Modeling of Liquid Phases

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.

This second volume in the set is devoted to the study of liquid phases.

PREFACE xi

NOTATIONS AND SYMBOLS xv

CHAPTER 1. PURE LIQUIDS 1

1.1 Macroscopic modeling of liquids 1

1.2. Distribution of molecules in a liquid 2

1.2.1. Molecular structure of a nonassociated liquid 3

1.2.2. The radial distribution function 4

1.2.3 The curve representative of the radial distribution function 6

1.2.4. Calculation of the macroscopic thermodynamic values 8

1.3. Models extrapolated from gases or solids 9

1.3.1. Guggenheim’s smoothed potential model 10

1.3.2. Mie’s harmonic oscillator model 13

1.3.3. Determination of the free volume on the basis of the dilation and the compressibility 15

1.4. Lennard-Jones and Devonshire cellular model 16

1.5. Cellular and vacancies model 25

1.6. Eyring’s semi-microscopic formulation of the vacancy model 29

1.7. Comparison between the different microscopic models and experimental results 32

CHAPTER 2. MACROSCOPIC MODELING OF LIQUID MOLECULAR SOLUTIONS 37

2.1. Macroscopic modeling of the Margules expansion 38

2.2. General representation of a solution with several components 39

2.3. Macroscopic modeling of the Wagner expansions 40

2.3.1. Definition of the Wagner interaction coefficients 40

2.3.2. Example of a ternary solution: experimental determination of Wagner’s interaction coefficients 41

2.4. Dilute ideal solutions 43

2.4.1. Thermodynamic definition of a dilute ideal solution 43

2.4.2. Activity coefficients of a component with a pure-substance reference 44

2.4.3. Excess Gibbs energy of an ideal dilute solution 44

2.4.4. Enthalpy of mixing for an ideal dilute solution 45

2.4.5. Excess entropy of a dilute ideal solution 46

2.4.6. Molar heat capacity of an ideal dilute solution at constant pressure 46

2.5. Associated solutions 46

2.5.1. Example of the study of an associated solution 47

2.5.2. Relations between the chemical potentials of the associated solution 49

2.5.3. Calculating the extent of the equilibrium in an associated solution 50

2.5.4. Calculating the activity coefficients in an associated solution 50

2.5.5. Definition of a regular solution 51

2.5.6. Strictly-regular solutions 52

2.5.7. Macroscopic modeling of strictly-regular binary solutions 53

2.5.8. Extension of the model of a strictly-regular solution to solutions with more than two components 56

2.6. Athermic solutions 57

2.6.1. Thermodynamic definition of an athermic solution 58

2.6.2. Variation of the activity coefficients with temperature in an athermic solution 58

2.6.3. Molar entropy and Gibbs energy of mixing for an athermic solution 58

2.6.4. Molar heat capacity of an athermic solution 59

CHAPTER 3. MICROSCOPIC MODELING OF LIQUID MOLECULAR SOLUTIONS 61

3.1. Models of binary solutions with molecules of similar dimensions 62

3.1.1. The microscopic model of a perfect solution 68

3.1.2. Microscopic description of strictly-regular solutions 70

3.1.3. Microscopic modeling of an ideal dilute solution 72

3.2. The concept of local composition 74

3.2.1. The concept of local composition in a solution 74

3.2.2. Energy balance of the mixture 76

3.2.3. Warren and Cowley’s order parameter 78

3.2.4. Model of Fowler & Guggenheim’s quasi-chemical solution 80

3.3. The quasi-chemical method of modeling solutions 87

3.4. Difference of the molar volumes: the combination term 92

3.4.1. Combinatorial excess entropy 92

3.4.2. Flory’s athermic solution model 97

3.4.3. Staverman’s corrective factor 98

3.5. Combination of the different concepts: the UNIQUAC model 101

3.6. The concept of contribution of groups: the UNIFAC model 107

3.6.1. The concept of the contribution of groups 108

3.6.2. The UNIFAC model 108

3.6.3. The modified UNIFAC model (Dortmund) 114

3.6.4. Use of the UNIFAC system in the UNIQUAC model 114

CHAPTER 4. IONIC SOLUTIONS 117

4.1. Reference state, unit of composition and activity coefficients of ionic solutions 119

4.2. Debye and Hückel’s electrostatic model 121

4.2.1. Presentation of the problem 122

4.2.2. Notations 123

4.2.3. Poisson’s equation 124

4.2.4. Electrical potential due to the ionic atmosphere 125

4.2.5. Debye and Hückel’s hypotheses 127

4.2.6. Debye and Hückel’s solution for the potential due to the ionic atmosphere 132

4.2.7. Charge and radius of the ionic atmosphere of an ion 134

4.2.8. Excess Helmholtz energy and excess Gibbs energy due to charges 136

4.2.9. Activity coefficients of the ions and mean activity coefficient of the solution 138

4.2.10. Self-consistency of Debye and Hückel’s model 141

4.2.11. Switching from concentrations to molalities 144

4.2.12. Debye and Hückel’s law: validity and comparison with experimental data 146

4.2.13. Debye and Hückel’s limit law 147

4.2.14. Extensions of Debye and Hückel’s law 148

4.3. Pitzer’s model 150

4.4. UNIQUAC model extended to ionic solutions 155

CHAPTER 5. DETERMINATION OF THE ACTIVITY OF A COMPONENT OF A SOLUTION 159

5.1. Calculation of an activity coefficient when we know other coefficients 160

5.1.1. Calculation of the activity of a component when we know that of the other components in the solution 160

5.1.2. Determination of the activity of a component at one temperature if we know its activity at another temperature 162

5.2. Determination of the activity on the basis of the measured vapor pressure 164

5.2.1. Measurement by the direct method 165

5.2.2. Method using the vaporization constant in reference II 166

5.3. Measurement of the activity of the solvent of the basis of the colligative properties 168

5.3.1. Use of measuring of the depression of the boiling point – ebullioscopy 168

5.3.2. Use of measuring of the depression of the freezing point – cryoscopy 170

5.3.3. Use of the measurement of osmotic pressure 172

5.4. Measuring the activity on the basis of solubility measurements 173

5.4.1. Measuring the solubilities in molecular solutions 174

5.4.2. Measuring the solubilities in ionic solutions 174

5.5. Measuring the activity by measuring the distribution of a solute between two immiscible solvents 176

5.6. Activity in a conductive solution 176

5.6.1. Measuring the activity in a strong electrolyte 176

5.6.2. Determination of the mean activity of a weak electrolyte on the basis of the dissociation equilibrium 180

APPENDICES 181

APPENDIX 1 183

APPENDIX 2 193

APPENDIX 3 207

BIBLIOGRAPHY 221

INDEX 225

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