Phase Modeling Tools Applications to Gases

This book is part of a set of books which offers advanced students successive characterization tool phases, the study of all types of phase (liquid, gas and solid, pure or multi-component), process engineering, chemical and electrochemical equilibria, and the properties of surfaces and phases of small sizes. Macroscopic and microscopic models are in turn covered with a constant correlation between the two scales. Particular attention has been given to the rigor of mathematical developments.

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 xiii

NOTATIONS xvii

SYMBOLS  xix

CHAPTER 1. THERMODYNAMIC FUNCTIONS AND VARIABLES  1

1.1. State variables and characteristic functions of a phase 2

1.1.1. Intensive and extensive conjugate variables 2

1.1.2. Variations in internal energy during a transformation 3

1.1.3 Characteristic function associated with a canonical set of variables  5

1.2. Partial molar parameters 7

1.2.1. Definition 7

1.2.2. Properties of partial molar variables 8

1.3. Chemical potential and generalized chemical potentials 8

1.3.1. Chemical potential and partial molar free enthalpy  8

1.3.2. Definition of generalized chemical potential  9

1.3.3. Variations in the chemical potential and generalized chemical potential with variables 10

1.3.4. Gibbs–Duhem relation 10

1.3.5. Generalized Helmholtz relations 11

1.3.6. Chemical system associated with the general system 12

1.4. The two modeling scales  14

CHAPTER 2. MACROSCOPIC MODELING OF A PHASE 15

2.1. Thermodynamic coefficients and characteristic matrices 15

2.1.1. Thermodynamic coefficients and characteristic matrix associated with the internal energy 15

2.1.2. Symmetry of the characteristic matrix  17

2.1.3. The thermodynamic coefficients needed and required to thermodynamically define the phase  17

2.1.4. Choosing other variables: thermodynamic coefficients and characteristic matrix associated with a characteristic function 19

2.1.5. Change in variable from one characteristic matrix to another  22

2.1.6. Relations between thermodynamic coefficients and secondary derivatives of the characteristic function 26

2.1.7. Examples of thermodynamic coefficients: calorimetric coefficients 27

2.2. Partial molar variables and thermodynamic coefficients 27

2.3. Common variables and thermodynamic coefficients  28

2.3.1. State equation  29

2.3.2. Expansion coefficients 30

2.3.3. Molar heat capacities  32

2.3.4. Young’s Modulus  34

2.3.5. Electric permittivity  34

2.3.6. Volumic and area densities of electric charge 34

2.4. Thermodynamic charts: justification of different types  35

2.4.1. Representation of a variable as a function of its conjugate 35

2.4.2. Representation of a characteristic function as a function of one of its natural variables 38

2.5. Stability of phases  39

2.5.1. Case of ensemble E0 of extensive variables 40

2.5.2. Coefficients associated with ensemble En  43

2.5.3. Case of other ensembles of variables  44

2.5.4. Conclusion: stability conditions of a phase in terms of thermodynamic coefficients  46

2.5.5. Example – applying stability conditions  46

2.6. Consistency of thermodynamic data 48

2.7. Conclusion on the macroscopic modeling of phases  49

CHAPTER 3. MULTI-COMPOUND PHASES – SOLUTIONS  51

3.1. Variables attached to solutions 51

3.1.1. Characterizing a solution  52

3.1.2. Composition of a solution 53

3.1.3. Peculiar variables and mixing variables 54

3.2. Recap of ideal solutions 57

3.2.1. Thermodynamic definition 57

3.2.2. Molar Gibbs energy of mixing of an ideal solution  57

3.2.3. Molar enthalpy of mixing of the ideal solution 57

3.2.4. Molar entropy of mixing of the ideal solution  58

3.2.5. Molar volume of mixing  58

3.2.6. Molar heat capacity of ideal solution: Kopp’s law 58

3.3. Characterization imperfection of a real solution 59

3.3.1. Lewis activity coefficients 60

3.3.2. Characterizing the imperfection of a real solution by the excess Gibbs energy  71

3.3.3. Other ways to measure the imperfection of a solution 74

3.4. Activity of a component in any solution: Raoult’s and Henry’s laws  76

3.5. Ionic solutions 77

3.5.1. Chemical potential of an ion  78

3.5.2. Relation between the activities of ions and the overall activity of solutes  80

3.5.3. Mean concentration and mean ionic activity coefficient 80

3.5.4. Obtaining the activity coefficient of an individual ion 82

3.5.5. Ionic strength 82

3.6. Curves of molar variables as a function of the composition in binary systems of a solution with two components  83

CHAPTER 4. STATISTICS OF OBJECT COLLECTIONS  87

4.1. The need to statistically process a system  87

4.1.1. Collections, system description – Stirling’s approximation  87

4.1.2. Statistical description hypothesis 88

4.1.3. The Boltzmann principle  89

4.2. Statistical effects of distinguishable non-quantum elements  89

4.2.1. Distribution law 90

4.2.2. Calculation of  91

4.2.3. Determining coefficient  92

4.2.4. Energy input to a system  95

4.2.5. The Boltzmann principle for entropy  96

4.3. The quantum description and space of phases 97

4.3.1. Wave functions and energy levels  97

4.3.2. Space of phases: discernibility of objects and states 98

4.3.3. Localization and non-localization of objects  98

4.4. Statistical effect of localized quantum objects 99

4.5. Collections of non-localized quantum objects 100

4.5.1. Eigen symmetrical and antisymmetric functions of non-localized objects  101

4.5.2. Statistics of non-localized elements with symmetrical wave functions 103

4.5.3. Statistics of non-localized elements with an asymmetric function  105

4.5.4. Classical limiting case 107

4.6. Systems composed of different particles without interactions 107

4.7. Unicity of coefficient  108

4.8. Determining coefficient in quantum statistics  110

CHAPTER 5. CANONICAL ENSEMBLES AND THERMODYNAMIC FUNCTIONS 113

5.1. An ensemble 113

5.2. Canonical ensemble 114

5.2.1. Description of a canonical ensemble 114

5.2.2. Law of distribution in a canonical ensemble  115

5.2.3. Canonical partition function  116

5.3. Molecular partition functions and canonical partition functions 117

5.3.1. Canonical partition functions for ensembles of discernable molecules  117

5.3.2. Canonical partition functions of indiscernible molecules  118

5.4. Thermodynamic functions and the canonical partition function 120

5.4.1. Expression of internal energy 120

5.4.2. Entropy and canonical partition functions 121

5.4.3. Expressing other thermodynamic functions and thermodynamic coefficients in the canonical ensemble 123

5.5. Absolute activity of a constituent 125

5.6. Other ensembles of systems and associated characteristic functions  127

CHAPTER 6. MOLECULAR PARTITION FUNCTIONS 131

6.1. Definition of the molecular partition function  131

6.2. Decomposition of the molecular partition function into partial partition functions 131

6.3. Energy level and thermal agitation 133

6.4. Translational partition functions  134

6.4.1. Translational partition function with the only constraint being the recipient 135

6.4.2. Translational partition function with the constraint being a potential centered and the container walls  137

6.5. Maxwell distribution laws  139

6.5.1. Distribution of ideal gas molecules in volume 139

6.5.2. Distribution of ideal gas molecules in velocity 140

6.6. Internal partition functions 142

6.6.1. Vibrational partition function 142

6.6.2. Rotational partition function  144

6.6.3. Nuclear partition function and correction of symmetry due to nuclear spin 146

6.6.4. Electronic partition function  149

6.7. Partition function of an ideal gas  149

6.8. Average energy and equipartition of energy 150

6.8.1. Mean translational energy 151

6.8.2. Mean rotational energy 152

6.8.3. Mean vibrational energy  152

6.9. Translational partition function and quantum mechanics 153

6.10. Interactions between species 155

6.10.1. Interactions between charged particles 155

6.10.2. Interaction energy between two neutral molecules  156

6.11. Equilibrium constants and molecular partition functions 161

6.11.1. Gaseous phase homogeneous equilibria  162

6.11.2. Liquid phase homogeneous equilibria 164

6.11.3. Solid phase homogenous equilibria 166

6.12. Conclusion on the macroscopic modeling of phases 167

CHAPTER 7. PURE REAL GASES 169

7.1. The three states of the pure compound: critical point  169

7.2. Standard state of a molecular substance  170

7.3. Real gas – macroscopic description  171

7.3.1. Pure gas diagram (P-V) 171

7.3.2. “Cubic” state equations 172

7.3.3. Other state equations  177

7.3.4. The theorem of corresponding states and the generalized compressibility chart  180

7.3.5. Molar Gibbs energy or chemical potential of a real gas 182

7.3.6. Fugacity of a real gas  183

7.3.7. Heat capacities of gases  186

7.4. Microscopic description of a real gas 188

7.4.1. Canonical partition function of a fluid  188

7.4.2. Helmholtz energy and development of the virial 195

7.4.3. Forms of the second coefficient of the virial  197

7.4.4. Macroscopic state equations and microscopic description  202

7.4.5. Chemical potential and fugacity of a real gas 203

7.4.6. Conclusion on microscopic modeling of a real gas  204

7.5. Microscopic approach of the heat capacity of gases  206

7.5.1. Classical theorem from the equipartition of energy  207

7.5.2. Quantum theorem of heat capacity at constant volume  208

CHAPTER 8. GAS MIXTURES  213

8.1. Macroscopic modeling of gas mixtures 213

8.1.1. Perfect solutions of perfect gases 213

8.1.2. Mixture of real gases  215

8.2. Characterizing gas mixtures  217

8.2.1. Method of the state equations of gas mixtures 218

8.2.2. The Beattie–Bridgeman state equation 218

8.2.3. Calculating the compressibility coefficient of a mixture 222

8.2.4. Method using activity coefficients of solutions  225

8.3. Determining activity coefficients of a solution from an equation of state  225

8.3.1. Methodology 226

8.3.2. Studying solutions using the PSRK method  227

8.3.3. VTPR Model 230

8.3.4. VGTPR Model 233

APPENDICES 237

APPENDIX 1 239

APPENDIX 2 243

APPENDIX 3 245

APPENDIX 4 253

APPENDIX 5 257

BIBLIOGRAPHY 261

INDEX 265

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