Modern Thermodynamics

by Arieh Ben-naim, Diego Casadei

Overview

This textbook introduces thermodynamics with a modern approach, starting from four fundamental physical facts (the atomic nature of matter, the indistinguishability of atoms and molecules of the same species, the uncertainty principle, and the existence of equilibrium states) and analyzing the behavior of complex systems with the tools of information theory, in particular with Shannon's measure of information (or SMI), which can be defined on any probability distribution. SMI is defined and its properties and time evolution are illustrated, and it is shown that the entropy is a particular type of SMI, i.e. the SMI related to the phase-space distribution for a macroscopic system at equilibrium. The connection to SMI allows the reader to understand what entropy is and why isolated systems follow the Second Law of Thermodynamics. The Second Llaw is also formulated for other systems, not thermally isolated and even open with respect to the transfer of particles. All the fundamental aspects of thermodynamics are derived and illustrated with several examples in the first part of the book. The second part addresses important applications of thermodynamics, covering phase transitions, mixtures and solutions (including the Kirkwood-Buff approach and solvation thermodynamics), chemical equilibrium, and the outstanding properties of water.This textbook is unique in two aspects. First, thermodynamics is introduced with a novel approach, based on information theory applied to macroscopic systems at equilibrium. It is shown that entropy is a particular case of Shannon's measure of information (SMI), and the properties and time evolution of the SMI are used to explain the Second Law of Thermodynamics. This represents a real breakthrough, as classical thermodynamics cannot explain entropy, nor clarify why systems should obey the Second Law. Second, this textbook offers the reader the possibility to get in touch with important and advanced applications of thermodynamics, to address the topics discussed in the second part of the book. Although they may go beyond the content of a typical introductory course on thermodynamics, some of them can be important in the curriculum chosen by the student. At the same time, they are of appealing interest to more advanced scholars.

Product Details

ISBN-13:	9789813200760
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/14/2016
Pages:	372
Product dimensions:	6.00(w) x 8.90(h) x 0.80(d)

Table of Contents

Preface vii

Fundamentals 1

1 Introduction and Overview 3

1.1 Ways of Teaching Thermodynamics 4

1.2 The First and Second Laws 5

1.3 Early Formulations of the Second Law 7

1.4 The Atomic Nature of Matter and Statistical Mechanics 10

1.5 The Birth of Information Theory 14

1.6 The Basic Ideas of Information Theory 16

1.7 Thermodynamic Equilibrium 20

2 The Historical Development of Thermodynamics 23

2.1 Thermodynamic Quantities 24

2.2 Volume and Pressure 25

2.3 Temperature, Pressure and Volume 27

2.4 Equilibrium States and Heat Engines 31

2.5 The Laws of Thermodynamics 33

2.6 The Atomic Nature of Matter 36

2.7 Statistical Thermodynamics 41

3 Elements of Probability Theory 43

3.1 The Axiomatic Approach to Probability 43

3.1.1 The classical definition 48

3.1.2 The frequentist definition 49

3.1.3 Probability as degree of belief 49

3.2 Independent Events and Conditional Probability 50

3.3 Random Variables, Average Variance and Correlation 53

3.4 Continuous Random Variables 56

3.5 The Binomial Distribution 57

3.6 The Normal Distribution 60

3.7 Multidimensional Distributions 62

4 Shannon's Measure of Information (SMI) 65

4.1 Introduction 65

4.2 Shannon's Mmeasure of Information (SMI) 68

4.3 The 20-Question Game Interpretation of H 73

4.4 Some Properties of the Function H 75

4.5 Examples 79

4.6 The Case of Infinite Number of Outcomes 82

4.7 Conditional and Mutual Information 83

4.8 The Various Interpretations of H 87

4.9 Summary of What Have We Learned in this Chapter 90

5 Three Theorems on Shannon's Measure of Information 93

5.1 The First Theorem: The Uniform Distribution 93

5.2 Comparison of Ddiscrete and Continuous SMI 95

5.3 Reinterpretation of f^*(x) as an Equilibrium Density 98

5.3.1 Specific versus generic configurations 99

5.3.2 Probability of a generic configuration 100

5.3.3 Different levels of detail 101

5.3.4 Probability of state distributions 102

5.4 The Second Theorem: The Exponential Distribution 107

5.5 Reinterpretation of f^* _i as an Equilibrium Distribution 110

5.6 The Third Theorem: The Normal Distribution 114

5.7 Summary of What We Have Learned in this Chapter 116

6 The Entropy Function of a Classical Ideal Gas 117

6.1 Some Comment on the Mathematical Notation 118

6.2 The Locational SMI of an Ideal Gas 120

6.3 The Mutual Information due to the Indistinguishability of the Particles 121

6.4 The Momentum SMI 123

6.4.1 The Maxwell-Boltzmann distribution 125

6.5 The Correction Due to the Uncertainty Principle 127

6.6 The Entropy of a Classical Ideal Gas 129

6.7 The Disorder and the Spreading Metaphors of the Entropy 131

6.8 Fundamental Properties of the Entropy Function S(E, V, N) 134

6.9 Summary of What We Have Learned so far 139

7 Thermodynamics of Ideal Gas 143

7.3 The Additivity of the Function S(E, V, N) 144

7.2 The Shape of the Function S(E, V, N) 145

7.2.1 Dependence on E, keeping V and N constant 145

7.2.2 Dependence on V, keeping E and N constant 147

7.2.3 Dependence on V, keeping E and V constant 149

7.2.4 The total change in entropy 151

7.3 Two Spontaneous Processes involving Ideal Gases 151

7.3.1 Expansion of an ideal gas 152

7.3.2 Heat transfer from a hot to a cold body 154

7.3.3 Clausius' definition of entropy 159

7.4 Spontaneous Mixing and Demixing 160

7.5 Summary of What. We Have Learned in this Chapter 164

8 The Fundamental Principles of Thermodynamics 167

8.1 Work and Heat: The First Law of Thermodynamics 168

8.2 Work in an Expansion Process 172

8.3 Isothermal Quasi-static Expansion Process 173

8.4 Work in a Quasi-static Adiabatic Process 178

8.5 Heat Capacity at Constant Volume and at Constant Pressure 180

8.6 Carnot Cycle and Efficiency of a float Engine 182

8.7 Entropy and The Second Law of Thermodynamics 186

8.8 Examples of Internal Parameters and Conditions of Equilibrium 191

8.8.1 Spontaneous transfer of heat between two subsystems: thermal equilibrium 191

8.8.2 Spontaneous "transfer" of volume between two subsystems: mechanical equilibrium 196

8.8.3 Spontaneous transfer of matter between two subsystems: matter equilibrium 197

8.8.4 Spontaneous transformation of molecules from one component to another: chemical equilibrium 199

8.9 Combining the First and The Second Laws 201

8.10 The Helmholtz and Gibbs Energies 206

8.10.1 The Gibbs-Duhem equation 208

8.10.2 The principle of maximum work 209

8.11 Reflections on the Meanings of Entropy and the Second Law 210

8.11.1 Name and interpretation 210

8.11.2 The concept 214

8.11.3 The role of time 218

8.12 Summary 220

Applications 223

9 The Phase Rule and Phase Diagrams 225

9.1 States of Matter and Phase Transitions 225

9.1.1 Non-classical states of matter 227

9.2 The Phase Rule 229

9.3 One-Component Systems 232

9.3.1 Coexistence of two phases of the same component 233

9.3.2 Coexistence of three phases of the same component 238

9.3.3 The critical point 238

9.3.4 Allotropy 242

9.3.5 The phase diagram of sulfur 243

9.3.6 The phase diagram of phosphorous 244

9.3.7 The phase diagram of carbon 246

9.4 Two-Component Systems 247

9.4.1 Two liquid phases at equilibrium 248

9.4.2 LCST and UCST 252

9.4.3 System with one liquid phase in equilibrium with solid phases 253

9.4.4 Systems with a congruent melting point 257

9.4.5 Two miscible liquids at equilibrium with a vapour phase 259

10 Mixtures and Solutions 263

10.1 Partial Molar Quantities 263

10.2 Pair Correlation Function and Kirk wood-Buff Integrals 264

10.2.1 The pair correlation function for simple one component liquid 265

10.2.2 The pair correlation function for multi component system of simple particles 269

10.2.3 The Kirkwood-Buff integrals 270

10.2.4 An exact, expression from the Kirkwood-Buff theory of solutions 272

10.3 The Three Reference Ideal Solutions 273

10.3.1 Ideal gas mixture. Experimental approach 274

10.3.2 Ideal gas mixture. Theoretical approach 275

10.3.3 Dilute ideal solutions. Experimental approach 276

10.3.4 Dilute ideal solutions. Theoretical approach 277

10.3.5 Symmetric ideal solution. Experimental approach 278

10.3.6 Symmetric ideal solution. Theoretical approach 279

10.4 Examples and Applications 280

10.4.1 Lowering the freezing temperature 280

10.4.2 Elevation of the boiling temperature 281

10.4.3 Osmotic pressure 281

10.5 Deviations from Ideal Behavior 284

10.5.1 Small deviations from ideal gas behavior 284

10.5.2 Small deviations from dilute ideal solutions 285

10.5.3 Small deviations from symmetrical ideal behavior 287

10.6 Global and Local Characterization of Mixtures 289

10.7 Large Deviations from SI Solutions and Stability Condition 290

10.8 Solvation Thermodynamics 293

10.8.1 Why do we need solvation thermodynamics? 293

10.8.2 Solvation process and solvation quantities 294

11 Chemical Equilibrium 301

11.1 The Simple Isomerization Reaction 301

11.2 A General Chemical Reaction 304

11.3 Chemical Equilibrium in a Solution 306

13.2 The Temperature Dependence of the Equilibrium Constant 307

11.2 The Pressure Dependence of the Equilibrium Constants 311

11.3 The Dependence of the Equilibrium Constant on the Concentration of a Component which is not Involved in the Reaction 312

12 Water and Aqueous Solutions 315

12.1 Relevance to Biology 315

12.2 Hydrogen Bonds 318

12.3 Water Molecules 321

12.4 Ice Structure 322

12.5 Mixture-Model Approach to Liquid Water 325

12.6 Exact Mixture-Model Approach to the Theory of Liquids 329

12.6.1 Coordination number 329

12.6.2 Two-component mixture 331

12.6.3 Dependence of the molar volume on the temperature 332

12.6.4 Dependence of the heat capacity on the temperature 334

12.6.5 Isothermal compressibility 336

Appendix A Solutions to Exercises 339

Appendix B Mathematics 345

B.1 Proof of log x ≤ x - 1 345

B.2 Euler's Theorem 345

Bibliography 347

Index 351