, the method here adopted may not be quite satisfactory however, the Libowitz method is understandable even to beginners who know only elementary thermodynamics and statistical mechanics. It goes without saying that the result calculated by the Libowitz method is essentially coincident with that calculated by the Fowler-Guggenheim method. 

Chapter 1 deals with classical non-stoichiometric compounds. By classical, the author means that the basic concept of the phase stability has been well established from a thermodynamical point of view, and does not mean that research in this field has been fully completed. In these compounds the origin of non-stoichiometry is point defects . In the first half of the chapter, the fundamental relation between point defects and non-stoichiometry is described in detail, based on (statistical) thermodynamics, and in the second half various examples, referred to the original papers, are shown. 

Most thermochemical calculations are made for closed thermodynamic systems, and the stoichiometry is most conveniently represented in terms of the molar quantities as determined from statistical calculations. In dealing with compressible flow problems in which it is essential to work with open thermodynamic systems, it is best to employ mass quantities. Throughout this text uppercase symbols will be used for molar quantities and lowercase symbols for mass quantities. 

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