**The most readable, understandable, and intuitively satisfying book on the thermodynamics of physical and chemical equilibrium**

Physical and chemical equilibrium and the calculation of the thermodynamic properties of mixtures is a topic of great interest and importance to engineers, but it is not always easy to master. Using the simplest mathematics possible, and providing extensive discussion of the theory and practice of equilibrium calculations, the Second Edition of *Physical and Chemical Equilibrium for Chemical Engineers* provides rigorous but readable, understandable, and intuitively satisfying coverage, showing the reader how to solve common and advanced problems by hand, and by using spreadsheets, and then explains how such problems are solved in computer process design programs.

Fully revised and expanded, this new edition explores physical and chemical equilibrium in detail, including a new appendix on the Bridgman Table and its uses, a new chapter on Equilibrium in Biochemical Reactions, and new sections on minimum work, eutectics and hydrates, adsorption, buffers, and the charge–balance method of computing them. Its appendix on Calculation of Fugacities from Pressure–Explicit EOSs clearly shows how modern computer equilibrium programs actually do their work using the SRK and related equations.

Providing a concise and highly accessible guide to the thermodynamic properties of a wide range of mixtures, and how these properties control physical and chemical equilibrium, this book is of interest not only to chemical, but also environmental and civil engineers. Large numbers of problems and examples worked in detail help the reader integrate the material in this book into his/her working technical toolkit. *Physical and Chemical Equilibrium for Chemical Engineers* is a must–have for anyone wanting to understand and then apply this important branch of thermodynamics.

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, gx 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 25C, 183

11.2.3 Partially Soluble Liquid Pairs at 25C, 183

11.2.4 Miscible Liquid Pairs at 25C, 183

11.2.5 Ternary LLE at 25C, 184

11.2.6 LLE at Temperatures Other Than 25C, 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 25C, 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

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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.