Thermodynamic Modeling of Geological Materials - Minerals, Fluids and Melts 0939950219

Table of contents :
Page 1
Titles
REVIEWS IN MINERALOGY
THERMODYNAMIC MODELING
MINERALS, FLUIDS AND MELTS
EDITORS: I.S.E. Carmichael & H.P. Eugster
Page 1
Titles
COPYRIGHT 1987
Volume 17: THERMODYNAMIC MODELING OF GEOLOGIC MATERIALS:
MINERALS, FLUIDS AND MELTS
~~.i~r~~r~:"!':::J~=r~~Yo"i':~~~~ ;X::d~~e;:::
ii
~~~:,.e.~:.: 1~'::1~. ~~':::y:~~~~~a~~:'~~~': J~=~~:'!
Page 1
Titles
REVIEWS IN MINERALOGY
FOREWORD
VOLUME 17
PREFACE
Page 2
Titles
geochemist of the century, we wish to dedicate this volume to the memory of V.M.
I.S.E. Carmichael
August 1987
H.P. Eugster
Page 1
Titles
Thermodynamic Modeling of Geological Materials:
Minerals, Fluids and Melts
TABLE OF CONTENTS
Chapter 1 Robert C. Newto
THERMODYNAMIC ANALYSIS OF PHASE EQUILIBRIA
IN SIMPLE MINERAL SYSTEMS
Chapter 2
Alexandra Navrotsk
MODELS OF CRYSTALLINE SOLUTIONS
Page 2
Titles
Chapter 3 Bernard J. Wood
Chapter 4
Page 3
Titles
121 Mixing Parameters
123 Solubilities of solids
123 Complex ion equilibria
123 Vapor-phase equilibria
125 Thermal properties
Chapter 5 John H. Wear
MODELS OF MINERAL SOLUBILITY IN CONCENTRATED
Chapter 6 Dimitri A. Sverjenskj
CALCULATION OF THE THERMODYNAMIC PROPERTIES
OF AQUEOUS SPECIES AND THE SOLUBILITIES OF
181 Standard molal Gibbs free energies of minerals
182 Standard molal Gibbs free energies of gases
182 Standard molal Gibbs free energies of aqueous species
188 Standard molal Gibbs free energies of aqueous complexes
188 Standard molal entropies, heat capacities and volumes of complexes at 25 DC
and 1 bar
191 Equation of state coefficients for aqueous complexes
195 Solute species
195 Charged species
vii
Page 4
Titles
Chapter 7
John R. Holloway
Page 5
Titles
Chapter 8
Page 6
Page 7
Titles
Chapter 9 John M. Ferry and Lukas Baumgartn
Page 8
Titles
Chapter 10 Hans P. Eugster and Lukas Baumgartner
MINERAL SOLUBILITIES AND SPECIATION IN
SUPER CRITICAL METAMORPHIC FLUIDS
Chapter 11
R. G. Berman and T. H. Brown
DEVELOPMENT OF MODELS FOR MULTICOMPONENT MELTS:
ANALYSIS OF SYNTHETIC SYSTEMS
Page 9
Titles
Chapter 12
Mark S. Ghiorso
Chapter 13 Mark S. Ghiorso and Ian S.E. Carmichael
Page 10
Page 1
Titles
Chapter 1 Robert C. Newton
Page 2
Page 3
Page 4
Page 5
Page 6
Tables
Table 1
Page 7
Tables
Table 1
Page 8
Page 9
Titles
40~--~~--~~--~~~~~
1600
Temperature Deg C
Page 10
Page 11
Page 12
Page 13
Titles
KYANITE
SILLIMANITE
\h
ok! , I , I ,~ , 1--1
200 400 600 800 ~~~
Temperature Oeg C
9
A
10
II
B
6S~-A J/K
Page 14
Titles
z=II'IH-6H'1 + II'IS-6s'l (18)
IT\ill'f ~
Page 15
Titles
8
10
12
Tables
Table 1
Page 16
Page 17
Titles
Temperature °c
-
-
I
Synthetic grossular
I
o t:, Natural grossulor
...
I
I
:'~.~ .. A·
.. :.
t'c?@
.,
.~
350f- .. ff
..
4001-
5001-
:r:
>-::>_ 4501-
I
I
I
I
400
600
800
1000
17
Page 18
Page 19
Titles
Anorthite
•
P(Bar)= 22.47T(oC)-587
/
500
Temperature
Temperatu re Oeg C
Page 20
Titles
.d·O 0
....
'"
'"
"
'"
'"
~B
~>
« ...,
...,
'"
°
...,
'"
..,.
'"
'"
i
u
.... '"
"''''
'"
'"
°
'"
°
20
Page 21
Page 22
Titles
I-
............. '"
I-
>;:_____~ 120
700
t.H;,s,298=231.82 kJ
, ,
4Z0 +QZ = 5AN+GR+2H20
Temperature (K)
22
Tables
Table 1
Page 23
Page 24
Page 25
Titles
~
Page 26
Titles
0.2 0.4 0.6 0.8 1.0
Page 27
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 1
Titles
Chapter 2
Alexandra Navrotsky
MODELS OF CRYSTALLINE SOLUTIONS
Page 2
Titles
0·--------------
+
o
B
Tables
Table 1
Page 3
Titles
(2)
Page 4
Titles
(5)
o .
(9)
(10)
Page 5
Titles
dSmix = -R[x In x + (I-x) In (I-x)]
dS(A} = -R In x ,
dS(B} = -R In (I-x) .
RT[x In x + (I-x) In (I-x)]
dpO(A} = RT In x. dp'(B} = RT In (I-x)
a(A) = x, a(B) = I-x
(14)
Page 6
Page 7
Titles
(25)
Page 8
Page 9
Page 10
Titles
(41)
(42)
Tables
Table 1
Page 11
Page 12
Page 13
Titles
0.5
0.4
0.3
6V
0.2
coo.s,o.' ,{ ·
0.1
0.5
6v
47
Tables
Table 1
Table 2
Page 14
Titles
70r--------.---------.--------.--------,--------,
60
48
Tables
Table 1
Page 15
Page 16
Titles
50
Tables
Table 1
Table 2
Page 17
Page 18
Page 19
Titles
(A) + [B] = (B) + [A] .
(58)
Page 20
Titles
ex + 2fJx •
Tables
Table 1
Page 21
Page 22
Tables
Table 1
Page 23
Titles
x 630 °C
N
0.4 0.6
1.0 r-----..--...,---,....-----r--___,
0.5
X= Fe3+
3+
X = FeT
1.0
0.5
0.2
0.4
0.6
57
Tables
Table 1
Page 24
Page 25
Titles
59
Tables
Table 1
Table 2
Table 3
Page 26
Page 27
Titles
~[ " \
iL ' , , 1 ~ , '
Tables
Table 1
Page 28
Page 29
Page 30
Page 31
Titles
a
b
.t~
c
d
Page 32
Page 33
Page 34
Page 35
Page 36
Page 1
Titles
Chapter 3 Bernard J. Wood
THERMODYNAMICS OF MULTICOMPONENT SYSTEMS
(1)
Page 2
Titles
l'
(C)
(8)
/f
(0)
t . "''-
" .
GBC = xlc µ(A) + xlc µ(B) ,
(2)
G(a) = XI µO(A,a) + xIf µO(B,a) + RT(XI In XI + XB" In XIf) (3)
G((3) = xl µO(A,(3) + xJ µO(B,(3) + RT(Xlln xl + xJ In XJ) ,
µ(A) = µO(A,a) + RT In XI . (4)
.,
µ(B) - µ(A) = µO(B,a) - µO(A,a) + RT In XIf - RT In XI .
(5)
Page 3
Titles
(8)
B n+1 B n f" (XJ) .
xJ xl
Page 4
Titles
-t.G12 X-J
-t.Gl~ (l-X-J)
Page 5
Page 6
Titles
(25)
Page 7
Titles
d1JI d1J-J d1JB" d1J-J
s
Page 8
Titles
s
Page 9
Titles
-S mix R I Xc 3
Page 10
Page 11
Page 12
Titles
82
Tables
Table 1
Table 2
Table 3
Page 13
Page 14
Page 15
Page 16
Titles
15 kbar
(A)
En
(B)
Fs
Tables
Table 1
Page 17
Titles
XM2·XM1 T
Page 18
Page 19
Titles
Table 6. Tests of Equilibrium at 11.5 kbar / 800' C
89
Tables
Table 1
Table 2
Page 20
Page 21
Titles
AS-1
W-2
91
Page 22
Page 23
Titles
ACKNOWLEDGEMENTS
REFERENCES
Page 24
Page 25
Page 26
Page 1
Titles
Chapter 4 Kenneth S. Pitzer
A THERMODYNAMIC MODEL FOR AQUEOUS
Page 2
Page 3
Page 4
Page 5
Page 6
Titles
q,-1
Page 7
Titles
, , ,
Page 8
Titles
,
Page 9
Titles
" ,
Page 10
Titles
+ I I
" ,
z
Page 11
Page 12
Tables
Table 1
Page 13
Page 14
Page 15
Titles
(44)
Page 16
Page 17
Titles
'"
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.....
, ,
'"
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e-,
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113
Page 18
Page 19
Titles
~
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, ,
e--
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~~~~~~~~
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sr
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184
Page 9
Titles
+ a . In (p + 1)1)
1 (_1_
- 1)
Page 10
Titles
-------~--.-
Page 11
Titles
-4
-6
- 8
o
o
0.5 KB
TEMPERATURE (DEG. C)
5KB
1.5 ALBITE + H+ = 0.5 PARAGONITE + 3.0 QUARTZ + Na+
500 600 700
TEMPERATURE (DEG. C)
187
Page 12
Page 13
Page 14
Titles
CHLORIDE COMPLEXES
b3
~
1u 0
A.
CHLORIDES
• ••
[c.
-I:
~
~
~:.
~
-,
o
'3
s~ABs·(CAL. MOLE-I.DEG-')
E.
o
ZnCI+
''_'; ~ AgCI2
B • /Z=-I
C~(ML'_I)Z+'
o
'Z 1
Tables
Table 1
Page 16
Titles
(67)
-----.----
S; - 71.5 I -, I
Wn=
Tables
Table 1
Page 17
Tables
Table 1
Page 18
Titles
.,;:
-'2
A.
B
c.
D.
or
'" -.
cQ
'" -.
'" ,
o
-I,
-'5
194
Page 19
Titles
*
Page 20
Titles
b I
*
Page 21
Page 22
Titles
TEMPERATURE (DEG C)
c.
,O~ i '0L:~~
.' ,! .
PRESSURE (KB) PRESSURE (KB)
Tables
Table 1
Page 23
Titles
pH
~
TEMPERATURE (DEG. C)
D.
TEMPERATURE (DEG. C)
~~~_::_ .. -
c.
Tables
Table 1
Table 2
Table 3
Page 24
Page 25
Page 26
Titles
B.
'"
Tables
Table 1
Page 27
Titles
TEMPERATURE (DEG. C)
Tables
Table 1
Page 28
Page 29
Page 30
Page 31
Page 32
Titles
208
Page 33
Page 34
Page 1
Titles
Chapter 7
John R. Holloway
Page 2
Page 3
Titles
F = - dl" I dz,
where r is the intermolecular separation.
Permanent dipole-permanent dipole forces. These are the easiest to visualize. They
213
Tables
Table 1
Table 2
Page 4
Titles
Tuu=
(1)
(2)
Page 5
Titles
(3)
Page 6
Titles
b
---------~ -
o
Tables
Table 1
Table 2
Page 7
Page 8
Page 9
Page 10
Titles
~~-------------------------,
~
l!!
-:
o
O+-~---.--~-.--~--.- ~
o
220
Page 11
Page 12
Titles
222
Tables
Table 1
Table 2
Page 13
Page 14
Titles
METHANE FORMATION ENERGY
1400
1200
1000
800
600
-20+-~--'---~-r--~-'--~~--~-4
400
14001600
10001200
800
O+-~-''-~-'--~~~--'-~--~~
600
10
20
...
224
Page 15
Page 16
Titles
c+m~=m 00
(7)
Page 17
Titles
(12)
Tables
Table 1
Page 18
Tables
Table 1
Page 19
Page 20
Tables
Table 1
Page 21
Tables
Table 1
Page 22
Page 23
Page 24
Page 1
Titles
Chapter 8
George H. Brimhall and David A. Crerar
Page 2
Page 3
Page 4
Titles
10
0.1
Sc \ I, ". "'1'; Zil .... G •
~:IIi' Mo· .. ··· ,_ \ / ''tot r., f
. \: y ~Ni
.Cr M "" .. !r··iCOMPATtBLE ELEMENTS)
IA 3B 58 78 8 18 3A 5A 7A
2A 48 68 28 4A 6A
GROUP in the Periodic Table
INCREASING INCOMPATtBILITY
~
~ 8ULK CONTINENTAL CRUST
"'\!!.r (tNCOMPATIBLE ELEMENTS)
C. Sa ~ U TI Pb )
,_ La h Ti .....
~ . - .... .tf ~ W Ag " ... ~ 1'Sb
K~' Sr ,,_... \ .., . ,
\"L .... :.JIi; M!\t,~' Au ~~.cd Go .!J.S.Bi
\ Y ..... _ V ." Cu ~ .. :,tt!..t iN
CO ....... Ti ~ Mn .... ~, 'J~1n .L'Si Se
j:::
~
Z
~ 0.01
U
w
E~,OOO
a: -
u:::
Tables
Table 1
Table 2
Page 7
Page 8
Titles
(b)
-A- - - - - - __ :::_ ..§_
I-

Citation preview

REVIEWS IN MINERALOGY VOLUME

17

THERMODYNAMIC MODELING OF GEOLOGICAL MATERIALS: MINERALS, FLUIDS AND MELTS EDITORS:

I.S.E. Carmichael & H.P. Eugster

AUTHORS: Robert C. Newton Dept of the Geophysical Sciences University of Chicago Chicago, Illinois 60637 David A. Crerar Alexandra Navrotsky Dept of Geological & Geophysical Sciences Princeton University Princeton, New Jersey 08544 Bernard J. Wood Dept of Geological Sciences Northwestern University Evanston, Illinois 60201 Kenneth S. Pitzer Department of Chemistry University of California, Berkeley Berkeley, California 94720 John H. Weare Department of Chemistry University of California, San Diego La Jolla, California 92093

SERIES

EDITOR:

Lukas Baumgartner Hans P. Eugster John M. Ferry Dimitri A Sverjensky Dept of Earth & Planetary Sciences The Johns Hopkins University Baltimore, Maryland 21218 John R. Holloway Depts of Chemistry and Geology Arizona State University Tempe, Arizona 85287 George H. Brimhall Ian S.E. Carmichael Dept of Geology & Geophysics University of California, Berkeley Berkeley, California 94720 Robert G. Berman Thomas H. Brown Dept of Geological Sciences University of British Columbia Vancouver, BC, Canada V6T 2B4 Mark S. Ghiorso Dept of Geological Sciences University of Washington Seattle, Washington 98195

Paul H. Ribbe Department of Geological Sciences Virginia Polytechnic Institute & State University Blacksburg, Virginia 24061

COPYRIGHT 1987 MINERALOGICAL

SOCIETY OF AMERICA

Second Printing. 1989. Printed by BcokCrafters,

Inc .•Chelsea. Michigan.

REVIEWS IN MINERALOGY ( Formerly: SHORT COURSE NOTES) ISSN 0275-0279 Volume

17: THERMODYNAMIC MINERALS,

MODELING FLUIDS

OF GEOLOGIC

MATERIALS:

AND MELTS

ISBN 0-939950-21-9 ADDITIONAL COPIES of this volume as well as those listed below may be obtained at moderate cost from the MINERALOGICAL SOCIETY OF AMERICA. 1625 I Street. N.W .• Suite 414. Washington. D.C. 20006 U.S.A. Volume 11: Carbonate.: Mlnaralogy and Chemi.try, 1983; R. J. Reader, Ed. 394 pp. Nine chapters on crystal chemistry. polymorphism, microstructures and phase relations of the rhombohedral and orthorhombic carbonates; the kinetics of CaCO, dissolution and precipitation; trace eJements. and isotopes in sedimentary car1lonates; the occurrence. solubility and solid solution behavior of Mg-calcites; geoIoQlc thermobarornetry using metamorphic carbonates. ISBN# O-M9950-15-4.

Volume 1: Sulfide Mineralogy, 1974; P. H. Rlbbe, Ed. 284 pp. Six chapters on the structures of sulfides and sulfosatts; the ~~l~~~~=~~~~~~~Synttesis.

phase

·Volume 2: Feld."ar Mineralogy, 2nd Edition, 1983; P. H. Rlbbe, Ed. 362 pp. Thirteen chapters on feldspar chemistry, structure and nomenclature: Al,Si order/disorder in relation to domain textures, diffraction patterns, lattice parameters and optical properties; determinative methods; subsoltdus phase refatlons, microstructures, kinetics and mechanisms of exsolution, and diffusion; color and interference colors; chemical properties; deformation. ISBN# 0-939950-14-6.

Volume 12: Auld Inclu.lon., 1984; by E. Roedder. 844 pp. Nineteen chapters providing an introduction to studies of all types of fluid inclusions, gas, liquid or meft, trapped in materials from the earth and space. and their application to the understanding of geological processes. ISBN# 0-939950-16-2.

Volume 4: Mineralogy and Geology of Natural Zeolite., 19n; F. A. Mumpton, Ed. 232 pp. Ten chapters on the crystal chemistry and structure of natural zeolites, their occurrence in sedimentary and Iow.grade metamcrphlc rocks and closed hydrc>~ic systems, their oommercial properties and utilization. ISBN# 0-939950-04-9.

Volume 13: Mlc8l,1984; S. W. Bailay, Ed. 584pp. Thirteen Charters on structures. crystal chemistry. spectroscopic and op:~~c%v"&"~:~~s~~~e;~~jg~'t~':g~s.

~~~:~x!~m:,~;u;;:e 18-9.

Volume 6: Marine Minerals, 1979; R. G. Bums, Ed. 380 pp. Ten chapters on manganese and iron oxkies, the silica po{ymorphs, zeolites, clay minerals, marine phosphorites, barites and placer minerals; evaporite mineralogy and chemistry. ISBN# 0-939950-06-5.

~~trolyles;

~~\~n~~~ 19-7.

Crystallography,

1985; by M. B. 8010-

1~~:xs~~n3~~f:~~~~~ and probfem sets, including

~~~~"{,~t~~[~P~t~~~:",~i~N~i~~9~~9ro:

~~~:,.e.~:.: 1~'::1~. ~~':::y:~~~~~a~~:'~~~': J~=~~:'!

Ed •. 570 pp. Starting with the theoretical, kinetic and experimental aspects of isotopic fractionation, 14 chapters deal with stable isotopes in the early solar system, in the mantle, and in the igneous and metamorphic rocks and ore deposits, as well as in magmatic volatiles, natural water, seawater, and in meteoric-hydrothermal systems. ISBN #0-939950-20-0.

irreversible thermcdyPyriboles-MlnSeven chapters

~~~~':'.~.~ 7A~~:;,n~r:~~~~~.e~~n'~~~:~~C~1 :,a~~~~:: michael, Ed •• 500 pp. Thermodynamic analysis of phase equilibria in simple and multi-component mineral systems, and thermcdynamlc models of crystalline solutions. igneous gases and fluid. ore fluid. metamcrphlc fluids. and silicate melts, are the subjects of this 14-chapter volume. ISBN # 0-939950-21-9.

~~.i~r~~r~:"!':::J~=r~~Yo"i':~~~~ ;X::d~~e;::: lations; amphibole and serpentine asbestos-mineralogy, occurrences, and health hazards. ISBN#

o=-~~i':'"4SB~;'~-~fg~i

~:~~~:~tdo~t~ ~?~·9=~~Bravai~ presented with numerous exam~es

Volume 8: Kinetics of Geochemical Processe., 1981; A. C. Lasaga and R. J. Kirkpatrick, Eds. 398 pp. Eight chapters on transition state theory and the rate laws of chemical reactions; kinetics of weatheri~ dfagenesis, igneous crystallization and goo-

Volume 9A: Amphiboles and Other Hydrous eralogy, 1981; D. R. Veblen, Ed. 372 pp.

and pe-

:~3

Volume 15: Mathematical

Volume 7: Pyroxenes, 1980; C. T. Prewitt, Ed. 525 pp. Ninll chapters on pyroxene crystal chemistry. spectroscopy. phase equilibria, subsondus phenomena and thermodynamics; cornposition and minerak)Qy of terrestrial, lunar, and meteoritic pyroxenes. ISBN# 0-939950-07-3.

~~~'1'~II~~~;0~~39~re-Ol

geochemistry

Volume 14: Microscopic to Macroscopic: Atomic Envlronmenll to Minerai Thermodynamic., 1985; S. W. Kieffer and A. Navrotsky, Ed •• 428 pp. Ektven chapters attempt to answer the question, "What minerals exist under given constraints of

Volume 5: Orthosilicates, 2nd Edition, 1982; P. H. Ribbe, Ed. 450 pp. Uebau's "Classification of Silicates" plus 12 chap. ters on silicate garnets, oIivines, spinels and humites; zircon and the actinide orthosilicates; titanite (sphene), chloritoid. staurolite. the aluminum silicates, topaz, and scores of miscellaneous orthosilicates. Indexed. ISBN# 0-939950-13-8.

0-939950-10-3.

Volume 9B: Amphiboles: Petrology and Experimental Phase Relations, 1982; D. R. Veblen and P. H. Ribbe, Ed •• 390 pp. Three chapters on phase relations of metamorphiC amphiboles (occurrences and theory); igneous amphiboles; experimental studies, ISBN# 0-939950-11-1.

Volume 18: Spectroscopic Methods In Mlnaralogy and Geology, 1988; F. C. Hawthome, Ed. 898 pp. Detailed explanations and encyciopedlc discussion of applications of spectroscopies of major importance to earth sciences. Included are IR. optical. Raman, Mossbauer. MAS NMR, EXAFS. XANES, EPR. ~~9:~~~~minescence. XRF. PIXE. RBS and EELS. ISBN #

Volume 10: Characterization of Metamorphllm through Minerai Equilibria, 1982; J. M. Feny, Ed. 397 pp. Nine chapters on an algebraic approach to composition and reaction spaces and their manipulation; the Gibbs' formulation of phase equilibria; ge0logic thermobarometry; buffering, infiltration, isotope fractionation, compositional zoning and inclusions; characterization of metamorphic fluids. ISBN# 0-939950-12-X.

Volume 19: Hydrous Phytioallicates (exclusiva of micas), 1988; S. W. Bailey, Ed. 698 pp. Seventeen chapters covering the crystal structures. crystal chemiStry. serpentine, kaolin. talc, pyrophyllite, chlorite. vermiculite, smectite. mixed-layer, sepiol~e. palygorskite, and mcdulated type hydrous phyllosilicate minerals.

ii

REVIEWS IN MINERALOGY

VOLUME 17

FOREWORD The editors and authors of this volume presented a short course, entitled "Thermodynamic Modeling of Geological Materials: Minerals, Fluids amd Melts," October 22-25, 1987, at the Wickenburg Inn near Phoenix, Arizona. This was the fourteenth in a series of such courses sponsored by the Mineralogical Society of America since 1974, and this is the eighteenth book published under the banner, Reviews in Mineralogy [Volume 9 was issued in two parts -- see list of available titles on the opposite page]. The text of this volume was assembled from author-prepared, camera-ready copy __ thus the wide variety in style and font types represented. Mrs. Marianne Stern patiently and skillfully did most of the paste-up of Volume 17.

Paul H. Ribbe Series Editor Blacksburg, VA

PREFACE When Van't Hoff calculated the effect of solution composition on the gypsum-anhydrite transition a century ago, he solved a significant geochemical problem (Hardie, 1967). Other well known examples of the early use of chemical thermodynamics in geology are Bowen's calculations of the plagioclase melting loop and the diopside-anorthite eutectic (Bowen, 1913, 1928). Except for a few specialists, however, these techniques were largely ignored by earth scientists during the first half of the 20th century. The situation changed dramatically by the 1950's when more and better thermodynamic data on geologic materials became available, and when thermodynamic arguments of increasing sophistication began to permeate the petrologic and geochemical literature. This rejuvenation was spearheaded by D.S. Korzhinskii, H. Ramberg, J.B. Thompson, J. Verhoogen and others. Today a graduating petrologist or geochemist can be expected to have a thorough grounding in geological thermodynamics. Rapid intellectual growth in a field brings with it the difficulty of keeping abreast of parallel and diverging specialties. In order to alleviate this problem, we asked a group of active researchers to contribute up-to-date summaries relating to their specialties in the thermodynamic modeling of geological materials, in particular minerals, fluids and melts. Whereas each of these topics could fill a book, by covering the whole range we hope to emphasize similarities as much as differences in the treatment of various materials. For instance, there are useful parallels to be noted between Margules parameters and Pitzer coefficients. The emphasis here is on modeling, after the required data have been collected, and the approach ranges form theoretical to empirical. We deliberately imposed few restrictions on the authors. Some chose to interpret modeling in the rigorous thermodynamic sense, while others approached their topics from more general geochemical viewpoints. We hope that any lack of unity and balance is compensated for by a collection of lively and idiosyncratic essays in which students and professionals will find new ideas and helpful hints. If the selection appears tilted towards fluids, it is because other recent summaries have emphasized minerals and melts. This volume could not have been assembled without the dedication, cooperation and understanding of every author and his or her typist(s), and without the unselfish efforts of the Series Editor and current President of MSA, Paul Ribbe. As a tribute to the foremost iii

geochemist of the century, we wish to dedicate this volume to the memory of V.M. Goldschmidt, on the eve of his 100th birthday. References: Bowen, N.L. (1913) Melting phenomena of the plagioclase feldspars. Amer. J. Sci. 35, 577590. Bowen, N.L. (1928) The Evolution of the Igneous Rocks. Princeton Univ. Press, Princeton, NJ, 322 p. Hardie, L.A. (1967) The gypsum-anhydrite equilibrium at one atmosphere pressure. Amer. Mineral. 52,171-200.

I.S.E. Carmichael

H.P. Eugster

Berkeley, California

Baltimore, Maryland August

iv

1987

Thermodynamic Modeling of Geological Materials: Minerals, Fluids and Melts TABLE

OF CONTENTS

Page ii

iii iii

COPYRIGHT; ADDITIONAL COPIES FOREWORD PREFACE

Chapter 1

Robert C. Newto

THERMODYNAMIC

ANALYSIS OF PHASE EQUILIBRIA

IN SIMPLE MINERAL SYSTEMS 1 5

5 5 8 9 10 11 12 16

23 27 27

28 28

lN1RODuCTIoN MgO-AlP3-Si02 PERIDOTITE MINERALS General approach Enstatite and forsterite Pyrope Spinel - a disordered phase MgTs - a fictive substance Cordierite in peridotites ALUMINUM SILICATES CALCIUM-ALUMINUM SILICATES CONTINUOUS DEHYDRATION REACTIONS -- HYDROUS CORDIERITE FERROUS IRON MINERALS SUMMARY ACKNOWLEDGMENTS REFERENCES

Chapter 2

Alexandra Navrotsk MODELS

35 37 39 42 44 46 51 51

52 60 63 66 66 67

OF CRYSTALLINE

SOLUTIONS

lN1RODuCTION SOME THERMODYNAMIC FORMALISMS THE IDEAL SOLUTION -- THE ENTROPY OF MIXING TERM REGULAR, SUBREGULAR AND GENERALIZED MIXING MODELS SYSTEMATICS IN MIXING PROPERTIES PHASES WITH DIFFERENT STRUCTURES ORDER-DISORDER IN SOLID SOLUTIONS General comments Cation interchange equilibria, especially in spinels Carbonates - calcite and dolomite structures Feldspar solid solutions CONCLUSIONS ACKNOWLEDGMENTS REFERENCES

v

Bernard J. Wood

Chapter 3 THERMODYNAMICS CONTAINING 71 78 78 79 79 80 80 80 81 81 81 83 84 84 84 84 84 85 85 85 85 85 88 90 92 93 93

OF MULTICOMPONENT SEVERAL

SOLID SOLUTIONS

COMPUTATIONOFMULTICOMPONENT, MULTIPHASEEQUILIBRIA METHODOF APPROACH TREATMENTOF SOLIDSOLUTIONS Partial molar entropy of mixing Excess free energies of mixing THE SYSTEMSAS, MAS, CAS ANDNAS Albite CaA12Si06 pyroxene and anorthite MgAl204 Spinel CMAS SYSTEM Pyroxenes Gamets FeO-Alp3-Si02 SYSTEM Fayalite Ferrosilite Almandine Hercynite COMPLEXSOLIDSOLUTIONS Olivine Gamet Plagioclase Pyroxenes CALCULATIONOF COMPLEXPHASEDIAGRAMS SUMMARY Adirondack granulites ACKNOWLEDGMENTS REFERENCES

Kenneth S. Pitzer

Chapter 4 A THERMODYNAMIC SOLUTIONS 97 98 100 100 103 105 108 109 111 112 112 112 116 117

SYSTEMS

MODEL

FOR AQUEOUS

OF LIQUID-LIKE

DENSITY

lN1RODUCTION NOTATION EXCESSGIBBSENERGY;ACTIVITYANDOSMOTICCOEFFICIENTS Basic equation Pure electrolytes Mixed electrolytes Neutral solutes Association equilibria TEMPERATUREANDPRESSUREEFFECTSON STANDARDSTATEPROPERTIES DATA BASE Standard-state values for 25°C Standard-state enthalpies, entropies, heat capacities, and volumes Pure-electrolyte parameters for 25°C Pure-electrolyte parameters for high temperatures vi

121 123 123 123 123 125 126 127 127 133 138

Mixing Parameters ApPLICATIONS

Solubilities of solids Complex ion equilibria Vapor-phase equilibria Thermal properties SUPPLEMENTARY COMMENTS ACKNOWLEDGMENTS ApPENDIX A: THEORETICAL ApPENDIX B: NUMERICAL EXPRESSIONS REFERENCES

BACKGROUND PARAMETERS

FOR TEMPERATURE

Chapter 5

John H. Wear

MODELS

OF MINERAL

SOLUBILITY

BRINES WITH ApPLICATION 143 145 148 153 155 155

160 162 166 171 171 174

DEPENDENCY

IN CONCENTRATED

TO FIELD OBSERVATIONS

lN1RODuCTION OVERVIEW OF THE MODEL MODELS FOR SYSTEMS SHOWING SPECIFIC INTERACTION

STRONG

ION PAIRS vs

ASSOCIATION:

INCLUSION OF TEMPERATURE AND PRESSURE AS VARIABLES MODELS FOR POORLY DETERMINED SYSTEMS COMPARISON OF HMW MODEL TO OTHER MODELS OVERVIEW OF THE APPLICATION OF MODELS TO NATURAL ENVIRONMENTS ApPLICATION TO PERMIAN AND MIOCENE EVAPORITES IN THE SEAWATER SYSTErAPPLICATION TO RECENT AND PRESENT DAY Ev APORA TION PROCESSES ACKNOWLEDGMENTS ApPENDIX REFERENCES

Chapter 6

Dimitri A. Sverjenskj

CALCULATION

OF THE THERMODYNAMIC

PROPERTIES

OF AQUEOUS SPECIES AND THE SOLUBILITIES OF MINERALS IN SUPERCRITICAL ELECTROLYTE 177

lN1RODUCTION

177

COMPUTATIONAL STRATEGY FOR MINERAL HYDROLYSIS CONSTANTS FOR MINERALS

181 181 182 182 186 188 188 191 195

195 195

SOLUBILITY

SOLUTIONS

CALCULATIONS

Standard molal Gibbs free energies of minerals Standard molal Gibbs free energies of gases Standard molal Gibbs free energies of aqueous species DISSOCIATION

CONSTANTS

OF AQUEOUS

SPECIES

Standard molal Gibbs free energies of aqueous complexes Standard molal entropies, heat capacities and volumes of complexes at 25 DC and 1 bar Equation of state coefficients for aqueous complexes ACTIVITY

COEFFICIENTS

OF SOLUTE

Solute species Charged species vii

SPECIES

AND ACTIVITY

OF THE SOLVENT

196 196 197 197 197 200 200 200 201 201 203 204 204

Neutral species Activity of the solvent COMPUTATIONAL APPROACH ILLUSTRATIVE EXAMPLES

Aqueous speciation of lead and chloride in supercritical chloride-bearing fluids Temperature dependence Pressure dependence Dependence on pH Dependence on HClo Solubility of galena in supercritical chloride-bearing solutions CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES

John R. Holloway

Chapter 7 IGNEOUS 211 212 212 212 213

214 214 215 215

217 217 218

219 219 219 221 221 221

223 223 223 223 225

225 225 226 229 229 229 231 231 231 232 232

FLUIDS

lN1RODUCTION PROPERTIES OF MOLECULAR SPECIES IN FLUIDS

Size and shape Attractive forces between molecules Permanent dipole-permanent dipole forces Dispersion forces Higher order permanent moments Potential energy relations Relative importance of attractive forces EQUATIONS OF STATE

Two-parameter equations Corresponding states Empirical equations Treatment of mixtures THE NATURE OF IGNEOUS FLUIDS

Properties of H20 Dissolved solutes Silica Alkali chlorides The nature of igneous fluids in the crust and upper mantle EQUILIBRIA IN C-O-H SYSTEMS

Free energy relations Methods of equilibrium calculation Minimization of total free energy Equilibrium constants and mass balance Solution for the C-O-H system Graphite undersaturatedjluid calculations Representation of results ADDITION OF OTHER COMPONENTS

Cl, F, and N Modeling fluid/melt systems IGNEOUS FLUID CALCULATIONS IN THE FuTURE ACKNOWLEDGMENTS REFERENCES

viii

George H. Brimhall

Chapter 8 ORE FLUIDS: 235

MAGMATIC

and David A. Crer

TO SUPERGENE

INTRODUCTION

Part I: The Generation of Magmas and Ore Fluids 236

PRE-METALLOGENIC

237 237 239 240 240

GENERATION OF MAGMAS AND PLUTONS AT SUBDUCTION ZONES

240 242 242 243 243

244 244 246

246 246 247 247 247 249 249 249 250

250 250 252

252 254 254

HISTORY OF MAGMAS

Oceanic zones Continental zones CLASSES OF ORE-FORMING PLUTONS

Magmatic source rocks Utility of biotite mineral chemistry Hornblende, biotite, and muscovite Classification by redox state and biotite halogen composition Classification of granitic plutons by intensive variables Oxygen fugacity HFIHp fugacity Correlation of ores with plutonic classes Physical implications of magmatic water Energy release Hydrothermal convection Lifetimes of hydrothermal systems SOURCES AND GENERAL COMPOSITIONS OF HYDROTHERMAL

SOLUTIONS

Sources of water Composition COMPOSITION OF MAGMATIC

WATER

Water solubility in silicate melts Partitioning of ore components between magmas and exsolving water Chloride and sulfur Cations and metals Magmatic to hydrothermal transition: the biotite sensor Compositions of hydrothermal biotites Early high temperature hydrothermal oxidation Relative importance of magmatic and meteoric waters Part II: Physical Chemistry of Hydrothermal Ore Fluids

255 255 255 257 257

257 257 259 259

259 260 261 261

261 261 263 263 264

SOL VENT -SOLUTE CONTROLS ON ORE SOLUTIONS

The water molecule Water structure, hydrogen bonding and polarity Dielectric constant of water Solvating power of water Coulomb's law Hydration Solvation energies Effects of temperature and pressure on the dielectric constant Temperature Pressure Effects of temperature and pressure on ionization Other effects of pressure and temperature on water-solute interactions Molecular vibration Ligand field stabilization Pressure-induced electron spin-pairing TRANSITION METAL COMPLEX IONS

Geologically important ligands ix

265 265 265 268 270 270 272 272 274 275 275 276 277 277 279 281 281

Chemical controls Hard-soft behavior Electonegativity, LFSE and ionic potential Why solubilities increase with temperature RECENT

EXAMPLES

AND APPLICATIONS

Molecular structures of complex ions Stoichiometries Stability constants Ore zoning Metal-organic complexing Activity coefficients Major salts Three main non-ideal effects Minor components in concentrated solutions Mineral solubility calculations from thermodynamic data How accurate are calculated solubilities? Estimating chemical conditions in mineral deposits Part III: Formation of Primary and Secondary Ore Deposits

283 283 284 284 285 286 287 287 287 287 289 289 290 290 292 292 294 296 296 296 297 297 298 302 306 308 308 308 308 310 310 310 310 311 311

PRIMARY

ORE DEPOSITION

Initial acidity Sulfur pH and alteration reactions pH buffer capacity Boiling Remaining deposition controls Temperature Dilution Oxygenfugacity Multi-stage mineralization and ore metal remobilization Ore metal remobilization versus introduction Relationships of wall rock alteration to mineralization Hypogene leaching Thermodynamic modeling of hypogene oxidation and sulfidation: effects of magmatic volatiles on hydrothermal fluids and proto res Destruction of wall rock buffer control: the role of biotite Feedback of chemical reaction andfluidflow: Fluid dominated threshold state and the importance of the advanced argillic alteration mineral assemblage Epithermal systems: Manifestations of deep porphyry mineralization? SECONDARY

ORE DEPOSITION

Atmosphere-dominated states Constitutive mass balance models and simplified chemical controls Residual enrichment Supergene enrichment Thermodynamic and fluid flow modeling of supergene enrichment Hypogene enrichment byferrolysis Internal factors Weathering paths in physical properties Primary permeability Available SUlfur External factors Geomorphic conditions Optimal conditions for secondary enrichment and preservation Steady state versus transient flow effects ACKNOWLEDGMENTS REFERENCES

x

Chapter 9

John M. Ferry and Lukas Baumgartn

THERMODYNAMIC

MODELS OF MOLECULAR FLUIDS

AT THE ELEVATED PRESSURES AND TEMPERATURES

OF

CRUSTAL METAMORPHISM 323 326 326 328 329 329 329 330 330 330 331 333 333 335 335 340 340 341 342 345 345 346 346 346 348 348 348 352 352 353 353 354 354 356 356 356 358 358 360 362 363 363

INTRODUCTION THERMODYNAMICS

OF FLUIDS WITH VARIABLES

V AND T

Intemal energy Entropy Helmholtz free energy Chemical potential Fugacity and fugacity coefficient EQUATIONS

OF STATE

Virial equation One-component fluids Fluid mixtures Redlich-Kwong equation One-component fluids Relationship between Redlich-Kwong and virial equations Fluid mixtures FuGACITIES

FROM EQUATIONS

OF STATE

Fugacities from virial equation of state Fugacities from Redlich-Kwong equation of state A note on the Lewis and Randall Rule EQUATIONS

OF STATE IN THE GEOCHEMICAL/PETROLOGICAL

LITERATURE

Virial equations of state, Redlich-Kwong equations of state Holloway-Flowers version Bottinga- Touret-Richet version Halbach-Chatterjee version Bowers-Helgeson version Kerrick-Jacobs version MINERAL-FLUID

EQUILIBRIA

AND EVALUATION

OF EQUATIONS

OF STATE

Basic equation for mineral-fluid equilibrium Equilibria and the thermodynamic data base Diagrams to evaluate equations of state Dolomite-quartz-talc-calcite Calcite-quartz-wollastonite Muscovite-calcite-quartz-sanidine-anorthite Zoisite-calclte-anorthite Discussion APPLICATIONS OF THE REDLICH-KWONG HIGHER-ORDER FLUID SOLUTIONS

The system C-O-H The system C-O-H-S The system C-O-H-S-N ACKNOWLEDGMENTS REFERENCES

xi

EQUATION

TO TERNARY

AND

Chapter 10

Hans P. Eugster and Lukas Baumgartner MINERAL SOLUBILITIES AND SPECIATION IN SUPER CRITICAL

367 368 369 371 371 373 375 376 376 376 377 377 377 377 379 379 379 381 381 381 383 383 385 385 387 387 389 389 391 391 391 394 397 398 398

METAMORPHIC

INTRODuCTIoN EQUATION of STATE FOR SOLUTES WATERASASOLVENT The solvent Quartz Corundum K-spar and muscovite Albite and paragonite Brucite and portlandite Magnetite DISSOCIATION CONSTANTS OF CHLORIDE COMPLEXES FROM CONDUCTIVITY DATA HCl NaCl KCl CaC12 MgC12 FeCl2 MINERAL SOLUBILITIES IN SUPERCRITICAL HzO-HCl MIXTURES Background Experimental methods METAL-CHLORIDE FREE ENERGIES AND SPECIATION KCl NaCl MgC12 CaC12 FeCl2 MnCl2 and NiC12 HzO-C02 MIXTURES AS SOLVENTS ACTIVITY COEFFICIENTS SPECIATION CALCULATIONS The methodology Speciation in the system MgO-CaO-Si02-HzO-HCl SUMMARY AND CONCLUSIONS ACKNOWLEDGMENTS REFERENCES

Chapter 11

R. G. Berman and T. H. Brown

DEVELOPMENT

OF MODELS

ANALYSIS 405 406 408

FLUIDS

FOR MULTICOMPONENT

OF SYNTHETIC

INTRODUCTION THEORETICAL CONSIDERATIONS Speciation models

xii

SYSTEMS

MELTS:

410 411 412 412 412 414 416 417 418 419 422 422 426 427 427 432 436 437 437

Stoichiometric solution models EXPERIMENTAL

CONSTRAINTS

Thermodynamic properties of melts Glass-liquid relationships Heat capacity of glasses Heat capacity of liquids Volumetric properties of liquids Enthalpy and entropy offormation and fusion Mixing properties of liquids Thermodynamic properties of minerals METHODOLOGY

Calibration of thermodynamic models Testing of calibrations APPLICATIONS

Speciation models Stoichiometric models CONCLUSIONS ACKNOWLEDGMENTS REFERENCES

Chapter 12

Mark S. Ghiorso

MODELING MAGMATIC SYSTEMS: 443 443 445 448 451 452 454 459 460 460 461 462 463

RELATIONS

INTRODUCTION

General constraints on the formulation of melt models Review of models that satisfy the thermodynamic requirements CALIBRATION CALCULATING

METHODS SOLID-LIQUID

EQUILIBRIA

Notation and mathematical statement of the problem An algorithm for finding the minimum of G Modeling irreversible reactions GENERALIZED

THERMODYNAMIC

POTENTIALS

Legendre transformations Minimization of generalized thermodynamic potentials SUMMARY REFERENCES

Chapter 13

Mark S. Ghiorso and Ian S.E. Carmichael

MODELING MAGMATIC SYSTEMS: 467 467 473 473 476 478 486 490 491 495 495 497

THERMODYNAMIC

PETROLOGIC ApPLICATIONS

INTRODUCTION THERMODYNAMIC CLASSIFICATION OF IGNEOUS ROCKS ACTIVITY OF SILICA AND DEPTH OF ORIGIN OF MAFIC MAGMAS

Silica activity and magmas Silica activity of basic magmas and a petrogenetic grid FRACTIONAL CRYSTALLIZATION OF OLIVINE THOLEIITIC ASSIMILATION OF PELITE INTO THOLEIITIC MAGMA ISOCHORIC CRYSTALLIZATION ISOBARIC VESICULATION SUMMARY APPENDIX REFERENCES

xiii

MAGMA

Chapter 1

Robert C. Newton

THERMODYNAMIC

ANALYSIS OF PHASE EQUILIBRIA

IN SIMPLE MINERAL

SYSTEMS

INTRODUCTION Knowledge of the thermodynamic properties of minerals has become of great importance in prediction of the physical and chemical conditions of the formation of mineral assemblages and as a guide to the synthesis of minerals. The three major sources of information are thermochemical and thermophysical measurements, derivation from experimental phase equilibrium data at elevated temperatures and pressures, and attempted reproduction of the expected temperatures and pressures of natural assemblages, under the assumption of "frozen-in" equilibrium (method of paragenetic analysis). All of these methods have made important contributions to the body of thermodynamic data, and comparison of results from the three independent sources is valuable in cross-checking, systemization and extending of the data sets. The most important properties of minerals for stability calculations are heat capacity, entropy, enthalpy of formation at one bar and molar volume as a function of temperature and pressure. This information is needed for end-member substances and for solid solutions. The volume properties are relatively easily and accurately measured by X-ray diffraction, but the thermal properties are much harder to evaluate accurately for complex substances. The most direct information is supplied by calorimetry, both thermophysical (heat capacity measurements) and thermochemical (enthalpy of solution measurements). Less direct but increasingly important sources of information are spectroscopy, such as infrared absorption, especially at low temperatures (Kieffer, 1981), electrochemical measurements, as in the solid electrolyte cell (Sato, 1971), and vapor pressure measurement, as with the high-temperature Knudsen cell (Rammansee and Fraser, 1982). The most extensively-used source of thermodynamic data of minerals remains derivation from experimental phase equilibrium diagrams, although the thermochemical and thermophysical methods approach the phase equilibrium method in comprehensiveness and potential accuracy. The precision of enthalpy of solution work is not quite at a level which allows generally accurate calculation of phase equilibrium. Nevertheless, constraints provided by thermochemical measurements are very important in initializing prase equilibrium calculations, and, in turn, phase equilibrium data are of utmost importance in optimizing thermodynamic parameters within the uncertainty limits of calorimetry, and in discriminating between conflicting sets of thermochemical measurements. The simplest kind of experimental equilibrium for deriving thermodynamic data is the univariant P-T curve of a reaction among pure substances. An example is the decarbonation of magnesite (MgC03) to periclase (MgO): MgC03

= MgO + CO2.

(A)

The general equation of equilibrium for simple devolatilization

is ( 1)

where dG is the free energy change, d V S the difference in volume between the solid products and reactants, n is the coefficient of the volatile species in the balanced equation, fv is the fugacity of the volatile species, in this case CO2, and R is the gas constant (8.3144 J/K). Experiments locating the P-T curve of Reaction (A) define, with the aid of equation (1), the most important property of magnesite, its free energy of formation 1

from the oxides (dGt) at one bar pressure by: dGt

(magnesite)

=

-dGA -,

(2)

The free energy of reaction may be expressed further as: (3) dW and dSo are, respectively, the changes in standard enthalpy and entropy of the reaction, likewise expressible as differences in standard enthalpies and entropies of formation of the products and reactants. The standard enthalpy of a substance at any temperature is related to that at any reference temperature, usually taken as 298.15 K, by integration of the heat capacity, Cp: H(T) - H(298.15) = fT298.1SCpdT.

(4)

Heat capacities above 298 K can usually be expressed as polynomial functions of temperature. A form convenient for representation of measured data is (Robie et aI., 1978) : Cp

a + bT + cIT2 + d/.yT + eT2.

=

(5)

Truncated forms lacking some of the terms are commonly used. The standard entropy of a substance may be expressed as a Third Law entropy plus a configurational term for those substances with possibility of significant residual atomic disorder at zero Kelvin, such as Al,Si disorder of feldspar: S(T)

= fo T(Cp/T)dT

+ S (conf.).

(6 )

The entropy of disorder of a mineral can sometimes be characterized by some physical measurement, such as X-ray diffraction, as ' S (conf.)

=

-RIXi

In Xi

(7)

for each atomic site, where Xi are the molar fractions of the atoms mixing on a site. The free energies of compression, -RT In fv, of H20 and CO2 are well measured in the ranges 0-1300 K and 1-10000 bars (Burnham et aI., 1969; Shmulovich and Shmonov, 1978). The fugacities of other volatiles, and of volatile mixtures, are much less well known, but can be approximated from theoretical considerations, such as the hard-sphere Modified Redlich-Kwong (MRK) equation of state (HOlloway, 1977; Kerrick and Jacobs, 1981). Extrapolation beyond measured pressure ranges by the MRK procedure may not be trustworthy (Haselton et aI., 1978). Very accurate work at elevated temperatures and pressures above a few kbar requires use of expansivities, a, and compressibilities, B, of the solids: V(T,P)

=

V(298.15)[l

+ a(T-298)

- BPI.

(8)

Since a is, in general, a function of pressure, and B of temperature, and since few high temperature and high pressure measurements of these quantities exist, some authors ignore a and B in derivation of thermodynamic data from phase equilibria (Connolly and Kerrick, 1984; Berman et aI., 1985). In a few propitious reactions, such as magnesite decarbonation, thermodynamic parameters of minerals may be determined accurately from one or a few experimental brackets of univariant equilibria. Where several or numerous experimental brackets of a 2

single equilibrium are available, various statistical treatments may be applied, and where a number of experimental brackets of several equilibria involving minerals in common are available, multiple regression (Gasparik and Newton, 1984) or Monte Carlo (Perkins et aI., 1981) methods may be used. In the latter method, large numbers of trial parameters are inserted into the thermodynamic equations, and the combination of parameters yielding minimal total deviation from experimental points or bracket midpoints is sought. In the method of linear parametric analysis ("linear programming"), as developed by Gordon (1973), allowable ranges of some parameters, such as standard entropy and enthalpy changes of a reaction, are defined by the composite restriction of experimental points. The most important thermodynamic parameter to be derived by analysis of phase equilibria is the enthalpy of formation, dWf 298' from the oxides or elements, because data obtainable from physicochemical measurements, principally solution calorimetry, are usually not precise enough for accurate calculations. Heat capacities are often fixed by thermophysical data, though they may sometimes be optimized within small ranges (Halbach and Chatterjee, 1984). Standard entropies are sometimes fixed by precise low-temperature heat capacity measurements, but are often uncertain to the extent of a configurational entropy. A practical problem is that samples too large to be synthesized by ordinary methods are often required for adiabatic calorimetry. Measurements on near-end-member natural samples require uncertain corrections for impurities. For these reasons standard entropies may be derived from phase equilibria even where heat capacity measurements exist. Molar volumes are almost always fixed by X-ray diffraction measurements. Thermodynamic analysis of univariant equilibria in oxide-silicate systems is conveniently referred to a generalization of expressions (1), (3), (4) and (6) (cf. Day and Kumin, 1980): L'lCp P dHof , S , 298 - Tdsof , S, 298 - fT298dTfT298 (9) Tf,s dT + f1 dVsdP + nRT In fv =

0 (a)

> 0 (b)

< 0 (c) dWf ~ 298 and SOf S.298 are, respectively, the differences in enthalpy and entropy of formation from the dxioes of the solid products and reactants: dH\s,298

= InidWf,298,i

with a similar coefficients.

expression

(solid products) - ImidWf,298,i for dsof

S 298' ' ,

where

nand

(solid

reactants),

(10)

m are the stoichiometric

At (T,P) points within the field of stability of the condensed (left-hand) assemblage, (9b) applies, and for the (T,P) region where devolatilization occurs, (sc) applies. Regression methods to obtain dHof S 298 and dsof S 298 with (9a) commonly use midpoints of isothermal or isobaric expehmental brackets or points on empirical curves drawn through the brackets. In this method, if dWf S 298 and dsoi$ 298 are best-fit quantities to all of the brackets, then, at a bracket mid-point (Ti,Pi): Equation (9a) becomes (11) where c)i is the free energy deviation of the bracket midpoint from the best-fit curve. Optimal dH\S,298 and dS\S,298 are found by minimizing the sum of the squared 3

deviations:

a(l:5r) (12)

0;

These expressions lead to two equations in the two unknowns dWf,s,298 and dS\s,298: pdH\s,298

- (ITi)dS\s,298

+ (Iki)

=

0, (13)

where p is the number of brackets. Methods using the inequalities (9b) and (9c), principally linear parametric analysis, make use of individual experimental data points. Examples of the use of these methods are given below. Since, in general, there can be at most j - 1 independent reactions among j phases, the number of conditions (10) will not be sufficient to determine all of the d H 298 quantities, one or more of which must be evaluated by other means, such as sotutlon calorimetry, if superior measurements exist. These "anchor phases" (Holland and Powell, 1985) are used to derive the less well known or unknown properties-of the other phases. Alternatively, a set of parameters from thermochemical measurements is used as initial input in multiple regression (Halbach and Chatterjee, 1984). The parameters are allowed to vary within the stated limits of the thermochemical measurements, and the set yielding the smallest composite residuals from experimental data is found. 0

Free energy, enthalpy and entropy of mixing in solid solutions are important in many calculations of mineral equilibria. These quantities are especially hard to define accurately by thermochemical and thermophysical methods, and are often derived from experimental phase equilibrium data. Discussion of solid solutions is deferred, for the most part, to Chapters 2 and 3 by A. Navrotsky and B.J. Wood. Serious problems with retrieval of thermodynamic parameters from phase equilibria have become apparent, especially where solid solutions are involved and where ordinary regression techniques are applied without regard to constraints from thermochemical measurements (see discussion by Cohen, 1986). Three major sources of error are: (a) too many parameters regressed from too few data points, (b) high correlation of derived parameters, as in the simultaneous retrieval of Fe,Mg mixing parameters of coexisting garnet and olivine (O'Neill and Wood, 1979), and (c) insufficient temperature, pressure, or composition baselines of experimental data. Temperature dependences of standard and solid solution free energies have characteristically been overestimated from the last source of error. Rigorous definition of uncertainties in derived thermodynamic properties is very difficult, or sometimes virtually impossible; the cumulative uncertainties in experimental observations, input thermochemical data, unknown validity of assumptions and approximations such as ideal solution behavior, uncertainties in volatile fugacities, especially where extended by theoretical equations of state, and neglect of solubility of the solids in the vapor, as in "pure water" calculations, tend to discourage rigorous treatments. Somewhat arbitrary criteria may be adopted, such as the extreme slopes of regression lines which can be drawn through plotted data points or the mean deviations of derived parameters from thermochemically measured ones (Day and Kumin, 1980; Halbach and Chatterjee, 1984). Discussions of error analysis in derivation of thermodynamic properties are given in Bird and Anderson (1973), Anderson (1977), Zen (1977) and Demarest and Haselton (1981). The last paper demonstrates that, for isothermal or isobaric pairs of runs

4

which bracket an equilibrium curve, if the half-width of the bracket is greater than the experimental temperature or pressure uncertainties of the runs, which is usually the case, the brackets are not subject to Gaussian error distribution and, hence, cannot be used in rigorous error propagation. Another problem is that error analysis based on individual brackets undoubtedly overestimates uncertainties in reaction parameters. The net constraint of many experimental brackets of a single univariant curve is often very stringent: a practically unique line may sometimes be demanded by a set of experimental brackets covering large P-T ranges, subject to the additional constraint of constant sign of curvature (no inflections). Thus, the composite constraint exacted by a number of brackets must considerably reduce the uncertainty in the free energy of the reaction. To the author's knowledge, no rigorous treatment of the multiple-bracket effect is available in the literature. For these reasons, the present paper, following the majority of authors (cf. Berman et aI., 1986), does not attempt definition of standard errors of derived parameters, but concentrates instead on internal consistency and optimization. The purpose of this chapter thermodynamic properties of some to show the correspondence between of the strengths and limitations of methods.

is to illustrate some common rock-forming measured and derived the thermochemical,

of the methods of deriving minerals, principally silicates, quantities, and to indicate some experimental and paragenetic

MgO-AI203-Si02 PERIDOTITE MINERALS General approach A comprehensive thermodynamic dataset is most conveniently built up by considering simple subsystems, such as MgO-AI203-Si02, for which there are many univariant equilibria with good experimental coverage. In this way complicating factors of solid solution are kept to a minimum. Standard properties of minerals may be derived by a step-wise process, in which the least ambiguous and most reliable data are selected to define the properties of a few minerals, and these properties in turn define those of other minerals through more complex reactions (Powell, 1978; Helgeson et aI., 1978). Alternatively the thermodynamic properties of a mineral subset are derived simultaneously by regression of experimental data available for all reactions in the subset (Holland and Powell, 1985). The former method has the advantage that the bestdetermined mineral properties serve as a foundation, whereas the latter method tends to distribute the uncertainties more broadly. The latter method has the advantage that it does not run the risk of biasing the entire dataset through misjudgment of the most reliable experimental data. The step-wise approach is illustrated in the Si02-undersaturated portion of the system MgO-AI203-Si02. This model peridotite system includes the phases forsterite, enstatite, pyrope, spinel and Mg-cordierite. Some of the most rigorously constraining data consist of single brackets, which favors the step-wise approach. Enstatjte and forsterjte A convenient starting point is the reactions of enstatite and forsterite with magnesite. Several workers have calculated the free energy of formation of magnesite from the reversed experimental data of Harker and Tuttle (1955) on Reaction (A), using Equation (1). The most comprehensive analysis is that of Robinson et al. (1982), which is accepted here. The input parameters d V sand fv are very accurately known at the temperatures (600°-900°C) and pressures (100-2800 bars) of the equilibrium and contribute almost negligible uncertainties. The uncertainty from all sources in dG ° f (magnesite) is of the order of 350 J. From dGof (magnesite), the corresponding quantities for enstatite and forsterite are obtained from the reactions:

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