Thermodynamics in Geochemistry: The Equilibrium Model

PREFACE......Page 8
TABLE OF CONTENTS......Page 12
1. THERMODYNAMICS—A MODEL SUBJECT......Page 24
1.1 Chemical Reactions......Page 25
1.2 Notation......Page 26
1.3 What's Ahead......Page 27
2.1 The Mathematical Side of Thermodynamics......Page 28
2.2.2 Variable Space......Page 29
2.2.3 Single-Valued and Continuous Functions......Page 30
2.2.4 Differentials and Derivatives......Page 31
2.2.5 Partial Derivatives and Total Differentials......Page 32
2.2.6 Exact and Inexact Differentials......Page 35
2.2.7 Simple Transformations of Partial Derivatives......Page 38
2.2.8 Integration......Page 40
2.2.9 Line Integrals......Page 41
2.3 Additional Mathematical Techniques......Page 42
2.3.1 Molar and Partial Molar Properties, Intensive and Extensive Variables, and Their Relation to Homogeneous Functions......Page 43
2.3.2 Homogeneous Functions......Page 44
2.3.3 Euler's Theorem for Homogeneous Functions......Page 45
2.3.4 Legendre Transforms......Page 47
2.3.5 Jacobian Transformations and Thermodynamic Partial Derivatives......Page 53
3.2 System......Page 58
3.3 Equilibrium......Page 59
3.3.1 Stable and Metastable Equilibrium......Page 60
3.3.2 Partial Equilibrium......Page 62
3.3.5 "Real" versus "Model" Equilibrium......Page 63
3.4 State Variables......Page 64
3.5 Degrees of Freedom......Page 65
3.6 Thermodynamic State Space......Page 66
3.6.1 Constraints and Metastable States......Page 67
3.7.1 Reversible Processes......Page 71
3.7.3 Virtual Processes......Page 72
3.8 Phases, Components, Species, and Constituents......Page 73
3.9.1 Using Properties as Variables......Page 74
3.9.3 Continuous Properties......Page 76
3.9.4 Thermodynamics as a Mathematical Model......Page 78
4.2 Zeroth Law of Thermodynamics......Page 81
4.3 Temperature Scales......Page 83
4.3.1 The Centigrade Scale......Page 84
4.3.2 Practical and Absolute Temperature Scales......Page 85
4.3.3 The Geophysical Laboratory Temperature Scale......Page 87
4.4 Internal Energy......Page 88
4.4.1 Energy......Page 89
4.4.2 The Internal Energy......Page 90
4.5 Energy Transfers......Page 91
4.6 The First Law of Thermodynamics......Page 93
4.6.1 Work......Page 95
4.6.2 Heat......Page 101
4.7 The Model Again......Page 102
5.2 The Kelvin and Clausius Formulations......Page 104
5.2.1 Absolute Temperature......Page 105
5.2.2 The Entropy......Page 106
5.2.3 Examples of Entropy Change......Page 107
5.3 The Fundamental Equation......Page 109
5.4 Thermodynamic Potentials......Page 111
5.4.2 Potentials......Page 112
5.4.4 Entropy as a Thermodynamic Potential......Page 116
5.4.5 The Legendre Transform......Page 118
5.5 Potentials For Work And Heat......Page 123
5.5.1 Heat Potentials......Page 127
5.6 Summary......Page 128
5.7 The Entropy of the Universe......Page 129
5.8 Differential Inequalities......Page 131
5.8.1 Controlled Metastable States......Page 132
5.8.2 The Affinity......Page 135
6.1 The Concept of Entropy......Page 137
6.2 Entropy, Probability, and Disorder......Page 138
6.2.1 Thermodynamic Probability......Page 140
6.2.2 Calculating Configurational Probabilities......Page 141
6.2.3 The Predominant Configuration......Page 144
6.3 The Boltzmann Distribution......Page 146
6.3.1 The Partition Function......Page 147
6.3.2 The Partition Function and Thermodynamic Properties......Page 150
6.3.3 Entropy and the Partition Function......Page 151
6.4.1 Entropy Increase in Isolated Systems......Page 152
6.4.2 dS versus Dq/T......Page 153
6.4.3 ΔS versus q/T......Page 156
6.4.4 Summary and Cautionary Note......Page 158
6.5.2 Tests of the Third Law......Page 159
6.5.4 Entropies of Ideal Mixing......Page 162
6.6.1 The Standard Model of the Universe......Page 164
6.6.2 The Expanding Universe versus the Second Law......Page 165
6.6.3 Conclusion......Page 166
7.2 Formation from the Elements......Page 167
7.3 Entropy Tabulation......Page 170
7.4.1 Standard Free Energies and Enthalpies of Formation at T and P......Page 175
7.4.2 Apparent Free Energies and Enthalpies of Formation at T and P......Page 176
7.4.4 Formation from the Oxides......Page 179
7.5.1 The Measurement of Thermodynamic Properties......Page 180
7.5.2 Calorimetry......Page 182
7.5.3 Entropies above 298 K......Page 188
7.5.5 Calculation of Δ[sub(f)]G°[sub(T[sub(r)])], Δ[sub(f)]G°[sub(T)], and Δ[sub(a)]G°[sub(T)]......Page 190
7.6 Properties of Solids at Elevated T and P......Page 192
7.6.1 Thermal Expansion and Compressibility......Page 193
7.6.2 Solid Volumes Not Considered Constant......Page 195
7.7 Sources of Data......Page 196
7.8 Simultaneous Evaluation of Data......Page 197
7.8.2 Derivation of Internally Consistent Data Bases Using Linear Programming......Page 198
7.9 Approximations......Page 201
8.2 Adiabatic Volume Changes......Page 205
8.2.1 Reversible Adiabatic or Isentropic Volume Changes......Page 206
8.2.2 Irreversible Adiabatic Volume Changes......Page 211
8.4 Calculation of Simple Phase Diagrams......Page 217
8.5 The Calcite - Aragonite Phase Diagram......Page 218
8.5.1 The Clapeyron Equation......Page 219
8.5.2 The Required Equations......Page 221
8.5.3 Effect of Approximations on Calculated Phase Boundaries......Page 226
9.2.1 Molar Properties as Derivatives......Page 231
9.2.2 Partial Molar Volume......Page 233
9.2.3 Total Volume as the Sum of the Partial Molar Volumes......Page 234
9.2.4 The Formal Definition......Page 236
9.2.5 Partial Molar Free Energy......Page 237
9.3 Apparent Molar Quantities......Page 239
9.3.1 Measurement of Apparent Molar Volume......Page 241
9.3.2 Integral and Differential Heats of Solution......Page 242
9.3.3 Heats of Solution from Experimental Data......Page 244
10.1 Solutions, Mixtures, and Models......Page 249
10.2.2 Ideal Liquid Solutions......Page 250
10.2.3 Ideal Solid Solutions......Page 251
10.3 Thermodynamics of Ideal Solutions......Page 252
10.4 Entropy of Ideal Solutions......Page 253
10.5 Enthalpy and Volume of Ideal Solutions......Page 258
10.6 Summary: Ideal Solutions......Page 259
10.7 Species, Components, and Constituents......Page 260
11.2 Definition of Fugacity......Page 265
11.3 Measurement of Fugacity......Page 266
11.4 The Compressibility Factor......Page 268
11.5 Fugacity in Equilibrium Calculations......Page 269
11.6 All Constituents Have a Fugacity......Page 270
11.7 Fugacities: a Summary......Page 272
11.8 Activity......Page 273
11.9 Activity and Mole Fraction......Page 274
11.9.1 Dalton's Law......Page 275
11.9.2 Henry's Law......Page 277
11.9.3 Raoult's Law......Page 278
11.9.4 The Lewis Fugacity Rule......Page 280
11.9.5 Ideal Mixing and Activity......Page 282
11.10 Activity Coefficients......Page 285
11.10.1 A Real Example......Page 287
12.2 Activities and Standard States......Page 290
12.3 Variable Temperature Standard States......Page 292
12.4.1 Standard States Based on Raoult's Law......Page 293
12.4.2 Standard States Based on Henry's Law......Page 295
12.4.3 Activities Independent of Units for a Given Standard State......Page 298
12.4.4 The Ideal One Molal Standard State......Page 299
12.4.5 Mole Fraction—Molality Conversion for Henryan Activity Coefficients......Page 301
12.4.6 Why One Molal? Why Ideal?......Page 303
12.5.1 Temperature......Page 304
12.5.2 Pressure......Page 305
12.6 Activities and Standard States: An Overall View......Page 306
12.7 Effect of Size of the Mole on Activities......Page 310
12.7.1 What is a Mole of Olivine?......Page 312
13.1 The Most Useful Equation in Thermodynamics......Page 316
13.2.1 K Equal to a Solubility......Page 319
13.2.2 K Equal to Fugacity of a Volatile Species......Page 321
13.2.3 Miscellaneous Special Equilibrium Constants......Page 322
13.3 Change of K with Temperature......Page 324
13.3.1 Δ[sub(r)]C°[sub(p)] = 0......Page 325
13.3.2 Δ[sub(r)]C°[sub(p)] = constant......Page 328
13.3.3 The Density Model......Page 329
13.3.4 Δ[sub(r)]C°[sub(p)] Known......Page 330
13.4 Change of K with Pressure......Page 331
13.5 Finding The Univariant Curve Using K......Page 337
14.2 Open Systems......Page 344
14.2.1 Thermodynamic Potentials in Open Systems......Page 345
14.2.2 Conditions of Equilibrium......Page 348
14.2.3 Equality of Chemical Potentials for All Constituents in All Phases......Page 350
14.2.4 Chemical Potentials Balanced in All Reactions......Page 351
14.3 Fundamental Equations......Page 352
14.3.1 Maxwell's Equations......Page 353
14.4 Conditions for Mass Transfer......Page 354
14.5 The Affinity and Chemical Potentials......Page 355
14.6.1 Phases and Components......Page 358
14.6.2 Derivation of the Phase Rule......Page 359
14.7 Buffered Systems......Page 360
14.7.1 Activities in Buffered Assemblages......Page 364
14.8 Osmotic Systems......Page 367
14.10 More on Components and Phases......Page 371
14.10.1 Components......Page 372
14.10.2 Phases......Page 375
14.11 Applications of the Phase Rule......Page 376
14.11.2 Al[sub(2)]O[sub(3)] – SiO[sub(2)] – H[sub(2)]O......Page 377
14.11.3 Pyrophyllite Example......Page 380
14.11.4 Marble Example......Page 382
15.1 Types of Solutions......Page 385
15.1.1 Regular Solutions......Page 386
15.1.3 The Spinodal Curve......Page 392
15.1.4 Further Thoughts on Non-ideal Enthalpies, Entropies, and Chemical Potentials......Page 393
15.2.1 Activities with Multi-site Mixing......Page 394
15.2.2 Substitution of the Same Cation on Several Non-equivalent Sites......Page 396
15.3 Excess Thermodynamic Functions......Page 398
15.3.1 Relationship Between Excess Properties and the Activity Coefficient......Page 399
15.4 The Margules Equations......Page 400
15.4.1 Margules Equations and Slightly Non-ideal, Symmetric Solutions......Page 401
15.4.2 Activity Coefficients in Symmetrical Solutions......Page 402
15.4.3 Margules Equations and Non-ideal, Asymmetric Solutions......Page 404
15.4.4 The Virial Equation......Page 405
15.4.5 Activity Coefficients in Asymmetric Systems......Page 407
15.4.6 Margules Equations for Ternary and Higher Order Systems......Page 408
15.5 Applying The Margules Equations......Page 409
15.5.1 Estimating Margules Parameters: Symmetrical Solvi......Page 410
15.5.2 Estimating Margules Parameters: Asymmetrical Solvi......Page 411
15.5.3 Calculating Solvi, Spinodal Curves, and Consolute Points......Page 412
15.5.4 Conclusion......Page 414
16.1 Methods of Describing Gases......Page 418
16.2.1 Ideal Gas......Page 419
16.2.2 The van der Waals Equation......Page 421
16.2.3 The Redlich-Kwong Equation and its Modifications......Page 424
16.2.4 Estimating the van der Waals and Redlich-Kwong Parameters from Critical Conditions......Page 425
16.3 Applying Equations of State to Gaseous Mixtures......Page 426
16.3.1 Calculating Fugacities from Gas Equations of State......Page 429
16.4 Some Concluding Comments......Page 437
17.2.1 Algebraic Example......Page 439
17.2.2 Activities of Electrolyte Components......Page 443
17.2.3 Choice of Solute Component......Page 446
17.2.4 Relationship of Solute Activity to Single Ion Activities......Page 447
17.2.5 Unsymmetrical Electrolytes......Page 449
17.2.6 General Relationships......Page 451
17.2.7 Stoichiometric versus Ionic Properties......Page 455
17.3 Numerical Values for Single-Ion Properties......Page 456
17.3.1 Possible Additional Conventions......Page 458
17.4 The Debye-Hückel Theory......Page 459
17.5 Measurement of Activity Coefficients......Page 461
17.5.1 Activity Coefficients of Neutral Molecules......Page 463
17.7 Estimating Activities of Individual Ions......Page 465
17.7.1 Activities of Minor Components in Concentrated Solutions......Page 466
17.8.1 A Statistical-Mechanical Approach to a Thermodynamic Problem......Page 467
17.8.2 The Pitzer Equations......Page 468
17.8.4 Example Applications to Concentrated Brines......Page 472
17.8.5 Application to Other Thermodynamic Properties......Page 473
17.9 The HKF Model for Aqueous Electrolytes......Page 474
17.9.2 The Born Functions......Page 475
17.9.3 The r[sub(e,,j)] Term......Page 478
17.9.5 The Model for ΔV° and ΔC°[sub(p)]......Page 480
17.9.6 Expressions for ΔS°, ΔH° and ΔG°......Page 481
17.9.8 Illustration of the Equivalence of the Properties of Aqueous HCl and Cl[sup(–)]......Page 483
17.10 Comparison of Pitzer and HKF Models......Page 484
18.2 Oxidation States of Natural Systems......Page 490
18.2.2 Cell Voltage and Free Energy of Reaction......Page 491
18.2.3 The Standard Hydrogen Electrode......Page 493
18.2.4 Oxidation and Reduction Sign Conventions......Page 495
18.2.5 The SHE Voltage Equals Zero Convention......Page 497
18.2.7 Measuring Thermodynamic Properties of Ions from Half- Cell Potentials......Page 498
18.2.9 Calculating Eh......Page 499
18.3.2 Sample Calculations......Page 501
18.3.3 Addition of Other Components......Page 504
18.3.4 Carbonate Equilibria......Page 505
18.3.6 pe – pH Diagrams......Page 508
18.3.8 Example Calculation, pe-pH......Page 510
18.3.9 On the Physical Meaning of pe......Page 511
18.4.1 Some Difficulties with Eh Field Measurements......Page 512
18.4.2 Alternative Measures of Oxidation Potential in Low- Temperature Environments......Page 513
18.5 Oxygen Fugacity......Page 514
18.5.1 On the Physical Significance of Oxygen Fugacity......Page 516
18.5.2 Oxygen Buffers in Field and Laboratory......Page 517
18.5.3 Solid Electrolyte Fugacity Sensors......Page 518
18.5.4 Calculation of Oxygen Fugacity — pH Diagrams......Page 520
18.5.6 Summary and Comment on Eh versus fo[sub(2)] as Parameters to Indicate Redox Conditions......Page 522
19.1 Rote Method for Simple Systems......Page 526
19.1.2 Example: Quartz Solubility in Alkaline Solutions......Page 529
19.2.1 Three Basic Approaches......Page 531
19.2.3 General Algorithms......Page 532
19.2.4 Newton-Raphson Solution......Page 533
19.2.5 Programs Designed for Specific Systems......Page 535
19.2.6 Methods Based on Free Energies......Page 538
19.2.7 Composition of the Early Solar Nebula......Page 539
19.2.8 Free Energy Minimization......Page 542
19.3 Reaction Path Calculations......Page 547
19.3.1 The Extent of Reaction or Progress Variable......Page 549
19.3.2 Multiple Reactions and Relative Rates......Page 551
19.3.3 Application to Mineral–Water Reactions......Page 553
19.3.4 Summary of the Numerical Model......Page 558
A. Constants and Numerical Values......Page 562
B. Expressions for ΔG°, ΔS°, ΔH°, In K as Function of Temperature for Various Versions of the Heat Capacity Function......Page 564
C. Standard Thermodynamic Properties of Selected Minerals and Other Compounds......Page 568
D. Electrochemical Cell Conventions......Page 574
E. EQBRM–A Fortran Speciation Program......Page 576
F. Answers to Problems......Page 589
BIBLIOGRAPHY......Page 592
C......Page 604
E......Page 605
H......Page 606
O......Page 607
S......Page 608
Z......Page 609