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### More About This Title Physical and Chemical Equilibrium for Chemical Engineers, Second Edition

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**NOEL de NEVERS**, PhD, followed five years of working for Chevron with thirty-seven years as a Professor in the Chemical Engineering Department of the University of Utah. His textbooks (and research papers) are in fluid mechanics, thermodynamics, and air pollution control engineering. He regularly consults as an expert on explosions, fires, and toxic exposures.

In addition to technical work, he has three "de Nevers's Laws" in a Murphy's Laws compilation and won the title "Poet Laureate of Jell-O Salad" in a Salt Lake City competition, with three limericks and a quatrain. He has climbed the Grand Teton, Mt. Rainier, Mt. Whitney, Kala Pattar, and Mt. Kilimanjaro, and is the official discoverer of Private Arch in Arches National Park.

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### English

Preface xiii

About the Author xv

Nomenclature xvii

**1 Introduction to Equilibrium 1**

1.1 Why Study Equilibrium? 1

1.2 Stability and Equilibrium 4

1.3 Time Scales and the Approach to Equilibrium 5

1.4 Looking Ahead Gibbs Energy 5

1.5 Units Conversion Factors and Notation 6

1.6 Reality and Equations 8

1.7 Phases and Phase Diagrams 8

1.8 The Plan of this Book 10

1.9 Summary 10

References 11

**2 Basic Thermodynamics 13**

2.1 Conservation and Accounting 13

2.2 Conservation of Mass 14

2.3 Conservation of Energy; the First Law of Thermodynamics 15

2.4 The Second Law of Thermodynamics 17

2.4.1 Reversibility 17

2.4.2 Entropy 18

2.5 Convenience Properties 19

2.6 Using the First and Second Laws 19

2.7 Datums and Reference States 21

2.8 Measurable and Immeasurable Properties 22

2.9 Work and Heat 22

2.10 The Property Equation 23

2.11 Equations of State (EOS) 24

2.11.1 EOSs Based on Theory 25

2.11.2 EOSs Based on Pure Data Fitting 25

2.12 Corresponding States 26

2.13 Departure Functions 28

2.14 The Properties of Mixtures 28

2.15 The Combined First and Second Law Statement; Reversible Work 29

2.16 Summary 31

References 33

**3 The Simplest Phase Equilibrium Examples and Some Simple Estimating Rules 35**

3.1 Some General Statements About Equilibrium 35

3.2 The Simplest Example of Phase Equilibrium 37

3.2.1 A Digression the Distinction between Vapor and Gas 37

3.2.2 Back to the Simplest Equilibrium 37

3.3 The Next Level of Complexity in Phase Equilibrium 37

3.4 Some Simple Estimating Rules: Raoult’s and Henry’s “Laws” 39

3.5 The General Two-Phase Equilibrium Calculation 43

3.6 Some Simple Applications of Raoult’s and Henry’s Laws 43

3.7 The Uses and Limits of Raoult’s and Henry’s Laws 46

3.8 Summary 46

References 48

**4 Minimization of Gibbs Energy 49**

4.1 The Fundamental Thermodynamic Criterion of Phase and Chemical Equilibrium 49

4.2 The Criterion of Equilibrium Applied to Two Nonreacting Equilibrium Phases 51

4.3 The Criterion of Equilibrium Applied to Chemical Reactions 53

4.4 Simple Gibbs Energy Diagrams 54

4.4.1 Comparison with Enthalpy and Entropy 55

4.4.2 Gibbs Energy Diagrams for Pressure-Driven Phase Changes 55

4.4.3 Gibbs Energy Diagrams for Chemical Reactions 57

4.5 Le Chatelier’s Principle 58

4.6 Summary 58

References 60

**5 Vapor Pressure the Clapeyron Equation and Single Pure Chemical Species Phase Equilibrium 61**

5.1 Measurement of Vapor Pressure 61

5.2 Reporting Vapor-Pressure Data 61

5.2.1 Normal Boiling Point (NBP) 61

5.3 The Clapeyron Equation 62

5.4 The Clausius–Clapeyron Equation 63

5.5 The Accentric Factor 64

5.6 The Antoine Equation and Other Data-Fitting Equations 66

5.6.1 Choosing a Vapor-Pressure Equation 67

5.7 Applying the Clapeyron Equation to Other Kinds of Equilibrium 67

5.8 Extrapolating Vapor-Pressure Curves 68

5.9 Vapor Pressure of Solids 69

5.10 Vapor Pressures of Mixtures 69

5.11 Summary 69

References 72

**6 Partial Molar Properties 73**

6.1 Partial Molar Properties 73

6.2 The Partial Molar Equation 74

6.3 Tangent Slopes 74

6.4 Tangent Intercepts 77

6.5 The Two Equations for Partial Molar Properties 78

6.6 Using the Idea of Tangent Intercepts 79

6.7 Partial Mass Properties 80

6.8 Heats of Mixing and Partial Molar Enthalpies 80

6.8.1 Differential Heat of Mixing 80

6.8.2 Integral Heat of Mixing 81

6.9 The Gibbs–Duhem Equation and the Counterintuitive Behavior of the Chemical Potential 82

6.10 Summary 84

References 87

**7 Fugacity Ideal Solutions Activity Activity Coefficient 89**

7.1 Why Fugacity? 89

7.2 Fugacity Defined 89

7.3 The Use of the Fugacity 90

7.4 Pure Substance Fugacities 90

7.4.1 The Fugacity of Pure Gases 91

7.4.2 The Fugacity of Pure Liquids and Solids 94

7.5 Fugacities of Species in Mixtures 95

7.6 Mixtures of Ideal Gases 95

7.7 Why Ideal Solutions? 95

7.8 Ideal Solutions Defined 96

7.8.1 The Consequences of the Ideal Solution Definition 96

7.9 Why Activity and Activity Coefficients? 98

7.10 Activity and Activity Coefficients Defined 98

7.11 Fugacity Coefficient for Pure Gases and Gas Mixtures 100

7.12 Estimating Fugacities of Individual Species in Gas Mixtures 100

7.12.1 Fugacities from Gas PvT Data 100

7.12.2 Fugacities from an EOS for Gas Mixtures 102

7.12.3 The Lewis and Randall (L-R) Fugacity Rule 102

7.12.4 Other Mixing Rules 103

7.13 Liquid Fugacities from Vapor-Liquid Equilibrium 104

7.14 Summary 104

References 105

**8 Vapor–Liquid Equilibrium (VLE) at Low Pressures 107**

8.1 Measurement of VLE 107

8.2 Presenting Experimental VLE Data 110

8.3 The Mathematical Treatment of Low-Pressure VLE Data 110

8.3.1 Raoult’s Law Again 111

8.4 The Four Most Common Types of Low-Pressure VLE 112

8.4.1 Ideal Solution Behavior (Type I) 114

8.4.2 Positive Deviations from Ideal Solution Behavior (Type II) 114

8.4.3 Negative Deviations from Ideal Solution Behavior (Type III) 115

8.4.4 Azeotropes 117

8.4.5 Two-Liquid Phase or Heteroazeotropes (Type IV) 118

8.4.6 Zero Solubility and Steam Distillation 120

8.4.7 Distillation of the Four Types of Behavior 121

8.5 Gas–Liquid Equilibrium Henry’s Law Again 122

8.6 The Effect of Modest Pressures on VLE 122

8.6.1 Liquids 123

8.6.2 Gases the L-R Rule 123

8.7 Standard States Again 124

8.8 Low-Pressure VLE Calculations 125

8.8.1 Bubble-Point Calculations 127

8.8.1.1 Temperature-Specified Bubble Point 127

8.8.1.2 Pressure-Specified Bubble Point 128

8.8.2 Dew-Point Calculations 129

8.8.2.1 Temperature-Specified Dew Point 129

8.8.2.2 Pressure-Specified Dew Point 129

8.8.3 Isothermal Flashes (T- and P-Specified Flashes) 130

8.8.4 Adiabatic Flashes 131

8.9 Traditional K-Factor Methods 132

8.10 More Uses for Raoult’s Law 132

8.10.1 Nonvolatile Solutes Boiling-Point Elevation 132

8.10.2 Freezing-Point Depression 135

8.10.3 Colligative Properties of Solutions 136

8.11 Summary 136

References 143

**9 Correlating and Predicting Nonideal VLE 145**

9.1 The Most Common Observations of Liquid-Phase Activity Coefficients 145

9.1.1 Why Nonideal Behavior? 145

9.1.2 The Shapes of ln g_x Curves 146

9.2 Limits on Activity Coefficient Correlations the Gibbs–Duhem Equation 147

9.3 Excess Gibbs Energy and Activity Coefficient Equations 148

9.4 Activity Coefficients at Infinite Dilution 150

9.5 Effects of Pressure and Temperature on Liquid-Phase Activity Coefficients 151

9.5.1 Effect of Pressure Changes on Liquid-Phase Activity Coefficients 151

9.5.2 Effect of Temperature Changes on Liquid-Phase Activity Coefficients 152

9.6 Ternary and Multispecies VLE 153

9.6.1 Liquid-Phase Activity Coefficients for Ternary Mixtures 154

9.7 Vapor-Phase Nonideality 155

9.8 VLE from EOS 158

9.9 Solubility Parameter 158

9.10 The Solubility of Gases in Liquids Henry’s Law Again 160

9.11 Summary 163

References 167

**10 Vapor–Liquid Equilibrium (VLE) at High Pressures 169**

10.1 Critical Phenomena of Pure Species 169

10.2 Critical Phenomena of Mixtures 170

10.3 Estimating High-Pressure VLE 174

10.3.1 Empirical K-Value Correlations 175

10.3.2 Estimation Methods for Each Phase Separately Not Based on Raoult’s Law 175

10.3.3 Estimation Methods Based on Cubic EOSs 176

10.4 Computer Solutions 178

10.5 Summary 178

References 179

**11 Liquid–Liquid Liquid–Solid and Gas–Solid Equilibrium 181**

11.1 Liquid–Liquid Equilibrium (LLE) 181

11.2 The Experimental Determination of LLE 181

11.2.1 Reporting and Presenting LLE Data 182

11.2.2 Practically Insoluble Liquid Pairs at 25_C 183

11.2.3 Partially Soluble Liquid Pairs at 25_C 183

11.2.4 Miscible Liquid Pairs at 25_C 183

11.2.5 Ternary LLE at 25_C 184

11.2.6 LLE at Temperatures Other Than 25_C 186

11.3 The Elementary Theory of LLE 187

11.4 The Effect of Pressure on LLE 190

11.5 Effect of Temperature on LLE 191

11.6 Distribution Coefficients 194

11.7 Liquid–Solid Equilibrium (LSE) 195

11.7.1 One-Species LSE 195

11.7.2 The Experimental Determination of LSE 195

11.7.3 Presenting LSE Data 195

11.7.4 Eutectics 197

11.7.5 Gas Hydrates (Clathrates) 199

11.8 The Elementary Thermodynamics of LSE 200

11.9 Gas–Solid Equilibrium (GSE) at Low Pressures 202

11.10 GSE at High Pressures 203

11.11 Gas–Solid Adsorption Vapor–Solid Adsorption 204

11.11.1 Langmuir’s Adsorption Theory 205

11.11.2 Vapor-solid Adsorption BET Theory 207

11.11.3 Adsorption from Mixtures 208

11.11.4 Heat of Adsorption 209

11.11.5 Hysteresis 210

11.12 Summary 211

References 215

**12 Chemical Equilibrium 217**

12.1 Introduction to Chemical Reactions and Chemical Equilibrium 217

12.2 Formal Description of Chemical Reactions 217

12.3 Minimizing Gibbs Energy 218

12.4 Reaction Rates Energy Barriers Catalysis and Equilibrium 219

12.5 The Basic Thermodynamics of Chemical Reactions and Its Convenient Formulations 220

12.5.1 The Law of Mass Action and Equilibrium Constants 222

12.6 Calculating Equilibrium Constants from Gibbs Energy Tables and then Using Equilibrium Constants to Calculate Equilibrium Concentrations 223

12.6.1 Change of Reactant Concentration Reaction Coordinate 224

12.6.2 Reversible and Irreversible Reactions 227

12.7 More on Standard States 227

12.8 The Effect of Temperature on Chemical Reaction Equilibrium 229

12.9 The Effect of Pressure on Chemical Reaction Equilibrium 234

12.9.1 Ideal Solution of Ideal Gases 235

12.9.2 Nonideal Solution Nonideal Gases 236

12.9.3 Liquids and Solids 237

12.10 The Effect of Nonideal Solution Behavior 238

12.10.1 Liquid-Phase Nonideality 238

12.11 Other Forms of K 238

12.12 Summary 239

References 242

**13 Equilibrium in Complex Chemical Reactions 243**

13.1 Reactions Involving Ions 243

13.2 Multiple Reactions 244

13.2.1 Sequential Reactions 244

13.2.2 Simultaneous Reactions 245

13.2.3 The Charge Balance Calculation Method and Buffers 246

13.3 Reactions with More Than One Phase 249

13.3.1 Solubility Product 249

13.3.2 Gas-Liquid Reactions 249

13.4 Electrochemical Reactions 252

13.5 Chemical and Physical Equilibrium in Two Phases 255

13.5.1 Dimerization (Association) 255

13.6 Summary 257

References 262

**14 Equilibrium with Gravity or Centrifugal Force Osmotic Equilibrium Equilibrium with Surface Tension 265**

14.1 Equilibrium with Other Forms of Energy 265

14.2 Equilibrium in the Presence of Gravity 266

14.2.1 Centrifuges 268

14.3 Semipermeable Membranes 269

14.3.1 Osmotic Pressure 270

14.4 Small is Interesting! Equilibrium with Surface Tension 271

14.4.1 Bubbles Drops and Nucleation 271

14.4.2 Capillary Condensation 275

14.5 Summary 275

References 278

**15 The Phase Rule 279**

15.1 How Many Phases Can Coexist in a Given Equilibrium Situation? 279

15.2 What Does the Phase Rule Tell Us? What Does It Not Tell Us? 280

15.3 What is a Phase? 280

15.4 The Phase Rule is Simply Counting Variables 281

15.5 More On Components 282

15.5.1 A Formal Way to Find the Number of Independent Equations 285

15.6 The Phase Rule for One- and Two-Component Systems 285

15.7 Harder Phase Rule Problems 288

15.8 Summary 288

References 291

**16 Equilibrium in Biochemical Reactions 293**

16.1 An Example the Production of Ethanol from Sugar 293

16.2 Organic and Biochemical Reactions 293

16.3 Two More Sweet Examples 294

16.4 Thermochemical Data for Biochemical Reactions 295

16.5 Thermodynamic Equilibrium in Large Scale Biochemistry 296

16.6 Translating between Biochemical and Chemical Engineering Equilibrium Expressions 296

16.6.1 Chemical and Biochemical Equations 297

16.6.2 Equilibrium Constants 297

16.6.3 pH and Buffers 298

16.6.4 Ionic Strength 298

16.7 Equilibrium in Biochemical Separations 298

16.8 Summary 299

References 300

**Appendix A Useful Tables and Charts 303**

A.1 Useful Property Data for Corresponding States Estimates 303

A.2 Vapor-Pressure Equation Constants 305

A.3 Henry’s Law Constants 306

A.4 Compressibility Factor Chart (z Chart) 307

A.5 Fugacity Coefficient Charts 307

A.6 Azeotropes 308

A.7 Van Laar Equation Constants 312

A.8 Enthalpies and Gibbs Energies of Formation from the Elements in the Standard States at T ¼ 298.15 K ¼ 25_C and P ¼ 1.00 bar 313

A.9 Heat Capacities of Gases in the Ideal Gas State 317

**Appendix B Equilibrium with other Restraints Other Approaches to Equilibrium 319**

**Appendix C The Mathematics of Fugacity Ideal Solutions**

Activity and Activity Coefficients 323

C.1 The Fugacity of Pure Substances 323

C.2 Fugacities of Components of Mixtures 324

C.3 The Consequences of the Ideal Solution Definition 326

C.4 The Mathematics of Activity Coefficients 326

**Appendix D Equations of State for Liquids and Solids Well Below their Critical Temperatures 329**

D.1 The Taylor Series EOS and Its Short Form 329

D.2 Effect of Temperature on Density 330

D.3 Effect of Pressure on Density 331

D.4 Summary 332

References 333

**Appendix E Gibbs Energy of Formation Values 335**

E.1 Values “From the Elements” 335

E.2 Changes in Enthalpy Entropy and Gibbs Energy 335

E.2.1 Enthalpy Changes 335

E.2.2 Entropy Changes 336

E.3 Ions 337

E.4 Presenting these Data 337

References 337

**Appendix F Calculation of Fugacities from Pressure-Explicit EOSs 339**

F.1 Pressure-Explicit and Volume-Explicit EOSs 339

F.2 f /P of Pure Species Based on Pressure-Explicit EOSs 339

F.3 Cubic Equations of State 340

F.4 fi /Pyi for Individual Species in Mixtures Based on Pressure-Explicit EOSs 342

F.5 Mixing Rules for Cubic EOSs 343

F.6 VLE Calculations with a Cubic EOS 344

F.7 Summary 345

References 346

**Appendix G Thermodynamic Property Derivatives and the Bridgman Table 347**

References 350

**Appendix H Answers to Selected Problems 351**

Index 353