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Phase equilibria in the system MgO-MgF2-SiO2-H2O Duffy, Clarence John 1977

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PHASE EQUILIBRIA IN THE SYSTEM MgO-MgF 2-Si0 2-H 20 by CLARENCE JOHN DUFFY B.Sc. Massachusetts I n s t i t u t e of Technology, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of G e o l o g i c a l Sciences We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1977 © Clarence John Duffy, 1977 In present ing th is thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 3 / < ^ . ^ J /?77 ABSTRACT The proportions of F and OH i n synthetic t a l c , clinohumite, chondro-d i t e , norbergite, b r u c i t e , and s e l l a i t e have been determined by X-ray mea-surement of interplanar spacings. Unit c e l l refinement has been c a r r i e d out on intermediate s e l l a i t e , approximate composition MgOHF. The composi-tions of coexisting phases have been determined i n fo r t y hydrothermal experiments y i e l d i n g information on eighteen d i f f e r e n t chemical e q u i l i b r i a . These data, combined with phase equilibrium and c a l o r i m e t r i c data from the l i t e r a t u r e , have been treated by the method of l e a s t squares to produce a thermodynamic model of the system. Computed e q u i l i b r i a based on the model are i n good agreement with data on natural assemblages. i i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES iv LIST OF FIGURES v ACKNOWLEDGEMENTS vi INTRODUCTION 1 SYMBOLS AND UNITS 1 EXPERIMENTAL METHODS Apparatus 2 Starting Materials 3 Fluorine Buffers 3 SOLID PHASES Synthesis and Identification 4 Characterization and Compositional Variation of Phases 5 Intermediate Sellaite, MgOHF 6 EXPERIMENTAL PHASE EQUILIBRIUM RESULTS " 7 THE THERMODYNAMIC MODEL Introduction 8 Formulation of the Least Squares Problem 10 Solution of the Least Squares Problem 18 Testing of the Alternative Models 22 APPLICATIONS 27 CONCLUSIONS 31 SELECTED REFERENCES 66 APPENDIX 70 i v LIST OF TABLES I. Symbols and Units 44 II . Compounds Considered 46 II I . Thermodynamic Data for the Fluorine Buffer 47 IV. Unit C e l l Parameters 48 V. V a r i a t i o n of Interplanar Spacings with Composition 50 VI. V a r i a t i o n of Molar Volumes with Composition 51 VII. Experimental Results 52 VIII. E q u i l i b r i a that Simultaneously Constrain the Thermodynamic Model 54 IX. Thermodynamic Data Evaluated P r i o r to Solution of the Least Squares Problem 55 X. Constraining Phase E q u i l i b r i a Data from other Sources 57 XI. Equations Constraining Single Model Parameters 58 XII. Comparison of Experimental Data with Thermodynamic Models 59 XIII. Thermodynamic Properties Determined by Least Squares (Preferred Model) 62 XIV. Matrix of C o r r e l a t i o n C o e f f i c i e n t s f o r Properties . i n Table XIII. 63 V LIST OF FIGURES 1. The system MgO-MgF2-Si02-H20 32 2. Comparison of the models with experimental data 34 3. Comparison of the models with experimental data 36 4. T - x section of MgO-MgF2-H20 and 2«ln(f If ) versus mole fraction fluoro-endmember 38 n2U rlr 5. Displacement of equilibria 40 6. Coexisting chondrodite or clinohumite and phlogopite 42 v i ACKNOWLEDGEMENTS This work was supported through National Research Council of Canada Grant A-4222 held by Dr. H.J. Greenwood and by a U.S. National Science Foundation Graduate Fellowship, .1969-1971. I am indebted to Dr. H.J. Greenwood for his excellent technical advice, but even more for his patience during periods of slow progress and for his constant enthusiasm which made this rather lengthy study a pleasure through-out. This thesis has benefited from my contact with many of the faculty, students, and staff of the Department of Geological Sciences, but particular mention should be made of Krista Scott, who assisted with the Weissenburg work on intermediate sel l a i t e , and of Ian Duncan, whose extensive knowledge of the geologic literature was of so much benefit. 1 INTRODUCTION The system Mg0-MgF2-Si02-H20 i s in many respects an ideal system for experimental study. High purity starting materials are readily available. There are no oxidation-reduction problems. Phase equilibrium and cal o r i -metric data are abundant on important subsystems. Synthesis experiments have been carried out on the solid solution phases (Van Valkenburg,'. 1955, 1961: Crane and Ehlers, 1969). A basis for theoretical treatment of such systems has been given by Thompson (1967) and Muan (1967). This study combines new experimental data on multiphase equilibria in the system with data previously available to form an overdetermined system of equations which describe experimentally observed chemical equilibria. The least squares solution of these equations i s presented as a model for chemical equilibrium in the system. Within limits, this describes the system under a wide range of conditions. Naturally occurring equilibria that can be represented by the model have received l i t t l e attention in the literature. This is probably due more to the d i f f i c u l t y of obtaining fluorine analyses than to the scarcity of suitable bulk compositions. What information does exist agrees well with the model. SYMBOLS AND UNITS The symbols used in this paper follow as closely as practicable the usage recommended by McGlashan (1970). Additional symbols are those in common use in the geologic literature. A l i s t of symbols and correspond-2 ing units is given in Table I. Table II l i s t s the symbols and formula units used for the various chemical compounds. EXPERIMENTAL METHODS Apparatus Experiments were conducted by enclosing reagent grade chemicals in sealed noble metal capsules and placing them in S t e l l i t e 25 or Rene 41 cold seal pressure vessels (Tuttle, 1949) and nichrome wound cylindrical fur-naces. Temperature was controlled by a fully proportional controller with a platinum resistance sensing element. Temperatures were constantly moni-tored on a Texas Instruments twenty-four point Multiwriter recorder, and were measured at intervals of one to three days on a Leeds and Northrup K-3 potentiometer. Steel sheathed Chromel-Alumel thermocouples were used for a l l temperature measurements.. A standard thermocouple was calibrated against the melting points of NaCl (1073.5 K) and CsCl (919 K). Working thermocouples were calibrated against the standard with the pressure vessel in a position of minimum gradient i n the furnace. Calibration is considered accurate, to ±2 K. Temperature variation over the bottom three centimeters of the pressure vessels was found to be less than 2 K. Pressures were measured on Ashcroft Maxisafe bourdon tube gauges c a l i -brated against a Heise Bourdon Tube gauge having a range of 0-7000 bar and an error of less than' 7 bar.. Individual pressure measurements are consid-ered to be accurate to ±20 bar. 3 Starting Materials The primary starting materials were Mallinckrodt Reagent MgF2 lot MRC, Baker and Adamson Reagent MgO lot W196, and Fisher Certified Reagent S i l i c i c Acid lot 730944. Before mixing, the MgF2 was fired at 673 K for one hour. MgO was fired for twenty-four hours at 1300 K. Both the MgF2 and MgO were allowed to cool in a desiccator with anhydrous CaSOit as the desiccant. The quantities of both materials required for the various mixes were then weigh-ed out immediately. The s i l i c i c acid was fired at 1500 K for twenty-four hours to produce cristobalite. This was then crushed in an agate mortar and passed through a 100 mesh screen and stored in an oven at 423 K unt i l weigh-ed. Bulk compositions made up from these starting materials were ground in d i s t i l l e d water for two hours in an agate mortar, and dried under a heat lamp before being loaded into run capsules. Attempts to use acetone as a grinding medium resulted in contamination of the mixes with a waxy residue which resulted in the generation of C0 2 during the experiments and in some cases in the crystallization of magnesite or graphite. Fluorine Buffers The fluorine buffers of Munoz and Eugster (1969) were examined for pos-sible application to this system. It was found that buffers involving quartz could be used to buffer talc, but that other minerals in the system reacted with quartz in the buffers. The graphite bearing buffers were gen-erally found unsatisfactory because their use resulted in the cry s t a l l i z a -tion of carbonates; or, as in the case of the humites, the approach to equilibrium was extremely slow. 4 The (graphite-methane)-(wollastonite-fluorite-quartz) buffer was suc-c e s s f u l l y used to buffer the F/OH r a t i o of t a l c . The f l u i d composition co-e x i s t i n g with t h i s buffer was calculated as outlined by Munoz and Eugster (1969). Fugacity c o e f f i c i e n t s f o r H 2 were ; taken from Shaw and Wones (1964), and for H2O from Burnham, Holloway, and Davis (1969). The remainder were obtained by use of the Redlich-Kwong equation as described by Edminster (1968). C r i t i c a l constants for methane were taken from American Society for Testing and Materials committee D-2 on petroleum products and lu b r i c a n t s and American Petroleum I n s t i t u t e research project 44 on hydrocarbons and re l a t e d compounds (1971). The remainder of the c r i t i c a l constants were taken from Mathews (1972). The remaining thermodynamic data used i n the buffer c a l c u l a t i o n s are l i s t e d , along with t h e i r sources, i n Table I I I . This data base d i f f e r s s i g n i f i c a n t l y from that used by Munoz and Ludington (1974) and Munoz and Eugster (1969) with respect to the enthalpy of formation of f l u o r i t e from the elements,. The value given by S t u l l and Prophet (1971) was adopted here over that of Robie and Waldbaum (1968) used by the above authors due to i t s larger data base and better agreement with the experimental work of Bratland et a l (1970). SOLID PHASES Synthesis and I d e n t i f i c a t i o n Synthesis of the pure fluoro-humite endmembers was achieved at approx-imately one atmosphere i n sealed platinum capsules. Attempts to synthesize such materials hydrothermally resulted i n hydroxyl bearing humites plus a d d i t i o n a l phases. A l l other phases were c r y s t a l l i z e d hydrothermally i n 5 sealed gold capsules. Solid phases were identified primarily by means of powder X-ray d i f -fraction. Standard patterns produced from single phase synthesis products were particularly useful in the analysis of products consisting of mixtures of humites or of humites and forsterite, since the patterns for these min-erals are quite similar. Optical o i l immersion methods were used as a secondary means of min-eral identification, but positive optical distinction between the various humite minerals is extremely d i f f i c u l t in fine grained run products. Op-t i c a l techniques were used mainly to inspect for small amounts of extrane-ous phases in :the run products. Characterization and Compositional Variation of Phases Cell dimensions were determined for a l l phases with variable fluorine-hydroxyl content except talc for which only dggn was determined. X-ray patterns of the materials used for the c e l l dimension determinations showed only peaks for the material in question. Optical examination in no case revealed more than 2% extraneous material. It should, however, be borne;'.in mind that a mixture of two humites could not reliably be detected optically. X-ray patterns were made at a scan speed of \ degree minute Silicon metal (ao=5.4305 ft) was used as the internal standard. Unit c e l l parameters were refined using the program of Evans, Appleman, and Handwerker (1963). Expressions for unit c e l l parameter as a function of composition were then derived on the basis of a least squares approximation. The coefficients for these equations are presented in Table IV. The data upon which they are based are presented in the appendix. Table IV also shows some endmember unit c e l l parameters calculated from these relations and published unit c e l l 6 parameters for comparison. Unit c e l l parameters of talc were not determined due to overlap of the various peaks in the powder diffraction pattern. Compositions of the phases in multiphase experiments were determined from the dependence on composition of interplanar spacing. These depend-encies are based on measured d-spacings in the case of talc and on calcu-lated d-spacings based on unit c e l l refinements for the other phases. Coefficients for these equations appear in Table V. Measurement of single peak positions was done by osci l l a t i o n between either a si l i c o n (aQ=5.4305 X) or a spinel (aQ=8.0833 X) standard peak and the peak in question. Four measurements of the peak position were made. Scan speed was \ degree minute Nickel f i l t e r e d Cu radiation was used. Least squares expressions for molar volume as a function of composition have been computed for the phases of variable fluorine hydroxyl content with the exception of talc. With the exception of brucite, a l l the volume func-tions are linear in composition. For sel l a i t e and the humites the d i f f e r -3 ence in volume between the hydroxyl and fluoro-endmembers is 2.9510.19 cm mol ^ of (OH)2. The volume of hydroxyl talc has been taken from Robie, Bethke, and Beardsly (1967) and that of fluoro-talc has been estimated as this value minus 2.95. The volume data are presented in Table VI. Intermediate Sellaite, MgOHF Crane and Ehlers (1969) synthesized a phase which they f e l t to be sto i -chiometric with a composition of MgOHF. The existence of this phase has been substantiated and i t s unit c e l l determined. Peaks were indexed on an ortho-rhombic unit c e l l from single crystal Weissenberg radiographs. Final c e l l refinements were done as described for the solid solution phases in the sys-tem. It is suggested that this compound has a structure similar to that of 7 sella i t e with ordering of the fluorine and hydroxyl resulting in a doubling of the a-dimension of the unit c e l l . It w i l l therefore be referred to here as intermediate s e l l a i t e . Cell dimensions for intermediate sel l a i t e coex-isting with sel l a i t e and with periclase, and those based on a refinement of the data of Crane and Ehlers (1969) are given in Table IV. The inference that s e l l a i t e and intermediate sellaite are both members of a single structurally continuous solid solution series i s based upon measurememt of c e l l parameters. Upon substitution of hydroxyl into the sel l a i t e structure, the symmetry changes from tetragonal to a lower sym-metry suggesting ordering of fluorine and hydroxyl. Cell parameters for sella i t e (x =0.50) are given in Table IV for comparison with those of intermediate s e l l a i t e . This evidence leads to description of se l l a i t e and intermediate se l l a -ite by the same set of functions in the thermodynamic modelling which f o l -lows . For this reason special note should be made of the volume function for the sellaites. The function given in Table VI is based only on the data for s e l l a i t e . The volume data for intermediate s e l l a i t e has not been in-cluded because the derivative of the volume with respect to composition is not closely constrained by the data at the intermediate s e l l a i t e composi-tion. The pressure effect upon equilibria involving intermediate s e l l a i t e is therefore somewhat uncertain. EXPERIMENTAL PHASE EQUILIBRIUM RESULTS Results of the experiments involving two or more phases for which the compositions have been determined are summarized in Table VII. With the exception of the data involving brucite, few of the equilibria depicted by 8 these data are closely bracketed. This fact combined with the generally long run times needed before apparent equilibrium was achieved creates some uncertainty as to the closeness of approach to equilibrium. There are also indications that metastability may be a problem. In particular experiment Chl00-C140-1R produced clinohumite considerably more fluorine rich than was fe l t to be truly stable. The experimental evidence does not resolve whether or not there exist stable talc-clinohumite t i e lines at 1023 K and 2000 bar. Experimental attempts to resolve this question were unsuccessful. The requirement that a l l the equilibrium data and a l l the thermodymanic data be internally consistent is a stringent one that can be tested in the form of a number of simultaneous equations. In order to analyze the experi-mental data and to provide a basis for calculating other equilibria in the system and to calculate equilibria at conditions other than those of the ex-periments, a thermodynamic model of the system has been constructed based on such a system of equations. THE THERMODYNAMIC MODEL Introduction The experimentally determined phase equilibria represent a large number of constraints upon the thermodynamic properties of the system. What f o l -lows i s the development of a model that is consistent both with these data and with other phase equilibria and calorimetric data. Some parameters are not closely constrained by the phase equilibrium data. These have been evaluated from existing thermochemical data. The remaining parameters have been determined by means of a linear least squares solution of a system of equations defined by the available phase equilibrium and thermochemical data. 9 The model is referenced to the components of the system, MgO, MgF2, Si0 2, and H20, at 1023 K and 1 bar. The temperature of 1023 K has been chosen because much of the experimental evidence has been collected near this temperature. This choice simplifies the form of some of the equations to be used. Choice of the components as a basis for the model rather than the elements avoids the unnecessary introduction of the uncertainties in the various thermochemical quantities of formation of the components from the elements. The treatment that follows evaluates thirty-three unknown thermodynamic parameters involved in the description and calculation of equilibria in this system. At a minimum, thirty-three equations would be needed to solve for these parameters. A l l of the models tested involve an array of more than seventy equations, so that there are many more equations than unknowns. Since a l l involve measured quantities, none w i l l be exact. The best approx-imation to simultaneously satisfy a l l of the equations has been sought by the method of least squares. This assumes that a l l of the measurements are normally distributed about the equilibrium values. This assumption w i l l only be true i f a l l the assemblages measured have reached equilibrium so that the only errors are those of measurement. Errors caused by failure of the experiments to reach equilibrium are d i f f i c u l t to treat since their error distributions w i l l be asymmetrical. The asymmetric nature of the error distributions arises because the experiments approach equilibrium from a particular direction and cannot pass through the equilibrium value. If only one equilibrium was studied and i t was only approached from one d i -rection, treatment by the method of least squares might well result in an undetectable bias in the results. 10 The data treated in this problem represent a large number of different equilibria which have been approached from different starting compositions. While lack of equilibrium may create problems i n weighting the data, the fact that a l l the data can be described by a single model i s strong evidence that a l l experiments considered have closely approached equilibrium. Pre-liminary experiments of short duration cannot be represented by any inter-nally consistent model and have been omitted both from Table VII and the thermodynamic analysis. Formulation of the Least Squares Problem Table VIII l i s t s eighteen chemical equilibria that have been used to constrain the thermochemical quantities for the phases in this system. These equations can be s p l i t into several classes, each having an equation of a general form. There follows a derivation of each of these general equations and remarks on any simplifying assumptions that have been made. Equilibria [1]. to [6] represent fluorine hydroxyl exchange between two solid solution phases, C and D. In equilibrium composition space these equilibria are represented by tie lines between coexisting solid solution phases. The basic equation describing such equilibria i s given by [ 1 9 ] H^C + yFD = yHD + yFC * From equation [19] i t follows that [20] A G H C + AGPD = AGHD + AG^  . It should be noted that these AG 'S refer to the species in the solid solu-.. tion and not to pure endmembers. A G may be reduced to a more tractable form through use of an activity term such that 11 [21] AG = A G * + RTlna . Referring to a standard state of 1023 K and 1 bar [22] AG = A G ° + AV°(P - 1) - AS°(T - 1023) + RTlna . This equation is an approximation assuming A V and AS" to be constant over the range of pressure and temperature considered. The approximation is use-ful because variation AV' over the pressure and temperature range of interest is small and variation for the several species involved in a given e q u i l i -bria may be expected to cancel each other to a large degree. The approxima-T tion that AS = AS° is adequate because a l l of the data for these equilibria were collected within five K of 1023 K. Combining equations [20] and [22] results i n [23] AG° + A f ° ( P - 1) - ASj(r - 1023) + RTln{ ( a ^ a ^ ) / ( a ^ a ^ ) } = 0 . Equilibrium [7] is similar to equilibria [1] through [6] except that i t re-presents an exchange equilibrium between a solid solution and the vapor phase. For equilibrium [7] [ 2 4 ] WHC + 2u R F = u F C + 2u H 2 0.. Following a development parallel to that leading to equation [23], and not-ing that for volatile species [25] AG = AG° - A S ° ( T - 1023) + RTln(fT'P/fT>1) T 1 and making the assumption that f ' =1, equation [24] becomes W A GR " Solids,R< P " X> " A S R ( T " 1 0 2 3> + K ^ F C 7 ^ + 2RTln(£ H 2 0/f H F) = 0 . Note that equation [26] contains the same approximations regarding constant A ^solids a n C^ ^ S a S e c l u a t : ' - o n [23]. Equation [26] involves two volatile spe-cies so that the assumption of constant A S is somewhat less satisfactory in equation [26], but over five K the error i s s t i l l t r i v i a l . Equilibria [8] to [16] may be written in the form [ 2 7 ] VHC H C + VHD H D + VHE H E + VH 20 H2° = ° ' From this i t follows that t 2 8 ] VHCyHC + V H D + W H E + VH 2O yH 20 = °. or [ 2 9 ] VHC A GHC + VHDAGHD + VHE A GHE + V H 2 0 A % 0 = ° ' In equilibria [8] through [12] a l l data are within five K of 1023 K, but for equilibria [13] through [16] data are available at substantially different temperatures. Consequently i t is necessary to include the heat capacity . terms for reactants and products. Thus for solids rT rT [30] AG = AG° + AV°(P - 1) - AS°(T -1023) + A C d T - T ( A c / r)dT 1023 1023 + RTlna .. For H20 [31] ^ - m n C ^ / ^ J ) - . The rather simple form of equation [31] is due to H20 being one of the re-ft ference compounds for the system. For this reason A G U = 0 by definition. H2u T 1 Combining equations [29], [30], and [31] and assuming f T ' = 1 gives H 2 U [ 3 2 ] A GR + A y s o l i d s , R ( P " 1 } " A S R ( r " 1 0 2 3 ) + 13 (T • rT AC^dT - T (AC„lT)dT K ! J K 1 0 2 3 1 0 2 3 + R r ( VHC l n aHC + VHDlnaHD + VHE l n aHE + VH 20 l n fH 20 ) = ° ' For equilibria [8] through [12], an adequate approximation to equation [32] may'be obtained by deleting the terms containing heat capacities. Equilibria [17] and [18], like [1] through [6] are exchange equilibria between solids. The identical compositions of HB and HS, and of FB and FS, require that two equations be used for each equilibrium. These are [ 3 3 ] yHC = H^D and C 3 4 ] yFC = yFD * Noting that equation [33] implies C35] A GHC " A GHD and combining this with [30] leads to [36] AG^ + AV°(P - 1) - A S ° ( T - 1023) + + RTln(a H D/a H C) = 0 A c ^ d r - T\ ( A c _ , / r ) d r K J R 1 0 2 3 1 0 2 3 Expanding equation [34] gives an equation equivalent to [36], but for the fluoro-endmembers. For both equations the assumption of constant AV was made. In the above general equations both heat capacity and activity are functions of the measurable quantities temperature, pressure, and composi-tion. Before the least squares problem can be solved i t is necessary to 14 adopt fu n c t i o n a l forms to approximate these q u a n t i t i e s . A su i t a b l e expres-s i o n f or heat capacity i s the function [37] C = a + br + c/T2 proposed by Maier and Kelley (1932). The a c t i v i t y term has been treated as outlined by Thompson (1967). Only d e t a i l s s p e c i f i c to t h i s study w i l l be given here. I t i s i n t e r e s t i n g to note that although the treatment used here i s s t r i c t l y empirical, the regular symmetric and asymmetric solutions (Thompson, 1967) are i d e n t i c a l i n fun c t i o n a l form to the zeroth order and quasi-chemical approximations of Guggenheim (1952), provided that the s u b s t i t u t i n g atoms or atomic groups are of approximately equal s i z e . Guggenheim's method of development of these mixing models i s u s e f u l i n that i t provides the reader with some f e e l i n g f or the p h y s i c a l s i g n i f i c a n c e of the form of the models. For the s o l i d s o l u t i o n phase C, the r e l a t i o n between composition and a c t i v i t y i s given by [38] - F C ^ l n a F C + = a x ^ R T l n x ^ + a ^ R T l n x ^ + A G102 3,1 + AV „dP -ex,C i ':i023 AS „dT . ex.C In the absence of evidence to the contrary, A S g x has been assumed to equal zero. (Most of the data presented here are for temperatures near 1023 K and thus give very poor co n t r o l on AS .) In addition, AV i s assumed to be GX SX constant for each phase. Thus equation [38] reduces to [39] * F C R T l n a F C + ^ l n a ^ = a x ^ R r l n x ^ + a x ^ R T l n x ^ + A G1°2 3,1 + A V 1 0 " ' ^ - 1) . ex,C 15 AC?1023 ,1 and A V 1 ' - , 2 3 > 1 remain functions of composition and have been ex-ex ex pressed as degree polynomials i n composition, e.g. n [40] A G 1 ' 0 2 3 ' 1 = J a.x1 1=0 1 X-ray measurement of the volumes of the s o l i d phases indicates that Av 1 0 2 3 »* ex i s zero f o r a l l phases except b r u c i t e (see Table VI). Even for b r u c i t e only the f i r s t three terms of the polymonial are necessary to represent the volume function adequately. This leads to T411 A V 1 0 2 3 ' 1 = x x W L 4 X J ex,C XFC HCV,C ' where W is. zero for a l l phases except b r u c i t e . V, C A fourth degree polynomial was used to approximate A G 1 0 2 3 ' 1 , although i t w i l l be seen l a t e r that such a complex function i s not necessary for most of the s o l i d s o l u t i o n phases. The r e s u l t of applying boundary constraints to the polynomial i s [42] A G 102 3, 1 = 2- w + 2'. w + „ 2 ' 2 E ex,C HC FC HC FC HC FC HC FC C In equation [42] the s are equivalent to the W%s of Thompson (1967). (W corresponds to the s o l u t i o n of HC i n a c r y s t a l of predominantly FC.) nC i s the c o e f f i c i e n t of the fourth degree term i n equation [40]. No theo-r e t i c a l s i g n i f i c a n c e has been associated with E. From the above i t follows that [43] R T l n a H C = a R T l n * ^ + x 2 ^ + Ix^ift -- ( l - 4 x F C + 3 4 c ) K c + ^ c ( P - l ) } and 16 [44] RTlna F C = aRrlnx^ + x^iw^ + 2* F C<tf H C - W^) ~ ( 1 - S I C + 3 XHC ) EC + ^ C ( P - 1 ) } ' The equations expressing the equilibria in Table VIII contain a large number of constants (AG°'s, AS°'s, A c's, w's, etc.)> which must be evaluated in order to describe the phase equilibria. In order to reduce the number of variables to be determined on the basis of least squares, those variables not well constrained by the phase equilibrium data and which could be e s t i -mated from other sources were^evaluated prior to seeking a least squares so-lution. It is now possible to write equations which express, within the ap-proximations stated, each of the equilibria in Table VIII. The equations which follow have been expanded and rearranged so that the parameters to be evaluated a l l appear in explicit form in the l e f t hand sides of the equa-tions. The exceptions to this are A s ° and A s ° . These quantities remain embedded in A S ° . The data base used to evaluate the right hand sides i s given in Table IX. The equations follow. For equilibria [1] through [6], [ 4 5 ] S c + S D " A GFC " AGHD + ( 34c " 2-FC ) WHC + ( x F C - 1)(1 - 3 x F C ) ^ F C - (2x F C - 6x|c + 4* F C)S C + ( 2 XFD " 3 4 D } V + ( 1 " *FD ) ( 1 " 3 W + (2*FD " 6*FD + : 4*FD ) BD = A l R ( 1 " P ) + A 5 R ( T ~ 1 0 2 3 ) + aRTln{(x F C - X F C X F D ) / ( X F D - x^x^)} + ( P - l ) { ( 2 x F D - l ) ^ D + ( l - 2 x F C ) ^ c } . 17 For equilibrium [7], [ 4 6 ] A GHC ' S c + (34c " 2 * F C % C + (*FC " " 3 V F C " <2*FC " 6*FC + 44c)£C - 2 A GH 20 " 2 A GHF + Solids, R*1 " P ) + AS°(T - 1023) + aRTln{x F C/(l - x^) } + 2Rrln(f- / f H F ) + ( P - D d - 2 x F C ) ^ c . For equilibria [8] through [16], [ 4 ? ] l V ^ G C + rL ^ C X F C { ( 2 X F C " 1 ) WKC + 2 ( 1 " *FC>*FC C Crn 2»J " ' 4*FC + 34c>*C}1 = A ^ o l i d s , R ( 1 " P ) + A S R ( r " 1 0 2 3 ) fT rT A c dr - T ( A c D / r ) d r - v u nR-rlnf - RrJ (av i n * ) K J K H2 U H 2 U r ^ u n ^ 0 1023 1023 CfH 20 - » v 4 c > V , c > . where the v are given by equation [27] . Pairs of equations corresponding to equations [33] and [34] of the form of [47] may be used to express equi-libria.[17] and {.18]. It is evident that, provided a l l f l u i d species are treated in a manner analogous to that for H20 and proper v are chosen, equation [47] w i l l represent a l l the equilibria considered in Table VIII. The a which appears in some of the above equations is the number of sites per formula unit upon which mixing is occurring. Using the formula units given in Table II, a = 2 for a l l solid solution phases. Equations [45], [46], and [47] have been applied to the data lis t e d i n Table VII and to the data of Crane and Ehlers (1969) and Chernosky (1974) listed in Table X. In order to u t i l i z e equilibrium data involving inter-18 mediate sell a i t e i t has been necessary to estimate the compositions of inter-mediate sell a i t e coexisting with periclase, brucite, and s e l l a i t e . This has been done by noting that a bulk composition of MgOHF can be crystallized to 100% intermediate sellaite while compositions of Mg(OH) F.. and Mg(OH)^ ^ F Q cannot, and that the c e l l parameters of intermediate sel l a i t e (Table IV) indicate very l i t t l e solid solution. Intermediate sel l a i t e coex-isting with periclase or brucite i s estimated to have a composition of x = Fo 0.49 while the composition of that coexisting with sel l a i t e i s estimated to be x„^ = 0.51. The estimated standard errors assigned to these quantities are FS 0.02. Since the data of Crane and Ehlers (1969) for coexisting periclase, intermediate s e l l a i t e , s e l l a i t e , and vapor do not include the composition of the sellaite only equilibrium [14] can be constrained with these data. For this equilibrium, only the data at 1000 bar were used here since these are the only data said by the authors to be reversed. To further constrain the model, several equations can be written which represent estimates of individual parameters to be evaluated. These equa-tions along with the sources of the data and their estimated standard errors are listed in Table XI. Solution of the Least Squares Problem The system of equations being used to describe the experimental data is highly overdetermined in the sense that there are many more equations than unknowns. Since, as already remarked, the equations involve measured quan-t i t i e s , i t may be expected that an exact solution to the equations w i l l not exist. For this reason an approximate solution must be sought. The approx-imation criterion to be used here i s that of least squares. A l l the equations to be used to describe this system are linear in the t i l unknowns. The i equation may be written in the form 19 n [51] Where the a. . and b. are constants and the x. are the unknown parameters, ij i 3 and n i s the number of unknowns. A system of m such equations may be writ-ten in the form [52] Ax = b where A is an m by n matrix containing the a^^> i is a column m-vector con-taining the b^ and x is a column n-vector containing the x^ .. The least squares solution is the n-vector x for which the euclidean length of the vector (Ax - b) is a minimum. If such a solution is found for the set of equations derived directly from the data, no account is taken of the varying degree of certainty to which each of the equations is known. In order to weight the equations ac-cording to their uncertainties a weight matrix L was applied making the sys-tem of equations to be solved The matrix L was derived from an estimate of the covariance matrix C of the equations. This was done by computing the Cholesky factorization of C to produce the lower triangular matrix F such that [53] L A:x = Lb eq eq [54] C = FF .T eq L is then defined by the relation [55] -1 L = F 20 The covariance matrix C was estimated by the relation eq 9f . .3f . [56] c. . = I f - 2 — J a2 In the above relation c . . is the covariance of equations i and j. The f. are defined by the relation [57] f ; = {la. .x.J - b. J The y are the experimentally determined quantities such as mole fraction, temperature, pressure, and volume. The Oy^ are the variances of the y^. In order to simplify calculation of the assumption has been made that only errors in the determination of mole fraction, temperature, and in the case of buffered experiments, f l u i d composition contribute significantly to the covariances. Further, even the errors in temperature measurement were found to be significant only for equilibria involving a fl u i d phase. In order to produce a solution vector that would reproduce the data i t was also necessary to arbitrarily overweight those equations which describe equilibrium between three solid phases, at least one of which is a solid solution phase, and the vapor. This was done by reducing ox for the solid solution phases involved by a factor of five. This has a secondary effect of making i t necessary that the covariances be set to zero between these equations and a l l others correlated to them by virtue of composition mea-surements . The least squares solution of equation [53] may be written as [58] x = [GA]+Gb where [GA]+ is the pseudoinverse of GA. The pseudoinverse of a matrix A 21 may be obtained from i t s singular value decomposition (Golub and Reinsch, 1970; Lawson and Hanson, 1974). The singular value decomposition was accom-plished by use of the University of British Columbia Computing Center sub-routine SOLSVD (Streat, 1973). The singular value decomposition yields T [59] A = USV :where i f A is a m by n matrix, then U is an m by m orthogonal matrix, V is an n by n orthogonal matrix and S i s an m by n diagonal matrix. A+ may be defined by the relation [60] A+ = VS+UT where s+ is an n by m diagonal matrix whose diagonal elements s + are given + = f l / s i for si>0 by [ 6 1 ] S i i 0 for S i=0 where the s^ are the diagonal elements S. The covariance matrix of the members of the solution vector x has been estimated as a2C, where the expression [62] a 2 = || Ax - b ||2/(m - n) is used to evaluate the scale factor a 2 and the unsealed covariance matrix C is defined by [63] C = VS+S+VT . The correlation matrix may be obtained from the covariance matrix by the relation [64] p. . = o2 ./a . .a . . ij iJ a JJ 22 where p. . is the correlation coefficient and a. . the covariance for x. and x .. J A more detailed account of the computational methods used and several useful fortran codes may be found in Lawson and Hanson (1974). An introduc-tion to least squares problems may be found in Bevington (1969). A more rigorous approach is presented by Plackett (1960). Testing of the Alternative Models A test of any thermodynamic model is that i t reproduce the experimental data within their uncertainties. Consequently i t is necessary to calculate the conditions of various equilibria from the thermochemical quantities which constitute the model. The simplest of these calculations involves finding the composition of one solution phase in equilibrium with another solution phase of specified composition. This may be done at any pressure and temperature by appropriate manipulation of equation [45] or [46] with addition of terms containing heat capacities. Calculation of the e q u i l i -brium position of a four phase f i e l d i s more complex. In general i t is nec-essary to solve a system of five equations simultaneously. Three indepen-dent equations, involving a l l three solid phases and the f l u i d phase, of the form of equation [45] or [46], with heat capacities, can be written. A fourth equation of the form of equation [47] can be written. Finally, since the f l u i d is present as a phase the equation [65] P = I f./y. i • where the i are the various species in the fl u i d phase, may be written. Only three species, H20, HF, and H 2 are known to be present in the f l u i d in 23 appreciable amounts. H2 is not involved in any of the equilibria considered and therefore i t s only influence is as an inert component in the flu i d phase. Simultaneous solution of this non-linear set of equations provides the com-position of a l l four coexisting phases. It should be noted that such cal-culations may produce either stable or metastable assemblages. The method used here to assess st a b i l i t y or metastability has been to inspect for four phase fields which have a compositional volume in common and calculate which assemblage has the lower Gibbs energy for a given composition within that compositional volume. The assemblages having the lowest Gibbs energies when this procedure is carried out for a l l the calculated assemblages are assumed to be stable. The success of this procedure depends upon the set of stable four phase assemblages being a subset of the set of assemblages being tested. The possibility that this condition may not be f u l f i l l e d exists because the method used to solve the system of equations outlined above requires a good i n i t i a l estimate to achieve convergence. If a solution cannot be achieved for a given four phase assemblage i t is not always obvious whether this is because a solution does not exist or because a suitable i n i t i a l estimate has not been made. Nevertheless, the correspondence between the calculated assemblages and those observed both in the laboratory and nature, provides some confidence that the most stable assemblages have been found. In order to evaluate the coefficients'of equation [47] for those equi-l i b r i a involving H2O i t has been assumed that the flu i d phase is an ideal mixture of H2O, HF, and H2 , and that the dilution of H20 by HF has a t r i v i a l effect upon AG . The model is consistent with these assumptions. Figure 1 indicates that for a l l equilibria considered here £ is less than two bar. HF £ H has been estimated as fourteen bar based on the buffering effects of the 24 b u f f e r i n g e f f e c t s o f the p r e s s u r e v e s s e l s . F o r a f l u i d so dominated by a s i n g l e s p e c i e s the assumption of i d e a l m i x i n g w i l l be q u i t e adequate f o r t h e m a n i p u l a t i o n s needed h e r e , s i n c e even l a r g e d e p a r t u r e s from i d e a l m i x i n g , w i l l have a t r i v i a l e f f e c t upon the c a l c u l a t e d f u g a c i t i e s o f the f l u i d s p e c i e s . Only i f a measurement o f the mole f r a c t i o n s of the f l u i d s p e c i e s c o u l d be made would the n o n - i d e a l i t y o f m i x i n g be a p p a r e n t . That t h e s e arguments may n o t be v a l i d f o r v e r y f l u o r i n e - r i c h b u l k c o m p o s i t i o n s i s d i s -c u ssed l a t e r . I n o r d e r t o f i n d the s i m p l e s t model t h a t w i l l d e s c r i b e the d a t a , a d d i -t i o n a l e q u a t i o n s were added t o the l e a s t squares problem which r e p r e s e n t l i n e a r c o m b i n a t i o n s among the parameters. These e q u a t i o n s were g i v e n s u f -f i c i e n t w e ight t h a t t h e i r r e s i d u a l s i n no case exceed 0.1 c a l moi I n i t i a l s o l u t i o n s o f the f u l l l e a s t s q uares problem produced e x c e s s parameters f o r t h e d i f f e r e n t humite phases which were e q u a l w i t h i n t h e i r un-c e r t a i n t i e s . T h i s f a c t c o u p l e d w i t h the s t r u c t u r a l s i m i l a r i t y o f the humites l e d to t h e a d d i t i o n o f c o n s t r a i n t s t h a t the excess parameters a r e identical f o r n o r b e r g i t e , c h o n d r o d i t e , and c l i n o h u m i t e . I t has a l s o been found p o s s i b l e to s e t the excess parameter ' j? ' e q u a l to zero f o r a l l phases except s e l l a i t e w i t h o u t d e t r a c t i n g from the q u a l i t y o f the model. S e t t i n g E e q u a l t o z e r o produces parameters t h a t deny the s t a b l e e x i s t e n c e o f i n t e r m e d i a t e s e l l a i t e and p r e d i c t t h a t the h y d r o x y l s e l l a i t e endmember i s more s t a b l e t h a n the h y d r o x y l b r u c i t e endmember i n c o n t r a d i c t i o n o f e x p e r i -m e n t a l e v i d e n c e and n a t u r a l o c c u r r e n c e s . A non-zero E has t h e r e f o r e been r e t a i n e d . Such a model produces an a c c e p t a b l e d e s c r i p t i o n o f the d a t a , b u t sug g e s t s t h a t W and Itf might be s e t e q u a l . Such a model, w i t h a symme-HB r B t r i e s o l u t i o n f o r b r u c i t e , has been chosen as the s i m p l e s t a c c e p t a b l e de-s c r i p t i o n o f the system. 25 Two simpler models have also been tested. In the f i r s t a l l solid solu-tions other than sel l a i t e are modeled as symmetric. In the second they are modeled as ideal. The three models w i l l be referred to as the preferred, symmetric, and ideal models. In Table XII and Figures 2 and 3 the models are compared to the experimental data. The experimental data depicted in Figure 2C and 2D (equilibria [15] and [16], x^, = 1) are largely from Chernosky (1976). These data were not available at the time the least squares problem was formulated. The data used in formulation of the least squares problem were from Chernosky (1974). Many of the experimental data are nearly as well reproduced by the sym-metric and ideal models as by the preferred. The major failures of the sim-pler models are in the prediction of the compositions of triplets of coex-isting solid phases and of the Gibbs energies of forsterite and enstatite. In some of these cases the calculated values based upon the simpler models differ from the data by several estimated standard errors of the data. Table XIII l i s t s the parameters that represent the solution vector for the least squares problem based on the preferred model. The values for these parameters have been given to the nearest units place in spite of their rather large estimated standard errors. This has been done due to the high correlations between these parameters which are evident from examina-;. tion of the matrix of correlation coefficients given in Table XIV. It i s extremely important to be aware of the role played by the matrix of correlation coefficients. Parameters found by solution of the least squares problem are not ful l y independent, being constrained to follow one another through their respective correlation coefficients. The important effect of this is that one is not free to arbitrarily specify values for 26 each parameter within i t s uncertantity because this w i l l deny the strong correlation that exists between the parameters and result in a much infer-ior set of computed equilibrium conditions. The parameters in Table XIII along with those list e d in Table IX may be used, within limits, to calculate phase equilibria in the system under a wide range of physical conditions. The most important limiting condition is probably the appearance of additional phases. If additional phases are sta-ble, some of the calculated equilibria w i l l be metastable. This may be a problem even at 1023 K and 2000 bar due to the possible stability of antho-phyllite. Discretion should also be exercised in using calculated e q u i l i -bria at conditions other than those of the experiments. This is particular-ly true of other temperatures due to the estimated nature of As° and C for many of the endmember phases and because of the assumption that AS = 0 f° r a l l solution phases. The position of the quartz-talc-sellaite-fluid and talc-norbergite-s e l l a i t e - f l u i d fields i s in doubt. No measurements of coexisting s e l l a i t e and talc compositions were possible due to the interference of s e l l a i t e d i f -fraction peaks with the (060) peak of talc. However, talc and norbergite more fluorine rich (x = 0.64, x = 0.94) than predicted by the model r Ic rN ( x ^ = 0.50, * T = 0.84) have been crystallized at 1023 K and 2000 bar. FTc FN Several explanations for this discrepancy are possible. There is some tex-tural evidence to suggest that a melt phase may have been present in these fluorine-rich experiments. The melt hypothesis offers an explanation only i f the observed norbergite and talc are quench phases since i f they coexist-ed with the melt at the conditions of the experiment they should be more hydroxyl-rich than i f no melt had been present. The textures are inconclu-sive regarding this possibility. 27 Two more probable possibilities are that the model does not adequately describe this portion of the system or that the high fluorine norbergites and talcs are metastable. Inadequacy of the model i n this region may arise from the treatment of the flui d phase as an ideal mixture of H20 and HF. The model predicts that at 1023 K and 2000 bar £ in a fl u i d coexisting HF with norbergite with x = 0.94 w i l l be approximately 10 bar. For a talc with x_ =0.64 £„_ is predicted to be about 70 bar. H90 at these condi-FTc HF /L 3 -1 tions has a specific volume of about 2 cm gram (Burnham, Holloway, and Davis, 1969). At this density there is a possibility of polymerization in the HF - H 2 O f l u i d which would lead to non-ideal mixing. With the increase in £„_, SiF^ may also become an important species in the f l u i d , especially r i r i f i t coexists with quartz. APPLICATIONS The thermodynamic model presented here is general enough to be of use in the analysis of a variety of mineralogic problems. There follows a small variety of applications to relatively simple problems. Figure 4A is a T-x section at 2000 bar of the system MgO-MgF2-H20 projected from H20. Such a projection is not s t r i c t l y legal (Greenwood, 1975) since H20 is not present as a pure phase in this system. However, HF is the only other f l u i d species of importance; and, as may been seen from examination of figure 4B, i t makes up a very minor proportion of the fl u i d , except at very fluorine-rich bulk compositions. For this reason projections from H20 on Mg0-MgF2-Si02 constitute adequate representations of the system MgO-MgF2-Si02-H20. 28 The effect of fluorine on the upper st a b i l i t y of brucite is illustrated in Figure 4A. Similar information is presented in an alternative form in Figure 5. Displacement of equilibria for talc and clinohumite are also pre-sented in Figure 5. The error limits shown in this figure represent 2a of the error derived from the error estimates of the least squares parameters. These should provide reasonably accurate estimates of uncertainty within the range of the experimental conditions, but may tend to underestimate errors outside this range. The error estimates have been presented in part to i l -lustrate the importance of the correlated nature of the least squares para-meters. If the model parameters are chosen within their own limits, but without regard for covariance, the error estimates are generally many times larger. The importance of these figures in the interpretation of natural assem-blages l i e s in the profound effect of fluorine substitution on the stability of hydroxyl-bearing minerals. It is v i t a l to know to what extent the hy-droxyl sites are occuppied by fluorine. With such knowledge, the assemblage can be as informative as the equivalent assemblage involving the hydroxyl endmember. Fluorine not only shifts the s t a b i l i t i e s of hydroxyl minerals, but is also distributed between a l l the available hydroxyl sites in an assemblage of phases. Figure 4B illustrates the composition of the f l u i d that coexists with solid solutions of definite compositions. Since two solid solution phases in equilibrium with the same flu i d must also be in equilibrium, at least metastably, with each other, this figure also provides information on the equilibrium compositions of coexisting solid solutions. The figure has been calculated for the specific conditions of 1023 K and 2000 bar. Numbers given for the f l u i d change markedly with temperature, but only slightly with 29 pressure. In contrast, the compositions of coexisting solid solutions are rather insensitive to changes in temperature and pressure, rendering them unsatisfactory as geothermometers or barometers, but valuable as a test of equilibration between natural mineral pairs. Bourne (1974) has analyzed coexisting pairs of phlogopite and clinohu-i mite or chondrodite from natural assemblages. Error estimates of 0.05 in mole fraction fluoro-endmember have been applied to these data, which are _ compared in Figure 6 to the calculated equilibrium distributions u t i l i z i n g the models presented here and an assumed ideal solution model for phlogopite, based on the data of Munoz and Ludington (1974). The two curves presented for each mineral pair reflect the different enthalpies of formation of fluorite discussed earlier and their effect on the fluorine buffer calcula-tion. The good correspondence between observed and calculated distributions indicates a close approach to equilibrium in the natural assemblages; but, because of the insensitivity of the distribution coefficients to pressure and temperature, i t provides no information on geothermometry or barometry. Bourne (1974) found no humite in any of the natural assemblages col-lected from a variety of locations. This finding i s in agreement with the experimental results of this study. The compositions of the one pair of coexisting norbergite and chondrodite ( x F N = 0.74, x^c^ = 0.72) agree well with the model. Stable assemblages in the system MgO-MgF2-Si02-H20 at 1023 and 873 K and 2000 bar are presented in Figure 1. These provide insight into the limited compositional ranges of naturally occurring minerals of the humite group. The combined data of Bourne (1974) and Jones, Ribbe, and Gibbs (1969) provide analyses for five norbergites, thirty-eight chondrodites, and twelve 30 titanium-poor clinohumites. From these data the most f l u o r i n e - r i c h norber-g i t e reported i s 98 mole percent fluoro-norbergite. According to the model, at temperatures le s s than 1300 K such a norbergite would not be stable at a pressure of H 20 greater than 100 bar. I t should be born i n mind, however, that t h i s p r e d i c t i o n rests on the p o t e n t i a l l y u n r e l i a b l e portion of the model. The most f l u o r i n e - r i c h natural chondrodite (x-, n u = 0.82) corresponds * Oh cl o s e l y to the model p r e d i c t i o n at 1100 K and 100 bar H 20 pressure. The i n --4 -1 s e n s i t i v i t y of maximum Xj?ch t 0 t e m P e r a t u r e a t constant pressure (2'10 K ) suggests that chondrodites with x„nu > 0.82 could c r y s t a l l i z e only with r Oh PT7 <1000 bar from very f l u o r i n e - r i c h bulk compositions. The non-occurrence H20 of such chondrodites i s probably due to the combined r a r i t y of these condi-tions and bulk compositions. Such chondrodites could not coexist with quartz due to the s t a b i l i t y of t a l c nor with magnesite or dolomite due to the s t a -b i l i t y of norbergite. The most f l u o r i n e - r i c h n atural clinohumite (x„n, = F LI 0.61) corresponds well to the predicted upper l i m i t of x , for the assem-r L I blage forsterite-chondrodite-clinohumite. The lower l i m i t s of x for the humite minerals predicted by the model F are generally lower than those of the observed natural minerals. For nor-bergi t e and chondrodite the lower l i m i t of f l u o r i n e content i s s p e c i f i e d by the brucite-norbergite-chondrodite and brucite-chondrodite-clinohumite equi-l i b r i a r e s p e c t i v e l y . These do not involve the f l u i d phase and are therefore r e l a t i v e l y i n s e n s i t i v e to pressure and temperature v a r i a t i o n . They explain reasonably w e l l the lower observed l i m i t s of x„„ and x„„, . The model pre-FN FCh diets the stable existence of hydroxyl-clinohumite, but t h i s phase has not been found i n natural assemblages, nor has i t been c r y s t a l l i z e d experiment-a l l y . This discrepancy i s probably a consequence of uncertainties i n the lea s t squares parameters. Within the error l i m i t s of the model, i t i s 31 possible that the dehydration temperature of hydroxyl clinohumite i s lower than that of hydroxyl b r u c i t e . I f th i s i s the case the lower l i m i t of FC. would be constrained by the b r u c i t e - c l i n o h u m i t e - f o r s t e r i t e equilibrium and pure hydroxyl-clinohumite would be unstable. CONCLUSIONS A model has been constructed f o r the system MgO-MgF2-Si02-H20 which ; i s i n good agreement with a v a i l a b l e experimental and f i e l d data, except perhaps at the very f l u o r i n e - r i c h bulk compositions. The correspondence between t h e o r e t i c a l and natural phase assemblages encourages confidence i n the model. The model, besides being of s p e c i f i c value to understanding the system under study, should provide an o u t l i n e of a general approach s u i t a b l e f o r dealing with dehydration and exhange e q u i l i b r i a i n a v a r i e t y of experimental and natural systems. Several e q u i l i b r i a present themselves as being f r u i t f u l areas for f u r -ther study. In p a r t i c u l a r the e q u i l i b r i a involving t a l c , norbergite, s e l l a -i t e , and quartz beg further study to c l a r i f y the nature of the f l u o r i n e - r i c h portion of the system. This information may not be of great immediate geo-l o g i c i n t e r e s t , but i t may contribute i n d i r e c t l y to the improvement of the model as a whole. The t a l c - e n s t a t i t e - q u a r t z - f l u i d and t a l c - e n s t a t i t e - f o r s t -e r i t e - f l u i d e q u i l i b r i a may contribute to refinement of the t a l c s o l i d s o l u -t i o n model. Substantial improvements might be made i n the d e s c r i p t i o n of clinohumite through study of the b r u c i t e - c l i n o h u m i t e - f o r s t e r i t e and p e r i -c l a s e - c l i n o h u m i t e - f o r s t e r i t e - f l u i d e q u i l i b r i a . Figure 1. The system MgO-MgF 2-Si0 2-H 20 projected from H 20 onto MgO-MgF 2-Si0 2. Numbers shown are f for each three phase f i e l d . A. T — 1023 K, P = 2000 bar. B. T = 873 K, P = 2000 bar. Figure 2. Comparison of the preferred, symmetric, and i d e a l models with experimental data. Rectangles i n A and B represent points within one a of the experimental data. Points i n C and D represent the maximum uncertainty i n the experimental bracket. Note: These data are only a part of the t o t a l data. For a model to be acceptable i t must conform to a l l the data. 35 Figure 3. Comparison of the preferred, symmetric, and i d e a l models with experimental data. Rectangles represent points within one a of the experimental data. Note: These data are only a part of the t o t a l data. For a model to be acceptable i t must conform to a l l the data. 37 Figure 4. A i s a T-x section at 2000 bar of the system MgO- '. MgF2-H.20 projected from H 20 onto Mg0-MgF2. B i s a plo t of f to H2O f r a t i o versus the compositions of the s o l i d s o l u t i o n phases i n Hr the system at 1023 K and 2000 bar. Figure 5. Displacement of equilibria by variation in fluorine content of solid solution phases. A. HB = P + H20. B. HCI = 4Fo + P + H20. C. HTc = 3En + Qtz + H20. D. HTc + Fo = 5En + H20. Error bars represent 2a in the temperature position of the e q u i l i -brium curves based upon the error estimates provided by the preferred model. Note: The Gibbs energy function for H20 given by Holloway, Eggler, and Davis (1971) was used in calculating these equilibria. P/kbar O M Ol P/kbar P/kbar / Figure 6. Partioning of fluorine and hydroxyl between phlogopite and chondrodite and between phlogopite and clinohumite. Squares represent points within 0.05 of the data for natural assemblages. The dashed lines are the relationships calculated using the data of Robie and Waldbaum (1968) for the enthalpy of fluorite in the buffer calcu-lation to characterize the phlogopite solid solution. The solid lines have been calculated using-the enthalpy of fluorite from Stull and Prophet (1971). 44 Table I. Symbols and Units Quantity Symbol Unit Equivalent SI Unit Thermodynamic temperature T Pressure P Molar volume of substance B v Molar enthalpy of substance B H E Molar entropy of substance B Molar Gibbs energy of substance B Molar heat capacity of sub-stance B at constant pressure Number of moles of substance B n a Mole fraction of substance B: x W i Chemical potential of substance B Fugacity of substance B f Activity of substance B a Jc Activity coefficient of sub- y stance B, mole fraction basis Stoichiometric coefficient of substance B K bar 3 ,K 10 5 Pa cm cm cal ,-1 mol 4.184 J T-1 mol cal mol" 1 K _ 1 4.184 J mol"1 K _ 1 cal ,-1 mol 4.184 J ,-1 mol cal mol ^ K ^  cal mol bar -1 4.184 J mol 1 K 1 4.184 J mol 105 Pa -1 Site population number Excess parameters (Gibbs energy) of substance B Excess parameter (volume) of substance B WHB' WFB B cal mol 3 cm -1 4.184 J mol 3 -1 cm Estimated standard error of quantity C General equation for a chemical reaction ' . 0 = V B B 45 Table I. (Concluded) Quantity Symbol Unit Equivalent SI Unit Gas constant R cal moi 4.184 J moi Superscripts ° i s used to denote a property of a pure substance at a temperature of 1023 K and a pressure of 1 bar. is used to denote a property of a pure substance at some temperature and/ or pressure other than 1023 K and 1 bar. A, except when subscripted by R (see below), is used to denote a quantity of formation from the compounds MgO, MgF2, Si0 2, H20, e.g. AG is .the Gibbs. energy df formation from the above compounds. Subscripts ex specifies an excess quantity, i.e. the difference between an actual quan-t i t y for a given solution phase and the value of that quantity in a corres-ponding ideal solution. R indicates that the A-quantity refers to the difference for a reaction, i.e. AG =£.v.AG.. R i i i t The symbol y has been used to represent the activity coefficient on a mole fraction basis rather than the recommended symbol f in order to avoid confusion between activity coefficient and fugacity. T,P denotes a property at temperature T and pressure P. Prefix Table I I . Compounds Considered Compound Symbol a Formula Unit P e r i c l a s e P MgO Brucite B Mg(0H,F) 2 Intermediate s e l l a i t e IS Mg(OH,F) 2 S e l l a i t e S Mg(OH,F) 2 Norbergite N Mg 3SiO t t(OH,F) 2 Chondrodite Ch Mg 5 S i 2 0 8 ( 0 H , F ) 2 Clinohumite C l M g 9Si t t0 1 6(0H,F) 2 F o r s t e r i t e Fo MgzSiOi, E n s t a t i t e En MgSi0 3 Talc Tc Mg 3Si^O 1 0(OH,F) 2 Quartz Qtz S i 0 2 Water H 20 H 20 Hydrogen f l u o r i d e HF HF a Prefixes of H and F are used with the symbols for the s o l i d phases i n order to denote the hydroxyl and f l u o r i n e endmembers res p e c t i v e l y , e.g. HS r e f e r s to h y d r o x y l - s e l l a i t e and FCh to fluoro-chondrodite. 47 Table III. Thermodynamic Data for the Fluorine Buffer Compound f,elements S298 C=a+b'; 10-3T+c'-:105/T2 V (cal moi 1) (cal moi 1 K 1) (cal moi 1 (cm ) a b c CaF2 -293,000 a 16.39 3 14.30 7.28 0.47 b 24.558 c CaSi0 3 -390,740 d 19.60 d 26.64 3.992 -6.517 e 39.93 c a-Si0 2 -217,700 a 9.91 a 10.495 9.277 -2.313 a 22.688 c B-Si0 2 14.080 2.400 0.0 a 23.718 c H20 -57,798 a 45.106 f 6.740 3.073 0.335 f HF -65,140 a 41.508 a 6.529 0.657. 0.231 a a Stull and Prophet (1971) b Naylor (1945) C Robie, Bethke, and Beardsly (1967) Robie and Waldbaum (1968) 6 Southard (1941). Friedman and Haar (1954) 48 Table IV. Unit Cell Parameters (X) Phase Unit c e l l parameter = a+b a •x Fluoro -endmember Space group Param. a b b a param. Calculated Published Sellaite ao 5.6185 -0.9929 0.3-10"2 4.626 4.623 c Sp. gr. ? b° 4.3774 0.2474 0.4-10"3 4.625 4.623 c Co 3.0507 -0.0006 -4 0.5-10 3.050 3.052 c Norbergite ao 4.7168 -0.0098 -4 0.2-10 4.707 4.709 d Pbnm bo 10.3188 -0.0543 o . i - i o - 2 10.265 10.271 d co 9.0440 -0.3200 0.8-10"5 8.724 8.727 d Chondrodite ao 4.7359 -0.0104 o . i - i o - 2 4.726 4.738 d,e P2i/b *o 10.2778 -0.0306 0.3-10-2 10.247 10.278 d,e co 7.9526 -0.1615 0.4-10"2 7.791 7.813 d,e a 108.84 0.37 0.8-10"2 109.21 109.30 d,e Clinohumite ao 4.7469 -0.0066 0.2-10"2 4.740 4.751 d,e P2i/b bQ 10.2481 -0.0223 o . i - i o " 2 10.226 10.236 d,e co 13.7163 -0.1342 o . i - i o " 2 13.582 13.587 d,e a 100.59 0.35 0.5-10-1 100.94 100.88 d,e Unit c e l l parameter = a4-b •x+c-x +d •x 3 a a b c d b a param. Brucite ao 3.1464 -0.1776 0.0920 0 .5 .10"* P3ml co 4.7694 -0.3525 1.7138 -2.5291 0 .2 -lO" 2 Description ao *o co Calculated (x=0.00 a) 3.147 4.770 Swanson et al (1956) 3.147 4.769 49 Table IV. (Concluded) Phase Space group Description Intermediate Coexisting with brucite 10.123(1) 4. 6861(6) 300780(8) sel l a i t e Coexisting with sel l a i t e 10.097(6) 4. 6812(19) 3.0738(25) Pbmn or Pbn21sr Crane and Ehlers (1969) e 10.122(2) 4. 6845(13) 3.0801(4) Sellaite Calculated (x=0.50 a) 5.122 4. 5011 3.0504 Note: ASTM Powder Diffraction F i l e card 14-9 gives 13.68 as aQ for FC1. This apparently should be modified to read 13.58. The Van Valkenburg (1961) FCh may well be a mixture of phases (probably two humites). The d-spacing of 3.897 list e d by Van Valkenburg (1961) does not correspond to any possible d-spacing for a chondrodite with the space group and c e l l d i -mensions li s t e d above. a x = mole fraction fluoro-endmember. •k The estimated standard errors have rather large uncertainties due to the small number of data points available (see appendix). C Swanson et a l (1955) Van Valkenburg (1961) Cell dimensions lis t e d were calculated by means of the program of Evans, Appleman, and Handwerker (1963) from published d-spacings. ^ Numbers in parentheses are one estimated standard error in the digit to their immediate l e f t . ^ Zero and f i r s t level c-axis Weissenburg diffraction patterns are con-sistent with an orthorhombic symmetry. The observed systematic extinctions are hOl, h+l=2n+l. and 0 k 1, k = 2n + 1. Table V. Variation of Interplanar Spacings with Composition Phase h k 1 d-spacing (ft) = a+b-x+c 2 a a b-10 cr.lO b , °d Brucite 1 1 0 1. 57318 -0.8873 0.4573 0 . 3 i l 0 _ 4 Sellaite 2 2 0 1. 76085 -1.2555 0.5*10~3 Norbergite 1 2 1 3. 11379 -0.5922 0.5*10_4 Norbergite 1 4 1 2. 00072 -0.5757 0.4-10-4 Chondrodite T 1 2 3. 04133 -0.3810 0.8-10-3 Clinohumite 5" 1 i 2. 36875 -0.1724 0.5-10-3 Clinohumite 9 0 0 1. 49801 -0.1628 0.9-10-4 Clinohumite 4 1 1 2. 55580 -0.2176 O.l-lO - 3 Talc 0 6 0 1. 52731 -0.0615 0.7-10-4 x = mole fraction fluoro-endmember. The estimated standard errors have rather large uncer-tainties due to the small number of data points available (see appendix). Table VI. V a r i a t i o n of Molar Volumes w i t h Composition r_ o o ~ XFA* yFA + XHA' yHA + XFA * *HA * \,A „o „ _ a B r u c i t e 25. 732 24. 621 -5.1897 0. 8-: 10" -2 S e l l a i t e 19. 649 22. 789 0. 1* 10" -1 Norbergite 63. 466 66. 293 0. 4': 10" -2 Chondrodite 107. 32 110. 32 0. 4* 10" •1 Clinohumite 194. 64 197. 49 0. 5* 10" -1 Talc 133. 30 136. 25 Note: A l l numbers are expressed i n cubic c e n t i -meters. The estimated standard e r r o r s have r a t h e r l a r g e u n c e r t a i n t i e s due to the small number of data p o i n t s a v a i l a b l e (see appendix). Table VII. Experimental Results Experiment T a P a Duration b,c Starting Products a ' b 2 . 1 n ( f H 2 0 / f H F ) number (K) (bar) (hour) materials TcO-3R : 1023(3) 2000 ( 15) 1871 TcO Tc43.3(20) 14.93 Tc60-6R 1020(4) 1990( 30) 1438 Tc60 Tc44.8(22) 14.99 N60-B25-1 1021(3) 2000( 15) 954 N60+B25 N63.4(5)+B21.6(2) N80-B20-1 d 1021(3) 2000( 15) 954 N80+B20 N71.6(4)+B22.9(4) N60-Chl00-1R 1023(3) 2000 ( 15) 1462 N60+Chl00 N76.6(8)+Ch73.9(15) N50-2 1024(3) 1990( 15) 984 N50 mix N63.4(15)+Ch61.1(35) N80-Tc40-1R 1022(3) 2000 ( 15) 1315 N80+Tc40 N83.9(6)+Tc46.4(31)+Ch N80-Tc20-1R 1022(3) 2000 ( 15) 1315 N80+T.c20 N82.2(3)-K:h:79.7(37)+Tc43.3(56) N8'0-Tc60-1R 1022(3) 2000( 15) 1315 N80+Tc60 N83.4(18)+Tc48.2(68)" N100-Tc40-1R 1022(5) 2000( 20) 1450 N100+Tc40 N89.1(15)+Tc50.4(34) Nl00-Tc60-1R 1022(3) 2000( 15) 1315 Nl00+Tc60 N93.2(6)+Tc59.5(29) Ch60-Tc40-1R 1022(5) 2000( 20) 1450 Ch60+Tc40 Ch64.7(8)+Tc34.6(23) N60-Tc60-1R 1022(3) 2000 ( 15) 1315 N60+Tc60 Ch77.9(26)+Tc47.3(33)+N C160-B0-2 1023(3) 2000( 15) 1665 Ch60+Fo +B0 mix Ch39.5(54)+C131.7(27)+P Chl00-C140-1R 1023(3) 2000( :i5) 1665 Chl00+C140 Ch69.2(37)+C167.8(40) C140-Tc40-1R 1022(3) 2000< 15) 1315 C140+Tc40 C157.4(28)+Tc30.3(40) C140-Tc20 1023(3) 2000 ( :i5) 1665 C140+Tc20 C154.3(18)+Tc22.1(49) -1R+C160 +Ch60+Fo C110-1 1024(3) 2000( :2o) 742 C110 mix C128.9(35)+Fo+P C160-Tc0-1R 1022(3) 2000 ( :i5) 1077 Ch60+Tc0+Fo Tc35.3(23)+Fo+En 2B5-4 970(2) 2000( :io) 1076 B17+P B12.7(2)+P 2B5-5 970(2) 2000 ( :io) 1076 B5 B13.5(2)+P 2B15-5 1023(3) 20001 :i5) 1459 B24+P B22.3(4)+P 1B15-18 1023(3) 2000< :i5) 1459 B15 B23.1(2)+P 2B15-7 1047(3) 1990< :i5) 1352 B15 B24.6(2)+P 2B15-8 1047(3) 1990 :i5) 1352 B25+P B24.3(3)+P 1B15-17 973(2) 10001 :i5) 1086 B15 B22.4(3)+P 2B15-4 973(2) 1000 :i5) 1086 B25+P B22.2(3)+P Table VII. (Concluded) Experiment T a P a Duration Starting Products number (K) (bar) (hour) materials 1B30-2 802(2) 2005(10) 423 B30 mix B22.1(2)+IS 1B30-5 970(2) 2000(10) 1076 B22+IS B24.1(2)+IS 1B30-6 970(2) 2000(10) 1076 B25+IS B24.5(2)+IS 1B30-3A 1022(3) 2000(15) 1077 B25+IS B24.3(2)+IS 1B30-1 1043(2) 1985(10) 146 B30 mix B24.4(2)+IS 1B30-3 1047(3) 1990(15) 1352 B30 mix B25.6(2)4-IS S75-1 827(2) 1990(10) 337 S75 mix S89.1(4)+IS S75-5 970(2) 2000(10) 1076 S75 mix S85.2(4)+IS S75-4 1022(3) 2000(15) 1077 S75 mix S84.3(4)+IS S75-3 1047(3) 1990(15) 1352 S75 mix S83.8(4)+IS S70-4 1070(2) 1990(10) 124 S70 mix S83.3(2)+IS S75-4A 1070(2) 1990(10) 124 S75 mix S83.0(2)+IS a Numbers in parentheses, are one estimated standard error in the digit to their immediate l e f t . ° The numbers which follow the mineral symbols are the mole percent fluoro-endmember. A l l starting materials include H20. A l l products i n -clude a vapor phase. c Mix referrs to a mixture of the primary starting materials of the bulk composition indicated. These experiments are not f e l t to have attained equilibrium, but they do bracket the composition of norbergite in equilibrium with brucite of composition x^—0.22. Table VIII. Equilibria that Simultaneously Constrain the Thermodynamic Model [1] HB+FN=FB+HN [2] HN+FCh=FN+HCh [3] HN+FTc=FN+HTc [4] HCh+FTc=FCh+HTc [5] HCh+FCl=FCh+HCl [6] HCl+FTc=FCl+HTc [7] HTc+2HF=FTc+2H20 [8] 2Fo+HCh=HCl [9] HCl=P+4Fo+H20 [10] 2HCh=P+HCl+H20 [11] 2HN=P+HCh+H20 [12] 14HN+HTc=9HCh+6H20 [13] HB=P+H20 [14] HIS=P+H20 [15] HTc=3En-t-Qtz+H20 [16] HTc+Fo=5En+H20 [17] HB+FIS=FB+HIS [18] HS+FIS=FS+HIS 55 Table IX. Thermodynamic Data Evaluated Prior to Solution of the Least Squares Problem Compound S298'1 C=a+b-T+c/T2 a V (cal mol"1 K_1) (cal mol"1 K"1) (cm3) a b-103 c-10 - 5 p 6.44 b 11.358 1.154 -2.585 c 11.248 d HB 15.09 e 26.117 1.501 -7.550 f 24.261 9 FB 16.511 2.931 -1.781 h 25.732 9 HS 26.117 1.501 -7.550 h 22.789 9 FS 13.68 2 16.511 2.931 -1.781 j 19.649 9 HN 36.52 * 64.182 6.410 -19.519 h 66.239 9 FN 36.39 * 54.576 7.840 -13.750 h 63.466 9 HCh 59.36 * 102.247 11.319 -31.488 h 110.32 9 FCh 59.13 k 92.641 12.749 -25.719 h 107.32 9 HCI 104.52 * 178.377 21.137 -55.426 h 197.49 9 FC1 104.38 * 168.771 22.567 -49.657 h 194.64 9 HTc 62.33 1 110.229 14.213 -39.916 h 136.25 d FTc 61.99 m 100.623 15.643 -34.147 h 133.30 9 Fo 22.7 n 38.065 4.909 -11.969 o 43.786 d En 16.2 n 24.245 5.020 -5.633 P 31.44 d a-Qtz 9.91 P 10.495 9.277 -2.313 P 22.688 d g-Qtz 14.080 2.400 0.000 P 23.718 d H20 45.106 q 6.740 3.073 0.335 1 HF 41.508 P 6.529 0.657 0.231 P . o , AG (cal mol . ) HF .10753 b,c,i,j,p,g,r Where the experimental data was available in the literature, a, b, and c were derived from that data by the method of least squares. 5 6 Table IX. (Concluded) Barron, Bergy and Morrison ( 1 9 5 9 ) V i c t o r and Douglas ( 1 9 6 3 ) d Robie, Bethke, and Beardsley ( 1 9 6 7 ) Giauque and Archibald ( 1 9 3 7 ) ^ King, Ferrante, and Pankratz ( 1 9 7 5 ) g This paper h Estimated as the appropriate summation of C_ , C T T„, C„„, r r r Fo HB FS and Cp 1 Todd ( 1 9 4 9 ) . 3 Naylor ( 1 9 4 5 ) ^ Estimated as. the appropriate summation of ^ H B ' A N < ^ Sj?s8 with a volume c o r r e c t i o n as described by Fyfe, Turner, and Verhoogen ( 1 9 5 8 ) 1 Robie and Stout ( 1 9 6 3 ) m Estimated as S ^ l ^ ^ H v ^ - V ^ + V ^ V ^ ) ( 0 . 6 ) n Kelley ( 1 9 4 3 ) , S2^8 estimated as equal to S 2 9 8 for c l i n o -e n s t a t i t e . ° Orr ( 1 9 5 3 ) ^ S t u l l and Prophet ( 1 9 7 1 ) , C^ , estimated as equal to that for c l i n o e n s t a t i t e . 9 Friedman and Haar ( 1 9 5 4 ) A t f 2 9 8 ' 1 of formation from the elements equals - 1 4 4 , 0 0 0 c a l moi , personal communication B. S. Hemingway (U.S. Geological Geological Survey, Reston, V i r g i n i a ) . 57 Table X. Constraining Phase Equilibrium Data from Other Sources Phases in equilibrium T (K) 3 P (bar) a Reference IS, S, P, H20 1038(15) 1000(15) Crane and Ehlers (1969) HTc, Fo, En, H20 890(5) 500 Chernosky (1974) 911(5) 1000 935(5) 2000 952(5) 3000 969(5) 4000 HTc, En, Qtz, H20 933(5) 500 Chernosky (1974) 970(5) 1000 1011(5) 2000 Numbers in parentheses are one estimated standard error in the digit to their immediate l e f t . 58 Table XI. Equations Constraining Single Model Parameters Equation a References 15550(226) c a l m o l _ 1 1, 2, 3, 4, 5, 6 -14213(338) c a l m o l - 1 4, 5, 7, 8, 9, 10 -7879(440) c a l mol" 1 4, 5, 7, 8, 10 "* Numbers i n parantheses are one estimated standard error i n the d i g i t to t h e i r immediate l e f t . 1. Taylor and Wells (1938) 2. Giauque and Archibald (1937) 3. King, Ferrante, and Pankratz (1975) 4. Vi c t o r and Douglas (1963) 5. Barron, Berg, and Morrison (1959) 6. Friedman and Haar (1954) 7. Charlu, Newton, and Kleppa (1975) 8. Kelley (1943) 9. Orr (1953) io. S t u l l and Prophet (1971) [48] [49] [50] AG, AG. AG. HB p Fo o En 59 Table XII. Comparison of Experimental Data with Thermodynamic Models Experiment Solid Composition of the solid solution phases number phases (mole fraction fluoro-endmember) Experimental Models Preferred Symmetric Ideal N80-Tc40-1R N Tc Ch 0.822(3) 0.433(56) 0.797(37) 0.819 0.462 0.788 0.822 0.421 0.749 0.857 0.498 0.817 N50-2 N Ch P 0.634(15) 0.611(35) 0.638 0.593 0.543 0.441 0.556 0.482 C160-B0-2 Ch Cl P 0.395(54) 0.317(27) 0.387 0.331 0.404 0.347 0.416 0.370 Ch60-C150-1R Ch Cl Fo 0.535(26) 0.483(9) 0.536 0.490 0.431 0.373 0.518 0.469 C110-1 Cl Fo P 0.289(35) 0.272 0.338 0.345 C160-Tc0-1R Tc . Fo En 0.353(23) 0.354 0.339 0.325 N60-B25-1 +N80-B20-1 B N , 0.220(0) 0.670(40) 0.220 0.662 0.220 0.650 0.220 0.647 2B5-4 B P 0.127(2) 0.130 0.126 0.089 2B5-5 B P 0.135(2) 0.130 0.126 0.089 2B15-5 B P 0.223(4) 0.214 0.214 0.229 1B15-18 B P 0.231(2) 0.214 0.214 0.229 2B15-7 B P 0.246(2) 0.249 0.248 0.287 Table XII. (Continued) Experiment number Solid phases Composition of the solid (mole fraction fluoro solution ph; -endmember) ases Experimental Preferred Models Symmetric Ideal 2B15-8 B P 0.243(3) 0.249 0.248 0.287 1B15-17 B P 0.224(3) 0.228 0.231 0.258 2B15-4 B P 0.222(3) 0.228 0.231 0.258 1B30-2 B 0.221(2) 0.221 0.221 0.222 IS 0.490(20) 0.489 0.493 0.497 1B30-5 B 0.241(2) 0.237 0.238 0.238 IS 0.490(20) 0.487 0.486 0.487 1B30-3A B 0.243(2) 0.243 0.244 0.243 IS 0.490(20) 0.487 0.484 0.485 1B30-6 B, 0.245(1) 0.237 0.238 0.238 IS 0.490(20) 0.487 0.486 0.487 1B30-1 B 0.244(2) 0.245 0.246 0.244 IS 0.490(20) 0.487 0.483 0.484 1B30-3 B 0.256(2) 0.246' 0.246 0.245 IS 0.490(20) 0.487 0.483 0.484 S75-1 S 0.891(4) 0.880 0.868 0.868 IS 0.510(20) 0.506 0.468 0.461 S75-5 S 0.852(4) 0.860 0.845 0.844 IS 0.510(20) 0.523 0.489 0.482 S75-4 S 0.843(4) 0.855 0.836 0.835 IS 0.510(20) 0.527 0.496 0.490 S75-3 s 0.838(4) 0.852 0.832 0.829 IS 0.510(20) 0.529 0.501 0.495 S70-4 S 0.833(2) 0.850 0.828 0.825 IS 0.510(20) 0.531 0.504 0.499 S74-4A s 0.830(2) 0.850 0.828 0.825 i s 0.510(20) 0.531 0.504: 0.499 61 Table XII. (Concluded) Phases i n Pressure Temperature equilibrium (bar) (K) Experimental Preferred Symmetric Ideal P, S, IS, f l u i d 1000 1038(15) 1042 978 957 Phase AG° ( c a l moi - 1) Calorimetric Models Preferred Symmetric . Ideal HB .15550(226) 15538 15536 15231 Fo -14213(338) -14249; -12553 -11954 En -7879(440) -7566 -6834 -6598 Note: Numbers i n parentheses are one estimated standard error i n the d i g i t to t h e i r immediate l e f t . Table XIII. Thermodynamic Properties Determined by Least Squares (Preferred Model) Parameter a Numerical Value Estimated standard error 4 GHB 15538 21 < 8466 1332 -31.7 1.6 b W =W HB FB -8396 541 44092 23719 <s 0.9 1.1 w HS -30341 32215 ^FS -145974 123278 V 141881 113066 AG° HN 5333 1017 AG-™, FN -15721 1423 -12031 '. . 775 -31161 1662 -41347 1124 -58853 1881 HN HCh HCI b -14125 2250 FN FCh FC1 b -5154 3669 4 CHTC -8509 401 Sic -2086 2813 HTc -56989 9276 -10239 2104 < Fo -14249 252 AG° En -7566 129 A-quantities are quantities of formation from MgO, MgF2, Si0 2, and H20. Excess parameters have been constrained to be equal. 63 FB ,,0 'FB Fo ,0 Table XIV. Matrix of Co r r e l a t i o n C o e f f i c i e n t s for Properties i n Table XIII. P a r a m e t 6 r A GHB A G F B A S F B *HB " A GHS A SHS ^HS ^FS AG° 1.000 0.212 0.009 -0.878 -0.010 0.068 0.009 0.010 -0.009 HB AG° 0.212 1.000 0.094 -0.246 -0.959 -0.001 0.915 0.951-0.900 A s ° 0.009 0.094 1.000 -0.009 -0.095 0.002 0.084 0.090 -0.082 Wv„ a -0.878 -0.246 -0.009 1.000 0.015 -0.077 -0.014 -0.015 0.013 HB v A G ° -0.010 -0.959 -0.095 0.015 1.000 -0.011 -0.980 -0.999 -0.970 rib A s ° 0.068 -0.001 0.002 -0.077 -0.011 1.000 0.011 0.011 -0.010 W„. 0.009 0.915 0.084 -0.014 -0.980 0.011 1.000 0.989-0.999 HS W ' 0.010 0.951 0.090 -0.015 -0.999 0.011 0.989 1.000 -0.982 E -0.009. -0.900 -0.082 0.013 0.970 -0.010 -0.999 -0.982 1.000 A G ° 0.000 -0.092 -0.009 0.001 0.095 0.002 -0.093 -0.095 0.092 HN A G ° -0.002 0.515 0.050 -0.002 -0.524 -0.011 0.500 0.519 -0.492 FN AG° , 0.000 -0.042 -0.004 0.000 0.044 0.001 -0.043 -0.044 0.043 HCh A G ° -0.001 0.438 0.043 -0.002 -0.446 -0.009 0.426 0.442 -0.419 FCh A G ° 0.000 -0.005 -0.001 0.000 0.006 0.000 -0.007 -0.006 0.007 HCl A G ° C 1 -0.001 0.372 0.036 -0.002 -0.378 -0.008 0.361 0.375 -0.355 'HN w a 0.000 0.126 0.012-0.001-0.130-0.002 0.125 0.129-0.123 w T a 0.000 0.090 0.D09- -0.001 -0.093 -0.001 0.090 0.092 -0.089 FN A GHTc 0.000 0.020 0.002 0.000 -0.020 0.000 0.019 0.020 -0.018 A G ° 0.000 0.128 0.012 -0.001 -0.130 -0.002 0.124 0.129 -0.122 FTc W„„ 0.000 0.087 0.009 0.000 -0.089 -0.002 0.084 0.088 -0.083 HTc ^ F T c 0.000 0.090 0.009 0.000 -0.091 -0.002 0.087 0.090 -0.085 AG° 0.000 0.022 0.002 0.000 -0.023 0.000 0.021 0.022 -0.021 AG° 0.000 0.021 0.002 0.000 -0.021 0.000 0.020 0.021 -0.020 64 Table XIV. (Continued) Parameter A G ° N A G ° N Acg^ A G ° C H A G ° C 1 A G ° C 1 3 a AG^ AG° 0.000 -0.002 0.000 -0.001 0.000 -0.001 0.000 0.000 0.000 l ib AG° -0.092 0.515 -0.042 0.438 -0.005 0.372 0.126 0.090 0.020 r D AS° -0.009 0.050 -0.004 0.043 -0.001 0.036 0.012 0.009 0.002 r JJ Wm a 0.001 -0.002 0.000 -0.002 0.000 -0.002 -0.001 -0.001 0.000 AG° 0.095 -0.524 0.044 -0.446 0.006 -0.378 -0.130 -0.093 -0.020 rib AS° 0.002 -0.011 0.001 -0.009 0.000 -0.008 -0.002 -0.001 0.000 rib >VU„ -0.093 0.500 -0.043 0.426 -0.007 0.361 0.125 0.090 0.019 rib W_.e -0.095 0.519 -0.044 0.442 -0.006 0.375 0.129 0.092 0.020 CO Es 0.092 -0.492 0.043 -0.419 .0.007 -0.355 -0.123 -0.089 -0.018 AG° 1.000 -0.243 0.842 -0.362 0.562 -0.171 -0.786 -0.959 0.213 AG° -0.243 1.000 -0.061 0.902 0.091 0.804 0.310 0.301 0.191 FN A G ° C H 0.842 -0.061 1.000 -0.076 0.906 0.168 -0.397 -0.717 0.618 AG° -0.362 0.902 -0.076 1.000 0.157 0.911 0.459 0.449 -0.307 bLn A G ° C 1 0.562 0.091 0.906 0.157 1.000 0.410 -0.092 -0.370 0.849 A G ° C 1 -0.171 0.804 0.168 0.911 0.410 1.000 0.355 0.308 0.517 Nmj a -0.786 0.310 -0.397 0.459 -0.092 0.355 1.000 0.774 0.131 HN & U T a -0.959 0.301 -0.717 0.449 -0.370 0.308 0.774 1.000 0.001 FN AG° 0.213 0.191 0.618 0.307 0.849 0.517 0.131 0.001 1.000 rlic AG° 0.330 0.211 0.181 0.026 0.087 -0.002 -0.448 -0.287 0.052 r l c -0.328 0.202 -0.107 0.383 0.032 0.414 0.500 0.315 0.097 HTc K__ -0.332 0.206 -0.111 0.388 0.029 0.422 0.499 0.320 0.112 FTc AG° 0.227 0.206 0.665 0.331 0.913 0.554 0.155 0.000 0.932 Fo AG° 0.220 0.199 0.642 0.320 0.882 0.537 0.141 0.001 0.991 En Table XIV. (Concluded) Parameter A G ° T C A G ° Q A G ° N HB AG° 0.128 0.087 0.090 0.022 0.021 r a AS° 0.012 0.009 0.009 0.002 0.002 FB Fo ,o 7En AG° 0.000 0.000 0.000 0.000 0.000 w„„ a -0.001 0.000 0.000 0.000 0.000 H B AG° -0.130 -0.089 -0.091 -0.023 -0.021 rib AS° -0.002 -0.002 -0.002 0.000 O.iOOO no W„. 0.124 0.084 0.087 0.021 0.020 rib 0.129 0.088 0.090 0.022 0.021 r o E g -0.122 -0.083 -0.085 --0.021 -0.020 AG° 0.330 -0.328 -0.332 0.227 0.220 HN ,o JFN ,o JHCh ,o 7FCh ,o JHC1 ,o 7FC1 'HN AG° 0.211 0.202 0.206 0.206 0.199 FF AG° 0.181 -0.107 -0.111 0.665 0.642 ril AG° 0.026 0.383 0.388 0.331 0.320 r t A6° , 0.087 0.032 0.029 0.913 0.882 nt AG°^ 1 -0.002 0.414 0.422 0.554 0.537 W„„ a -0.448 0.500 0.499 0.155 0.141 Wmi a -0.287 0.315 0.320 0.000 0.001 FN AG° 0.052 0.097 0.112 0.932 0.991 H l C AG° 1.000 -0.882 -0.851 0.016 0.044 FTc W„„ -0.882 1.000 0.974 0.124 0.105 HTc -0.851 0.974 .1.000 0.123 0.120 FTc AG° 0.016 0.124 0.123 1.000 0.967 AG° 0.044 0.105 0.120 0.967 1.000 a W =w w =w =w w =w =w . see Table XIII. HB FB' HN HCh HCl' FN FCh FC1' 66 SELECTED REFERENCES American Society for Testing and Materials committee D-2 on petroleum products and lubricants, and American Petroleum Institute research project 44 on hydrocarbons and related compounds., 1971. Physical constants of hydrocarbons Ci to C I Q . American Society for Testing and Materials, 72 p. Barron, T.H.K., Berg, W.T., and Morrison, J.A. 1959. On the heat . capacity of crystalline magnesium oxide. Proceddings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 250, pp. 70-83. Bevington, P.R. 1969. Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill Book Company, New York, 336 p. Bourne, J.H. 1974. The petrogensis of the humite group minerals in regionally metamorphosed marbles of the grenville supergroup. Ph.D., Thesis, Queen's University, 159 p. Bratland, D., Fekri, A., Grjotheim, K., and Motzfeldt, K. 1970. Vapor pressure of s i l i c o n tetrafluoride above mixtures of fluorides and s i l i c a , I. The system CaF2~Si02. Acta Chemica Scandinavica, 24, pp. 864-870. Burnham, C.W., Holloway, J.R., and Davis, N.F. 1969. Thermodynamic properties of water to 1000 °C and 10,000 bars. Geological Society of America, Special Paper, 132, 96 p. Charlu, T.V., Newton, R.C, and Kleppa, O.J. 1975. Enthalpies of forma-tion at 970 K of compounds in the system Mg0-Al203-Si02 from high temp-erature: solution calorimetry. Geochimica et Cosmd.chimica Acta, 39, pp. 1487-1497. Chernosky, J.V., Jr. 1974. The stability f i e l d of anthophyllite - an , experimental redetermination, (abstract). Geological Society of America. Abstracts with Programs, 6, p. 687. 1976. The stability of anthophyllite - a reevaluation based on new experimental data. American Mineralogist, 61, pp. 1145-1155. Crane, R.L. and Ehlers, E.G. 1969. The system MgF2-Mg0-H20. American Journal of Science, 267, pp. 1105-1111. Edmister, W.C 1968. Applied hydrocarbon thermodynamics, part 32, com-. p o s s i b i l i t y factors and fugacity coefficients from the Redlich-Kwong equation of state. Hydrocarbon Processing, 47, pp. 239-244. 67 Evans, H.T., Jr., Appleman, D.E., and Handwerder, D.S. 1963. The least squares refinement of crystal unit cells with powder diffraction data by an automatic computer indexing method. American Crystallographic Association Annual Meeting, Cambridge Massachusetts, Program Abstracts, pp. 42-43. Friedman, A.S. and Haar, L. 1954. High-speed machine computation of ideal gas thermodynamic functions. I. Isotopic water molecules. Journal of Chemical Physics, 22, pp. 2051-2058. Fyfe, J.R., Turner, F.J., and Verhoogen, J. 1958. Metamorphic reactions and metamorphic facies. Geological Society of America, Memoir 73, 259 p. Giauque, W.F. and Archibald, R.C. 1937. The entropy of water from the third law of thermodynamics. The dissociation pressure and calorimetric heat of the reaction Mg(0H)2 = MgO + H20. The heat capacities of Mg(0H)2 and MgO from 20 to 300 °K. Journal of the American Chemical Society, 59, pp. 561-569. Golub, G.H. and Reinsch, C. 1970. Singular value decomposition and least squares solutions. Numerische Mathematik, 14, pp. 403-420. Greenwood, H.J. 1963. The synthesis and stability of anthophyllite. Journal of Petrology, 4, pp. 317-351. 1975. Thermodynamically valid projections of extensive phase rela-tionships. American Mineralogist, 60, pp...1—8. Guggenheim, E.A. 1952. Mixtures. Oxford, Clarendon Press, London, 275 p. Holloway, J.R., Eggler, D.H., and Davis, N.F. 1971 Analytical expression for calculating the fugacity and free energy of H20 to 10,000 bars and 1300 °C. Geological Society of America Bulletin, 82, pp. 2639-2642. Jones, N.W., Ribbe, P.H., and Gibbs, G.V. 1969. Crystal chemistry of the humite minerals. American Mineralogist, 54, pp. 391-411. Kelley, K.K. 1943. Specific heats at low temperature of magnesium ortho-s i l i c a t e and magnesium metasilicate. Journal of the American Chemical Society; 65, pp. 339-341. King, E.G., Ferrante, M.J., and Pankratz, L.B. 1975. Thermodynamic data for Mg(0H)2 (burcite). United States, Bureau of Mines, Report of Investigations, 8041, 16 p. Lawson, C.L. and Hanson, R.J. 1974. Solving Least Squares Problems. Prentice-Hall, Incorporated, Englewood C l i f f s , New Jersey, 340 p. Maier, CB. and Kelley, K.K. 1932. An equation for the representation of high temperature heat content data. Journal of the American Chemical Society, 54, pp. 3243-3246. 68 Mathews, J.F. 1972. The c r i t i c a l constants of inorganic substances. Chemical Reviews, 72, pp. 71-100. McGlashan, M.L. (Chairman, commission on symbols, terminology and units). 1970. Manual of symbols and terminology for physicochemical quantities and units. Pure and Applied Chemistry, 21, pp. 1-44. Muan, A. 1967. Determination of thermodynamic properties of silicates from locations of conjugation lines in ternary systems. American Mineralogist, 52, pp. 797-804. Munoz, J.L. and Eugster, H.P. 1969. Experimental control of fluorine reactions in hydrothermal systems. American Mineralogist, 54, pp. 943-959. Munoz, J.L. and Ludington, S.D. 1974. Fluorine-hydroxyl exchange in bio t i t e . American Journal of Science, 274, pp. 396-413. Naylor, B.F. 1945. Heat contents at high temperatures of magnesium and calcium fluorides. Journal of the American Chemical Society, 67, pp. 150-152. Orr, R.L. 1953. High temperature heat contents of magnesium orthosilicate and ferrous orthosilicate. Journal of the American Chemical Society, 75, pp. 528-529. Plackett, R.L. 1960. Principles of Regression Analysis. Oxford, Clarendon Press, London, 173 p. Robie, R.A., Bethke, P.M., and Beardsly, K.M. 1967. Selected X-ray crystal-lographic data, molar volumes, and densities of minerals and related sub-stances. United States, Geological Survey, Bulletin, 1248, 87 p. Robie, R.A. and Stout, J.W. 1963. 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APPENDIX Measured Cell Parameters of Solid Solution Phases Phase a X a G (ft) 2>b c Q (ft) a ( ) 3 V (cm ) Sellaite 1.00 4.6249(4) 4.6249(4) 3.0502(3) 19.646(3) Sp. gr. ? 0.90 4.7270(7) 4.5997(6) 3.0502(4) 19.971(4) 0.85 4.7731(9) 4.5879(6) 3.0503(3) 20.115(4) Brucite 0.00 3.1464(2) 4.7695(7) 24.627(4) P3ml 0.05 3.1377(3) 4.7556(6) 24.419(4) 0.15 3.1218(2) .4.7466(5) 24.126(3) 0.15 3.1218(3) 4.7469(6) 24.129(4) 0.20 3.1146(3) 4.7469(9) 24.018(5) 0.25 3.1077(2) 4.7490(5) 23.921(4) Norbergite 1.00 4.7072(5) 10.2650(9) 8.7240(10) 63.469(8) Pbnm 0.90 4.7085(10) 10.2691(21) 8.7562(22) 63.746(18) 0.70 4.7108(10) 10.2808(15) 8.8200(19) 64.315(15) Chondrodite 1.00 4.7247(9) 10.2490(9) 7.7880(18) 109.2(2) 107.232(23) P2x/b 1.00 4.7264(8) 10.2477(8) 7.7907(13) 109.2(2) 107.297(17) 0.80 4.7263(7) 10.2512(9) 7.8279(13) 109.1(2) 107.896(18) 0.80 4.7285.(14) 10.2542(16) 7.8220(27) 109.1(3) 107.900(37) 0.70 4.7287(13) 10.2519(15) 7.8398(22) 109.1(3) 108.148(30) 0.60 .4.7293(4) 10.2727(4) 7.8573(15) 109.1(2) 108.620(18) Clinohumite 1.00 4.7396(12) 10.2259(33) 13.5819(31) 100.9(5) 194.630(74) P2j/b 0.50 4.7450(19) 10.2356(121) 13.6497(50) 100.8(8) 196.099(210) - 0.40 4.7428(17) 10.2396(50) 13.6619(34) 100.7(5) 196.316(92) Note: Numb ers in parentheses are one estimated standard error in the digit to their immediate l e f t . x = mole fraction fluoro-endmember. 

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