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Thermodynamics in Earth and Planetary Sciences

Second Edition

123 Jibamitra Ganguly Department of Geosciences University of Arizona Tucson, AZ, USA

ISSN 2510-1307 ISSN 2510-1315 (electronic) Springer Textbooks in Earth Sciences, Geography and Environment ISBN 978-3-030-20878-3 ISBN 978-3-030-20879-0 (eBook) https://doi.org/10.1007/978-3-030-20879-0

1st edition: © Springer-Verlag Berlin Heidelberg 2008 2nd edition: © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland A theory is the more impressive the greater the simplicity of its premises, the more different kind of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts. Albert Einstein Dedicated to the Pioneers who led the way and Students, Colleagues and Mentors who helped me along the way Preface to the Second Edition

The first edition of the book was written when I had to carry a full load of teaching, maintain well-funded research programs and carry out many other activities that are typically expected of someone holding an academic position in USA. There was hardly any free time to write a book of this nature, and conse- quently writing of the book became a stressful undertaking. This, however, is a common situation with active scientists who write books. After my retirement a few years ago that allowed me considerable freedom of how I use my time, I felt that I should work on the book again for a second edition that would give me the opportunity to correct many typographical errors that I have spotted in the meanwhile (these were corrected in the Chinese translation that was published in 2015), improve the clarity of presentation in several places and add new materials. In the last category, I have added a new chapter on Statistical Thermodynamics and also significant amount of new materials in many of the existing chapters; most of these additions are in Chaps. 4 – 8, 10, 12, 13 and Appendices A and C. Additionally, there is a new Appendix (D) containing solutions of selected prob- lems that are marked by asterisks in the chapters. Answers and hints for solutions have been provided for the problems for which solutions are not included in this Appendix. As in the first edition, I have inserted the problems into appropriate places within the text that the problems relate to instead of following the usual practice of collecting them at the end of each chapter. I am thankful to Dr. Manga Venkateswara Rao for helpful discussions and reviews of selected chapters. In the Introduction of their ground breaking work on paleothermometry using Statistical Thermodynamics in 1951, Harold Urey and co-workers remarked:

Geologists have drawn many conclusions from the purely qualitative evi- dence of geological studies in regard to the past climatic conditions on the earth. These deductions are based upon a great variety of evidence, and the ability of the geologists to deduce as much as they have in regard to these conditions excites the wonder and admiration of all the uninitiated who examine their work even casually.

Although this statement is about the geologists’ early contribution to paleocli- mate studies, it also applies, at least in my judgment, to geologists’, and in a broader

ix x Preface to the Second Edition sense Earth scientists’ contributions to a variety of large-scale problems. However, as in the paleoclimate studies, one of the most important developments in Geo- logical and related aspects of Planetary sciences has been the integration of quantitative analysis using thermodynamics with the “qualitative evidence” that the Geologists are used to dealing with and their ability to extract major new insights from that to address large-scale natural processes. Hopefully, the extent and scope of application of thermodynamics to the quantitative analysis of complex natural processes would continue to grow.

Tucson, AZ, USA Jibamitra Ganguly March 2019 Preface to the First Edition

When the knowledge is weak and the situation is complicated, thermodynamic relations are really the most powerful Richard Feynman

Thermodynamics has played a major role in improving our understanding of natural processes and would continue to do so for the foreseeable future. In fact, a course in thermodynamics has now become a part of Geosciences curriculum in many Institutions despite the fact that a formal thermodynamics course is taught in every other department of physical sciences, and also in departments of Chemical Engineering, Materials Sciences, and Biological Sciences. The reason thermody- namics is taught in a variety of departments, probably more so than any other subject, is that its principles have wide ranging applications but the teaching of thermodynamics also needs special focus depending on the problems in a par- ticular field. There are numerous books in thermodynamics that have usually been written with particular focus to the problems in the traditional fields of Chemistry, Physics, and Engineering. In recent years, several books have also been written that emphasized applications to Geological problems. Thus, one may wonder why there is yet another book in thermodynamics. The primary focus of the books that have been written with Geosciences audience in mind has been chemical thermody- namics or Geochemical thermodynamics. Along with expositions of fundamental principles of thermodynamics, I have tried to address a wide range of problems relating to geochemistry, petrology, mineralogy, geophysics, and planetary sci- ences. It is not a fully comprehensive effort but is a major attempt to develop a core material that should be of interest to people with different specialties in the Earth and Planetary Sciences. The conditions of the systems in the Earth and Planetary Sciences to which thermodynamics have been applied cover a very large range in pressure-temperature space. For example, the P-T conditions for the processes at the Earth’s surface are 1 bar, 25 °C, whereas those for the processes in the deep interior of the Earth are at pressures of the order of 106 bars and temperatures of the order of 103 °C. The pressures for processes in the solar nebula are 10−3–10−4 bars. The extreme

xi xii Preface to the First Edition range of conditions encompassed by natural processes requires variety of manipu- lations and approximations that are not readily available in the standard textbooks on thermodynamics. Earth scientists have made significant contributions in these areas that have been overlooked in the standard texts since the expected audience of these texts rarely deal with the conditions that Earth scientists have to. I have tried to highlight the contributions of Earth scientists that have made possible meaningful applications of thermodynamics to natural problems. In order to develop a proper appreciation of thermodynamic laws and thermo- dynamic properties of matter, it is useful to look into their physical picture by relating them to the microscopic descriptions. Furthermore, in geological problems, it is often necessary to extrapolate thermodynamic properties of matter way beyond the conditions at which these have been measured, and also to be able to estimate thermodynamic properties because of lack of adequate data to address a specific problem at hand. These efforts require an understanding of the physical or micro- scopic basis of thermodynamic properties. Thus, I have occasionally digressed to the discussion of thermodynamics from microscopic viewpoints, although the formal aspects of the subject of thermodynamics can be completely developed without appealing to the microscopic picture. On the other hand, I have not spent too much effort to discuss how the thermodynamic laws were developed, as there are many excellent books dealing with these topics, but rather focused on exploring the implications of these laws after discussing their essential contents. In several cases, however, I have chosen to provide the derivations of equations in consid- erable detail in order to convey a feeling of how thermodynamic relations are manipulated to derive practically useful relations. This book has been an outgrowth of a course on thermodynamics that I have been teaching to graduate students of Earth and Planetary Sciences at the University of Arizona for over a decade. In this course, I have meshed the development of the fundamental principles with applications, mostly to natural problems. This may not be the most logical way of presenting the subject, but I have found it to be an effective way to keep the interest of the students alive, and answer “why am I doing this?” In addition, I have put problems within the text in appropriate places, and in many cases posed the derivation of some standard equations as problems, with hints wherever I felt necessary based on the questions that I have received from my students when they were given these problems to solve. I have tried to write this book in a self-contained way, as much as possible. Thus, the introductory chapter contains concepts from mechanics and quantum chemistry that were used later to develop concepts of thermodynamics and an understanding of some of their microscopic basis. The Appendix B contains a summary of some of the mathematical concepts and tools that are commonly used in classical thermodynamics. Selected sections of the book have been reviewed by a number of colleagues: Sumit Chakraborty, Weji Cheng, Jamie Connolly, Mike Drake, Charles Geiger, Ralph Kretz, Luigi Marini, Denis Norton, Giulio Ottonello, Kevin Righter, Surendra Saxena, Rishi Narain Singh, Max Tirone, and Krishna Vemulapalli. Preface to the First Edition xiii

I gratefully acknowledge their help but take full responsibility for the errors that might still be present. In addition, feedbacks from the graduate students, who took my thermodynamics course, have played an important role in improving the clarity of presentation, and catching errors, not all of which were typographical. I will be grateful if the readers draw my attention to errors, typographical or otherwise, that might have still persisted. All errors will be posted on my web page that can be accessed using the link http://www.geo.arizona.edu/web/Ganguly/JG_page.html. I started writing the book seriously while I was in the Bayerisches Geoinstitüt, Bayreuth, and University of Bochum, both in Germany, during my sabbatical leave in 2002–2003 that was generously supported by the Alexander von Humboldt Foundation through a research prize (forschungspreis). I gratefully acknowledge the support of the AvH foundation, and the hospitality of the two institutions, espe- cially those of the hosts, Profs. Dave Rubie and Sumit Chakraborty. Research grants from the NASA Cosmochemistry program to investigate thermodynamic and kinetic problems in the planetary systems provided significant incentives to explore planetary problems, and also made my continued involvement in thermodynamics through the period of writing this book easier from a practical standpoint. I am also very grateful for these supports. I hope that this book would be at least partly successful in accomplishing its goal of presenting the subject of thermodynamics in a way that shows its power in the development of quantitative understanding of a wide variety of geological and planetary processes. And finally, as remarked by the noted thermodynamicist, Kenneth Denbigh (1955) Thermodynamics is a subject which needs to be studied not once but several times over at advancing levels

October 2007 Jibamitra Ganguly Tucson, Arizona, USA 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 Crystal Field Effects ...... 14 1.7.1 Electronic Shells, Subshells and Orbitals ...... 14 1.7.2 Crystal or Ligand Field Effects ...... 16 1.8 Some Useful Physical Quantities and Units ...... 18 References ...... 19 2 First and Second Laws ...... 21 2.1 The First Law ...... 22 2.2 Second Law: The Classic Statements ...... 24 2.3 Carnot Cycle: Entropy and Absolute Temperature Scale ..... 26 2.4 Entropy: Direction of Natural Processes and Equilibrium ..... 29 2.5 Microscopic Interpretation of Entropy: Boltzmann Relation ... 31 2.6 Black Hole and Generalized Second Law of Thermodynamics ...... 35 2.7 Entropy and Disorder: Mineralogical Applications ...... 36 2.7.1 Configurational Entropy ...... 37 2.7.2 Vibrational Entropy ...... 42 2.7.3 Configurational Versus Vibrational Entropy ...... 43 2.8 First and Second Laws: Combined Statement ...... 47 2.9 Condition of Thermal Equilibrium: An Illustrative Application of the Second Law ...... 49 2.10 Limiting Efficiency of a Heat Engine and Heat Pump ...... 50 2.10.1 Heat Engine ...... 50 2.10.2 Heat Pump ...... 52 2.10.3 Heat Engines in Nature ...... 53 References ...... 56

xv xvi Contents

3 Thermodynamic Potentials and Derivative Properties ...... 57 3.1 Thermodynamic Potentials ...... 57 3.2 Equilibrium Conditions for Closed Systems: Formulations in Terms of the Potentials ...... 60 3.3 What Is Free in Free Energy? ...... 62 3.4 Maxwell Relations ...... 63 3.5 Thermodynamic Square: A Mnemonic Tool ...... 64 3.6 Vapor Pressure and Fugacity ...... 65 3.7 Derivative Properties ...... 68 3.7.1 Thermal Expansion and Compressibility ...... 68 3.7.2 Heat Capacities ...... 70 3.8 Grüneisen Parameter ...... 73 3.9 P–T Dependencies of Coefficient of Thermal Expansion and Compressibility ...... 76 3.10 Summary of Thermodynamic Derivatives ...... 77 References ...... 77 4 Third Law and Thermochemistry ...... 79 4.1 The Third Law and Entropy ...... 79 4.1.1 Observational Basis and Statement ...... 79 4.1.2 Third Law Entropy and Residual Entropy ...... 81 4.2 P-T Dependence of Heat Capacity Functions ...... 82 4.3 Non-lattice Contributions to Heat Capacity and Entropy of Pure Solids ...... 87 4.3.1 Electronic Transitions ...... 87 4.3.2 Magnetic Transitions ...... 89 4.4 Unattainability of Absolute Zero ...... 91 4.5 Thermochemistry: Formalisms and Conventions ...... 92 4.5.1 Enthalpy of Formation ...... 92 4.5.2 Hess’s Law ...... 94 4.5.3 Gibbs Free Energy of Formation ...... 94 4.5.4 Thermochemical Data ...... 95 References ...... 98 5 Critical Phenomenon and Equations of States ...... 101 5.1 Critical End Point ...... 101 5.2 Near- and Super-Critical Properties ...... 105 5.2.1 Divergence of Thermal and Thermo-Physical Properties ...... 105 5.2.2 Critical Fluctuations ...... 107 5.2.3 Super- and Near-Critical Fluids ...... 109 5.3 Near-Critical Properties of Water and Magma-Hydrothermal Systems ...... 109 Contents xvii

5.4 Equations of State ...... 113 5.4.1 Gas ...... 114 5.4.2 Solid and Melt ...... 121 References ...... 128 6 Phase Transitions, Melting, and Reactions of Stoichiometric Phases ...... 131 6.1 Gibbs Phase Rule: Preliminaries ...... 131 6.2 Phase Transformations and Polymorphism ...... 133 6.2.1 Thermodynamic Classification of Phase Transformations ...... 133 6.3 Landau Theory of Phase Transition ...... 136 6.3.1 General Outline ...... 136 6.3.2 Derivation of Constraints on the Second Order Coefficient ...... 140 6.3.3 Effect of Odd Order Coefficient on Phase Transition ...... 140 6.3.4 Order Parameter Versus Temperature: Second Order and Tricritical Transformations ...... 141 6.3.5 Landau Potential Versus Order Parameter: Implications for Kinetics ...... 142 6.3.6 Some Applications to Mineralogical and Geophysical Problems ...... 144 6.4 Reactions in the P-T Space ...... 146 6.4.1 Conditions of Stability and Equilibrium ...... 146 6.4.2 P-T Slope: Clapeyron-Clausius Relation ...... 148 6.5 Temperature Maximum on Dehydration and Melting Curves ...... 149 6.6 Extrapolation of Melting Temperature to High Pressures ..... 153 6.6.1 Kraut-Kennedy Relation ...... 153 6.6.2 Lindemann-Gilvarry Relation...... 155 6.7 Calculation of Equilibrium P-T Conditions of a Reaction .... 156 6.7.1 Equilibrium Pressure at a Fixed Temperature ...... 156 6.7.2 Effect of Polymorphic Transition ...... 160 6.8 Evaluation of Gibbs Energy and Fugacity at High Pressure Using Equations of States ...... 164 6.8.1 Birch-Murnaghan Equation of State ...... 165 6.8.2 Vinet Equation of State ...... 165 6.8.3 Redlich-Kwong and Related Equations of State for Fluids ...... 166 6.9 Schreinemakers’ Principles ...... 168 6.9.1 Enumerating Different Types of Equilibria ...... 168 6.9.2 Self-consistent Stability Criteria ...... 170 xviii Contents

6.9.3 Effect of an Excess Phase ...... 171 6.9.4 Concluding Remarks ...... 171 References ...... 172 7 Thermal Pressure, Earth’s Interior and Adiabatic Processes ..... 175 7.1 Thermal Pressure ...... 176 7.1.1 Thermodynamic Relations ...... 176 7.1.2 Core of the Earth ...... 177 7.1.3 Magma-Hydrothermal System ...... 180 7.2 Adiabatic Temperature Gradient ...... 182 7.3 Temperature Gradients in the Earth’s Mantle and Outer Core ...... 184 7.3.1 Upper Mantle ...... 184 7.3.2 Lower Mantle and Core ...... 186 7.4 Isentropic Melting in the Earth’s Interior ...... 189 7.5 The Earth’s Mantle and Core: Linking Thermodynamics and Seismic Velocities ...... 193 7.5.1 Relations Among Elastic Properties and Sound Velocities ...... 193 7.5.2 Radial Density Variation ...... 195 7.5.3 Transition Zone in the Earth’s Mantle ...... 198 7.6 Horizontal Adiabatic Flow at Constant Velocity ...... 201 7.6.1 Joule-Thompson Experiment and Coefficient ...... 201 7.6.2 Entropy Production in Joule-Thompson Expansion ...... 204 7.7 Adiabatic Flow with Change of Kinetic and Potential Energies ...... 205 7.7.1 Horizontal Flow with Change of Kinetic Energy: Bernoulli Equation ...... 206 7.7.2 Vertical Flow ...... 207 7.8 Ascent of Material Within the Earth’s Interior ...... 209 7.8.1 Irreversible Decompression and Melting of Mantle Rocks ...... 210 7.8.2 Thermal Effect of Volatile Ascent: Coupling Fluid Dynamics and Thermodynamics ...... 213 References ...... 214 8 Thermodynamics of Solutions ...... 217 8.1 Chemical Potential and Chemical Equilibrium ...... 217 8.2 Partial Molar Properties ...... 222 8.3 Determination of Partial Molar Properties ...... 224 8.3.1 Binary Solutions ...... 224 8.3.2 Multicomponent Solutions ...... 226 8.4 Fugacity and Activity of a Component in a Solution ...... 229 Contents xix

8.5 Determination of Activity of a Component Using Gibbs-Duhem Relation ...... 232 8.6 Molar Properties of a Solution ...... 234 8.6.1 Formulations ...... 234 8.6.2 Entropy of Mixing and Choice of Activity Expression ...... 236 8.7 Ideal Solution and Excess Thermodynamic Properties ...... 236 8.7.1 Thermodynamic Relations ...... 236 8.7.2 Ideality of Mixing: Remark on the Choice of Components and Properties ...... 238 8.8 Solute and Solvent Behaviors in Dilute Solution ...... 239 8.8.1 Henry’s Law ...... 240 8.8.2 Raoult’s Law ...... 242 8.9 Speciation of Water in Silicate Melt ...... 245 8.10 Standard States: Recapitulations and Comments ...... 248 8.11 Stability of a Solution ...... 250 8.11.1 Intrinsic Stability and Instability of a Solution ..... 251 8.11.2 Extrinsic Instability: Decomposition of a Solid Solution ...... 254 8.12 Spinodal, Critical and Binodal (Solvus) Conditions ...... 255 8.12.1 Thermodynamic Formulations ...... 255 8.12.2 Upper and Lower Critical Temperatures ...... 262 8.13 Effect of Coherency Strain on Exsolution ...... 264 8.14 Spinodal Decomposition ...... 266 8.15 Solvus Thermometry ...... 268 8.16 Chemical Potential in a Field ...... 269 8.16.1 Formulations ...... 269 8.16.2 Applications ...... 271 8.17 Osmotic Equilibrium ...... 276 8.17.1 Osmotic Pressure, Reverse Osmosis ...... 276 8.17.2 Natural Salinity Gradients and Power Generation ... 277 8.17.3 Osmotic Coefficient ...... 279 8.17.4 Determination of Molecular Weight of a Solute .... 280 References ...... 281 9 Thermodynamic Solution and Mixing Models: Non-electrolytes ... 283 9.1 Ionic Solutions ...... 283 9.1.1 Single Site, Sublattice and Reciprocal Solution Models ...... 284 9.1.2 Disordered Solutions ...... 288 9.1.3 Coupled Substitutions ...... 289 9.1.4 Ionic Melt: Temkin and Other Models ...... 290 xx Contents

9.2 Mixing Models in Binary Systems ...... 291 9.2.1 Guggenheim or Redlich-Kister, Simple Mixture and Regular Solution Models ...... 291 9.2.2 Subregular Model ...... 294 9.2.3 Darken’s Quadratic Formulation ...... 295 9.2.4 Quasi-chemical and Related Models ...... 298 9.2.5 Athermal, Flory-Huggins and NRTL (Non-random Two Liquid) Models ...... 301 9.2.6 Van Laar Model ...... 303 9.2.7 Associated Solutions ...... 306 9.3 Multicomponent Solutions ...... 309 9.3.1 Power Series Multicomponent Models ...... 310 9.3.2 Projected Multicomponent Models ...... 311 9.3.3 Comparison Between Power Series and Projected Methods ...... 312 9.3.4 Estimation of Higher Order Interaction Terms ..... 313 9.3.5 Solid Solutions with Multi-site Mixing...... 314 9.3.6 Concluding Remarks ...... 314 References ...... 315 10 Equilibria Involving Solutions and Gaseous Mixtures ...... 319 10.1 Extent and Equilibrium Condition of a Reaction ...... 319 10.2 Gibbs Free Energy Change and Affinity of a Reaction ...... 321 10.3 Gibbs Phase Rule and Duhem’s Theorem ...... 323 10.3.1 Phase Rule ...... 323 10.3.2 Duhem’s Theorem ...... 326 10.4 Equilibrium Constant of a Chemical Reaction ...... 327 10.4.1 Definition and Relation with Activity Product ..... 327 10.4.2 Pressure and Temperature Dependencies of Equilibrium Constant ...... 329 10.5 Solid-Gas and Homogeneous Gas Speciation Reactions ...... 331 10.5.1 Condensation of Solar Nebula ...... 331 10.5.2 Surface-Atmosphere Interaction in Venus ...... 334 10.5.3 Metal-Silicate Reaction in Meteorite Mediated by Dry Gas Phase ...... 336 10.5.4 Effect of Vapor Composition on Equilibrium Temperature: T Versus Xv Sections ...... 338 10.5.5 Volatile Compositions and Oxidation States of Natural Systems ...... 342 10.6 Equilibrium Temperature Between Solid and Melt ...... 348 10.6.1 Eutectic and Peritectic Systems ...... 348 10.6.2 Systems Involving Solid Solution ...... 350 10.7 Azeotropic Systems ...... 353 Contents xxi

10.8 Reading Solid-Liquid Phase Diagrams ...... 356 10.8.1 Eutectic and Peritectic Systems ...... 356 10.8.2 Crystallization and Melting of a Binary Solid Solution ...... 358 10.8.3 Intersection of Melting Loop and a Solvus ...... 359 10.8.4 Ternary Systems ...... 361 10.9 Natural Systems: Granites and Lunar Basalts ...... 363 10.9.1 Granites ...... 363 10.9.2 Lunar Basalts ...... 365 10.10 Pressure Dependence of Eutectic Temperature and Composition ...... 366 10.11 Reactions in Impure Systems ...... 369 10.11.1 Reactions Involving Solid Solutions ...... 370 10.11.2 Reactions Involving Solid Solutions and Gaseous Mixture ...... 378 10.12 Retrieval of Activity Coefficient from Phase Equilibria ...... 381 10.13 Equilibrium Abundance and Compositions of Phases ...... 383 10.13.1 Closed System at Constant P-T ...... 383 10.13.2 Closed System at Constant V-T ...... 390 10.13.3 Minimization of Korzhinskii Potential ...... 393 References ...... 395 11 Element Fractionation in Geological Systems ...... 399 11.1 Fractionation of Major Elements ...... 399 11.1.1 Exchange Equilibrium and Distribution Coefficient ...... 399 11.1.2 Temperature and Pressure Dependence of KD ...... 401 11.1.3 Compositional Dependence of KD ...... 402 11.1.4 Thermometric Formulation ...... 405 11.2 Trace Element Fractionation Between Mineral and Melt ..... 406 11.2.1 Thermodynamic Formulations ...... 406 11.2.2 Illustrative Applications ...... 411 11.2.3 Estimation of Partition Coefficient ...... 412 11.3 Metal-Silicate Fractionation: Magma Ocean and Core Formation ...... 416 11.3.1 Pressure Dependence of Metal-Silicate Partition Coefficients ...... 418 11.3.2 Pressure Dependence of Metal-Silicate Distribution Coefficients ...... 421 xxii Contents

11.3.3 Pressure Dependencies of Ni Versus Co Partition- and Distribution-Coefficients: Depth of Terrestrial Magma Ocean ...... 423 11.4 Effect of Temperature and f(O2) on Metal-Silicate Partition Coefficient ...... 424 References ...... 426 12 Electrolyte Solutions and Electrochemistry ...... 429 12.1 Chemical Potential ...... 430 12.2 Activity and Activity Coefficients: Mean Ion Formulations ... 431 12.3 Mass Balance Relation ...... 432 12.4 Standard State Convention and Properties ...... 432 12.4.1 Solute Standard State ...... 432 12.4.2 Standard State Properties of Ions ...... 434 12.5 Equilibrium Constant, Solubility Product and Ion Activity Product: Survival of Marine Carbonate Organisms ...... 435 12.6 Ion Activity Coefficients and Ionic Strength ...... 437 12.6.1 Debye-Hückel and Related Methods ...... 437 12.6.2 Mean-Salt Method ...... 439 12.7 Multicomponent High Ionic Strength and High P-T Systems ...... 441 12.8 Activity Diagrams of Mineral Stabilities ...... 444 12.8.1 Method of Calculation ...... 445 12.8.2 Illustrative Applications ...... 448 12.9 Electrochemical Cells, Nernst Equation and f(O2) Measurement by Solid Electrolyte ...... 452 12.9.1 Electrochemical Cell and Half-Cells ...... 452 12.9.2 Emf of a Cell and Nernst Equation ...... 453 12.9.3 Oxygen Fugacity Measurement Using Solid Electrolyte Sensor ...... 454 12.9.4 Standard Emf of Half-Cell and Full-Cell Reactions ...... 456 12.10 Hydrogen Ion Activity in Aqueous Solution: pH and Acidity ...... 456 12.11 Eh-pH Stability Diagrams ...... 457 12.12 Chemical Model of Sea Water ...... 462 References ...... 465 13 Surface Effects ...... 467 13.1 Surface Tension and Energetic Consequences ...... 467 13.2 Surface Thermodynamic Functions and Adsorption ...... 469 13.3 Temperature, Pressure and Compositional Effects on Surface Tension ...... 472 13.4 Langmuir Isotherm ...... 474 Contents xxiii

13.5 Crack Propagation ...... 477 13.6 Equilibrium Shape of Crystals ...... 478 13.7 Contact and Dihedral Angles ...... 481 13.8 Dihedral Angle and Interconnected Melt or Fluid Channels ... 486 13.8.1 Connectivity of Melt Phase and Thin Melt Film in Rocks ...... 488 13.8.2 Core Formation in Earth and Mars...... 488 13.9 Surface Tension and Grain Coarsening ...... 491 13.10 Effect of Particle Size on Solubility and Melting ...... 494 13.11 Coarsening of Exsolution Lamellae ...... 500 13.12 Nucleation ...... 503 13.12.1 Theory ...... 503 13.12.2 Microstructures of Metals in Meteorites ...... 504 13.13 Effect of Particle Size on Mineral Stability ...... 506 References ...... 510 14 Statistical Thermodynamics Primer ...... 513 14.1 Boltzmann Distribution and Partition Function ...... 513 14.2 Thermodynamic Properties ...... 515 14.3 Expressions of Partition Functions ...... 518 14.4 Heat Capacity of Solids ...... 525 14.5 Chemical Equilibria and Stable Isotope Fractionation ...... 527 14.5.1 General Treatment of Chemical Reaction ...... 529 14.5.2 Stable Isotope Fractionation: Theoretical Foundation ...... 530 14.5.3 Stable Isotope Fractionation: Some Geochemical Applications ...... 536 References ...... 541

Appendix A: Rate of Entropy Production and Kinetic Implications .... 543 Appendix B: Review of Some Mathematical Relations...... 557 Appendix C: Estimation of Thermodynamic Properties of Solids ...... 567 Appendix D: Solutions of Selected Problems...... 585 References ...... 593 Author Index...... 597 Subject Index...... 605 Commonly Used Symbols

(Usual meanings, unless specified otherwise)

a a a ai or a(i) Activity of a component i in a phase 0 fi Cp & CP Isobaric-molar and isobaric-speci c heat capacity, respectively Cv Heat capacity at constant volume a=b Partition coefficient of i between the phases a and b Di a=b ¼ð a= bÞ Di Xi Xi dZ An exact differential or total derivative of Z dY An inexact differential @X A partial derivative of X a ðaÞ a fi , fi Fugacity of a component i in a phase f Reduced partition function ratio (RPFR) F Helmholtz free energy F0 Faraday constant G, Gm Total and molar Gibbs free energy Ã; o o Gi Gi Gibbs free energy of i in the standard (*) and pure ( ) state, respectively DGmix; DGxs Gibbs free energy of mixing and excess (xs) Gibbs free energy mixing of a solution, respectively à DrG; DrG Gibbs energy change and standard state (*) Gibbs energy change of a reaction DGf;e; DGf;o Gibbs free energy of formation of a compound from the constituent elements and oxides, respectively x Gf Gibbs free energy of formation of a solute ion in a hypothetical “solute standard state” at unit molality in an electrolyte solution (Chap. 12.4.1) gi Partial molar Gibbs free energy of the component i in a solution g Acceleration due to gravity H, Hm Total and molar enthalpy of a solution, respectively Ã; o o Hi Hi Enthalpy of i in the standard (*) and pure ( ) state, respectively DHmixðalso DHxsÞ Enthalpy of mixing of a solution

xxv xxvi Commonly Used Symbols

DHf;e & DHf;o Heat of formation of a compound from elements and oxides, respectively DrH Enthalpy change of a reaction ; 0 Partial and specific(′) molar enthalpy of the component i in a hi hi solution h Planck constant, unless specified as height K Equilibrium constant of a reaction kT Isothermal bulk modulus kS Adiabatic bulk modulus 0 KH; K H Henry’s law constant in fugacity-composition and activity-composition relations, respectively, of a dilute component −23 kB Boltzmann constant (1.38065  10 J/K) KD(i-j) Distribution coefficient of the components i and j between a pair of phases KT,ks Isothermal and adiabatic bulk modulus, respectively L Avogadro’s number (6.02217  1023 mol−1) mi Molality of a component in a solution N Total number of moles ni Number of moles of the component i Qeq Equilibrium activity ratio of a reaction (¼K) Q Canonical partition function in Chap. 14 q Molecular partition function R Gas constant (8.314 J/mol-K) S, Sm Total, molar entropy of a solution, respectively Ã; o Si Si Entropy of the component i in the standard and pure state, respectively DSmix; DSxs Entropy of mixing and excess (xs) entropy of mixing of a solution DrS Entropy change of a reaction si Partial molar entropy of the component i in a solution Tc Temperature of a critical point Tc(sol) Critical temperature of mixing of a solution U; u0 Internal energy and specific internal energy (′), respectively V, Vm Total and molar volume of a solution, respectively vi or Vi Partial molar volume of the component in a solution DVmixðalso DVxsÞ Volume of mixing of a solution DrV Volume change of a reaction þ W þ ; dw Total and infinitesimal work done by a system, respectively À WÀ; dw Total and infinitesimal work done on a system, respectively dx þ ; dxÀ Infinitesimal non-PV work done by and on a system, respectively Xi or X(i) Atomic or mole fraction of the component i in a solution p xi Atomic fraction of i in the sublattice p of a solution Commonly Used Symbols xxvii yi Partial molar property of a component i in a solution with a total property Y (e.g., vi: partial molar volume of the component i in a solution with a total volume V) a Coefficient of thermal expansion bT Isothermal bulk modulus b S Adiabatic bulk modulus a ci Activity coefficient of a component i in a phase a c Dihedral angle in Chap. 13 Ci Concentration of component i per unit surface area of an interface k Lagrangian multiplier a à o li ; li & li Chemical potential of i in a phase a, in the standard state, and in the pure state, respectively l fi JT Joule-Thompson coef cient l Reduced mass of a diatomic molecule p0 Shear modulus Cth Thermodynamic Grüneisen parameter U, / Fugacity and Osmotic coefficient, respectively r Surface tension; also rate of entropy production r0 Symmetry number in rotational partition function Physical and Chemical Constants

Avogadro’s number L ¼ 6:022ðÞ 1023 molÀ1 ¼ : ðÞÀ23 À1 Boltzmann constant kB 1 38065ÀÁ 10 JK Faraday constant F0 ¼ 9:6485 104 C molÀ1 ðÞC: Coulomb ¼ J=V Gas constant R ¼ 8:314 JmolÀ1KÀ1 ¼ 1:9872 cal molÀ1 KÀ1 ¼ 83:14 bar cm3molÀ1KÀ1 Planck constant h ¼ 6:62607ðÞ 10À34 Js Acceleration of gravity g ¼ 9:80665 m sÀ2 at the Earth’s surface

xxix Some Commonly Used Physical Quantities: SI Units and Conversions

Quantity SI unit Some conversions − Force Newton (N): kg m s 2 1N ¼ 105 dyne − Pressure Pascal (Pa): N m 2 1 bar ¼ 105 Pa ¼ À1 À2 ¼ : ¼ kg m s 1 atmosphereÀ 1 01325 bar 760 mm of Hg 1 GPa GigapascalÞ¼10 kbarðÞ¼ kilobars 104 bar Energy Joule (J): Nm 1 cal ¼ 4:184 J ¼ kg m2 sÀ2 1J ¼ 107 ergs ¼ 10 cm3 bar ¼ 1Pam3 1 eVðÞ electron volt =atom ¼ 96:475 kJ=mol ¼ 23:058 kcal=mol 1eV ¼ 1:602ðÞ 10À19 J watt Js−1 (W) Length meter (m) 1cm ¼ 104 lmðÞ micron 1 nmðÞ¼ nanometer 10 A˚ ðÞangstrom 1 inch ¼ 2:54 cm

xxxi About the Author

Jibamitra Ganguly was educated in India 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 appointment to a faculty position at the University of Arizona, where he is currently a Professor Emeritus of Geosciences. The author has made major contributions in a wide range of areas in the Earth and Planetary sciences relating to phase equilibria, thermodynamics, and diffusion kinetics that reflect an effective blend of experimental and theoretical studies with observational data. Besides the first edition of this book, the author has written a book entitled Mixtures and Mineral Reactions (co-author: S. K. Saxena) and edited a volume entitled Diffusion, Atomic Ordering and Mass Transport, all published by Springer-Verlag. In addition Prof. Ganguly is also the author of a book on the life and work of the Indian astrophysicist, Meghnad Saha (Meghnad Saha: His Science and Persona Through Letters and Writings), published by the Indian National Science Academy. In addition Prof. Ganguly is also the author of a book on the life and work of the Indian astrophysicist, Meghnad Saha (Meghnad Saha: His Science and Persona Through Letters and Writings), published by the Indian National Science Academy. He is a Fellow of the Mineralogical Society of America and the American Geophysical Union, and a recipient of the Alexander von Humboldt research prize.

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