Thermodynamics of the Earth and Planets

Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1 Energy in planetary processes......Page 14
1.1 Some necessary definitions......Page 15
1.3 Mechanical energy. An introduction to dissipative and non-dissipative transformations......Page 17
1.3.1 Gravitational potential energy......Page 20
1.3.2 Kinetic energy......Page 23
1.4.1 The concept of expansion work......Page 26
1.4.2 Quasi-static, reversible and irreversible processes......Page 28
1.4.3 Measuring expansion work. Material properties and equations of state......Page 29
1.5 Isothermal and adiabatic processes. Dissipative vs. non-dissipative transformations redux......Page 36
1.6 Elastic energy......Page 37
1.7 Two complementary descriptions of nature: macroscopic and microscopic......Page 39
1.8.1 Electrostatic forces......Page 40
1.8.2 Atomic bonding......Page 43
1.8.3 Magnetic forces......Page 45
1.9 Thermal energy and heat capacity......Page 48
1.10 The First Law of Thermodynamics......Page 52
1.11 Independent variables and material properties......Page 53
1.12.1 Discontinuous phase transitions and latent heat......Page 54
1.12.2 Adiabatic expansion of gases......Page 59
1.12.3 Frictional heating in faults and shear zones......Page 60
1.13.1 Enthalpy of reaction and enthalpy of formation......Page 62
1.13.2 Enthalpy of reaction as a function of temperature......Page 64
1.14.1 Internal energy of a monatomic ideal gas......Page 68
1.14.2 Degrees of freedom and heat capacities of polyatomic gases......Page 73
1.14.3 Heat capacities of solids......Page 76
1.15 An overview of the properties of matter and equations of state......Page 77
Exercises for Chapter 1......Page 80
2 Energy sources in planetary bodies......Page 83
2.1 Planetary heat flows......Page 84
2.2 Dissipation of gravitational potential energy......Page 86
2.3 Gravitational binding energy......Page 88
2.4 Accretion......Page 91
2.4.1 The effect of large impacts......Page 98
2.5 Contraction......Page 102
2.6 Differentiation......Page 109
2.7 Tidal dissipation of mechanical energy......Page 116
2.8 Dissipation of electrical energy......Page 125
2.9 Radioactive heating......Page 129
Exercises for Chapter 2......Page 133
3 Energy transfer processes in planetary bodies......Page 135
3.1 Transport processes......Page 137
3.2.1 Overview of heat transfer processes......Page 139
3.2.2 The diffusion equation......Page 140
3.2.3 The physical meaning of diffusivity......Page 146
3.2.4 The diffusive thermal boundary layer......Page 148
3.3.1 Lord Kelvin and the thermal structure of the Earth......Page 150
3.3.2 Can the conductive model for the Earth be saved?......Page 153
3.4 Convection as a heat engine......Page 154
3.5 Planetary adiabats......Page 158
3.6.1 The diffusion--advection equation......Page 161
3.6.2 A velocity scale for advection......Page 162
3.7 Convection as a heat transport mechanism......Page 166
3.7.1 Dynamics of thermal convection......Page 167
3.7.2 The thermal boundary layer......Page 170
3.7.3 Scaling of heat transport by convection......Page 173
3.7.4 Energy conservation in a convecting fluid......Page 175
3.8.1 Convection with moving plates......Page 178
3.8.2 Convection with a stagnant lid......Page 182
3.9 Convection and cooling of solid planetary interiors......Page 186
Exercises for Chapter 3......Page 191
4.1 An intuitive approach to entropy......Page 194
4.2 The entropy postulate and the Second Law of Thermodynamics......Page 196
4.3 The First Law of Thermodynamics revisited......Page 198
4.4 Entropy generation and energy dissipation......Page 199
4.5.1 Thermodynamic efficiency......Page 202
4.5.2 Carnot's cycle......Page 204
4.6.1 Microstates and macrostates......Page 209
4.6.2 Boltzmann's postulate......Page 215
4.7.1 Statement of the Third Law of Thermodynamics......Page 219
4.7.2 An absolute entropy scale......Page 220
4.8.1 The entropy maximum principle revisited......Page 222
4.8.2 The energy minimum principle......Page 226
4.8.3 Internal energy as a thermodynamic potential......Page 229
4.8.5 The Legendre transform......Page 231
4.8.6 The four fundamental thermodynamic potentials......Page 235
4.9 Gibbs free energy......Page 236
4.9.1 Derivatives of the Gibbs free energy......Page 237
4.9.2 The Gibbs free energy surface......Page 239
Exercises for Chapter 4......Page 240
5.1.1 Fundamental concepts......Page 242
5.1.2 Gibbs free energy surfaces and phase boundaries......Page 248
5.1.3 Properties of phase boundaries and phase transitions......Page 249
5.2 Equilibrium among pure chemical species......Page 251
5.2.1 An important digression: uncertainties in calculated equilibrium conditions......Page 256
5.3.1 Equilibrium among chemical species in an arbitrary state......Page 258
5.3.2 Chemical potential as a driving force for mass transfer......Page 260
5.4 Partial molar properties......Page 261
5.5.1 Definition of activity......Page 266
5.5.2 The equilibrium constant......Page 268
5.6.1 Definition of ideal solution and ideal activity......Page 271
5.6.2 Ideal activity and configurational entropy......Page 272
5.6.3 Ideal activity--composition relationships in fluids and in simple crystalline solids......Page 273
5.7 The geometric view of activity and Gibbs free energy of mixing......Page 278
5.8.1 Crystalline solutions with multiple ionic sites......Page 279
5.8.2 Crystalline solutions with non-vanishing configurational entropyin the end-member species......Page 284
5.9 Non-ideal solutions......Page 287
5.9.1 Excess mixing functions......Page 288
5.9.2 Raoult's law and Henry's law......Page 290
5.9.3 Polynomial expansion of excess mixing functions......Page 292
5.9.4 Perils and tribulations of excess mixing functions......Page 294
Exercises for Chapter 5......Page 298
6.1.1 The Gibbs--Duhem equation......Page 300
6.1.2 Gibbs' phase rule......Page 301
6.1.3 Choosing and switching components......Page 303
6.2 Analysis of phase equilibrium among phases offixed composition......Page 308
6.2.1 Some fundamental concepts and terminology......Page 309
6.2.2 Schreinemakers' rule......Page 311
6.2.3 Chemography......Page 315
6.2.4 Compositional degeneracy......Page 323
6.3.1 Externally controlled chemical potentials......Page 328
6.3.2 Slopes of pseudo-univariant phase boundaries......Page 331
6.4 Equilibrium among phases of variable composition......Page 338
6.5.1 Condensed phases......Page 339
6.5.2 Phase transitions involving a gas at low pressure......Page 341
6.6 Discontinuous phase transitions in phases of variable composition......Page 343
6.6.1 Complete miscibility in both phases......Page 344
6.6.2 Complete immiscibility in the low temperature or high pressure phase......Page 351
Exercises for Chapter 6......Page 360
7.1 An intuitive approach to critical phase transitions......Page 362
7.2 Location of the critical mixing point......Page 368
7.3 Calculation of non-dimensional solvi......Page 372
7.4 Order--disorder phase transitions in crystalline solids......Page 374
7.5 Analogies with other phase transitions......Page 382
7.6.1 Critical phase transitions......Page 385
7.6.2 Lambda phase transitions......Page 388
7.6.3 Discontinuous phase transitions......Page 390
7.6.4 Some comments of Landau theory......Page 392
Exercises for Chapter 7......Page 397
8.1 An introduction to equations of state for solids......Page 399
8.2.1 The Murnaghan isothermal EOS......Page 401
8.2.3 Bulk modulus at zero pressure......Page 403
8.2.4 Pressure derivative of the bulk modulus......Page 407
8.2.5 The Birch--Murnaghan isothermal EOS......Page 411
8.3 Isothermal equations of state from interatomic potentials:the Born--Mie EOS......Page 418
8.4 Thermal pressure......Page 420
Exercises for Chapter 8......Page 432
9.1 Fugacity and standard state fugacity......Page 433
9.2.1 The van der Waals equation of state......Page 441
9.2.2 The critical point of a van der Waals fluid......Page 443
9.2.3 The phase diagram of a van der Waals fluid......Page 445
9.3 The principle of corresponding states......Page 453
9.4 Equations of state for real fluids at P--T conditions typical of the crusts and upper mantles of the terrestrial planets......Page 455
9.4.1 Cubic equations: the van der Waals EOS revisited......Page 456
9.4.2 Cubic equations: the Redlich--Kwong EOS......Page 457
9.4.3 Cubic equations: the Kerrick--Jacobs modified Redlich--Kwong EOS......Page 458
9.4.4 Expansion of the residual Helmholtz free energy: the Pitzer--Sterner EOS......Page 459
9.4.5 The Brodholt--Wood EOS for H2O at high pressure......Page 460
9.4.6 Summary......Page 461
9.5.1 Standard state fugacity of pure fluids......Page 463
9.5.2 Fugacities in mixed fluids......Page 467
9.6.1 The chemical equilibrium approach to fluid speciation......Page 472
9.6.2 Calculation of fluid speciation by Gibbs free energy minimization......Page 478
9.7 Fluids at the conditions of giant planet interiors......Page 486
Exercises for Chapter 9......Page 488
10.1.1 Melting of simple solids......Page 490
10.2 Melting point depression. Eutectics, cotectics and peritectics......Page 494
10.2.1 A revision of the melting point depression equation......Page 495
10.2.2 Multicomponent eutectics and cotectics......Page 496
10.3 Partitioning of trace components between solids and melts......Page 497
10.4.1 Melting point depression revisited (again!)......Page 500
10.4.2 Effect of partitioning of incompatible trace components on melt generation......Page 501
10.4.3 Volatiles and melting......Page 502
10.4.4 Melting point depression by trace volatiles......Page 504
10.6.1 Fundamental relations......Page 507
10.6.2 Batch decompression melting of a one-component system......Page 510
10.6.3 Fractional decompression melting of a one-component system......Page 513
10.6.4 Decompression melting in multicomponent systems......Page 515
10.6.5 Departures from the constant-entropy assumption......Page 522
10.7 Open system melting......Page 525
10.8 The nature of solid--melt equilibrium in icy satellites......Page 530
Exercises for Chapter 10......Page 534
11.1 Some properties of dilute solutions......Page 535
11.1.1 Concentration scale in dilute solutions......Page 536
11.1.2 Standard states in dilute solutions......Page 537
11.1.3 Activity coefficient of a dilute solute and excess Gibbs free energy of mixing......Page 538
11.1.4 Gases as low-concentration solutes......Page 539
11.2 Effects of dilute solutes on the properties of the solvent......Page 542
11.2.1 The osmotic coefficient......Page 543
11.2.3 Relationship between osmotic coefficient of the solvent and activity coefficients of the solutes......Page 544
11.2.4 Freezing and boiling of dilute solutions......Page 545
11.2.5 Solubility of a pure solute......Page 547
11.2.6 Expansion of the stability field of liquid water by dilute solutes......Page 548
11.3 Electrolyte dissociation......Page 553
11.3.1 The dielectric constant and the structure of polar liquids......Page 554
11.3.2 Free energy of ions in dielectric materials......Page 556
11.4 Thermodynamic formulation of electrolyte solutions......Page 557
11.4.1 Thermodynamic functions of ionic species......Page 558
11.4.2 Standard states and activities of strong and weak electrolytes......Page 559
11.4.3 The reference for standard state properties of ionic species......Page 561
11.4.4 Dissociation of water and pH......Page 562
11.4.5 Equilibrium between electrolyte solutions and solid phases......Page 565
11.5 Speciation in ionic solutions. Iron solubility in ocean wateras an example......Page 567
11.6 Activity coefficients in electrolyte solutions......Page 575
11.6.1 Debye--Hückel theory......Page 576
11.6.2 Activity coefficients in concentrated electrolyte solutions......Page 585
Exercises for Chapter 11......Page 588
12.1.1 Fundamental concepts......Page 590
12.1.2 Heat diffusion revisited and the principle of minimum entropy production rate......Page 593
12.2.1 Fundamental relationships......Page 594
12.2.2 Fick's laws of chemical diffusion......Page 595
12.2.3 Interdiffusion......Page 599
12.2.4 Coupling of mass and heat transfer......Page 602
12.3.1 Fundamental concepts......Page 605
12.3.2 Thermodynamics, kinetics and entropy production by chemical reactions......Page 607
12.3.3 Differential and integral rate laws......Page 611
12.3.4 Some simple composite reactions......Page 613
12.4.1 Thermally activated processes. Arrhenius law......Page 622
12.4.2 Photochemical processes......Page 623
12.5 An introduction to kinetics of heterogeneous processes......Page 624
Exercises for Chapter 12......Page 628
13.1 Gravitational binding of planetary atmospheres......Page 629
13.2 Equilibrium thermodynamics in a gravitational field......Page 633
13.2.1 Pressure in a one-component isothermal atmosphere.Atmospheric scale height......Page 635
13.2.2 Compositional stratification in a fluid immersed in a gravitational field......Page 636
13.3.1 Fundamental concepts and equations of thermal radiation......Page 638
13.3.2 Radiant energy exchange......Page 640
13.3.3 Molecular absorption and emission of electromagnetic radiation......Page 645
13.3.4 Absorption and emission of electromagnetic radiation:the macroscopic description......Page 647
13.3.5 Absorption cross section and mean free path......Page 649
13.3.6 A radiative toy model of greenhouse warming......Page 650
Exercises for Chapter 13......Page 656
14.1 Chemical evolution of post-nebular atmospheres......Page 658
14.2.1 Carbon fixing and respiration of reduced organic carbon......Page 670
14.2.2 Respiration without reduced organic carbon......Page 672
14.2.3 Speculations about the origin and evolution of metabolism on Earth......Page 673
14.3.1 Extraterrestrial life based on free energy gradients other thanchemical potentials......Page 678
14.3.2 Extraterrestrial life based on chemical potential gradientsuncommon or non-existent in Earth......Page 679
14.4 Entropy and life......Page 681
Exercise for Chapter 14......Page 682
Appendix 1 Physical constants and other useful numbers and conversion factors......Page 684
Appendix 2 Derivation of thermodynamic identities......Page 685
References......Page 688
Index......Page 703