Phase Modeling Tools

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 xiiiNOTATIONS xviiSYMBOLS  xixCHAPTER 1. THERMODYNAMIC FUNCTIONS AND VARIABLES  11.1. State variables and characteristic functions of a phase 21.1.1. Intensive and extensive conjugate variables 21.1.2. Variations in internal energy during a transformation 31.1.3 Characteristic function associated with a canonical set of variables  51.2. Partial molar parameters 71.2.1. Definition 71.2.2. Properties of partial molar variables 81.3. Chemical potential and generalized chemical potentials 81.3.1. Chemical potential and partial molar free enthalpy  81.3.2. Definition of generalized chemical potential  91.3.3. Variations in the chemical potential and generalized chemical potential with variables 101.3.4. Gibbs–Duhem relation 101.3.5. Generalized Helmholtz relations 111.3.6. Chemical system associated with the general system 121.4. The two modeling scales  14CHAPTER 2. MACROSCOPIC MODELING OF A PHASE 152.1. Thermodynamic coefficients and characteristic matrices 152.1.1. Thermodynamic coefficients and characteristic matrix associated with the internal energy 152.1.2. Symmetry of the characteristic matrix  172.1.3. The thermodynamic coefficients needed and required to thermodynamically define the phase  172.1.4. Choosing other variables: thermodynamic coefficients and characteristic matrix associated with a characteristic function 192.1.5. Change in variable from one characteristic matrix to another  222.1.6. Relations between thermodynamic coefficients and secondary derivatives of the characteristic function 262.1.7. Examples of thermodynamic coefficients: calorimetric coefficients 272.2. Partial molar variables and thermodynamic coefficients 272.3. Common variables and thermodynamic coefficients  282.3.1. State equation  292.3.2. Expansion coefficients 302.3.3. Molar heat capacities  322.3.4. Young’s Modulus  342.3.5. Electric permittivity  342.3.6. Volumic and area densities of electric charge 342.4. Thermodynamic charts: justification of different types  352.4.1. Representation of a variable as a function of its conjugate 352.4.2. Representation of a characteristic function as a function of one of its natural variables 382.5. Stability of phases  392.5.1. Case of ensemble E0 of extensive variables 402.5.2. Coefficients associated with ensemble En  432.5.3. Case of other ensembles of variables  442.5.4. Conclusion: stability conditions of a phase in terms of thermodynamic coefficients  462.5.5. Example – applying stability conditions  462.6. Consistency of thermodynamic data 482.7. Conclusion on the macroscopic modeling of phases  49CHAPTER 3. MULTI-COMPOUND PHASES – SOLUTIONS  513.1. Variables attached to solutions 513.1.1. Characterizing a solution  523.1.2. Composition of a solution 533.1.3. Peculiar variables and mixing variables 543.2. Recap of ideal solutions 573.2.1. Thermodynamic definition 573.2.2. Molar Gibbs energy of mixing of an ideal solution  573.2.3. Molar enthalpy of mixing of the ideal solution 573.2.4. Molar entropy of mixing of the ideal solution  583.2.5. Molar volume of mixing  583.2.6. Molar heat capacity of ideal solution: Kopp’s law 583.3. Characterization imperfection of a real solution 593.3.1. Lewis activity coefficients 603.3.2. Characterizing the imperfection of a real solution by the excess Gibbs energy  713.3.3. Other ways to measure the imperfection of a solution 743.4. Activity of a component in any solution: Raoult’s and Henry’s laws  763.5. Ionic solutions 773.5.1. Chemical potential of an ion  783.5.2. Relation between the activities of ions and the overall activity of solutes  803.5.3. Mean concentration and mean ionic activity coefficient 803.5.4. Obtaining the activity coefficient of an individual ion 823.5.5. Ionic strength 823.6. Curves of molar variables as a function of the composition in binary systems of a solution with two components  83CHAPTER 4. STATISTICS OF OBJECT COLLECTIONS  874.1. The need to statistically process a system  874.1.1. Collections, system description – Stirling’s approximation  874.1.2. Statistical description hypothesis 884.1.3. The Boltzmann principle  894.2. Statistical effects of distinguishable non-quantum elements  894.2.1. Distribution law 904.2.2. Calculation of  914.2.3. Determining coefficient  924.2.4. Energy input to a system  954.2.5. The Boltzmann principle for entropy  964.3. The quantum description and space of phases 974.3.1. Wave functions and energy levels  974.3.2. Space of phases: discernibility of objects and states 984.3.3. Localization and non-localization of objects  984.4. Statistical effect of localized quantum objects 994.5. Collections of non-localized quantum objects 1004.5.1. Eigen symmetrical and antisymmetric functions of non-localized objects  1014.5.2. Statistics of non-localized elements with symmetrical wave functions 1034.5.3. Statistics of non-localized elements with an asymmetric function  1054.5.4. Classical limiting case 1074.6. Systems composed of different particles without interactions 1074.7. Unicity of coefficient  1084.8. Determining coefficient in quantum statistics  110CHAPTER 5. CANONICAL ENSEMBLES AND THERMODYNAMIC FUNCTIONS 1135.1. An ensemble 1135.2. Canonical ensemble 1145.2.1. Description of a canonical ensemble 1145.2.2. Law of distribution in a canonical ensemble  1155.2.3. Canonical partition function  1165.3. Molecular partition functions and canonical partition functions 1175.3.1. Canonical partition functions for ensembles of discernable molecules  1175.3.2. Canonical partition functions of indiscernible molecules  1185.4. Thermodynamic functions and the canonical partition function 1205.4.1. Expression of internal energy 1205.4.2. Entropy and canonical partition functions 1215.4.3. Expressing other thermodynamic functions and thermodynamic coefficients in the canonical ensemble 1235.5. Absolute activity of a constituent 1255.6. Other ensembles of systems and associated characteristic functions  127CHAPTER 6. MOLECULAR PARTITION FUNCTIONS 1316.1. Definition of the molecular partition function  1316.2. Decomposition of the molecular partition function into partial partition functions 1316.3. Energy level and thermal agitation 1336.4. Translational partition functions  1346.4.1. Translational partition function with the only constraint being the recipient 1356.4.2. Translational partition function with the constraint being a potential centered and the container walls  1376.5. Maxwell distribution laws  1396.5.1. Distribution of ideal gas molecules in volume 1396.5.2. Distribution of ideal gas molecules in velocity 1406.6. Internal partition functions 1426.6.1. Vibrational partition function 1426.6.2. Rotational partition function  1446.6.3. Nuclear partition function and correction of symmetry due to nuclear spin 1466.6.4. Electronic partition function  1496.7. Partition function of an ideal gas  1496.8. Average energy and equipartition of energy 1506.8.1. Mean translational energy 1516.8.2. Mean rotational energy 1526.8.3. Mean vibrational energy  1526.9. Translational partition function and quantum mechanics 1536.10. Interactions between species 1556.10.1. Interactions between charged particles 1556.10.2. Interaction energy between two neutral molecules  1566.11. Equilibrium constants and molecular partition functions 1616.11.1. Gaseous phase homogeneous equilibria  1626.11.2. Liquid phase homogeneous equilibria 1646.11.3. Solid phase homogenous equilibria 1666.12. Conclusion on the macroscopic modeling of phases 167CHAPTER 7. PURE REAL GASES 1697.1. The three states of the pure compound: critical point  1697.2. Standard state of a molecular substance  1707.3. Real gas – macroscopic description  1717.3.1. Pure gas diagram (P-V) 1717.3.2. “Cubic” state equations 1727.3.3. Other state equations  1777.3.4. The theorem of corresponding states and the generalized compressibility chart  1807.3.5. Molar Gibbs energy or chemical potential of a real gas 1827.3.6. Fugacity of a real gas  1837.3.7. Heat capacities of gases  1867.4. Microscopic description of a real gas 1887.4.1. Canonical partition function of a fluid  1887.4.2. Helmholtz energy and development of the virial 1957.4.3. Forms of the second coefficient of the virial  1977.4.4. Macroscopic state equations and microscopic description  2027.4.5. Chemical potential and fugacity of a real gas 2037.4.6. Conclusion on microscopic modeling of a real gas  2047.5. Microscopic approach of the heat capacity of gases  2067.5.1. Classical theorem from the equipartition of energy  2077.5.2. Quantum theorem of heat capacity at constant volume  208CHAPTER 8. GAS MIXTURES  2138.1. Macroscopic modeling of gas mixtures 2138.1.1. Perfect solutions of perfect gases 2138.1.2. Mixture of real gases  2158.2. Characterizing gas mixtures  2178.2.1. Method of the state equations of gas mixtures 2188.2.2. The Beattie–Bridgeman state equation 2188.2.3. Calculating the compressibility coefficient of a mixture 2228.2.4. Method using activity coefficients of solutions  2258.3. Determining activity coefficients of a solution from an equation of state  2258.3.1. Methodology 2268.3.2. Studying solutions using the PSRK method  2278.3.3. VTPR Model 2308.3.4. VGTPR Model 233APPENDICES 237APPENDIX 1 239APPENDIX 2 243APPENDIX 3 245APPENDIX 4 253APPENDIX 5 257BIBLIOGRAPHY 261INDEX 265