Michel SOUSTELLE is a chemical engineer and Emeritus Professor at Ecole des Mines de Saint-Etienne in France. He taught chemical kinetics from postgraduate to Master degree level while also carrying out research in this topic.

• PREFACE ixNOTATIONS AND SYMBOLS xiiiCHAPTER 1. PURE CRYSTALLINE SOLIDS 11.1. Characteristic values of a solid 11.2. Effect of stress and Young’s modulus 21.3. Microscopic description of crystalline solids 41.4. Partition function of vibration of a solid 51.4.1. Einstein’s single-frequency model 51.4.2. Debye’s frequency distribution model 61.4.3. Models with more complex frequency distributions 91.5. Description of atomic solids 101.5.1. Canonical partition function of an atomic solid 101.5.2. Helmholtz energy and internal energy of an atomic solid 111.6. Description of molecular solids 131.6.1. Partition function of molecular crystals 131.6.2. Thermodynamic functions of molecular solids 141.7. Description of an ionic solid 151.7.1. Crosslink energy of an ionic solid 151.7.2. Born/Haber cycle 221.7.3. Vibrational partition function and internal energy of an ionic solid 231.8. Description of a metallic solid 261.8.1. Sommerfeld’s electron perfect gas model 271.8.2. The metallic bond and band theory 371.9. Molar specific heat capacities of crystalline solids 461.9.1. Contribution of the vibrational energy to the specific heat capacity at constant volume 461.9.2. Specific heat capacity of an atomic solid at constant volume 501.9.3. Specific heat capacity of a molecularor ionic-solid at constant volume 541.9.4. Conclusion as to the specific heat capacity of a crystalline solid 541.10. Thermal expansion of solids 551.10.1. Expansion coefficients 551.10.2. Origin of thermal expansion in solids 581.10.3. Quantum treatment of thermal expansion. Grüneisen parameter 621.10.4. Expansion coefficient of metals 68CHAPTER 2. SOLID SOLUTIONS 712.1. Families of solid solutions 712.1.1. Substitutional solid solutions 722.1.2. Insertion solid solution 752.2. Order in solid solutions 822.2.1. Short-distance order 832.2.2. Long-distance order 872.3. Thermodynamic models of solid solutions 942.3.1. Determination of the Gibbs energy of mixing 942.3.2. The microscopic model of the perfect solution 1002.3.3. Microscopic model of strictly-regular solutions 1022.3.4. Microscopic model of the ideal dilute solution 1042.3.5. Fowler and Guggenheim’s quasi-chemical model of the solution 1062.4. Thermodynamic study of the degree of order of an alloy 1112.4.1. Hypotheses of the model: configuration energy 1122.4.2. Expression of the configuration partition function 1132.4.3. The Gorsky, Bragg and Williams model 1142.4.4. The quasi-chemical model 1202.4.5. Comparison of the models against experimental results 1272.5. Determination of the activity of a component of a solid solution 1322.5.1. Methods common to solid solutions and liquid solutions 1342.5.2. Methods specific to solid solutions 140CHAPTER 3. NON-STOICHIOMETRY IN SOLIDS 1473.1. Structure elements of a solid 1473.1.1. Definition 1483.1.2. Symbolic representation of structure elements 1493.1.3. Building unit of a solid 1513.1.4. Description and composition of a solid 1513.2. Quasi-chemical reactions in solids 1533.2.1. Definition and characteristics of a quasi-chemical reaction between structure elements 1533.2.2. Homogeneous quasi-chemical reactions in the solid phase 1563.2.3. Inter-phase reactions 1583.3. Equilibrium states between structure elements in solids 1583.4. Thermodynamics of structure elements in unary solids 1593.4.1. Structure elements of a unary solid 1593.4.2. Global equilibrium of an isolated crystal – influence of temperature 1623.5. Thermodynamics of structure elements in stoichiometric binary solids 1653.5.1. Symmetrical disorders in stoichiometric binary solids 1663.5.2. Asymmetrical disorders in stoichiometric binary solids1673.6. Thermodynamics of structure elements in non-stoichiometric binary solids 1693.6.1. Deviations from stoichiometry and point defects 1693.6.2. The predominant defect method – the Wagner classification 1713.6.3. Equilibrium of a Wagner solid with one of its gaseous elements 1743.6.4. General equilibrium of a non-stoichiometric binary solid with one of its gaseous elements 1753.7. Representation of complex solids – example of metal oxy-hydroxides 1803.7.1. The pseudo-binary approximation 1803.7.2. The predominant-defect generalization 1803.8. Determination of the equilibrium constants of the reactions involving structure elements 1813.8.1. Recap on calculating the equilibrium constants using statistical thermodynamics 1823.8.2. Examination of the pre-exponential term in the quasi-chemical equilibrium constants 1843.8.3. Determination of the internal energy of transformation of quasi-chemical reactions 187CHAPTER 4. SOLID SOLUTIONS AND STRUCTURE ELEMENTS 1954.1. Ionic solid solutions 1954.1.1. Introduction of foreign elements into stoichiometric binary solids 1974.1.2. Influence of foreign elements introduced into a non-stoichiometric binary solid 2004.2. Thermodynamics of equilibria between water vapor and saline hydrates: non-stoichiometric hydrates 2044.2.1. Experimental demonstration of non-stoichiometry of a hydrate 2044.2.2. Equilibria between stoichiometric hydrates 2074.2.3. Equilibrium reactions in non-stoichiometric hydrates 2074.2.4. The limits of the domains of divariance 213APPENDICES 217APPENDIX 1. THE LAGRANGE MULTIPLIER METHOD 219APPENDIX 2. SOLVING SCHRÖDINGER’S EQUATION 223BIBLIOGRAPHY 227INDEX 231

Michel SOUSTELLE is a chemical engineer and Emeritus Professor at Ecole des Mines de Saint-Etienne in France. He taught chemical kinetics from postgraduate to Master degree level while also carrying out research in this topic.

• PREFACE ixNOTATIONS AND SYMBOLS xiiiCHAPTER 1. PURE CRYSTALLINE SOLIDS 11.1. Characteristic values of a solid 11.2. Effect of stress and Young’s modulus 21.3. Microscopic description of crystalline solids 41.4. Partition function of vibration of a solid 51.4.1. Einstein’s single-frequency model 51.4.2. Debye’s frequency distribution model 61.4.3. Models with more complex frequency distributions 91.5. Description of atomic solids 101.5.1. Canonical partition function of an atomic solid 101.5.2. Helmholtz energy and internal energy of an atomic solid 111.6. Description of molecular solids 131.6.1. Partition function of molecular crystals 131.6.2. Thermodynamic functions of molecular solids 141.7. Description of an ionic solid 151.7.1. Crosslink energy of an ionic solid 151.7.2. Born/Haber cycle 221.7.3. Vibrational partition function and internal energy of an ionic solid 231.8. Description of a metallic solid 261.8.1. Sommerfeld’s electron perfect gas model 271.8.2. The metallic bond and band theory 371.9. Molar specific heat capacities of crystalline solids 461.9.1. Contribution of the vibrational energy to the specific heat capacity at constant volume 461.9.2. Specific heat capacity of an atomic solid at constant volume 501.9.3. Specific heat capacity of a molecularor ionic-solid at constant volume 541.9.4. Conclusion as to the specific heat capacity of a crystalline solid 541.10. Thermal expansion of solids 551.10.1. Expansion coefficients 551.10.2. Origin of thermal expansion in solids 581.10.3. Quantum treatment of thermal expansion. Grüneisen parameter 621.10.4. Expansion coefficient of metals 68CHAPTER 2. SOLID SOLUTIONS 712.1. Families of solid solutions 712.1.1. Substitutional solid solutions 722.1.2. Insertion solid solution 752.2. Order in solid solutions 822.2.1. Short-distance order 832.2.2. Long-distance order 872.3. Thermodynamic models of solid solutions 942.3.1. Determination of the Gibbs energy of mixing 942.3.2. The microscopic model of the perfect solution 1002.3.3. Microscopic model of strictly-regular solutions 1022.3.4. Microscopic model of the ideal dilute solution 1042.3.5. Fowler and Guggenheim’s quasi-chemical model of the solution 1062.4. Thermodynamic study of the degree of order of an alloy 1112.4.1. Hypotheses of the model: configuration energy 1122.4.2. Expression of the configuration partition function 1132.4.3. The Gorsky, Bragg and Williams model 1142.4.4. The quasi-chemical model 1202.4.5. Comparison of the models against experimental results 1272.5. Determination of the activity of a component of a solid solution 1322.5.1. Methods common to solid solutions and liquid solutions 1342.5.2. Methods specific to solid solutions 140CHAPTER 3. NON-STOICHIOMETRY IN SOLIDS 1473.1. Structure elements of a solid 1473.1.1. Definition 1483.1.2. Symbolic representation of structure elements 1493.1.3. Building unit of a solid 1513.1.4. Description and composition of a solid 1513.2. Quasi-chemical reactions in solids 1533.2.1. Definition and characteristics of a quasi-chemical reaction between structure elements 1533.2.2. Homogeneous quasi-chemical reactions in the solid phase 1563.2.3. Inter-phase reactions 1583.3. Equilibrium states between structure elements in solids 1583.4. Thermodynamics of structure elements in unary solids 1593.4.1. Structure elements of a unary solid 1593.4.2. Global equilibrium of an isolated crystal – influence of temperature 1623.5. Thermodynamics of structure elements in stoichiometric binary solids 1653.5.1. Symmetrical disorders in stoichiometric binary solids 1663.5.2. Asymmetrical disorders in stoichiometric binary solids1673.6. Thermodynamics of structure elements in non-stoichiometric binary solids 1693.6.1. Deviations from stoichiometry and point defects 1693.6.2. The predominant defect method – the Wagner classification 1713.6.3. Equilibrium of a Wagner solid with one of its gaseous elements 1743.6.4. General equilibrium of a non-stoichiometric binary solid with one of its gaseous elements 1753.7. Representation of complex solids – example of metal oxy-hydroxides 1803.7.1. The pseudo-binary approximation 1803.7.2. The predominant-defect generalization 1803.8. Determination of the equilibrium constants of the reactions involving structure elements 1813.8.1. Recap on calculating the equilibrium constants using statistical thermodynamics 1823.8.2. Examination of the pre-exponential term in the quasi-chemical equilibrium constants 1843.8.3. Determination of the internal energy of transformation of quasi-chemical reactions 187CHAPTER 4. SOLID SOLUTIONS AND STRUCTURE ELEMENTS 1954.1. Ionic solid solutions 1954.1.1. Introduction of foreign elements into stoichiometric binary solids 1974.1.2. Influence of foreign elements introduced into a non-stoichiometric binary solid 2004.2. Thermodynamics of equilibria between water vapor and saline hydrates: non-stoichiometric hydrates 2044.2.1. Experimental demonstration of non-stoichiometry of a hydrate 2044.2.2. Equilibria between stoichiometric hydrates 2074.2.3. Equilibrium reactions in non-stoichiometric hydrates 2074.2.4. The limits of the domains of divariance 213APPENDICES 217APPENDIX 1. THE LAGRANGE MULTIPLIER METHOD 219APPENDIX 2. SOLVING SCHRÖDINGER’S EQUATION 223BIBLIOGRAPHY 227INDEX 231

Michel SOUSTELLE is a chemical engineer and Emeritus Professor at Ecole des Mines de Saint-Etienne in France. He taught chemical kinetics from postgraduate to Master degree level while also carrying out research in this topic.

• PREFACE ixNOTATIONS AND SYMBOLS xiiiCHAPTER 1. PURE CRYSTALLINE SOLIDS 11.1. Characteristic values of a solid 11.2. Effect of stress and Young’s modulus 21.3. Microscopic description of crystalline solids 41.4. Partition function of vibration of a solid 51.4.1. Einstein’s single-frequency model 51.4.2. Debye’s frequency distribution model 61.4.3. Models with more complex frequency distributions 91.5. Description of atomic solids 101.5.1. Canonical partition function of an atomic solid 101.5.2. Helmholtz energy and internal energy of an atomic solid 111.6. Description of molecular solids 131.6.1. Partition function of molecular crystals 131.6.2. Thermodynamic functions of molecular solids 141.7. Description of an ionic solid 151.7.1. Crosslink energy of an ionic solid 151.7.2. Born/Haber cycle 221.7.3. Vibrational partition function and internal energy of an ionic solid 231.8. Description of a metallic solid 261.8.1. Sommerfeld’s electron perfect gas model 271.8.2. The metallic bond and band theory 371.9. Molar specific heat capacities of crystalline solids 461.9.1. Contribution of the vibrational energy to the specific heat capacity at constant volume 461.9.2. Specific heat capacity of an atomic solid at constant volume 501.9.3. Specific heat capacity of a molecularor ionic-solid at constant volume 541.9.4. Conclusion as to the specific heat capacity of a crystalline solid 541.10. Thermal expansion of solids 551.10.1. Expansion coefficients 551.10.2. Origin of thermal expansion in solids 581.10.3. Quantum treatment of thermal expansion. Grüneisen parameter 621.10.4. Expansion coefficient of metals 68CHAPTER 2. SOLID SOLUTIONS 712.1. Families of solid solutions 712.1.1. Substitutional solid solutions 722.1.2. Insertion solid solution 752.2. Order in solid solutions 822.2.1. Short-distance order 832.2.2. Long-distance order 872.3. Thermodynamic models of solid solutions 942.3.1. Determination of the Gibbs energy of mixing 942.3.2. The microscopic model of the perfect solution 1002.3.3. Microscopic model of strictly-regular solutions 1022.3.4. Microscopic model of the ideal dilute solution 1042.3.5. Fowler and Guggenheim’s quasi-chemical model of the solution 1062.4. Thermodynamic study of the degree of order of an alloy 1112.4.1. Hypotheses of the model: configuration energy 1122.4.2. Expression of the configuration partition function 1132.4.3. The Gorsky, Bragg and Williams model 1142.4.4. The quasi-chemical model 1202.4.5. Comparison of the models against experimental results 1272.5. Determination of the activity of a component of a solid solution 1322.5.1. Methods common to solid solutions and liquid solutions 1342.5.2. Methods specific to solid solutions 140CHAPTER 3. NON-STOICHIOMETRY IN SOLIDS 1473.1. Structure elements of a solid 1473.1.1. Definition 1483.1.2. Symbolic representation of structure elements 1493.1.3. Building unit of a solid 1513.1.4. Description and composition of a solid 1513.2. Quasi-chemical reactions in solids 1533.2.1. Definition and characteristics of a quasi-chemical reaction between structure elements 1533.2.2. Homogeneous quasi-chemical reactions in the solid phase 1563.2.3. Inter-phase reactions 1583.3. Equilibrium states between structure elements in solids 1583.4. Thermodynamics of structure elements in unary solids 1593.4.1. Structure elements of a unary solid 1593.4.2. Global equilibrium of an isolated crystal – influence of temperature 1623.5. Thermodynamics of structure elements in stoichiometric binary solids 1653.5.1. Symmetrical disorders in stoichiometric binary solids 1663.5.2. Asymmetrical disorders in stoichiometric binary solids1673.6. Thermodynamics of structure elements in non-stoichiometric binary solids 1693.6.1. Deviations from stoichiometry and point defects 1693.6.2. The predominant defect method – the Wagner classification 1713.6.3. Equilibrium of a Wagner solid with one of its gaseous elements 1743.6.4. General equilibrium of a non-stoichiometric binary solid with one of its gaseous elements 1753.7. Representation of complex solids – example of metal oxy-hydroxides 1803.7.1. The pseudo-binary approximation 1803.7.2. The predominant-defect generalization 1803.8. Determination of the equilibrium constants of the reactions involving structure elements 1813.8.1. Recap on calculating the equilibrium constants using statistical thermodynamics 1823.8.2. Examination of the pre-exponential term in the quasi-chemical equilibrium constants 1843.8.3. Determination of the internal energy of transformation of quasi-chemical reactions 187CHAPTER 4. SOLID SOLUTIONS AND STRUCTURE ELEMENTS 1954.1. Ionic solid solutions 1954.1.1. Introduction of foreign elements into stoichiometric binary solids 1974.1.2. Influence of foreign elements introduced into a non-stoichiometric binary solid 2004.2. Thermodynamics of equilibria between water vapor and saline hydrates: non-stoichiometric hydrates 2044.2.1. Experimental demonstration of non-stoichiometry of a hydrate 2044.2.2. Equilibria between stoichiometric hydrates 2074.2.3. Equilibrium reactions in non-stoichiometric hydrates 2074.2.4. The limits of the domains of divariance 213APPENDICES 217APPENDIX 1. THE LAGRANGE MULTIPLIER METHOD 219APPENDIX 2. SOLVING SCHRÖDINGER’S EQUATION 223BIBLIOGRAPHY 227INDEX 231