THERMODYNAMICS Principles and Applications to Earth and Planetary Sciences Jibamitra Ganguly 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 Springer-Verlag PREFACE 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 thermodynamics 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 particular 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 thermodynamics or Geohemical 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 sciences. 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 oC, 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 oC. The pressures for processes in the solar nebula are 10-3 – 10-4 bars. The extreme range of conditions encompassed by natural processes requires variety of manipulations and approximations that are not readily available in the standard text books 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 thermodynamic 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 microscopic basis of thermodynamic properties. Thus, I have occasionally digressed to the discussion of thermodynamics from microscopic view points, 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 considerable 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 II 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, Jamie Connolly, Charles Geiger, Ralph Kretz, Luigi Marini, Denis Norton, Giulio Ottonello, Surendra Saxena, Rishi Narayan Singh, Max Tirone and Krishna Vemulapalli. 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 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 Alexender von Humboldt Foundation through a research prize (forschungspreis). I gratefully acknowledge the support of the AvH foundation, and the hospitality of the two institutions, especially those of the hosts, Professors 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 25, 2007 Jibamitra Ganguly CONTENTS CHAPTER 1 Introduction 1. Nature and Scope of Thermodynamics 2. Irreversible and Reversible Processes 3. Thermodynamic Systems, Walls and Variables 4. Work 5. Stable and Metastable Equilibrium 6. Lattice Vibrations 7. Electronic Configurations and Crystal Field Effects 8. Some Useful Units and Conversions CHAPTER 2 First and Second Laws of Thermodynamics 1. The First Law 2. The Second Law: Classic Statements 3. Carnot Cycle or Heat Engine: Entropy and Absolute Temperature Scale 4. Entropy: Direction of Natural Processes and Equilibrium 5. Microscopic Interpretation of Entropy: Boltzmann relation and controversy 6. Entropy and Disorder: Mineralogical applications 7. First and Second Laws: Combined statement 8. Condition of Thermal Equilibrium: An illustrative application of second law 9. Limiting Efficiency of a Heat Engine 10. Some Heat Engines in Nature CHAPTER 3 Thermodynamic Potentials and Derivative Properties 1. Thermodynamic Potentials 2. Equilibrium Conditions 3. What is Free in Free energy? 4. Maxwell Relations 5. Thermodynamic Square: A mnemonic tool 6. Vapor pressure and Fugacity 7. Derivative Properties Thermal expansion and compressibilities Heat capacities 8. Grüneissen Parameter 9. P-T Dependencies of Coefficient of Thermal Expansion and Compressibility 10. Summary of Partial Derivatives CHAPTER 4 Third Law and Thermochemistry 1. The Third Law & Entropy 2. Behavior of the Heat Capacity Functions 3. Non-lattice Contribution to the Heat Capacity and Entropy of Solids 4. Unattainability of Absolute Zero 5. Thermochemistry: Formalisms and conventions CHAPTER 5 Critical phenomenon and Equations of State 1. Critical End Point 2. Near- and Super-critical properties: Geological applications 3. Near-critical Properties of Water and Magma-hydrothermal Systems 4. Equations