Thermodynamics in Earth and Planetary Sciences

Preface to the Second Edition
Preface to the First Edition
Contents
Commonly Used Symbols
Physical and Chemical Constants
Some Commonly Used Physical Quantities: SI Units and Conversions
About the Author
1 Introduction
	1.1 Nature and Scope of Thermodynamics
	1.2 Irreversible and Reversible Processes
	1.3 Thermodynamic Systems, Walls and Variables
	1.4 Work
	1.5 Stable and Metastable Equilibrium
	1.6 Lattice Vibrations
	1.7 Electronic Configurations and Crystal Field Effects
		1.7.1 Electronic Shells, Subshells and Orbitals
		1.7.2 Crystal or Ligand Field Effects
	1.8 Some Useful Physical Quantities and Units
	References
2 First and Second Laws
	2.1 The First Law
	2.2 Second Law: The Classic Statements
	2.3 Carnot Cycle: Entropy and Absolute Temperature Scale
	2.4 Entropy: Direction of Natural Processes and Equilibrium
	2.5 Microscopic Interpretation of Entropy: Boltzmann Relation
	2.6 Black Hole and Generalized Second Law of Thermodynamics
	2.7 Entropy and Disorder: Mineralogical Applications
		2.7.1 Configurational Entropy
			2.7.1.1 Random Atomic Distribution: Complete Disorder
			2.7.1.2 Ordering with Random Atomic Distribution Within Each Sublattice
			2.7.1.3 Solved Problem: Change of Configurational Entropy Due to Random Mixing of Gases
			2.7.1.4 Constrained Random Atomic Distribution Within a Sublattice
		2.7.2 Vibrational Entropy
		2.7.3 Configurational Versus Vibrational Entropy
	2.8 First and Second Laws: Combined Statement
	2.9 Condition of Thermal Equilibrium: An Illustrative Application of the Second Law
	2.10 Limiting Efficiency of a Heat Engine and Heat Pump
		2.10.1 Heat Engine
		2.10.2 Heat Pump
		2.10.3 Heat Engines in Nature
	References
3 Thermodynamic Potentials and Derivative Properties
	3.1 Thermodynamic Potentials
	3.2 Equilibrium Conditions for Closed Systems: Formulations in Terms of the Potentials
	3.3 What Is Free in Free Energy?
	3.4 Maxwell Relations
	3.5 Thermodynamic Square: A Mnemonic Tool
	3.6 Vapor Pressure and Fugacity
	3.7 Derivative Properties
		3.7.1 Thermal Expansion and Compressibility
		3.7.2 Heat Capacities
	3.8 Grüneisen Parameter
	3.9 P–T Dependencies of Coefficient of Thermal Expansion and Compressibility
	3.10 Summary of Thermodynamic Derivatives
	References
4 Third Law and Thermochemistry
	4.1 The Third Law and Entropy
		4.1.1 Observational Basis and Statement
		4.1.2 Third Law Entropy and Residual Entropy
	4.2 P-T Dependence of Heat Capacity Functions
	4.3 Non-lattice Contributions to Heat Capacity and Entropy of Pure Solids
		4.3.1 Electronic Transitions
		4.3.2 Magnetic Transitions
	4.4 Unattainability of Absolute Zero
	4.5 Thermochemistry: Formalisms and Conventions
		4.5.1 Enthalpy of Formation
		4.5.2 Hess’s Law
		4.5.3 Gibbs Free Energy of Formation
		4.5.4 Thermochemical Data
	References
5 Critical Phenomenon and Equations of States
	5.1 Critical End Point
	5.2 Near- and Super-Critical Properties
		5.2.1 Divergence of Thermal and Thermo-Physical Properties
		5.2.2 Critical Fluctuations
		5.2.3 Super- and Near-Critical Fluids
	5.3 Near-Critical Properties of Water and Magma-Hydrothermal Systems
	5.4 Equations of State
		5.4.1 Gas
			5.4.1.1 Van der Waals and Reduced Equations of State
			5.4.1.2 Principle of Corresponding States and Compressibility Factor
			5.4.1.3 Redlich-Kwong and Related Equations of State
			5.4.1.4 Virial and Virial-Type EoS
		5.4.2 Solid and Melt
			5.4.2.1 Birch-Murnaghan Equations
			5.4.2.2 Further Developments for High P-T Conditions
	References
6 Phase Transitions, Melting, and Reactions of Stoichiometric Phases
	6.1 Gibbs Phase Rule: Preliminaries
	6.2 Phase Transformations and Polymorphism
		6.2.1 Thermodynamic Classification of Phase Transformations
	6.3 Landau Theory of Phase Transition
		6.3.1 General Outline
		6.3.2 Derivation of Constraints on the Second Order Coefficient
		6.3.3 Effect of Odd Order Coefficient on Phase Transition
		6.3.4 Order Parameter Versus Temperature: Second Order and Tricritical Transformations
		6.3.5 Landau Potential Versus Order Parameter: Implications for Kinetics
		6.3.6 Some Applications to Mineralogical and Geophysical Problems
	6.4 Reactions in the P-T Space
		6.4.1 Conditions of Stability and Equilibrium
		6.4.2 P-T Slope: Clapeyron-Clausius Relation
	6.5 Temperature Maximum on Dehydration and Melting Curves
	6.6 Extrapolation of Melting Temperature to High Pressures
		6.6.1 Kraut-Kennedy Relation
		6.6.2 Lindemann-Gilvarry Relation
	6.7 Calculation of Equilibrium P-T Conditions of a Reaction
		6.7.1 Equilibrium Pressure at a Fixed Temperature
			6.7.1.1 Solved Problem: Depth of Diamond Formation
		6.7.2 Effect of Polymorphic Transition
	6.8 Evaluation of Gibbs Energy and Fugacity at High Pressure Using Equations of States
		6.8.1 Birch-Murnaghan Equation of State
		6.8.2 Vinet Equation of State
		6.8.3 Redlich-Kwong and Related Equations of State for Fluids
	6.9 Schreinemakers’ Principles
		6.9.1 Enumerating Different Types of Equilibria
		6.9.2 Self-consistent Stability Criteria
		6.9.3 Effect of an Excess Phase
		6.9.4 Concluding Remarks
	References
7 Thermal Pressure, Earth’s Interior and Adiabatic Processes
	7.1 Thermal Pressure
		7.1.1 Thermodynamic Relations
		7.1.2 Core of the Earth
		7.1.3 Magma-Hydrothermal System
	7.2 Adiabatic Temperature Gradient
	7.3 Temperature Gradients in the Earth’s Mantle and Outer Core
		7.3.1 Upper Mantle
		7.3.2 Lower Mantle and Core
	7.4 Isentropic Melting in the Earth’s Interior
	7.5 The Earth’s Mantle and Core: Linking Thermodynamics and Seismic Velocities
		7.5.1 Relations Among Elastic Properties and Sound Velocities
		7.5.2 Radial Density Variation
			7.5.2.1 Williamson-Adams Equation
		7.5.3 Transition Zone in the Earth’s Mantle
	7.6 Horizontal Adiabatic Flow at Constant Velocity
		7.6.1 Joule-Thompson Experiment and Coefficient
		7.6.2 Entropy Production in Joule-Thompson Expansion
	7.7 Adiabatic Flow with Change of Kinetic and Potential Energies
		7.7.1 Horizontal Flow with Change of Kinetic Energy: Bernoulli Equation
		7.7.2 Vertical Flow
			7.7.2.1 Change of Temperature with Pressure
			7.7.2.2 Geyser Eruption
	7.8 Ascent of Material Within the Earth’s Interior
		7.8.1 Irreversible Decompression and Melting of Mantle Rocks
		7.8.2 Thermal Effect of Volatile Ascent: Coupling Fluid Dynamics and Thermodynamics
	References
8 Thermodynamics of Solutions
	8.1 Chemical Potential and Chemical Equilibrium
	8.2 Partial Molar Properties
	8.3 Determination of Partial Molar Properties
		8.3.1 Binary Solutions
		8.3.2 Multicomponent Solutions
			8.3.2.1 Darken Equation
			8.3.2.2 Hillert Equation
	8.4 Fugacity and Activity of a Component in a Solution
	8.5 Determination of Activity of a Component Using Gibbs-Duhem Relation
	8.6 Molar Properties of a Solution
		8.6.1 Formulations
		8.6.2 Entropy of Mixing and Choice of Activity Expression
	8.7 Ideal Solution and Excess Thermodynamic Properties
		8.7.1 Thermodynamic Relations
		8.7.2 Ideality of Mixing: Remark on the Choice of Components and Properties
	8.8 Solute and Solvent Behaviors in Dilute Solution
		8.8.1 Henry’s Law
		8.8.2 Raoult’s Law
	8.9 Speciation of Water in Silicate Melt
	8.10 Standard States: Recapitulations and Comments
	8.11 Stability of a Solution
		8.11.1 Intrinsic Stability and Instability of a Solution
		8.11.2 Extrinsic Instability: Decomposition of a Solid Solution
	8.12 Spinodal, Critical and Binodal (Solvus) Conditions
		8.12.1 Thermodynamic Formulations
		8.12.2 Upper and Lower Critical Temperatures
	8.13 Effect of Coherency Strain on Exsolution
	8.14 Spinodal Decomposition
	8.15 Solvus Thermometry
	8.16 Chemical Potential in a Field
		8.16.1 Formulations
		8.16.2 Applications
			8.16.2.1 Variation of Pressure and Composition in the Earth’s Atmosphere
			8.16.2.2 Solution in a Gravitational Field
			8.16.2.3 Variation of Isotopic Ratios with Height
			8.16.2.4 A Case Study of Equilibrium Distribution in a Gravitational Field
	8.17 Osmotic Equilibrium
		8.17.1 Osmotic Pressure, Reverse Osmosis
		8.17.2 Natural Salinity Gradients and Power Generation
		8.17.3 Osmotic Coefficient
		8.17.4 Determination of Molecular Weight of a Solute
	References
9 Thermodynamic Solution and Mixing Models: Non-electrolytes
	9.1 Ionic Solutions
		9.1.1 Single Site, Sublattice and Reciprocal Solution Models
		9.1.2 Disordered Solutions
		9.1.3 Coupled Substitutions
		9.1.4 Ionic Melt: Temkin and Other Models
	9.2 Mixing Models in Binary Systems
		9.2.1 Guggenheim or Redlich-Kister, Simple Mixture and Regular Solution Models
		9.2.2 Subregular Model
		9.2.3 Darken’s Quadratic Formulation
		9.2.4 Quasi-chemical and Related Models
		9.2.5 Athermal, Flory-Huggins and NRTL (Non-random Two Liquid) Models
		9.2.6 Van Laar Model
		9.2.7 Associated Solutions
	9.3 Multicomponent Solutions
		9.3.1 Power Series Multicomponent Models
		9.3.2 Projected Multicomponent Models
		9.3.3 Comparison Between Power Series and Projected Methods
		9.3.4 Estimation of Higher Order Interaction Terms
		9.3.5 Solid Solutions with Multi-site Mixing
		9.3.6 Concluding Remarks
	References
10 Equilibria Involving Solutions and Gaseous Mixtures
	10.1 Extent and Equilibrium Condition of a Reaction
	10.2 Gibbs Free Energy Change and Affinity of a Reaction
	10.3 Gibbs Phase Rule and Duhem’s Theorem
		10.3.1 Phase Rule
			10.3.1.1 General Derivation
			10.3.1.2 Special Case: Externally Buffered Systems
		10.3.2 Duhem’s Theorem
	10.4 Equilibrium Constant of a Chemical Reaction
		10.4.1 Definition and Relation with Activity Product
		10.4.2 Pressure and Temperature Dependencies of Equilibrium Constant
	10.5 Solid-Gas and Homogeneous Gas Speciation Reactions
		10.5.1 Condensation of Solar Nebula
		10.5.2 Surface-Atmosphere Interaction in Venus
		10.5.3 Metal-Silicate Reaction in Meteorite Mediated by Dry Gas Phase
		10.5.4 Effect of Vapor Composition on Equilibrium Temperature: T Versus Xv Sections
			10.5.4.1 Binary Vapor Phase
			10.5.4.2 Ternary Vapor Phase
		10.5.5 Volatile Compositions and Oxidation States of Natural Systems
			10.5.5.1 Oxidation States of Planetary Systems
			10.5.5.2 Volatile Compositions: Metamorphic and Magmatic Systems
	10.6 Equilibrium Temperature Between Solid and Melt
		10.6.1 Eutectic and Peritectic Systems
		10.6.2 Systems Involving Solid Solution
	10.7 Azeotropic Systems
	10.8 Reading Solid-Liquid Phase Diagrams
		10.8.1 Eutectic and Peritectic Systems
		10.8.2 Crystallization and Melting of a Binary Solid Solution
		10.8.3 Intersection of Melting Loop and a Solvus
		10.8.4 Ternary Systems
	10.9 Natural Systems: Granites and Lunar Basalts
		10.9.1 Granites
		10.9.2 Lunar Basalts
	10.10 Pressure Dependence of Eutectic Temperature and Composition
	10.11 Reactions in Impure Systems
		10.11.1 Reactions Involving Solid Solutions
		10.11.2 Reactions Involving Solid Solutions and Gaseous Mixture
			10.11.2.1 Thermodynamic Formulations
	10.12 Retrieval of Activity Coefficient from Phase Equilibria
	10.13 Equilibrium Abundance and Compositions of Phases
		10.13.1 Closed System at Constant P-T
		10.13.2 Closed System at Constant V-T
		10.13.3 Minimization of Korzhinskii Potential
	References
11 Element Fractionation in Geological Systems
	11.1 Fractionation of Major Elements
		11.1.1 Exchange Equilibrium and Distribution Coefficient
		11.1.2 Temperature and Pressure Dependence of KD
		11.1.3 Compositional Dependence of KD
		11.1.4 Thermometric Formulation
	11.2 Trace Element Fractionation Between Mineral and Melt
		11.2.1 Thermodynamic Formulations
		11.2.2 Illustrative Applications
			11.2.2.1 Trace Element Pattern of Basalt Derived from Garnet-Peridotite
			11.2.2.2 Highly Incompatible Trace Element as Indicator of Source Region of Melt
		11.2.3 Estimation of Partition Coefficient
	11.3 Metal-Silicate Fractionation: Magma Ocean and Core Formation
		11.3.1 Pressure Dependence of Metal-Silicate Partition Coefficients
		11.3.2 Pressure Dependence of Metal-Silicate Distribution Coefficients
		11.3.3 Pressure Dependencies of Ni Versus Co Partition- and Distribution-Coefficients: Depth of Terrestrial Magma Ocean
	11.4 Effect of Temperature and f(O2) on Metal-Silicate Partition Coefficient
	References
12 Electrolyte Solutions and Electrochemistry
	12.1 Chemical Potential
	12.2 Activity and Activity Coefficients: Mean Ion Formulations
	12.3 Mass Balance Relation
	12.4 Standard State Convention and Properties
		12.4.1 Solute Standard State
		12.4.2 Standard State Properties of Ions
	12.5 Equilibrium Constant, Solubility Product and Ion Activity Product: Survival of Marine Carbonate Organisms
	12.6 Ion Activity Coefficients and Ionic Strength
		12.6.1 Debye-Hückel and Related Methods
		12.6.2 Mean-Salt Method
	12.7 Multicomponent High Ionic Strength and High P-T Systems
	12.8 Activity Diagrams of Mineral Stabilities
		12.8.1 Method of Calculation
		12.8.2 Illustrative Applications
			12.8.2.1 Spring Waters
			12.8.2.2 Stability of Magnesium Silicates
	12.9 Electrochemical Cells, Nernst Equation and f(O2) Measurement by Solid Electrolyte
		12.9.1 Electrochemical Cell and Half-Cells
		12.9.2 Emf of a Cell and Nernst Equation
		12.9.3 Oxygen Fugacity Measurement Using Solid Electrolyte Sensor
		12.9.4 Standard Emf of Half-Cell and Full-Cell Reactions
	12.10 Hydrogen Ion Activity in Aqueous Solution: pH and Acidity
	12.11 Eh-pH Stability Diagrams
	12.12 Chemical Model of Sea Water
	References
13 Surface Effects
	13.1 Surface Tension and Energetic Consequences
	13.2 Surface Thermodynamic Functions and Adsorption
	13.3 Temperature, Pressure and Compositional Effects on Surface Tension
	13.4 Langmuir Isotherm
	13.5 Crack Propagation
	13.6 Equilibrium Shape of Crystals
	13.7 Contact and Dihedral Angles
	13.8 Dihedral Angle and Interconnected Melt or Fluid Channels
		13.8.1 Connectivity of Melt Phase and Thin Melt Film in Rocks
		13.8.2 Core Formation in Earth and Mars
	13.9 Surface Tension and Grain Coarsening
	13.10 Effect of Particle Size on Solubility and Melting
	13.11 Coarsening of Exsolution Lamellae
	13.12 Nucleation
		13.12.1 Theory
		13.12.2 Microstructures of Metals in Meteorites
	13.13 Effect of Particle Size on Mineral Stability
	References
14 Statistical Thermodynamics Primer
	14.1 Boltzmann Distribution and Partition Function
	14.2 Thermodynamic Properties
	14.3 Expressions of Partition Functions
	14.4 Heat Capacity of Solids
	14.5 Chemical Equilibria and Stable Isotope Fractionation
		14.5.1 General Treatment of Chemical Reaction
		14.5.2 Stable Isotope Fractionation: Theoretical Foundation
		14.5.3 Stable Isotope Fractionation: Some Geochemical Applications
			14.5.3.1 “Clumped Isotope” Thermometry
			14.5.3.2 Mineral-Hydrogen/Water Stable Isotope Fractionation
	References
Appendix_1
A.4. Onsager Reciprocity Relation and Thermodynamic Applications
B.7. Sterling’s Approximation
C.3.3. Ab Initio Calculation of Thermodynamic Properties
	References
Author Index
Subject Index