N. D. Chatterjee

Applied Mineralogical Thermodynamics Selected Topics

With 79 Figures

Springer-Verlag Berlin Heidelberg GmbH

Prof. Dr. NIRANJAN D. CHATTERJEE

Ruhr-Universităt Bochum

Institut fUr Mineralogie

Postfach 10 21 48

D-4630 Bochum 1

ISBN 978-3-540-53215-6

Library ofCongress CataIoging·in·Publication Data. Chatterjee, N. D. (Niranjan D.) Applied mineralogical thennodynamics: selected topicsjN. D. Chatterjee. p. cm. Includes biblio­graphical references (p. ) Includes index. ISBN 978-3-540-53215-6 ISBN 978-3-662-02716-5 (eBook) DOI 10.1007/978-3-662-02716-5

1. Mineralogy. 2. Thennodynamics. I. Title. QE364.2.T45C49 1990 549-dc20 90-10394 CIP

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© Springer-Verlag Berlin Heidelberg 1991 Originally published by Springer-V erlag Berlin Heidelberg New York in 1991

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Preface

Thermodynamic treatment of mineral equilibria, a topic central to mineralogical thermodynamics, can be traced back to the tum of the century, when J.H. Van't Hoff and his associates pioneered in applying thermodynamics to the mineral assemblages observed in the Stassfurt salt deposit. Although other renowned researchers joined forces to develop the subject - H.E.Boeke even tried to popularize it by giving an overview of the early developments in his "Grundlagen der physikalisch-chemischen Petrographie", Berlin, 1915 - it remained, on the whole, an esoteric subject for the majority of the contemporary geological community.

Seen that way, mineralogical thermodynamics came of age during the last four decades, and evolved very rapidly into a mainstream discipline of geochemistry. It has contributed enormously to our understanding of the phase equilibria of mineral systems, and has helped put mineralogy and petrology on a firm quantitative basis. In the wake of these developments, academic curricula now require the students of geology to take a course in basic thermodynamics, traditionally offered by the departments of chemistry. Building on that foundation, a supplementary course is generally offered to familiarize the students with diverse mineralogical applications of thermo­dynamics. This book draws from the author's experience in giving such a course, and has been tailored to cater to those who have had a previous exposure to the basic concepts of chemical thermodynamics. The presentation has much benefited through the feedback received from the students at the Universities of Bochum, FRG, and Allahabad, India. In response to their needs, I may have occasionally gone into length, may be even to the point of tedium, to reiterate certain concepts.

Applications of thermodynamics to mineral systems are too many to be done justice to within the framework of one course. That has led to my selecting a set of topics. Those included cover numerous aspects of heterogeneous equilibria between solids and fluids, the glaring omissions being ionic equilibria and the thermodynamics of silicate melt; the title of the book emphasizes this. The approach is to handle the selected topics explicitly, and

VI Preface

in a step-by-step manner, such that the beginners have little difficulty in working on their own.

The book has been structured, and held at a level appropriate for use as course material for the first year graduate class. Its purpose is to serve as a practical guide toward the application of thermodynamics to solid-solid and solid-fluid phase relations. The emphasis is mostly on the methods of mineralogical thermodynamics. Consequently, the worked examples have been chosen to demonstrate the methods, less so because of their geological importance. It is believed that the knowledge of the methods involved may help raise the critical awareness with which the currently published papers are absorbed by the students. Numerous worked examples are given with supportive details, some of which can be solved by hand-held calculators.

I thank Klaus-Dieter Grevel and Helmut Rimbach for assistance with some worked examples. Ms. Almut Fischer and Renate Lehmann's help in drafting the figures is gratefully acknowledged. Thanks are due to the Institute of Mineralogy, Ruhr University, Bochum, for supporting my effort. Critical reviews of the earlier drafts of this manuscript by a few friends and colleagues including Bernard W. Evans (Seattle, USA), Alok K. Gupta (Allahabad, India), Darrell J. Henry, (Baton Rouge, USA), Walter V. Maresch (Miinster, FRG), Yastami Oka (Yokohama, Japan), and Konstantin K. Podlesskii (Cher­nogolovka, USSR), were most helpful. I express my sincere appreciation for their constructive comments and at the same time exonerate them for any error or oversight. And last but not least, I thank Springer-Verlag for patient cooperation in producing this book.

February 1990 NIRANJAN D. CHATIERJEE

Contents

1

1.1 1.2 1.2.1

1.2.2 1.2.2.1

1.2.2.2 1.2.2.3 1.2.2.4 1.2.2.5 1.2.2.6 1.3

2

2.1 2.2 2.2.1 2.2.2 2.3 2.4

2.5

3

3.1 3.2

Summary of Basic Thermodynamic Concepts: A Refresher

Introduction. . . . . . . . . . . . . . . . . . . .. 1 Gibbs Energy as a State Function. . . . . . . . .. 2 Temperature and Pressure Dependence of Gibbs Energy, and the Choice of a Standard State. 2 Composition Dependence of Gibbs Energy. . .. 7 Chemical Potential and the Gibbs-Duhem Equation. . . . . . . . . . . . . .. 7 Property Changes on Mixing. . . . . . . . . . .. 14 Activity, Activity Coefficient, and Standard State. 16 Ideal, Excess, and Total Molar Properties. . . .. 20 Some Properties of Ideal Solutions. . . . . . . .. 22 Excess Molar Properties of Mixing. . . . . . . . . 24 Mineral Equilibria, Equilibrium Constant. . . 30

Measurement, Evaluation, and Tabulation of Thermodynamic Properties

Introduction. . . . . . . . . . . . . . . . . . . .. 33 Outlines of Some Calorimetric Methods. . . . .. 34 Calorimetry of Non-Reacting Systems. . . . . .. 34 Calorimetry of Reacting Systems. . . . . . . . . . 39 Outlines of Electrochemical Cell Measurements . 42 Evaluation and Tabulation of Thermodynamic Data. . . . . . . . . . . . .. 43 Worked Examples of Evaluation and Manipulation of Data . . . . . . . . . . . . . . . . . . . . . . .. 49

Equations of State for Fluids and Fluid Mixtures

Introduction. . . . . . . . . . . . . . . . . . . .. 55 Some Fundamental Concepts: Standard States, Fugacity, Activity. . . . . . . . . . . . . . . . .. 55

VIII Contents

3.3 3.4 3.4.1 3.4.2 3.4.3

3.4.4

3.4.4.1 3.4.4.2

3.5

3.5.1 3.5.2

3.6

4

4.1 4.2 4.3 4.3.1 4.3.2

4.4 4.5

5

5.1 5.2 5.3

5.4 5.5

5.5.1 5.5.2 5.5.3

Utility of the Volume Equations of State. . . . . . 59 Volume Equations of State for Pure Fluids. . .. 60 General Comments and Literature Overview. .. 60 Some Examples of Analytical Equations of State. 64 A Modified Redlich-Kwong (MRK) Equation of State for H20. . . . . . . . . . . . . . 70 Calculation of Phase Relations and Thermodynamic Properties of H20 from MRK. . . . . . . . . . . . 73 Calculation of the Saturation Curve of H20. . .. 73 Calculation of Fugacity and Molar Gibbs Energy of H20 . . . . . . . . . . . . . . . . . . . . . . .. 75 Modified Redlich-Kwong Equation of State for a Fluid Mixture. . . . . . . . . . . . . . . . . .. 78 Theoretical Background. . . . . . . . . . . . . . . 78 A Modified Redlich-Kwong Equation for the H20-C02 Mixture. . . . . . . . . . . . . . . . .. 80 Concluding Remarks. . . . . . . . . . . . . . .. 83

Phase Relations Among End-Member Solids

Introduction. . . . . . . . . . . . . . . . . . .. 85 Thermodynamic Formalism. . . . . . . . . . . . . 85 Error Propagation Calculation. . . . . . . . . .. 88 Basic Formalism. . . . . . . . . . . . . . . . . . . 88 Covariance of the Tabulated Thermodynamic Data. . . . . . . . . . . . . . .. 91 Worked Examples. . . . . . . . . . . . . . . . .. 93 Concluding Remarks. . . . . . . . . . . . . . . . 105

Phase Relations Among End-Member Solids and a Pure Fluid

Introduction. . . . . . . . . . . . . . . . . . . . . 107 Theoretical Background. . . . . . . . . . . . . . . 107 Error Propagation Formalism for Solid-Fluid Equilibria. . . . . . . . . . . . . . . . . . . . .. 109 Sample Calculations of Dehydration Equilibria. . 111 Thermodynamic Interpolation and Extrapolation of Reaction Reversal Data, and Linear Summation ofIndependent Equilibria. . . . . . . . . . . . . . 122 Theoretical Basis. . . . . . . . . . . . . . . . .. 122 Worked Examples of Dehydration Reactions. . . 126 Worked Examples of Redox Reactions. . . . . . . 135

6

6.1 6.2 6.3

6.4

6.5

7

7.1 7.2 7.3

7.4 7.4.1 7.4.2 7.5 7.5.1 7.5.2 7.5.2.1 7.5.2.2 7.5.2.3 7.5.2.4 7.5.2.5 7.5.2.6

7.6

8

8.1 8.2

8.2.1 8.2.2

8.2.2.1 8.2.2.2

Contents IX

Phase Relations Among End-Member Solids and a Binary Fluid Mixture

Introduction. . . . . . . . . . . . . . . . . . . . . 141 Thermodynamic Background. . . . . . . . . . . . 141 Sample Calculation of Mixed-Volatile (H20-C02) Equilibria. . . . . . . . . . . . .. 145 Interpolation, Extrapolation, and Linear Summation of Reaction Reversal Data. 154 Concluding Remarks. . . . . . . . . . . . . . .. 164

Derivation of an Internally Consistent Thermodynamic Dataset by Mathematical Programming

Introduction. . . . . . . . . . . . . . . . . . . 165 The Nature of the Problem. . . . . . . . . . . . . 166 A Thermodynamic Model for Mathematical Programming. . . . . . . . . . . . . . . . . . .. 169 Outlines of Mathematical Programming. . . . . . 170 Input Constraints. . . . . . . . . . . . . . . . .. 170 Basic Formalisms and Definitions. . . . . . . . . 172 Worked Example: The System AI2SiOs . . . . . . 173 Input Database. . . . . . . . . . . . . . . . . . . . 173 Thermodynamic Analysis. . . . . . . . . . . . . . 174 Intr.oduction. . . . . . . . . . . . . . . . . . . . . 174 Phase Equilibria Constraints. . . . . . . . . . . . 176 Application of Linear Programming. . . . . . . . 179 Phase Property Constraints ............. 181 Application of Quadratic Programming. . 182 An Internally Consistent Thermodynamic Dataset for the Three Al2SiOs Phases by Quadratic Programming. . . . . . . . . . . . . 184 Future Perspectives and Concluding Remarks. . . 190

Thermodynamics of Crystalline Solutions

Introduction, Scope, Definitions. . . . . . . 193 Extension of the Thermodynamic Theory of Molecular Solutions to Crystalline Solutions. .. 194 Introduction. . . . . . . . . . . . . . . . . . . . . 194 Derivation of Equations for Simple Crystalline Solutions ....................... 195 Solutions with Site Mixing on One Sublattice. . . 195 Simple Crystalline Solutions with Charge-Coupled Site Mixing on Two Sub lattices . . . . . . . . .. 204

X Contents

8.2.3

8.3

8.4 8.4.1 8.4.2 8.5

8.5.1 8.5.2

8.5.2.1 8.5.2.2 8.5.2.3

8.5.3

8.5.3.1 8.5.3.2 8.5.3.3 8.5.4

8.5.4.1

8.5.4.2

8.5.4.3

8.5.4.4

8.6.

9

9.1 9.2

9.2.1 9.2.2

9.2.2.1 9.2.2.2

Thennodynamic Treatment of Complex Crystalline Solutions. . . . . . . . . . . . . . . . . . . . . . . 207 Some Equations for Excess Properties of Crystalline Solutions. . . . . . . . . . . . . . . . 211 Phase Relations in a Binary Solution ........ 213 Isostructural Solution ................ 213 Non-Isostructural Solution. . . . . . . . . . . . . 218 Fonnulation of Equations of State for Crystalline Solutions. . . . . . . . . . . . . . . . 219 Introduction. . . . . . . . . . . . . . . . . . . . . 219 Manipulation of Molar Quantity vs Composition Data. . . . . . . . . . . . . . . . . . 220 General Fonnalisms. . . . . . . . . . . . . . . . . 220 An Example of Processing Calorimetric Data. . . 223 Examples of Fitting V (X) Data for Crystalline Solutions. . . . . . . . . . . . . . . . 227 Handling Experimental Data on the Composition Dependence of Partial Molar Quantities of Mixing of a Component. . . . . . . . . . . . . . . 230 Fonnalisms of Gibbs-Duhem Integration. . . . . 230 Worked Examples of Gibbs-Duhem Integration .. 232 An Alternative to Gibbs-Duhem Integration .... 237 More on Equations of State for Crystalline Solutions: Some Worked Examples. 240 An Equation of State for the Halite-Sylvite Crystalline Solution, (Na,K)CI. . . . . . . . .. 240 Excess Mixing Properties of the Monticellite­Forsterite Crystalline Solution, (Ca,Mg)MgSi04 . 244 Thennodynamic Mixing Properties of Zn(AI,Cr)204 Spinel ................. 247 Solution Modeling of Grossular-Almandine Garnets ........................ 252 An Epilogue. . . . . . . . . . . . . . . . . . . . . 256

Phase Equilibria Involving Nonideal Solutions and Outlines of Geothermometry and Geobarometry

Introduction and Scope. . . . . . . . . . . . . . . 257 Calculation of Heterogeneous Phase Equilibria Involving Nonideal Solutions. . . . . . . . . . . . 258 General Considerations. . . . . . . . . 258 A Qualitative Look at the Gibbs Energy Minimization Method. . 258 Theoretical Basis. . . . . . . . . . . . . . . . . . 258 Graphical Analysis of Isobaric-Isothennal G-X Sections. . . . . . . . . . . . . . . . . . . . . 259

9.2.2.3

9.2.3

9.2.3.1

9.2.3.1.1

9.2.3.1.2

9.2.3.2

9.2.3.3

9.3 9.3.1 9.3.2 9.3.2.1

9.3.2.2

9.3.2.3

9.3.3 9.3.3.1 9.3.3.2 9.3.4

9.3.4.1 9.3.4.2 9.3.5

9.3.5.1

9.3.5.2

9.3.6

Contents XI

From Isobaric-Isothermal G-X Sections to Phase Diagrams. . . . . . . . . . . . . . . 261 Alternative Techniques for Phase Diagram Calculations with Nonidea1 Solutions, and Some Worked Examples. . . . . . . . . . . . . .. . 262 Calculation of P-T -X Phase Diagram for the NaAlSi30 s-KAlSi30 s Binary. . . . . . . . 263 Calculation of the Isobaric T -Xkf Section for the NaAISi30 s-KAlSi30 s Binary: the Analbite-Sanidine Solvus. . . . . . . . . . . . 263 Calculation of the P-Xkf Section Through the NaAISi30s-KAlSi30 s Pseudobinary, and the P-T Diagram. . . . . . . . . . . . . . . . . . . . .. 268 NaAI2[AlSi3010(OII)2]-KAl2[AlSi3010(OII)2] Pseudobinary: Computation of Phase Relations for MUltiple Equilibria. . . . . . . . . . . . .. 272 Computation of a T -XC02 Phase Diagram with a Crystalline Solution ................ 276 Outlines of Geothermometry and Geobarometry. 279 Scope ......................... 279 Some Fundamental Considerations. . . 280 Identification of Equilibrium Mineral Assemblages. . . . . . . . . . . . . . . 280 Thermodynamic Basis for the Evaluation of the Intensive Variables. . . . . . . .. ... 281 Choice of Equilibria Appropriate for Geothermometry and G~obarometry. . . . . . 281 Setting up Geobarometers and Geothermometers. 282 Calibration of Geobarometers. . . . . . . . . .. 284 Calibration of Exchange Geothermometers . 288 Limits of Applicability and Evaluation of Uncertainties in Geothermometry and Geobarometry. . . . . . . . . . . . . . . . . . . . 294 Limits of Applicability. . . . . . . . . . . . . . . 294 Evaluation of Uncertainties. . . . . . . . . . . . . 295 Two Worked Examples of Geothermometry, Geobarometry, and Geohygrometry ........ 297 Staurolite- and Sillimanite-Bearing Metapelitic Rocks, Rangeley Quadrangle, Maine. . . . . . . . 297 Staurolite-Kyanite-Bearing Metapelites from the Great Smoky Mountains, North Carolina 301 Concluding Remarks. . . . . . . . . . . . . . . . 304

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Subject Index . ......................... 319

Symbols, Abbreviations, and Constants

A Helmholtz energy as an extensive quantity A A Redlich-Kister parameter, unless otherwise stated A, A' Coefficients of virial or corresponding state equations a A constant of a vdW, RK, or MRK equation related to

the attractive intermolecular interactions in a fluid amix a for a fluid mixture a A coefficient of the Co p(T) function ai Activity of the component i in a given phase aidi Ideal activity of the component i in a given phase

B Redlich-Kister parameter, unless otherwise stated B, B' Coefficients of virial or corresponding state equations b A constant of a vdW, RK, or MRK equation related to

the repulsive intermolecular interactions in a fluid bmix b for a fluid mixture b A coefficient of the Co p(T) function

C A Redlich-Kister parameter, unless otherwise stated C, C' Coefficients ofvirial or corresponding state equations "c Degree Celsius c A coefficient of the Co p(T) function

d A coefficient of the Co p(T) function

E Electromotive force e A coefficient of the Co p(T) function

F Faraday constant, 96487 J Volt-1 mol-1

ri Standard state [1 bar, T] fugacity of the species i fi Fugacity of the pure species i at specified T and P Ii A dimensionless quantity defined asNri f\ Fugacity of the component i in a fluid mixture at a given

set of T, P, and Xi J'Vi A dimensionless quantity defined asf'lri f A coefficient of the Co p(T) function

XIV Symbols, Abbreviations, and Constants

Gm Gidm Gexm

gi

gm,i gid .

m,l gexm,i

K K KD

n

q

R

Gibbs energy as an extensive quantity Gibbs energy as an intensive (molar) quantity Molar Gibbs energy of formation of i at 1 bar and T Molar Gibbs energy of i at specified P and T Apparent molar Gibbs energy of formation of i at P and T Integral molar Gibbs energy of mixing Integral molar ideal Gibbs energy of mixing Integral molar excess Gibbs energy of mixing Partial molar Gibbs energy of i Partial molar Gibbs energy of mixing of i Partial molar ideal Gibbs energy of mixing of i Partial molar excess Gibbs energy of mixing of i

Enthalpy as an extensive quantity Enthalpy as an intensive (molar) quantity Enthalpy of solution Molar enthalpy of formation of i at 1 bar and T Molar enthalpy of i at specified P and T Apparent molar enthalpy of formation of i at P and T Integral molar enthalpy of mixing Integral molar ideal enthalpy of mixing Integral molar excess enthalpy of mixing Partial molar enthalpy of i Partial molar enthalpy of mixing of i Partial molar ideal enthalpy of mixing of i Partial molar excess enthalpy of mixing of i

Kelvin Equilibrium constant Distribution coefficient of species between a set of phases in a specified reaction Product of mole fractions of species in a given reaction Product of activity coefficients of species for a given reaction

Number of moles

Pressure (bar or kbar, as specified) Critical pressure Reduced pressure (= PIPe)

Site multiplicity of a specified sublattice of a crystalline solution

Gas constant. 8.3143 J K-l mol-1

s S SOT So

f,T,i S\ Sm Sidm

sexm

Si

Smi Sid' .

m,1

Sexm,i

T

u u

v Y YOT

Ym yidm yexm

Vi

Vmi Vid' .

m,1 Vex .

m,1

WG ·· ,IJ WU" ,IJ WH ·· ,IJ

WS" ,IJ WV " ,IJ Wep" ,IJ

X-I

Symbols, Abbreviations, and Constants XV

Entropy as an extensive quantity Entropy as an intensive (molar) quantity Standard molar entropy at 1 bar and T Molar entropy of formation of i at 1 bar and T Molar entropy of i at specified P and T Integral molar entropy of mixing Integral molar ideal entropy of mixing Integral molar excess entropy of mixing Partial molar entropy of i Partial molar entropy of mixing of i Partial molar ideal entropy of mixing of i Partial molar excess entropy of mixing of i

Temperature in Kelvin (K), unless indicated as degree Celcius Cc) Critical temperature Reduced temperature (= T!Tc)

Internal energy as an extensive quantity Internal energy as an intensive (molar) quantity

Volume as an extensive quantity Volume as an intensive (molar) quantity Standard molar volume at 1 bar and T Integral molar volume of mixing Integral molar ideal volume of mixing Integral molar excess volume of mixing Partial molar volume of i Partial molar volume of mixing of i Partial molar ic\eal volume of mixing of i Partial molar excess volume of mixing of i

Margules Gibbs energy parameter for i in i-j solution Margules internal energy parameter for i in i-j solution Margules enthalpy parameter for i in i-j solution Margules entropy parameter for i in i-j solution Margules volume parameter for i in i-j solution Margules heat capacity parameter for i in i-j solution

Mole fraction of i

Any extensive quantity Any intensive (molar) quantity Any integral molar quantity of mixing Any integral molar ideal quantity of mixing Any integral molar excess quantity of mixing

XVI Symbols, Abbreviations, and Constants

Yi Any partial molar quantity of i y m,i Any partial molar quantity of mixing of i yid. Any partial molar ideal quantity of mixing of i m,l yex m,i Any partial molar excess quantity of mixing of i

Z Compressibility factor Zc Critical compressibility factor

a Isobaric thermal expansion ~ Isothermal compressibility Yi Activity coefficient of component i in a given phase <l>i Fugacity coefficient of a pure fluid species i <1>'\ Fugacity coefficient of i in a fluid mixture ~i Chemical potential of component i at P,T, and Xl;t:l) ~ \ Standard chemical potential of i at P and T ~oi Standard chemical potential of i at 1 bar and T Vi Stoichiometric coefficient of i in a reaction cri Standard deviation of i cr2i Variance of i Pij Correlation coefficient of i and j