Thermodynamics in Earth and Planetary Sciences / Edition 1

by Jibamitra Ganguly

Overview

This book presents the fundamental principles of classical thermodynamics and their ramifications, and explores their applications to a wide variety of problems in the Earth and Planetary sciences spanning the range from the Earth's core, mantle and crust, aqueous solutions, fluid-rock interactions to solar nebula, terrestrial planets and asteroids. Both geochemical and geophysical aspects of thermodynamics have been integrated within a general framework. The topics covered include the laws of classical thermodynamics and some of their microscopic basis, critical phenomenon and near-critical processes, equations of states, phase transitions and phase equilibria, pressure effect on melting, Earth's interior and adiabatic processes, thermodynamics of solutions, element fractionation, planetary magma oceans, electrolyte solutions, surface effects, properties of very small and nanoparticles, entropy production and kinetic implications, and estimation of thermochemical properties of solids. There is also a review of mathematical methods of classical thermodynamics. Many solved problems have been included to illustrate how thermodynamic analyses lead to quantitative understanding of natural systems.

Product Details

ISBN-13:	9783642095993
Publisher:	Springer Berlin Heidelberg
Publication date:	11/19/2010
Edition description:	Softcover reprint of hardcover 1st ed. 2008
Pages:	501
Product dimensions:	6.10(w) x 9.20(h) x 1.20(d)

About the Author

Jibamitra Ganguly was educated in India (Calcutta University and Jadavpur University) and the University of Chicago, where he received his Ph.D. degree in Geophysical Sciences. This was followed by post-doctoral research at the Yale University and the University of California, Los Angeles, and an appointment to a faculty position at the University of Arizona, where he is currently a Professor Emeritus of Geosciences. His research, which covers a wide range of areas in the Earth and Planetary sciences, has been characterized by an effective blend of experimental, theoretical and observational studies to develop quantitative understanding of natural processes. Prof. Ganguly co-authored (with Prof. S. K. Saxena) the book Mixtures and Mineral Reactions, and edited the volume Diffusion, Atomic Ordering and Mass Transport. He is a fellow of the Mineralogical Society of America and the American Geophysical Union, and received the Alexander von Humboldt research prize for his contributions to thermodynamic and diffusion kinetic studies of natural systems.

Table of Contents

1 Introduction 1

1.1 Nature and Scope of Thermodynamics 1

1.2 Irreversible and Reversible Processes 3

1.3 Thermodynamic Systems, Walls and Variables 4

1.4 Work 5

1.5 Stable and Metastable Equilibrium 9

1.6 Lattice Vibrations 10

1.7 Electronic Configurations and Effects of Crystal Fields 13

1.7.1 Electronic Shells, Subshells and Orbitals 13

1.7.2 Crystal or Ligand Field Effects 15

1.8 Some Useful Physical Quantities and Units 17

2 First and Second Laws 19

2.1 The First Law 20

2.2 Second Law: The Classic Statements 22

2.3 Carnot Cycle: Entropy and Absolute Temperature Scale 23

2.4 Entropy: Direction of Natural Processes and Equilibrium 27

2.5 Microscopic Interpretation of Entropy: Boltzmann Relation 29

2.5.1 Summary of the Important Relations in the First and Second Laws 33

2.6 Entropy and Disorder: Mineralogical Applications 33

2.6.1 Configurational Entropy 33

2.6.2 Vibrational Entropy 38

2.6.3 Configurational vs. Vibrational Entropy 40

2.7 First and Second Laws: Combined Statement 43

2.8 Condition of Thermal Equilibrium: An Illustrative Application of the Second Law 44

2.9 Limiting Efficiency of a Heat Engine and Heat Pump 46

2.9.1 Heat Engine 46

2.9.2 Heat Pump 47

2.9.3 Heat Engines in Nature 49

3 Thermodynamic Potentials and Derivative Properties 53

3.1 Thermodynamic Potentials 53

3.2 Equilibrium Conditions for Closed Systems: Formulations in Terms of the Potentials 56

3.3 What is Free in Free Energy? 58

3.4 Maxwell Relations 58

3.5 Thermodynamic Square: A Mnemonic Tool 59

3.6 Vapor Pressure and Fugacity 61

3.7 Derivative Properties 63

3.7.1 Thermal Expansion and Compressibility63

3.7.2 Heat Capacities 65

3.8 Gruneisen Parameter 68

3.9 P-T Dependencies of Coefficient of Thermal Expansion and Compressibility 71

3.10 Summary of Thermodynamic Derivatives 71

4 Third Law and Thermochemistry 73

4.1 The Third Law and Entropy 73

4.1.1 Observational Basis and Statement 73

4.1.2 Third Law Entropy and Residual Entropy 75

4.2 Behavior of the Heat Capacity Functions 76

4.3 Non-Lattice Contributions to Heat Capacity and Entropy of End-member Solids 80

4.3.1 Electronic Transitions 80

4.3.2 Magnetic Transitions 82

4.4 Unattainability of Absolute Zero 84

4.5 Thermochemistry: Formalisms and Conventions 85

4.5.1 Enthalpy of Formation 85

4.5.2 Hess' Law 87

4.5.3 Gibbs Free Energy of Formation 87

4.5.4 Thermochemical Data 88

5 Critical Phenomenon and Equations of States 91

5.1 Critical End Point 91

5.2 Near- and Super-Critical Properties 95

5.2.1 Divergence of Thermal and Thermo-Physical Properties 95

5.2.2 Critical Fluctuations 96

5.2.3 Super- and Near-Critical Fluids 98

5.3 Near-Critical Properties of Water and Magma-Hydrothermal Systems 99

5.4 Equations of State 102

5.4.1 Gas 103

5.4.2 Solid and Melt 111

6 Phase Transitions, Melting and Reactions of Stoichiometric Phases 115

6.1 Gibbs Phase Rule: Preliminaries 115

6.2 Phase Transformations and Polymorphism 116

6.2.1 Thermodynamic Classification of Phase Transformations 117

6.3 Landau Theory of Phase Transition 119

6.3.1 General Outline 119

6.3.2 Derivation of Constraints on the Second Order Coefficient 123

6.3.3 Effect of Odd Order Coefficient on Phase Transition 124

6.3.4 Order Parameter vs. Temperature: Second Order and Tricritical Transformations 124

6.3.5 Landau Potential vs. Order Parameter: Implications for Kinetics 126

6.3.6 Illustrative Application to a Mineralogical Problem 127

6.4 Reactions in the P-T Space 129

6.4.1 Conditions of Stability and Equilibrium 129

6.4.2 P-T Slope: Clayperon-Classius Relation 130

6.5 Temperature Maximum on Dehydration and Melting Curves 131

6.6 Extrapolation of Melting Temperature to High Pressures 135

6.6.1 Kraut-Kennedy Relation 136

6.6.2 Lindemann-Gilvarry Relation 138

6.7 Calculation of Equilibrium P-T Conditions of a Reaction 138

6.7.1 Equilibrium Pressure at a Fixed Temperature 138

6.7.2 Effect of Polymorphic Transition 143

6.8 Evaluation of Gibbs Energy and Fugacity at High Pressure Using Equations of States 145

6.8.1 Birch-Murnaghan Equation of State 146

6.8.2 Vinet Equation of State 146

6.8.3 Redlich-Kwong and Related Equations of State for Fluids 147

6.9 Schreinemakers' Principles 148

6.9.1 Enumerating Different Types of Equilibria 149

6.9.2 Self-consistent Stability Criteria 150

6.9.3 Effect of an Excess Phase 151

6.9.4 Concluding Remarks 151

7 Thermal Pressure, Earth's Interior and Adiabatic Processes 153

7.1 Thermal Pressure 153

7.1.1 Thermodynamic Relations 153

7.1.2 Core of the Earth 155

7.1.3 Magma-Hydrothermal System 157

7.2 Adiabatic Temperature Gradient 159

7.3 Temperature Gradients in the Earth's Mantle and Outer Core 161

7.3.1 Upper Mantle 161

7.3.2 Lower Mantle and Core 163

7.4 Isentropic Melting in the Earth's Interior 165

7.5 The Earth's Mantle and Core: Linking Thermodynamics and Seismic Velocities 169

7.5.1 Relations among Elastic Properties and Sound Velocities 169

7.5.2 Radial Density Variation 171

7.5.3 Transition Zone in the Earth's Mantle 175

7.6 Joule-Thompson Experiment of Adiabatic Flow 177

7.7 Adiabatic Flow with Change of Kinetic and Potential Energies 180

7.7.1 Horizontal Flow with Change of Kinetic Energy: Bernoulli Equation 181

7.7.2 Vertical Flow 182

7.8 Ascent of Material within the Earth's Interior 184

7.8.1 Irreversible Decompression and Melting of Mantle Rocks 185

7.8.2 Thermal Effect of Volatile Ascent: Coupling Fluid Dynamics and Thermodynamics 187

8 Thermodynamics of Solutions 189

8.1 Chemical Potential and Chemical Equilibrium 189

8.2 Partial Molar Properties 193

8.3 Determination of Partial Molar Properties 195

8.3.1 Binary Solutions 195

8.3.2 Multicomponent Solutions 197

8.4 Fugacity and Activity of a Component in Solution 200

8.5 Determination of Activity of a Component using Gibbs-Duhem Relation 203

8.6 Molar Properties of a Solution 205

8.6.1 Formulations 205

8.6.2 Entropy of Mixing and Choice of Activity Expression 207

8.7 Ideal Solution and Excess Thermodynamic Properties 207

8.7.1 Thermodynamic Relations 207

8.7.2 Ideality of Mixing: Remark on the Choice of Components and Properties 209

8.8 Solute and Solvent Behaviors in Dilute Solution 210

8.8.1 Henry's Law 210

8.8.2 Raoult's Law 213

8.9 Speciation of Water in Silicate Melt 215

8.10 Standard States: Recapitulations and Comments 219

8.11 Stability of a Solution 221

8.11.1 Intrinsic Stability and Instability of a Solution 221

8.11.2 Extrinsic Instability: Decomposition of a Solid Solution 225

8.12 Spinodal, Critical and Binodal (Solvus) Conditions 226

8.12.1 Thermodynamic Formulations 226

8.12.2 Upper and Lower Critical Temperatures 232

8.13 Effect of Coherency Strain on Exsolution 234

8.14 Spinodal Decomposition 236

8.15 Solvus Thermometry 237

8.16 Chemical Potential in a Field 239

8.16.1 Formulations 239

8.16.2 Applications 240

8.17 Osmotic Equilibrium 244

8.17.1 Osmotic Pressure and Reverse Osmosis 244

8.17.2 Osmotic Coefficient 245

8.17.3 Determination of Molecular Weight of a Solute 246

9 Thermodynamic Solution and Mixing Models: Non-electrolytes 249

9.1 Ionic Solutions 249

9.1.1 Single Site, Sublattice and Reciprocal Solution Models 250

9.1.2 Disordered Solutions 254

9.1.3 Coupled Substitutions 255

9.1.4 Ionic Melt: Temkin and Other Models 256

9.2 Mixing Models in Binary Systems 256

9.2.1 Guggenheim or Redlich-Kister, Simple Mixture and Regular Solution Models 257

9.2.2 Subregular Model 259

9.2.3 Darken's Quadratic Formulation 261

9.2.4 Quasi-Chemical and Related Models 263

9.2.5 Athermal, Flory-Huggins and NRTL (Non-random Two Site) Models 266

9.2.6 Van Laar Model 268

9.2.7 Associated Solutions 270

9.3 Multicomponent Solutions 273

9.3.1 Power Series Multicomponent Models 274

9.3.2 Projected Multicomponent Models 275

9.3.3 Comparison Between Power Series and Projected Methods 277

9.3.4 Estimation of Higher Order Interaction Terms 277

9.3.5 Solid Solutions with Multi-Site Mixing 278

9.3.6 Concluding Remarks 278

10 Equilibria Involving Solutions and Gaseous Mixtures 281

10.1 Extent and Equilibrium Condition of a Reaction 281

10.2 Gibbs Free Energy Change and Affinity of a Reaction 283

10.3 Gibbs Phase Rule and Duhem's Theorem 284

10.3.1 Phase Rule 285

10.3.2 Duhem's Theorem 287

10.4 Equilibrium Constant of a Chemical Reaction 289

10.4.1 Definition and Relation with Activity Product 289

10.4.2 Pressure and Temperature Dependences of Equilibrium Constant 291

10.5 Solid-Gas Reactions 292

10.5.1 Condensation of Solar Nebula 292

10.5.2 Surface-Atmosphere Interaction in Venus 296

10.5.3 Metal-Silicate Reaction in Meteorite Mediated by Dry Gas Phase 297

10.5.4 Effect of Vapor Composition on Equilibrium Temperature: T vs. X[subscript v] Sections 299

10.5.5 Volatile Compositions: Metamorphic and Magmatic Systems 303

10.6 Equilibrium Temperature Between Solid and Melt 305

10.6.1 Eutectic and Peritectic Systems 305

10.6.2 Systems Involving Solid Solution 308

10.7 Azeotropic Systems 310

10.8 Reading Solid-Liquid Phase Diagrams 312

10.8.1 Eutectic and Peritectic Systems 312

10.8.2 Crystallization and Melting of a Binary Solid Solution 314

10.8.3 Intersection of Melting Loop and a Solvus 315

10.8.4 Ternary Systems 317

10.9 Natural Systems: Granites and Lunar Basalts 319

10.9.1 Granites 319

10.9.2 Lunar Basalts 321

10.10 Pressure Dependence of Eutectic Temperature and Composition 322

10.11 Reactions in Impure Systems 324

10.11.1 Reactions Involving Solid Solutions 324

10.11.2 Solved Problem 329

10.11.3 Reactions Involving Solid Solutions and Gaseous Mixture 331

10.12 Retrieval of Activity Coefficient from Phase Equilibria 335

10.13 Equilibrium Abundance and Compositions of Phases 337

10.13.1 Closed System at Constant P-T 337

10.13.2 Conditions Other than Constant P-T 342

11 Element Fractionation in Geological Systems 347

11.1 Fractionation of Major Elements 347

11.1.1 Exchange Equilibrium and Distribution Coefficient 347

11.1.2 Temperature and Pressure Dependence of K[subscript D] 349

11.1.3 Compositional Dependence of K[subscript D] 350

11.1.4 Thermometric Formulation 353

11.2 Trace Element Fractionation Between Mineral and Melt 354

11.2.1 Thermodynamic Formulations 354

11.2.2 Illustrative Applications 359

11.2.3 Estimation of Partition Coefficient 360

11.3 Metal-Silicate Fractionation: Magma Ocean and Core Formation 363

11.3.1 Pressure Dependence of Metal-Silicate Partition Coefficients 367

11.3.2 Pressure Dependence of Metal-Silicate Distribution Coefficients 369

11.3.3 Pressure Dependencies of Ni vs. Co Partition- and Distribution-Coefficients 370

11.4 Effect of Temperature and f(O[subscript 2]) on Metal-Silicate Partition Coefficient 372

12 Electrolyte Solutions and Electrochemistry 375

12.1 Chemical Potential 376

12.2 Activity and Activity Coefficients: Mean Ion Formulations 377

12.3 Mass Balance Relation 378

12.4 Standard State Convention and Properties 378

12.4.1 Solute Standard State 378

12.4.2 Standard State Properties of Ions 380

12.5 Equilibrium Constant, Solubility Product & Ion Activity Product 381

12.6 Ion Activity Coefficients and Ionic Strength 382

12.6.1 Debye-Huckel and Related Methods 382

12.6.2 Mean-Salt Method 384

12.7 Multicomponent High Ionic Strength and High P-T Systems 385

12.8 Activity Diagrams of Mineral Stabilities 389

12.8.1 Method of Calculation 389

12.8.2 Illustrative Applications 392

12.9 Electrochemical Cells and Nernst Equation 396

12.9.1 Electrochemical Cell and Half-cells 396

12.9.2 Emf of a Cell and Nernst Equation 397

12.9.3 Standard Emf of Half-Cell and Full-Cell Reactions 398

12.10 Hydrogen Ion Activity in Aqueous Solution: pH and Acidity 399

12.11 Eh-pH Stability Diagrams 399

12.12 Chemical Model of Sea Water 403

13 Surface Effects 409

13.1 Surface Tension and Energetic Consequences 409

13.2 Surface Thermodynamic Functions and Adsorption 411

13.3 Temperature, Pressure and Compositional Effects on Surface Tension 414

13.4 Crack Propagation 415

13.5 Equilibrium Shape of Crystals 416

13.6 Contact and Dihedral Angles 418

13.7 Dihedral Angle and Interconnected Melt or Fluid Channels 423

13.7.1 Connectivity of Melt Phase and Thin Melt Film in Rocks 423

13.7.2 Core Formation in Earth and Mars 425

13.8 Surface Tension and Grain Coarsening 428

13.9 Effect of Particle Size on Solubility 430

13.10 Coarsening of Exsolution Lamellae 432

13.11 Nucleation 434

13.11.1 Theory 434

13.11.2 Microstructures of Metals in Meteorites 436

13.12 Effect of Particle Size on Mineral Stability 438

Appendix A Rate of Entropy Production and Kinetic Implications 443

A.1 Rate of Entropy Production: Conjugate Flux and Force in Irreversible Processes 443

A.2 Relationship Between Flux and Force 447

A.3 Heat and Chemical Diffusion Processes: Comparison with the Empirical Laws 448

A.4 Onsager Reciprocity Relation and Thermodynamic Applications 450

Appendix B Review of Some Mathematical Relations 453

B.1 Total and Partial Differentials 453

B.2 State Function, Exact and Inexact Differentials, and Line Integrals 454

B.3 Reciprocity Relation 456

B.4 Implicit Function 457

B.5 Integrating Factor 458

B.6 Taylor Series 459

Appendix C Estimation of Thermodynamic Properties of Solids 461

C.1 Estimation of Cp and S of End-Members from Constituent Oxides 461

C.1.1 Linear Combination of Components 461

C.1.2 Volume Effect on Entropy 462

C.1.3 Electronic Ordering Effect on Entropy 462

C.2 Polyhedral Approximation: Enthalpy, Entropy and Volume 463

C.3 Estimation of Enthalpy of Mixing 466

C.3.1 Elastic Effect 466

C.3.2 Crystal-Field Effect 468

References 471

Author Index 491

Subject Index 497