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Carbon dioxide and carbon dioxide-water mixtures : |b P-V-T properties and fugacities to high pressure.. 1990
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Title | Carbon dioxide and carbon dioxide-water mixtures : |b P-V-T properties and fugacities to high pressure and temperature constrained by thermodynamic analysis and phase equilibrium experiments |
Creator |
Mäder, Urs Karl |
Publisher | University of British Columbia |
Date Created | 2011-01-13T20:58:29Z |
Date Issued | 2011-01-13T20:58:29Z |
Date | 1990 |
Description | The thermophysical properties of supercritical CO₂ and H₂O-CO₂ mixtures are reviewed and their computation and prediction improved through theory and experiment. A resolution is attempted among inconsistencies between and within data sets, including P-V-T measurements, phase equilibrium experiments and equations of state. Pure carbon dioxide: Equations of state for CO₂ (Kerrick & Jacobs, 1981; Bottinga & Richet, 1981; Holloway, 1977) are based solely on P-V-T data up to 8 kbar and lead to deviations from phase equilibrium data at pressures greater than 10-20 kbar. Mathematical programming analysis has been applied to the fitting of parameters for an equation of state using simultaneously constraints from phase equilibrium and P-V-T data. Phase equilibrium data up to 42 kbar are used to define a feasible region for the adjustable parameters in free energy space. Each half-bracket places an inequality constraint on the fugacity of CO₂ provided the thermophysical properties of the solid phases are known. Except for magnesite thermophysical data from the mineral data base of Berman (1988) were used. A least squares objective function served to optimize parameters to P-V-T data. The enthalpy of formation of magnesite was revised on the basis of recent low pressure phase equilibrium experiments by Philipp (1988) to —1112.505 kj/mole. Piston-cylinder experiments were performed to constrain the equilibrium magnesite ⇌ periclase + CO₂ at high pressure. The equilibrium boundary is located at 12.1(±1) kbar, 1173-1183 °C (±10), and at 21.5(±1) kbar, 1375-1435 °C (±10). A van der Waals type equation of state with five adjustable parameters has been developed for CO₂. The function is smooth and continous above the critical region, behaves well in the high and low pressure limits, and the calculation of ʃ VdP for free energy does not require numerical integration. Computed free energies are consistent with all phase equilibrium data at high pressure, and computed volumes agree reasonably with P-V-T measurements. The proposed equation is: [ Equation omitted ] with B₁ = 28.0647, B₂ = 1.7287.10⁻⁴, B3 = 83653, A₁ = 1.0948.10⁹, A₂ = 3.3 7 47.10⁹, and R = 83.147, in units of Kelvin, bar and cm³/mole. The equation is recommended up to 50 kbar and above 400 K with reasonable extrapolation capabilities. A FORTRAN source code to evaluate the volume and fugacity is provided. Thermophysical properties for the calcium carbonate polymorphs calcite-I, IV, V, and aragonite were derived that are consistent with phase equilibrium experiments. Data required for further improvement include high pressure phase equilibria involving CO₂, constraints on the thermal expansion of magnesite, and P-V-T data to resolve inconsistencies among existing measurements. Water-carbon dioxide mixtures: The two widely used equations of state for H₂O-CO₂ mixtures are those proposed by Kerrick & Jacobs (1981) and by Holloway (1977)-Flowers (1979). Evaluation of existing equations and data is difficult due to inconsistencies among experimental studies. P-V-T-X data by Franck ＆ Todheide (1959) are inconsistent with data by Greenwood (1973) and Gehrig (1980), and cannot be reconciled with measured phase equilibria in H₂O-CO₂ fluid mixtures. Data by Greenwood and Gehrig are in loose agreement but extend only to 600 bar and do not constrain activities at higher pressures. A procedure is developed for using experimental phase equilibrium constraints to put limits on the fugacities of components of the fluid mixture. Inconsistencies among phase equilibrium studies are discussed. It is concluded that the data base available is not yet adequate to derive a reliable equation of state for H₂O-CO₂ mixtures. Future work must include P-V-T-X measurements to 8 kbar and phase equilibrium studies to resolve inconsistencies. These can constrain deviations from ideal mixing in the fluid phase, and constrain specific volumes at high pressures where P-V-T-X data connot be obtained. |
Subject |
Carbon Dioxide -- Thermal Properties Thermodynamics Phase Rule And Equilibrium Equations Of State |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2011-01-13T20:58:29Z |
DOI | 10.14288/1.0052688 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geological Science |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/30629 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0052688/source |
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CARBON DIOXIDE AND CARBON DIOXIDE - WATER MIXTURES: P - V - T PROPERTIES AND FUGACITIES TO HIGH PRESSURE AND T E M P E R A T U R E CONSTRAINED BY THERMODYNAMIC ANALYSIS AND PHASE EQUILIBRIUM EXPERIMENTS By Urs Karl Mader Dipl. Natw. E T H , Swiss Federal Institute of Technology, Zurich M.Sc. (Geology), The University of British Columbia A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES GEOLOGICAL SCIENCES We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C O L U M B I A August 1990 © Urs Karl Mader, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or [her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ICH The University of British Columbia Vancouver, Canada Date / f < - y u ^ / /<5, / S ^ O DE-6 (2/88) Abstract The thermophysical properties of supercritical CO2 and H2O-CO2 mixtures are reviewed and their computation and prediction improved through theory and experiment. A reso- lution is attempted among inconsistencies between and within data sets, including P-V-T measurements, phase equilibrium experiments and equations of state. Pure carbon dioxide: Equations of state for CO2 (Kerrick & Jacobs, 1981; Bottinga & Richet, 1981; Holloway, 1977) are based solely on P-V-T data up to 8 kbar and lead to deviations from phase equilibrium data at pressures greater than 10-20 kbar. Mathematical programming analysis has been applied to the fitting of parameters for an equation of state using simultaneously constraints from phase equilibrium and P-V- T data. Phase equilibrium data up to 42 kbar are used to define a feasible region for the adjustable parameters in free energy space. Each half-bracket places an inequality constraint on the fugacity of C 0 2 provided the thermophysical properties of the solid phases are known. Except for magnesite thermophysical data from the mineral data base of Berman (1988) were used. A least squares objective function served to optimize parameters to P-V-T data. The enthalpy of formation of magnesite was revised on the basis of recent low pressure phase equilibrium experiments by Philipp (1988) to —1112.505 kj/mole. Piston-cylinder experiments were performed to constrain the equilibrium magnesite periclase + CO2 at high pressure. The equilibrium boundary is located at 12.1(±1) kbar, 1173-1183 °C (±10), and at 21.5(±1) kbar, 1375-1435 °C (±10). A van der Waals type equation of state with five adjustable parameters has been developed for CO2. The function is smooth and continous above the critical region, ii behaves well in the high and low pressure limits, and the calculation of / VdP for free energy does not require numerical integration. Computed free energies are consistent with all phase equilibrium data at high pressure, and computed volumes agree reasonably with P-V-T measurements. The proposed equation is: V-b(V,T) TV2 V4 ' 1 ' ' V3 + C ' Bt+B2T with Bx = 28.0647, B2 = 1.7287-10"4, B3 = 83653, A x = 1.0948-109, A2 = 3.3 7 47-109, and R = 83.147, in units of Kelvin, bar and cm3/mole. The equation is recommended up to 50 kbar and above 400 K with reasonable extrapolation capabilities. A FORTRAN source code to evaluate the volume and fugacity is provided. Thermophysical properties for the calcium carbonate polymorphs calcite-T, IV, V, and aragonite were derived that are consistent with phase equilibrium experiments. Data required for further improvement include high pressure phase equilibria involving CO2, constraints on the thermal expansion of magnesite, and P-V-T data to resolve inconsistencies among existing measurements. Water-carbon dioxide mixtures: The two widely used equations of state for H2O-CO2 mixtures are those proposed by Kerrick & Jacobs (1981) and by Holloway (1977)-Flowers (1979). Evaluation of existing equations and data is difficult due to inconsistencies among experimental studies. P-V-T-X data by Franck k. Todheide (1959) are inconsistent with data by Greenwood (1973) and Gehrig (1980), and cannot be reconciled with measured phase equilibria in H2O-CO2 fluid mixtures. Data by Greenwood and Gehrig are in loose agreement but extend only to 600 bar and do not constrain activities at higher pressures. A procedure is developed for using experimental phase equilibrium constraints to put limits on the fugacities of components of the fluid mixture. Inconsistencies among phase equilibrium studies are discussed. iii It is concluded that the data base available is not yet adequate to derive a reliable equation of state for H2O-CO2 mixtures. Future work must include P-V-T-X measure- ments to 8 kbar and phase equilibrium studies to resolve inconsistencies. These can constrain deviations from ideal mixing in the fluid phase, and constrain specific volumes at high pressures where P-V-T-X data connot be obtained. iv Table of Contents Abstract i i List of Tables v i i i List of Figures ix Acknowledgement xi i Preface xi i i 1 A n Equation of State for C 0 2 to high P and T 1 1.1 Introduction 1 1.2 Method 4 1.3 Equation of State 8 1.4 P - V - T Data 11 1.5 Phase Equilibrium Data 13 1.6 New Experiments on Magnesite Periclase + CO2 16 1.7 Magnesite Properties 16 1.8 Results 18 1.9 Calcite(I-IV-V) - Aragonite 23 1.10 Conclusions 28 Bibliography 30 2 Properties of C Q 2 - H 2 Q Mixtures at High Pressure and Temperature 38 v 2.1 Introduction 38 2.2 The System C 0 2 - H 2 0 39 2.3 Constraints on Equations of State 42 2.4 Existing Equations of State 52 2.5 Thermodynamic Relationships for Fluid Mixtures 60 2.6 P - V - T - X Properties Constrained by Phase Equilibrium Data 70 2.7 P - V - T - X Data on H 2 0 - C 0 2 Mixtures 71 2.8 Phase Equilibrium Data Involving H 2 0 - C 0 2 Mixtures 74 2.9 Results, Discussion and a Possible Approach 82 2.10 Conclusions and Future Work 88 Bibliography 96 Appendices 106 A Integration of the Equation of State 106 B F O R T R A N - 7 7 Subroutine 109 C Piston-Cylinder Experiments 114 C . l Piston-Cylinder Apparatus 114 C.2 Sample Assembly 114 C.3 Temperature Calibration 116 C.4 Pressure Calibration 121 C.5 Run Procedures for Phase Equilibrium Experiments 131 C.6 Equilibrium magnesite r = ± periclase + C 0 2 131 D Comparison of Calculated Volumes with Measured Volumes 141 vi E P - V - T - X Measurements: Data and Experimental Procedures 155 vii List of Tables 1.1 Experimentally measured P-V-T properties of C 0 2 , 11 1.2 List of phase equilibrium studies involving CO2 14 1.3 Constraints on CO2 fugacities imposed by experimental brackets 17 1.4 Thermodynamic properties of calcite polymorphs 25 2.5 Excluded volumes for various equations of state 45 2.6 Notation for thermodynamic equations 61 C.7 Temperature measurements for thermal gradient calibration 117 C.8 Effect of pressure on the emf of the thermocouple 119 C.9 Melting point of gold under pressure 127 C.10 Melting point of silver under pressure 127 C . l l Pressure corrections due to friction 130 C.12 List of experiments on magnesite decarbonation 134 C.13 Summary of run products and textures of magnesite decarbonation exper- iments 137 viii List of Figures 1.1 Comparison oi equations of state with phase equilibrium experiments . . 2 1.2 Mathematical behaviour of the Mader & Berman equation of state . . . . 9 1.3 Isobaric V — T diagram of high pressure P-V-T measurements 12 1.4 Isobaric V — T diagram with Mader & Berman equation of state con- strained by P-V-T data only 19 1.5 Isobaric V — T diagram comparing Mader &; Berman equation of state with P-V-T measurements 20 1.6 P — T diagram of phase equilibria involving magnesite 24 1.7 P — T diagram of phase equilibria involving calcite polymorphs 29 2.8 Geometric interpretation of partial molar excess volume 65 2.9 Schematic T — X diagram of a single fluid species equilibrium 67 2.10 Equilibrium calcite+andalusite+quartz anorthite-f CO2 75 2.11 Equilibrium quartz+dolomite+EkO calcite + talc+C02 78 2.12 Equilibrium magnesite+talc forsterite+H^O+CC^ 80 2.13 Graphs of In 7 versus pressure 87 2.14 Equilibria involving margarite, andalusite and corundum 93 C.15 Pyrex sample assembly for piston-cylinder apparatus 115 C.16 Sample assembly for thermal gradient calibration 116 C. 17 Thermal gradients within the piston-cylinder apparatus 118 C.18 Sample assembly for melting point calibration of gold and silver 123 C.19 Thermal analysis traces of melting of gold under pressure 125 ix C.20 Thermal analysis traces of melting of silver under pressure 126 C.21 Melting of gold and silver under pressure 128 C.22 Experimental brackets on the equilibrium magnesite ^ periclase + carbon dioxide 132 C.23 Plate I of textures of experimental run products 138 C.24 Plate II of textures of experimental run products 139 C. 25 Plate III of textures of experimental run products 140 D. 26 Comparison of computed volumes with data of Shmonov & Shmulovich . 142 D.27 Comparison of computed volumes with data of Kennedy 144 D.28 Comparison of computed volumes with data of Juza et al 145 D.29 Comparison of computed volumes with data of Tsiklis et al 146 D.30 Comparison of computed volumes with data of Michels et al . 148 D.31 Comparison of computed volumes with data of Vukalovich et al 149 D.32 Comparison of computed volumes with data of Amagat 150 D.33 Comparison of computed volumes with data of Michels & Michels . . . . 151 D.34 Comparison of computed volumes with data of Vukalovich et al 153 D. 35 Comparison of computed volumes with data of Vukalovich et al 154 E. 36 P - Ve graphs: 400 °C, XCo2 =0.1-0.4 162 E.37 P - Ve graphs: 400 °C, XCo2 =0.6-0.9 163 E.38 P - Ve graphs: 450 °C, XCo2 =0.1-0.4 164 E.39 P-Ve graphs: 450 °C, XCo2 =0.6-0.9 165 E.40 P~Ve graphs: 500 °C, XCo2 =0.1-0.4 166 E.41 P - Ve graphs: 500 °C, XCo2 =0.6-0.9 167 E.42 P-Ve graphs: 600 °C, XCo2 =0.1-0.4 168 E.43 P - Ve graphs: 600 °C, XCo2 =0.6-0.9 169 x E.44 P - Ve graphs: 700 °C, XCo2 =0.1-0.4 170 E.45 P - Ve graphs: 700 °C, A r C o 2 =0.6-0.9 171 E.46 P-Ve graphs: 800 °C, XCo2 =0.1-0.4 172 E.47 P - V e graphs: 800 °C, XCo2 =0.6-0.9 173 E.48 X - Ve graphs: 400 °C, 200 k 300 bar 174 E.49 X - Ve graphs: 400 °C, 400 k 500 bar 175 E.50 X - Ve graphs: 500 °C, 200 k 400 bar 176 E.51 X -Ve graphs: 500 °C, 500 k 1000 bar 177 E.52 X - Ve graphs: 600 °C, 200 k 400 bar 178 E.53 X - Ve graphs: 600 °C, 500 k 1000 bar 179 E.54 X - Ve graphs: 700 °C, 200 k 500 bar 180 xi Acknowledgement This thesis could not have been produced without the help of many friends and colleagues. I was lucky to be able to learn and profit from many outstanding scientists, particularly from Hugh Greenwood, Rob Berman, Tom Brown, 'Capi' DeCapitani, Kelly Russell, Pat Meagher and John Ross. I am indebted to Hugh Greenwood for his influence and inspiration as a scientist, a colleague and a thoughtful person. What I have achieved in the laboratory I owe to him. Rob Berman gave more of his time and help than I could possibly ask for. I always enjoyed those rather intense work sessions with Rob on the computer, in the lab, or just paddling down a river on occasion. Tom Brown stopped me thinking every now and then with one of his subtle but fundamental questions. Equally important were all the people who kept operations running smoothly: the machinists Doug Poison and Ray Rodway, John Knight who helped with electronics and performed the microprobe analyses, Stanya Horsky who spent time translating from Russian and assisting with analytical matters, Bryon Cranston, Yvonne Douma, Ed Montgomery, Mark Baker, the office staff and the crew from the inter-library loans office. John Holloway kindly provided computer codes for his equation of state. My fellow graduate students, ice hockey mates and ski bums provided stimulus and a balance to matters scientific. The ever so important funding was arranged with grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada to Hugh Greenwood and Rob Berman (Geological Survey of Canada, Ottawa), and through a grant from the Canada- British Columbia Mineral Development Agreement administerd by the B.C. Geological Survey Branch with the help of Dan Hora. The Unversity Graduate Fellowship program at UBC provided financial assistance for more than four years for which I am grateful. My parents who did not like to see me leave Switzerland, and Ursula who did not like to see me work late nights, but married me in spite of it, deserve my deep appreciation. Stephan, ten months of age, crawling, and definitly very human set the importance of my thesis into much needed perspective during the final leg. xii Preface This thesis is a contribution towards the computation and prediction of thermophysi- cal properties of supercritical fluids, specifically pure carbon dioxide and water-carbon dioxide mixtures. The methodology tightly interweaves theory and experiment, rele- vant to both synthetic and natural chemical systems. The thesis is part of an effort by the UBC-petrology group over more than two decades to create a reliable data base of thermophysical properties of minerals and fluids applicable to natural systems. This mul- tidisciplinary approach first culminated in the publication of a comprehensive data base by Berman in 1988, and software to compute, phase equilibria by Brown et al. (1988). Since then no petrology meeting has passed without presentations making extensive use of these powerful tools. An impressive example was this year's 'quantitative methods in petrology' symposium in Vancouver hosted by the Mineralogical Society of Canada in honour of Hugh Greenwood. The system CO2-H2O is chemically simple. The large range in pressure and tem- perature relevant to geological problems require moving beyond rigorous theory into the world of empiricism. This empirical element introduces arbitrarily made assumptions that cannot be tested as being 'right' or 'wrong', but only as being 'practical' or 'imprac- tical'. As a logical consequence many of the numerical values offered in this thesis are transitional in character and will change with the continuing refinement and enlargement of the experimental and natural data base, the advancement in computer technology and the development of better theories. The experimental contributions, the methodology and the assessment of existing work are expected to form a permanent contribution. The insight gained from thermodynamic modeling forms a firm basis to recommend further xiii experimental work that would put the most stringent constraints on the thermophysical properties of minerals and fluids. Most aspects and results of this thesis were presented orally at meetings in Denver and Vancouver (Mader et al. 1988,1990). The first chapter without appendices was submitted to the Journal of Petrology in August, 1990 (Mader &c Berman, in review) and parts of the second chapter will appear in a special volume of the Canadian Mineralogist combining contributions to the Greenwood Symposium on 'quantitative methods in petrology' held in Vancouver, May 1990 (Mader, in preparation). The experimental part (appendix C) will form part of a separate publication combined with an analysis of calibration and friction relevant to piston-cylinder work (Mader, in preparation). The two chapters are organized as stand-alone papers each having a list of references. Both chapters include its own introduction and conclusions. The table of contents, the lists of figures and tables, and the pagination combine the two parts into a single docu- ment in order to fulfill library requirements. xiv to Ursula and my parents Chapter 1 An Equation of State for Carbon Dioxide to High Pressure and Temperature 1.1 Introduction Volumetric properties of CO2 at high pressures are needed for the solution of a number of problems in metamorphic and igneous petrology, one of the most important being the computation of phase equilibria (i.e. Perkins et al, 1986; Brown et al., 1988). Be- cause volumes have been measured up to only 8 kbar (Shmonov k Shmulovich, 1974), extrapolation using an equation of state is required. Existing equations of state for carbon dioxide suitable for computation of geological phase equilibria include those by Holloway (1977, 1981a, 1981b), Touret &; Bottinga (1979), Bottinga k Richet (1981), Kerrick k Jacobs (1981), Powell k Holland (1985, Holland k Powell, 1990), Saxena k Fei (1987a, 1987b), Shmulovich k Shmonov (1975), Mel'nik (1972), and Ryzhenko k Volkov (1971). These equations are based on a variety of theories and incorporate adjustable parameters to fit experimentally measured volumes (see Ferry k Baumgartner, 1987, Holloway, 1987 for detailed reviews, and Prausnitz et al., 1986 for theory). Although most equations fit P-V-T data adequately, it has been pointed out that all of them are inconsistent with phase equilibrium data at pressures above 10 to 20 kbar (figure 1.1) (Haselton et al, 1978; Berman, 1988; Mader et al. 1988; Chernosky k Berman, 1989). Equations that use empirically combined parameters, and polynomial equations in particular, may achieve excellent agreement with observed data, commonly at the expense of reasonable extrapolation. Even equations based on 1 Chapter 1. An Equation of State for C02 to high P and T 2 TEMPERATURE (DEG C) Figure 1.1: Experimental brackets on the equilibrium Mst+En ^ F0+CO2 and reaction boundaries computed with the database of Berman (1988) with revised magnesite data (Chernosky k Berman, 1989, and this study) and equations of state for CO2 by Kerrick k Jacobs, 1981 (K&J), Bottinga k Richet, 1981 (B&R), Holloway, 1977, 1981b (dotted curve, Hoi), and Saxena k Fei, 1987a (S&F). The dashed curve was computed with the B&R equation with magnesite properties perturbed by its estimated uncertainties as discussed in the text. Abbreviations: En: ortho enstatite, Fo: forsterite, Mst: magnesite. The diagram was produced with GEO-Calc (Brown et a/., 1988). Chapter 1. An Equation of State for C02 to high P and T 3 sound theory may show unconstrained behaviour if they contain some empirical element. For example, Kerrick & Jacob's (1981) modified Redlich-Kwong equation (Redlich & Kwong, 1949) has no positive volume defined at progressively higher temperatures with rising pressure, due to the mathematical form of the a(P,T) parameter. The equation proposed by Bottinga & Richet (1981) shows the best extrapolation properties to 42 kbar (Chernosky &; Berman, 1989, and below), but contains discontinuities due to separate parameter fitting for different volume intervals (figure 1.4). The modified Redlich-Kwong equation adopted by Holloway (1977) from de Santis et al. (1974) has a b parameter (excluded volume) derived from corresponding states systematics which leads to volumes at high pressures that are too large. Powell L Holland (1985, Holland & Powell, 1990) fit a simple polynomial function directly to the logarithm of fugacities derived from the equation of state of Shmonov & Shmulovich (1974) and Bottinga Sz Richet (1981), which is computationally efficient but limited in its applications, i.e volumes cannot be derived reliably and extrapolation is not recommended. To circumvent the problem of inadequate P-V-T data at high pressures, alternate approaches utilize shock wave data, although uncertainties are large. Because measur- ments on pure fluid CO2 do not seem to exist, one must rely on approximations based on data for similar fluids related to one another through corresponding states systematics (e.g. Saxena L Fei, 1987a; Helffrich & Wood, 1989). The equations chosen by these au- thors to fit the data are based on some formulation of the intermolecular potential (e.g. Lennard-Jones), but have the disadvantage of incorporating discontinuities where they require a different function for pressures below several kilobars. Saxena & Fei (1987a) fit a virial-like equation to shock wave data valid at pressures above 5 kbar. The purpose of the present paper is to provide metamorphic and igneous penologists with an equation of state for carbon dioxide that is compatible with existing experimental data, both phase equilibrium and P-V-T, that extrapolates reliably to upper mantle Chapter 1. An Equation of State for C02 to high P and T 4 conditions, and that is mathematically tractable. Preliminary results were reported by Mader et al, 1988. A FORTRAN-77 coded subroutine is supplied (appendix B) that computes fugacitiy and volume at specified pressure and temperature. 1.2 Method Provided a reliable data base of thermophysical properties of minerals is available, ther- modynamic properties of CO2 may be tested by comparing calculated and experimentally determined phase equilibria (Haselton et al., 1978; Ferry & Baumgartner, 1987; Berman, 1988; Chernosky h Berman, 1989). Inspection of figure 1.1 indicates a general failure of existing equations of state for CO2 calibrated on P-V-T measurements only, to extrapo- late to high pressures. An alternate approach incorporates phase equilibrium experiments as constraints rather than merely as tests of thermodynamic parameters. This is espe- cially important in the present context as phase equilibrium data for CO2 extend to much higher pressures (42 kbar) than P-V-T data (8 kbar). Before proceeding, it is important to demonstrate that the inconsistencies illustrated in figure 1.1 are due to inadequate equations of state for CO2 and not to errors in the thermodynamic properties of the minerals. The standard state properties, AfHPr'Tr, gPrJr a n ( i yPT.,Tr^ Q f gjj minerals involved in the equilibria of table 1.2 are tightly con- strained by calorimetry and low pressure phase equilibrium data (see Berman, 1988, for details). The enthalpy of formation for magnesite has been ajusted to a less neg- ative value compared to Chernosky & Berman (1989) for reasons discussed in section 'Magnesite'. The most poorly constrained properties are the heat capacity and thermal expansivity of magnesite, for which the data extend only to 750 K (Kelly, 1960) and 773 K (Markgraf & Reeder, 1985), respectively. The sensitivity of the calculated phase Chapter 1. An Equation of State for CO2 to high P and T 5 equilibria to possible errors in extrapolation of these magnesite properties can be ex- amined by adjusting the functions given by Berman (1988) and Chernosky & Berman (1989) within experimental uncertainties in such a way that the inconsistencies between existing equations of state and phase equilibrium data are minimized. The heat capacity function of Berman (1988), adjusted from Berman & Brown (1985), is already aimed at minimizing inconsistencies with the Kerrick & Jacobs (1981) equation of state at high pressure, i.e. to render magnesite less stable. In order to decrease the stability of magne- site the thermal expansion was maximised while maintaining average %-deviations from measured values within 100 % of those computed with the best fit of Berman (1988). This results in an increase of 25 % of the v4 term, and a decrease of 21 % of the v3 term (to adjust for increased curvature) compared to Berman (1988) using his equation (5) Vp'T/Vp'Tr = 1 + v3(T - TT) + v4(T - Trf. Phase equilibria computed with this perturbed volume function for magnesite are still inconsistent with all equations of state for CO2 (figure 1.1). It is concluded that inconsistencies cannot be attributed entirely to errors in the properties of minerals, and, instead, phase equilibrium data may be used to constrain the high pressure properties of CO2. Linear programming offers an appropriate mathematical formalism for such an ap- proach, introduced to geologists by Greenwood (1967a), applied to thermodynamic anal- ysis of phase equilibria by Gordon (1973,1977), and extended to mathematical program- ming analysis by Berman et al. (1986) and Berman (1988). Mathematical programming is a tool to handle large optimization problems with a variety of linear or nonlinear equal- ity or inequality constraints.The optimal solution is found by minimizing or maximizing an objective function such as a conventional least squares formulation. An optimal solu- tion, i.e. 'best fit', is consistent with respect to all constraints and renders the smallest (or largest) possible value for the objective function. A survey of nonlinear programming codes may be found in papers by Schittkowski (1980) and Wasil et al. (1989). Chapter 1. An Equation of State for C02 to high P and T 6 In the present study parameters for the equation of state for CO2 are optimized with CO2 fugacities constrained by phase equilibrium half-brackets up to pressures of 42 kbar. The objective function achieves a best fit to P-V-T data up to 8 kbar. An acceptable mathematical solution for the parameters of the equation of state is consistent with all phase equilibrium constraints and fits volumetric data as closely as the data warrant. For the mathematical programming analysis phase equilibrium constraints are written in such a way as to separate the unknown contributions of CO2 to the Gibbs potential from the known contributions of all other phases involved: solids,C02 solids, C O 2 AfHPr,Tr - T • SfT'Tr + £ CPidT - T • J* dT Vp-Tr(P - Pr) + [P(Vf'T - V P r J r ) dP JPr ^ c o 2 j Pv££dP (1.1) J Pr solids + 5 > + In equation (1.1) ARGP'T denotes the change in the Gibbs free energy of a reaction, A&GP'T the apparent free energy of a pure phase as defined in Berman (1988), AfHPr'Tr the enthalpy of formation from the elements, S P r , T r , VPR'TR, the third law entropy and molar volume at standard pressure (P r = 1 bar) and temperature (Tr = 298.15 K), and Cp and v refer to the heat capacity at constant pressure and the stoichiometric coefficient. Given a reliable, internally consistent thermodynamic database for minerals, the only unknown part of equation (1.1) is the last term, / VQQ dP. From the relationship A\\GP'T = 0 at equilibrium, it follows that each experimental half-bracket leads to an inequality of the form A R G P ' T < 0 or A R G P , T > 0 depending whether products or reactants are stable, respectively. We can thus rearrange equation (1.1) accordingly and Chapter 1. An Equation of State for CO2 to high P and T 7 impose an inequality constraint on / VQQ dP for each half-bracket: ^co, \l V££ dP 5 - A R G S L B (1-2) J Pr where ARGIOTOWH denotes all the terms of equation (1.1) but the last one, and the 'less than' refers to the case where products are stable. The fugacity of CO2 is related to the volume integral by the relationship JET I n / £ £ = fvigdP (1.3) J Pr choosing Pr = 1 bar and assuming approximate equality of unit pressure and unit fugacity at 1 bar. R denotes the gas constant. Each inequality constraint of the form of equation (1.2) requires integration of the equation of state and is therefore nonlinear with respect to parameters of the equation of state. For this study the internally consistent thermodynamic database of Berman (1988), including compressibility and expansivity terms, is used to compute AR.G^O W I 1 for about 120 phase equilibrium half brackets (table 1.2). Note that in deriving the above data base high pressure equilibria involving CO2 were not used as constraints in order to avoid introducing errors from existing equations of state. The properties of the solid phases at low pressures are biased through being forced to be consistent with the Kerrick & Ja- cobs (1981) equation used by Berman (1988). Incorporating the same phase equilibrium constraints will force the new equation to be consistent with Kerrick & Jacobs (1981) at low pressures within experimental uncertainties. A set of about 440 measured volumes serves to optimize the difference between computed and observed volumes (table 1.1). The nonlinear optimization problem is solved with a mainframe program using a gen- eral reduced gradient strategy (GRG2, Lasdon & Waren, 1982). An interactive interface implemented by the UBC computing center (Vaessen, 1984), together with home-grown routines for graphical and statistical progress monitoring makes this package a powerful Chapter 1. An Equation of State for CO2 to high P and T 8 tool for large optimization problems. Typical runtimes for equations with 5 adjustable parameters and 100 inequality constraints that do not require numerical integration are 20 to 200 seconds on an Amdahl 5860. Numerical integration increases computing time by a factor of 10 or more. Stability problems do not occur if the mathematical form of the equation of state is chosen carefully. An initial guess of the parameters need not be feasible with respect to the constraints. 1.3 Equation of State There are essentially four types of equations of state from which to choose: i) van der Waals type (Redlich k Kwong, 1949, Holloway, 1977, Touret k Bottinga, 1979, Kerrick k Jacobs, 1981, Bottinga k Richet, 1981), ii) virial type (Saxena k Fei, 1987a, 1987b), iii) molecular potential formulations (Helffrich k Wood, 1989) and iv) empirical functions to directly fit the logarithm of the fugacity (Powell k Holland, 1985, Holland k Powell, 1990). A van der. Waals type of equation was chosen for the following reasons: simple mathematical form, favorable behaviour in the limits (P —> 00, P —> 0), potentially small number of adjustable parameters, and the imitation of subcritical behaviour in the simplest way. Furthermore, empirical polynomials dependent on pressure or temperature were avoided to ensure reasonable extrapolation. Only continuous functions were consid- ered in order to render derivatives meaningful. An equation discussed but not adopted by Bottinga k Richet (1981) served as a starting point for development: V-b TV2 V4 K ' where b, A \ , A2 are adjustable parameters, and R is the gas constant. The last term of equation (1.4) becomes important at small volumes, i.e. high pressures, resulting in increased compressibility at high pressures. The b parameter, a measure of the 'incom- pressible volume' is commonly assumed to be a constant. This approximation is valid Chapter 1. An Equation of State for CO2 to high P and T J I I L " I 1 1 1 1 1 1 1 1 00 o - c j _ | (_) 300 K V ( v a p ) V ( l i q ) o I I I I I I I I I I -40 160 (BAR) 00 i 1 r 1 1 1 1—r -20 0 20 40 60, 80 PRESSURE (KBRR) (X10 1 ) Figure 1.2: Mathematical behaviour of the Mader & Berman equation of state. The volume at infinite pressure is labeled V(inf). Inset figure depicts the van der Waals like behaviour at subcritical pressure and temperature with volumes of the liquid, V(liq), and the vapour, V(vap), connected by a tie line. Chapter 1. An Equation of State for CO2 to high P and T 10 • up to 50 kbar as suggested by Holloway (1987). Corresponding states theory applied j to the van der Waals equation predicts a b parameter of 43 cm3/mole. De Santis et al. j (1974) derived a value for b of 29.7 cm3/mole for their modified Redlich-Kwong equation 1 j based on P-V-T data up to 1400 bar (the same value was adopted by Holloway, 1977). The smallest measured volume is 31 cm3/mole at 7.1 kbar and 100 °C (Tsiklis et al., 1971) and shock wave data (Zubarev & Telegen, 1962) on solid CO2 indicate a volume of 18 cm3/mole (with large uncertainties) at 180 kbar and 900 °C (as revised by Ross & Ree, 1980). It seems obvious that an equation successful at high pressures requires a compressible 'incompressible volume' which may be achieved mathematically by mak- ing b dependent on volume (see also Kerrick & Jacobs, 1981; Bottinga & Richet, 1981; Touret & Bottinga, 1979). An inverse power of the volume dependency of the 6-term was determined empirically subject to the restriction of obtaining an integrable equation. A small temperature dependence of b also proved to be advantageous, resulting in the final equation: P = R T J * i _ + ^ ( 1 5 } V - ( £ 1 + B2T - TV2 V4 K ' } with C = Bsl(B\ + B2T) and 5 independently adjustable parameters B\, B2, B3, A\, A2. Note that the A1/TV2 term yields a better fit to the data than the Redlich-Kwong term a/\/TV(V + b). The function converges to the ideal gas law at very low pressures and has a smallest volume at infinite pressure dependent on temperature and parameter values (figure 1.2). The C-term ensures continuity of the equation to infinite pressure and renders the fourth-order polynomial in V of the denominator integrable (c.f. appendix A). The volume at specified pressure and temperature is determined iteratively and the integration JVdP may be performed analytically (cf. appendix A). At subcritical pressures and temperatures equation (1.5) behaves like the van der Waals equation (figure 1.2, inset). The equation is continuous between 0 bar and infinite pressure, and at Chapter 1. An Equation of State for CO2 to high P and T 11 Author, Year F T [°C] P [kbar] # Average % deviation M&B PVT K&J B&R Shmonov Sz Shmulovich, 74 y 400-700 1.0-8.0 56 2.07 1.62 0.89 1.11 Tsiklis et al., 1971 y 100-400 2.0-7.0 44 0.95 0.80 0.68 1.42 Juza et al., 1965 y 150-475 0.7-4.0 40 1.17 0.69 0.95 0.86 Michels et al, 1935 y 125-150 0.07-3.1 46 2.14 1.33 - 1.14 Kennedy, 1954 y 200-1000 0.03-1.4 72 0.79 0.55 1.17 0.47 Amagat, 1891 y 137-258 0.05-1.0 60 2.20 1.77 - 0.87 Vukalovich et al., 1962 y 200-750 0.01-0.6 104 0.68 0.61 0.39 0.44 Vukalovich et al., 1963a n 125-150 0.02-0.6 43 1.54 1.24 - 0.83 Vukalovich et al., 1963b n 650-800 0.02-0.2 22 0.10 0.08 0.24 0.23 Michels & Michels, 1935 n 100-150 0.03-0.07 35 0.58 0.66 1.33 0.24 Table 1.1: Experimentally measured P-V-T properties of CO2 used to constrain or test the equation of state. Average percent deviations of calculated volumes from measured volumes are quoted for several equations of state: M&B: Mader k Berman equation (1.5) of this study; PVT: equation (1.5) with parameters based on P-V-T data only (see text); K&J: Kerrick k Jacobs (1981); B k R : Bottinga k Richet (1981). Column 'F ' indicates data sets that were used to constrain equation (1.5) (y: yes) and data that were used for comparison only (n: no). Column indicates the number of data points. temperatures above 0 K for volumes larger than the discontinuity at V(inf) in figure 1.2. 1.4 P - V - T Data Experimentally measured volumes were used to minimize the difference between com- puted and measured volumes during optimization procedures. Some data served to test the equation, particularly near the critical region. Data sets range from Amagat's (1891) classical work to Shmonov k Shmulovich's (1974) high pressure measurements (table 1.1). The more important data sets at higher pressures include Shmonov k Shmulovich (1974), Tsiklis et al. (1971), Juza et al. (1965), Michels et al. (1935) and Kennedy (1954). The accuracy of the P-V-T data is almost impossible to establish. Precision at pres- sures below 1 kbar is generally better than 0.2 % and about 0.5 % at higher pressures Chapter 1. An Equation of State for C02 to high P and T 12 300 440 580 720 860 1000 TEMPERATURE (K) Figure 1.3: Isobaric volume-temperature diagram of high pressure P-V-T measurments (S&S: Shmonov & Shmulovich; T,L&T: Tsiklis et al). Trends within individual data sets are delineated for clarity. See text for discussion. Chapter 1. An Equation of State for CO2 to high P and T 13 depending on the type of equipment used. A complete review of equipment and quality of data sets up to 1972 is provided by Angus et al. (1973). At lower pressures and high temperatures possible problems arise from the presence of species other than CO2, such as CO, O2, or from the precipitation of graphite and from oxidation of the pressure vessel walls. Graphite coatings were reported by Vukalovich et al. (1963b) at temper- atures above 800 °C. High pressure equipment is prone to uncertainties due to sealing problems, calibration, thermal gradients, pressure and temperature measurment, and de- formation under pressure. The nonexistent or small areas of overlap between individual high pressure data sets make the detection of inconsistencies difficult. If, however, one extrapolates measurements from one data set into an adjacent one by some conservative method such as isobaric sections, inconsistencies of the order of 2 % or more are common (figure 1.3). One of the most disturbing discrepancies is found between the 1 kbar data of Shmonov & Shmulovich (1974) and those of Kennedy (1954) (figure 1.4). This allows for the possibility that parts of entire data sets are systematically in error by more than 2 %. Some more details are discussed in appendix D. In summary, we estimate that the accuracy of high pressure P-V-T data is no better than 2 % and possibly much worse in some cases. 1.5 Phase Equilibrium Data About 120 phase equilibrium experiments from 25 data sets of 16 laboratory studies (table 1.2) were used to put bounds on / Vco2 dP according to equation (1.2). Al l phase equilibria used include magnesite or calcite as C02-bearing phases and only stoichiomet- ric phases. High pressure data sets (> 25 kbar) involve magnesite only. Studies including dolomite were avoided because of insufficient thermodynamic data on its ordering state. Similarly, phase equilibria including geikielite, meionite, spurrite and tilleyite were not Chapter 1. An Equation of State for CO2 to high P and T 14 Author Equilibrium P [kbar] T [°C] M F MB S&A, 1923 Cc =̂ Lm+C0 2 0.00-0.03 890-1210 GA,DT n y H&T, 1955 Mst Pe+C0 2 0.01-2.7 650-900 CS y y Cc ^ Lm+C0 2 0.01-0.02 980-1070 CS n n H&T, 1956 Cc+/?Qz ^ W0+CO2 0.3-3.1 590-800 GA n y G&H, 1961 Cc ^ Lm+C0 2 0.01-0.08 840-1200 GA n n Mst ^ Pe+C0 2 1.0-5.0 810-1010 GA n n Mst ^ Pe+C0 2 1.0-10 660-1090 GA n n B, 1962 Cc ^ Lm+C0 2 0.00-0.03 900-1210 GA n y W, 1963a Cc+Fo+Di ^ M0+CO2 0.07-0.54 710-890 CS y y Ak+Fo+Cc =̂ M0+CO2 0.5-0.7 900-930 CS n (y) Cc+Di 5=* Ak+C0 2 0.08-0.7 720-930 CS y y W, 1963b Cc+Fo ^ Mo+Pe+C0 2 0.08-0.7 710-920 CS n n JfcM, 1968 Mst ^ Pe+C0 2 0.5-1.0 700-760 CS y y J,1969 En+Mst ^ F0+CO2 2.0 560. CS y y S, 1974 Cc+An ^ Wo+Ge+C0 2 0.5-0.7 840-890 PC n (y) Cc+An+Co ^ Ge+C0 2 0.5-0.7 790-860 PC y y I&W, 1975 Mst =̂ Pe+C0 2 20-36 1350-1600 PC y y N&S, 1975 En+Mst ^ F0+CO2 19-41 1000-1500 PC y y H, 1978 Cc+/?Qz ^ W0+CO2 10-19 1000-1330 PC n y Mst+Cs ^ En+Fo+C0 2 37-42 1130-1200 PC y y En+Mst ^ F0+CO2 24-25 1125 PC y y Mst+Ru ^ Gk+C0 2 13-37 950-1250 PC n i E , 1979 En+Mst ^ F0+CO2 25-29 1110-1250 PC y y P, 1988 Mst ^ Pe+C0 2 0.4-1.2 700-760 GA,PA y y M, 1990 Mst 5=4 Pe+C0 2 12-21 1170-1435 PC y y Table 1.2: List of phase equilibrium studies including pure C 0 2 and stoichiometric phases. Column ' M ' : experimental method, CS: cold-seal, GA: gas apparatus (internally heated), PC: piston-cylinder, DT: differential thermal analysis, PA: pressure analysis. Column 'F ' indicates data sets used for fitting (y: yes, n: no). Column ' M B ' indicates consis- tency with Mader & Berman equation of state of this study (y: yes, n: no, i : insufficient thermodynamic data on geikielite). See text for discussion of inconsistent data sets. Abbreviations of minerals: Cc: calcite, Lm: lime, Mst: magnesite, Pe: periclase, /?Qz: /?-quartz, Fo: forsterite, Di: diopside, Mo: monticellite, An: anorthite, Co: corun- dum, Wo: wollastonite, Ge: gehlenite, Cs: coesite, En: ortho enstatite, Ru: rutile, Gk: geikielite. Abbreviations of authors: S&A: Smith & Adams, H&T: Harker & Tuttle, G&H: Goldsmith & Heard, B: Baker, W: Walter, J&M: Johannes & Metz, J: Johannes, S: Shmulovich, I&W: Irving & Wyllie, N&S: Newton & Sharp, H: Haselton et al, E: Eggler et al., P: Philipp, M: Mader, this study. Chapter 1. An Equation of State for CO2 to high P and T 15 considered. A thermodynamic analysis of the system CaO-Si02-C02 at high tempera- tures is provided by Treimann & Essene (1983) including larnite, rankinite, spurrite and tilleyite. Reactions including calcite were only used below the calcite-I - calcite-IV tran- sition (ca. 790 °C, Mirwald 1976) because of poorly constrained properties of calcite-IV and calcite-V (see section below). The equilibrium En + Mst Fo + CO2, constrained between 19 and 41 kbar by three studies (Newton k Sharp, 1975; Haselton et al., 1978; Eggler et al., 1979), forms one of the cornerstones of the present work (cf. figure 1.6). Equally important are tight constraints on the equilibrium magnesite T=* periclase + CO2 at low pressures (Philip, 1988) and at high pressures (new experiments, this study). Experimental uncertainties were accounted for by displacing positions of the half- brackets away from the equilibrium based on best estimates of uncertainties in pressure and temperature. These uncertainties are indicated by tails leading to the uncorrected pressure-temperature coordinates from the symbols plotted in figures 1.1, 1.6 and 1.7. Table 1.3 shows an example of how one particular data set (Newton k Sharp, 1975) is treated to impose constraints on the fugacity of CO2 between 19 and 41 kbar and 1373-1773 K. Many of the experimental studies used as constraints in this paper do not properly demonstrate reversibility of the phase equilibria studied. This is accounted for by increasing the range of uncertainty on that side of the equilibrium boundary ap- proximated by stability or synthesis runs rather than reversals. This is in most cases the reactant-stable (low temperature) side. The overall consistency of the entire data set seems to indicate that reaction kinetics worked in favor of the 'abridged' approach to 'bracketing' phase equilibria in this case. Mathematical programming analysis forces one to scrutinize every experimental data point extremely carefully. Inconsistencies be- tween data sets appear unequivocally, directing attention to problematic data. A result of this enforced scrutiny emphasizes the unfortunate fact that many experimentalists present their data inadequately, lacking tabulated run conditions, with insufficent data Chapter 1. An Equation of State for C02 to high P and T 16 to estimate uncertainties (i.e. pressure calibration in piston-cylinder work). 1.6 New Experiments on Magnesite Periclase + C O 2 Existing experiments (Irving & Wyllie, 1975) on the equilibrium magnesite ^ periclase + CO2 do not put stringent constraints on the equilibrium position at high pressure (figure 1.6). The equilibrium was therefore reversed at 12.1 kbar between 1173 and 1183 °C, and at 21.5 kbar between 1375 and 1435 °C in a piston-cylinder apparatus using 3/4-inch talc-pyrex assemblies. Friction corrections were calibrated against the melting curve of gold and silver (Mirwald at al., 1975) and amounted to 2.9 kbar at 15 kbar and 3.5 kbar at 25 kbar nominal pressure. Thermal gradients were measured and the effect of pressure on the electromotive force of the Pt-PtlO%Rh thermocouples (Getting & Kennedy, 1970) was accounted for. Uncertainties are estimated at ±1.0 kbar and ±10 K. Details are reported in appendix C. 1.7 Magnesite Properties The thermophysical properties of magnesite are crucial to this study and require close examination. At pressures below 10 kbar and temperatures below 800 °C magnesite properties are well constrained and any consistent set analysis (e.g. Berman, 1988) is not hampered by uncertainties in CO2 properties (all existing equations of state do not deviate noticeably at pressures below 10 kbar). It is therefore possible to derive standard state thermodynamic properties of magnesite prior to the optimization of CO2 properties at higher pressures. Chernosky &z Berman (1989) revised the enthalpy of formation for magnesite (—1114.505 kJ/mole) based on experimental constraints on phase equilibria in systems with a H 2 0 - C 0 2 fluid phase. Trommsdorff & Connolly (1990) propose an increase of the Gibbs free energy of formation for magnesite (an enthalpy Chapter 1. An Equation of State for C02 to high P and T 17 St. P T P T RT In fCo2 Exp. Exp. Adj. Adj.- Obs. M&B B&R K&J S&F Hoi [kbar] [K] [kbar] [K] [kJ] [kJ] [kJ] [kJ] [kJ] [kJ] Re 19.0 1273 20.5 1248 > 154.6 158.9 161.7 163.2 156.6 162.5 Pr 19.0 1298 17.5 1313 < 158.2 154.7 157.8 159.0 153.3 157.4 Re 32.0 1523 33.5 1498 > 213.6 220.6 224.6 227.6 213.0 228.3 Pr 32.0 1553 30.5 1568 < 219.4 217.9 222.4 225.1 211.2 224.3 Re 41.0 1723 42.5 1698 > 260.0 265.2 270.1 274.4 253.0 275.9 Pr 41.0 1773 39.5 1788 < 268.9 264.7 270.4 274.3 253.2 274.0 Table 1.3: Constraints on CO2 fugacities imposed by experimental brackets on the reac- tion Mst + En ^ Fo + C 0 2 (Newton k Sharp, 1975). Column 'St.' indicates whether reactants (Re) or products (Pr) are stable. Columns 'Adj.' contain pressures and temper- atures adjusted for experimental uncertainties. Column 'Obs.' (observed) was computed with equation (1.2) and the data base of Berman (1988) with revised magnesite proper- ties (Berman k Brown, 1985, Chernosky k Berman, 1989, and this study). The reminder of the columns were computed with various equations of state: M&B: Mader k Berman equation (1.5) of this study; B&R: Bottinga k Richet (1981); K&J : Kerrick k Jacobs (1981); S&F: Saxena k Fei (1987a); Hoi: Holloway (1977). Numbers printed in bold face are inconsistent with experimental brackets (column 'Obs.'). of about —1111 kJ/mole) based on field evidence on phase diagram topologies (CaO- MgO-Si0 2-C02-H 2 0) and new phase equilibrium experiments by Philipp (1988) on the equilibrium magnesite periclase + CO2. A value of —1112.505 kJ/mole for the enthalpy of formation from the elements for magnesite was derived by linear programming analysis (Berman et al., 1986, Berman, 1988). This value is consistent with Philipp's (1988) accurate pressure analysis exper- iments, as well as with the brackets on the same equilibrium determined by Harker k Tuttle (1955) and Johannes k Metz (1968) using conventional cold-seal techniques. This value used in conjunction with the data base of Berman (1988) produces the essential features of the phase diagram topologies suggested from natural mineral assemblages as outlined by Trommsdorff k Connolly (1990). The destabilization of magnesite was minimized in order not to deviate more than necessary from experimental constraints in systems including magnesite and a H 2 0 - C 0 2 fluid mixture (e.g. magnesite + talc Chapter 1. An Equation of State for C02 to high P and T 18 forsterite + H 2 0 + CO2, Greenwood, 1967b). Any such discrepancy has to be counter bal- anced by more non-ideal mixing of H2O-CO2 (larger excess volume on mixing) compared to the mixing model of Kerrick k Jacobs (1981) if all other thermodynamic parameters are well constrained. The heat capacity coefficients for magnesite of Berman k Brown (1985) were adopted, combined with the thermal expansion function of Berman (1988) and the compressibility function as revised by Chernosky k Berman (1989). The uncertainty on extrapolating heat capacity and thermal expansion (cf. section 'Method') is the largest single contri- bution to the uncertainty of the CO2 properties derived in this study. 1.8 Results As a first step the ability of equation (1.5) to fit P-V-T data in the absence of additional constraints from phase equilibrum experiments (figure 1.4) is demonstrated. The follow- ing equation of state parameters are derived: B\ = 29.5713, B2 = 3.16418 • 10~4, B3 = 10.2554 • 104, A x = 1.10002 • 109, A 2 = 2.54456 • 109, in units of cm3/mole, K and bar. The results are difficult to compare to other equations of state because each was cali- brated with different weights given to various sets of data. Figure 1.4 shows only the high pressure subset of all constraining P-V-T measurments, with the overall quality of fit of the three equations being comparable (cf. also table 1.1). The equations share one particular feature: volumes extrapolated towards high pressures are larger than those inferred from the high pressure P-V-T data of Shmonov k Shmulovich (1974). Secondly, it is important to know which phase equilibrium constraints influence the equation of state parameters. Phase equilibrium constraints at pressures below 8 kbar are fully compatible with P-V-T data and the equation of state (see also section on calcite polymorphs). The high pressure phase equilibrium constraints (table 1.2), however, are Chapter 1. An Equation of State for C02 to high P and T 19 450 550 650 750 850 950 1050 TEMPERATURE ( K ) Figure 1.4: Isobaric volume-temperature diagram showing comparison of different equa- tions of state and experimentally measured volumes. SkS: Shmonov k Shmulovich; T,L&T: Tsiklis et al; Kkl: Kerrick k Jacobs (1981) equation of state; B k R : Bottinga k Richet (1981) equation; M&B: Mader k Berman equation (1.5) of this study con- strained by P-V-T data only. The discontinuities in the slopes at 47.22 cm3/mole of the B k R equation are a result of using separate fit parameters for different volume intervals. Chapter 1. An Equation of State for C02 to high P and T 20 TEMPERRTURE (K) (X10- Figure 1.5: Isobaric volume-temperature diagram comparing the Mader & Berman equa- tion (1.5) constrained by P-V-T measurments only (MkB(P-V-T)) with equation (1.5) constrained additionally by phase equilibrium experiments (M&B). Measured volumes are overlain for comparison: S&S: Shmonov k Shmulovich; T,L&T: Tsiklis et al.; M,M&W: Michels et al.. SkS (calc) depict volumes extrapolated by Shmonov k Shmulovich (1974) based on their own experimental data. The 9 kbar data points and curves are not shown for clarity. Chapter 1. An Equation of State for C02 to high P and T 21 not compatible. The prominent feature observed is, as would be expected from figures 1.1 and 1.4, that the RTln fco2 terms and thus the volumes are forced to become smaller (more stable CO2) compared to predictions based on P-V-T data alone (figure 1.5). The final equation of state parameters are given below in units of cm3/mole, K and bar: BX = 28.0647 B2 = 1.72871 • 10"4 B3 = 8.36534 • 104 A x = 1.09480 • 109 A 2 = 3.37475 • 109 Table 1.3 lists the magnitudes of mismatch for several equations of state compared to experimental data by Newton k Sharp (1975). Numerical convergence to a global minimum is difficult to demonstrate, but one can monitor the convergence behaviour from different initial estimates for the parameters and test solutions against those of different algorithms. The most important criterion, however, is whether the solution is acceptable in terms of fulfilling the desired task, i.e. reliable computation of geological phase equilibria and consistency with experimentally measured properties within their uncertainties. The first criterion is met by virtue of the method used: only parameters that are consistent with the constraining phase equilibria are acceptable to the algorithm. The second one, the agreement with measured volumes, can easily be tested. Table 1.2 summarizes the consistency of the Mader k Berman equation of state with phase equilibrium experiments involving pure CO2 and stoichiometric phases. The two data sets inconsistent with the equation include some experiments by Goldsmith k Heard (1962) with Pco2 likely less than Ptotai and one set by Walter (1936b) that is totally inconsistent (> 100 K) with any other data. Berman (1988) attributes the latter inconsistency to Walter's failure to recognize periclase in any run products. Figure 1.6 compares computed high pressure phase equilibria including magnesite with experimental data. Reactions including high temperature polymorphs of calcite, calcite-IV and calcite- V, are discussed in a separate section below. The equation of state thus performs reliably Chapter 1. An Equation of State for CO2 to high P and T 22 up to at least 42 kbar. Figure 1.5 and table 1.1 document the comparison of measured and calculated vol- umes. Agreement at low pressures and supercritical temperatures is excellent because the equation approximates the ideal gas law. The equation was not constrained by data below 373 K and therefore the shape of the subcritical area is only approximate and solely a result of its van der Waals like mathematical form. The critical point deduced from the equation is at 335.60 (±0.05) K, 89.22 (±0.05) bar and 115.5 (±0.5) cm3/mole. Sug- gested true values (Angus et al., 1973) based on experimental work, at 304.20 (±0.05) K, 73.858 (±0.05) bar and 94.07 (±0.1) cm3/mole deviate substantially from our computed ones, but nevertheless the presence of a critical region in approximately the correct area improves the behaviour of the equation in the near-critical region (1-800 bar, 300-700 K) tremendously. Rigorous comparisons of computed CO2 properties with experimental phase equilib- ria have previously been hampered by insufficient thermophysical data of solid phases, magnesite in particular (Haselton et al, 1978; Bottinga & Richet, 1981). The properties of solids chosen by Saxena & Fei (1987a) lead to good agreement with CO2 properties obtained with their equation of state, and render the Bottinga & Richet (1981) equation grossly inconsistent at 40 kbar pressure. This is in contrast with our computations (cf. figure 1.1) which render CO2 far too stable with the Saxena & Fei (1987a) equation. This discrepancy is at least in part due to Saxena & Fei's choice of heat capacity func- tions (Robie et al., 1979) that are not suitable for extrapolation (i.e. beyond 750 K for magnesite). The agreement with volumes measured at high pressures is reasonable and compa- rable with that of other equations of state (figures 1.4, 1.5). Equation (1.5) is able to fit measured volumes significantly better without the additional constraints from phase equilibria (figure 1.5), with the noteable exception of the volumes measured at the highest Chapter 1. An Equation of State for C02 to high P and T 23 pressures, which show excellent agreement. This may indicate too little flexibility of the equation, some systematic problems with high pressure experimental P-V-T equipment, or inaccurate mineral properties used to constrain the parameters. It seems that a more complex equation is not justified with the amount, extent and quality of P-V-T data available. Additional volumetric constraints imposed by shock wave data could possibly improve the extrapolation properties of equation (1.5) towards pressures above 100 kbar. Shock wave measurments on solid CO2 (or more likely a mixture of solid and liquid at initial conditions) (Zubarev & Telegin, 1962) are difficult to apply, and measurments on liquid CO2 appear to be nonexistent. The equation of state based on shock data of similar fluids and corresponding states systematics of Saxena & Fei (1987a) is inconsistent with phase equilibrium experiments (cf. table 1.3, figure 1.1). A new approach taken by Helffrich & Wood (1989) appears to bridge shock wave data and phase equilibrium data more successfully. In summary, the region where the Mader & Berman equation is demonstrated to per- form reliably spans 400-1773 K and 1 bar to 42 kbar. Extrapolation to higher pressures and temperatures is expected to yield useful results, possibly to 80 kbar and 2300 K. 1.9 Calcite(I-IV-V) - Aragonite In the process of calibrating the equation of state inconsistencies became evident between constraints on CO2 fugacities imposed by magnesite phase equilibria and those by calcite reactions. The well constrained high pressure brackets on En + Mst r=* Fo + CO2 and those on Mst ^ Pe + C 0 2 (figure 1.6) demand smaller CO2 fugacities (more stable CO2) than high pressure brackets on Cc + /?Qz Wo -j- C 0 2 (figure 1.7). One likely explanation is the presence of the more stable calcite polymorphs calcite-IV and calcite-V Chapter 1. An Equation of State for CO2 to high P and T 24 1 r i 1 1 1 1 1 1 r 800 1000 1200 1400 1600 1800 T E M P E R A T U R E (DEG C) Figure 1.6: Pressure-temperature diagram with phase equilibria involving magnesite computed with the equation of state for CO2 of this study compared to phase equilibrium experiments. A friction correction of —3 kbar was applied to the brackets by Irving & Wyllie (1975) based on calibrations by Huang & Wyllie (1975). Abbreviations of minerals: Cs: coesite, Mst: magnesite, En: ortho enstatite, Fo: forsterite, Pe: periclase, aQz: o>quartz. The diagram was produced with GEO-Calc (Brown et al., 1988). Chapter 1. An Equation of State for C02 to high P and T 25 Std. state prop. AfHPr'Tr [kJ/mole] SPr,Tr [J/mole/K] [J/bar] Reference Aragonite Calcite(-I) Calcite-IV Calcite-V -1207.597 -1206.697 -1204.580 -1199.768 87.490 91.893 94.100 97.895 3.415 3.690 3.683 3.689 (*),(*),(!) (*),(*),(B) (*),(*),(*) (*),(*),(*) Cp coefficients k0 Jfcx (xl0~ 2) fc2(xl0-5) M x i o - 7 ) Calcite(-I) Ar, Cc-IV, Cc-V 178.19 178.19 -16.577 -16.577 -4.827 -4.827 16.660 16.660 (2,3,4,B) (a) V coefficients vt (xlO 6) v2(xld12) t>3(xl05) v 4 (x l0 1 0 ) Aragonite Calcite(-I) Cc-IV, Cc-V -1.620 -1.400 -1.400 0.0080 0.0060 0.0060 3.670 0.897 0.897 227.4 227.4 227.4 (5,6,7),(5,8) (B),(a),(B),(B) (a) Table 1.4: Thermodynamic properties of calcite polymorphs. Standard state thermo- dynamic properties of calcite polymorphs at 1 bar and 298.15 K, isobaric heat capacity function coefficients (ko — k3) and volume function coefficients (vi — v4). The heat capac- ity function of Berman k Brown (1985) is used: CP = k0+ ktT-°-5+ k2T~2+ k3T~3, and the volume function of Berman (1988): VP'T'/VPr<Tr = 1 + vx(P - Pr)+ v2(P - Pr)2+ v3(T - Tr)+ v4(T - Tr)2. CP is in units of [J/mole/K], V in [J/bar], T in [K] and P in [bar]. References: (B) Berman, 1988; (*) values derived in this study; (a) assumed, see text for discussion; (1) Robie et al., 1979; (2) Stavely k Linford, 1969; (3) Jacobs et al., 1981; (4) Kelly, 1960; (5) Salje k Viswanathan, 1976; (6) Madelung k Fuchs, 1921; (7) Singh k Kennedy, 1974; (8) Kozu k Kani, 1934. (cf. Mirwald, 1976, for nomenclature and summary) at high temperatures which would tend to shift the above equilibrium to higher temperatures. As a first step, all experimental brackets including calcite at high temperatures were not considered as constraints on the properties of CO2 in order to remove the possibly significant effects of calcite-IV and calcite-V. In order to test the above hypothesis, ther- mophysical properties of the calcite polymorphs were derived from constraints imposed by the aragonite-calcite-I-calcite-IV-calcie-V phase diagram. The equilibrium Cc(I/IV/V) + /3Qz Wo + CO2 may then be recomputed with the obtained thermodynamic data to compare with experimental brackets. Unfortunately few quantitative data are available on the unquenchable polymorphs Chapter 1. An Equation of State for CO2 to high P and T 26 calcite-IV, calcite-V and the polymorphic transitions. The transition boundaries have been the subject of numerous studies (Boeke, 1912; Eitel, 1923; Clark, 1957; Crawford &; Fyfe, 1964; Boettcher k Wyllie, 1968; Johannes k Puhan, 1971; Crawford k Hoersch, 1972; Cohen k Klement, 1973; Irving k Wyllie, 1975; Mirwald, 1976, 1979a, 1979b). The phase transitions between calcite-I-IV-V are most likely related to order-disorder within the anion sublattice (Lander, 1949; Salje k Viswanathan, 1976; Mirwald, 1979a, 1979b; Dove & Powell, 1989). There is, however, still some debate about the exact nature and the existence of discrete phase transitions in calcite (Markgraf k Reeder, 1985). In order to minimize the number of thermophysical parameters to be derived, all polymorphic transitions were treated as first order phase transitions with small volume discontinuities at the phase boundaries. This should lead to a reasonable approximation of the energetics of these phase transitions. In view of the lack of data on thermal and volumetric properties the following assump- tions were made. Al l polymorphs share the same heat capacity function. Experimental data by Kobayashi (1950b) that suggest a lower heat capacity for aragonite than calcite were disregarded due to its imprecision and inconsistency of his aragonite and calcite (Kobayashi, 1950a) data with other studies (Stavely k Linford, 1969; Jacobs et al., 1981; Kelly, 1960). Calcite—IV and V are assumed to share the same expansivity and compress- ibility as calcite-I. X-ray data by Mirwald (1979a) suggest a larger thermal expansion for Cc-IV than for Cc-I, which is, however, not observed in Markgraf & Reeder's (1985) measurments. The extremely large thermal expansion of aragonite relative to calcite suggested by measurments up to 450 °C (Salje k Viswanathan, 1976; Kozu k Kani, 1934) was reduced by about 15 % during fitting in order to constrain the curvature of the Cc T=* Ar equilibrium to the phase equilibrium data. The difference in compressibility between calcite and aragonite is derived from studies that include measurments on both minerals by the same technique (Salje k Viswanathan, 1976; Madelung k Fuchs, 1921). Chapter 1. An Equation of State for C02 to high P and T 27 Al l other thermodynamic properties were fitted to measurements as summarized in table 1.4. Calcite—I properties are nearly identical to those in Berman (1988). The standard state properties (AfHPr'Tr, S P r ' T r , VPr'Tr) of Cc-IV and Cc-V were refined by a linear programming technique (Berman et al, 1986, Berman, 1988) using phase equilibrium constraints (Cc-I ^ Ar, Cc-IV ^ Ar, Cc-V ^ Ar, Cc-I ^ Cc-IV, Cc-IV Cc-V; figure 1.7). The derived thermophysical properties are consistent with most phase equilibrium studies with the exception of the Ar T=* Cc-I transition at low temperatures (< 200°C) and the Ar ^ Cc-V transition. At 100 °C the computed Arag ^ Cc-I boundary is located 0.5 kbar too low compared to experiments by Crawford &: Fyfe (1964). Data of Mirwald (1979a) on the Ar ^ Cc-V equilibrium are internally inconsistent and not compatible with the brackets obtained by Irving k Wyllie (1975) (figure 1.7). This boundary is therefore poorly constrained and allows some latitude in the properties of Cc-V. Computed phase equilibria involving Cc-I, Cc-IV and Cc-V using the above refined properties (table 1.4) and the Mader k Berman equation (1.5) for C 0 2 (figure 1.7) are consistent with all phase equilibrium experiments (table 1.2) with the exception of small inconsistencies (<4 K) with one half bracket on the reaction Ak + Cc-IV + Fo ^ Mtc + C 0 2 (Walter, 1963a) and An + Cc-IV ^ Wo + Ge + C 0 2 (Shmulovich, 1974). Most importantly, however, the computed equilibria are compatible with the high pressure brackets on the reaction Cc-IV/V + /?Qz ^ Wo + C 0 2 (Haselton et al, 1978) and the high temperature brackets on the equilibrium Cc-(IV/V) ^ Lime -f C 0 2 (Smith k Adams, 1923; Baker, 1962). It appears that only the experimental data of Smith k Adams (1923) and Baker (1962) for the calcite decarbonation are reconcilable with the calorimetric data of lime and the above treatment, which are inconsistent with data by Harker k Tuttle (1955) and Goldsmith k Heard (1961). Smith's differential thermal analysis experiments are, however, not reversed in both directions and constrain only the Chapter 1. An Equation of State for C02 to high P and T 28 product-stable assemblage. The above treatment of Cc-IV and Cc-V does not put any constraints on the properties of CO2, but allows one to estimate the effects of more stable calcite polymorphs. Additional data on the calcite polymorphs are required to permit a more accurate thermodynamic treatment that accounts explicitly for the observed order- disorder phenomena. It is concluded that the high temperature calcite polymorphs Cc-IV and Cc-V need to be considered for computation of phase equilibria at high temperatures. The energetics of the polymorphic transitions result in an increase of the carbonate stability field that leads to good agreement with CO2 properties calculated with the new Mader Sz Berman equation of state. 1.10 Conclusions The following points are emphasized: i) the mathematical programming technique allows significant improvements of the equation of state for CO2, ii) phase equilibrium brackets put important constraints on the fugacity of CO2 at pressures not accessible by conven- tional P-V-T measurments, and iii) some experimentally measured volumes of CO2 at high pressures might have much larger uncertainties than previously estimated. The data needed for further improvements include: i) more high quality phase equilib- rium brackets at high pressures on reactions involving C 0 2 , ii) a set of carefully measured C 0 2 volumes up to 12 kbar, iii) shock wave data on fluid CO2, iv) measurments on the expansivity of magnesite to high temperatures under pressure, v) more quantitative data on the complex phase transitions undergone by calcite, and vi) progress in providing mathematically practical equations of state with a theoretical basis. Chapter 1. An Equation of State for CO2 to high P and T 29 40000 35000 - _ 30000 - CO CC g 25000 LU CrT 20000 ZD CO Lu 15000 CC CL 10000 -4 5000 - I I • Mirwald (1976,1979) o I rv ing £ Uyllie (1975) v Lewis Z Cohen (1973) C c - V flrag V C c - I V x C c x Harker Z Tuttle (1956 A Haselton et al. (1978) 1 1 400 600 800 1000 1200 1400 T E M P E R A T U R E ( D E G C) 1600 Figure 1.7: Pressure-temperature diagram with phase equilibria involving calcite poly- morphs computed with thermodynamic properties of table 1.4 and Berman (1988) and the Mader k Berman equation of state for CO2 of this study. The dashed curve depicts the effect of omitting calcite-IV and calcite-V in order to compute the wollastonite re- action. The dotted curve was computed with the Kerrick k Jacobs (1981) equation of state not considering any of the high temperature calcite polymorphs. Abbreviations of minerals: Arag: aragonite, Cc: calcite, bQz: beta quartz, Wo: wollastonite. The diagram was produced with GEO-Calc (Brown et al., 1988). Bibliography Akella, J. , 1979, Quartz ^ coesite transition and the comparative friction measurements in piston-cylinder apparatus using talc-alsimag-glass (TAG) and NaCl high-pressure cells. N. Jh. Miner. Monatshefte, 217-224. 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Fugacities of gaseous solutions. Chem. Reviews, 44, 233-244. Bibliography 36 Robie, R., Hemingway, B.S., k Fisher, J.R., 1979, Thermodynamic Properties of Min- erals and Related Substances at 298.15 K and 1 bar (105 Pascals) Pressure and Higher Temperatures. U.S. Geol. Surv. Bull., 1452, 456 p. Ross, M . , k Ree, F.H., 1980, Repulsive forces of simple molecules and mixtures at high density and temperature. J. Chem. Phys., 73, 6146-6152. Ryzhenko, B.N., k Volkov, V.P., 1971, Fugacity coefficients of some gases in a broad range of temperatures and pressures. Geochem. Int., 8, 468-481. Translated from Geokhimiya, 7, 760-773, 1971. Salje, E . , & Viswanathan, K., 1976, The phase diagram calcite-aragonite as derived from the crystallographic properties. Contr. Miner. Petrol., 55, 55-67. de Santis, R., Breedveld, G.J.F., k Prausnitz, J.M., 1974, Thermodynamic properties of aqueous gas mixtures at advanced pressures. Ind. Eng. Chem. Proc. Des. Devel., 13, 374-377. Saxena, S.K., k Fei, Y., 1987a, High pressure and high temperature fluif fugacities. Geochim. Cosmochim. Acta, 51, 783-791. Saxena, S.K., 1987b, & Fei, Y., 1987b, Fluids at crustal pressures and temperatures, I. Pure species. Contr. Miner. Petrol., 95, 370-375. Schittkowski, K., 1980, Nonlinear Programming Codes - Information, Tests and Perfor- mance. Lecture Notes in Economics and Mathematical Systems, No. 183, Springer-Verlag, Berlin, Heidelberg, New York. 242 pp. Shmonov, V . M . , 1977, The apparatus for measuring P-V-T properties of gases up to 1000 K and 8000 bar (in Russian). Och. Fiz. Khem. Petrol, 1, 236-245. Shmonov, V . M . , k Shmulovich, K.I., 1974, Molar volumes and equation of state of CO2 at temperatures from 100-1000°Cand pressures from 2000-10000 bars (in Russian). Akad. Nauk SSSR Doklady, 217, 935-938. Shmulovich, K.I. 1974, Phase equilibria in the system CaO-Al 203-Si02-C02. Geochem. Int., 11, 883-887. Shmulovich, K.I., & Shmonov, V .M. , 1975, Fugacity coefficients for C 0 2 from 1.0132 to 10000 bar and 450-1300 K. Geochem. Int., 12, 202-206. Translated from Geokhimiya, 9, 551-555, 1975. Singh, A.K. , k Kennedy, G.C., 1974, Compression of calcite to 40 kbar. J. Geophys. Res., 79,2615-2622. Smith, F.H. , k Adams, L.H., 1923, The system calcium oxide-carbon dioxide. J. Am. Chem. Soc, 45, 1167-1184. Bibh.ogia.phy 37 [98] Stavely, L.A.K. , & Linford, R.G., 1969, The heat capacity and entropy of calcite and aragonite, and their interpretation. J. Chem. Therm., 1, 1-11. [99] Touret, J . , & Bottinga, Y., 1979, Equation d'etat pour le CO2; application aux inclusions carboniques. Bull. Mineral., 102, 577-583. [100] Treiman, A.H. , & Essene, E.J. , 1983, Phase equilibria in the system CaO-Si02-C02- Am. J. Sci., 283-A, 97-120. [101] Trommsdorff, V., & Connolly, J.A.D., 1990, Constraints on phase diagram topology for the system CaO-MgO-Si02-CU2-H20. Contr. Miner. Petrol., in print. [102] Tsiklis, D.S., Linshits, L.R., & Tsimmerman, S.S., 1971, Measurment and calculation of molar volumes of carbon dioxide at high pressure and temperature (in Russian). Teplofiz. Svoistva Vesch. Mater, 3, 130-136. [103] Vaessen, W., 1984, UBC NLP: Nonlinear Function Optimization. Computing Center, The University of British Columbia, users manual, unpublished, 167 pp. [104] Vukalovich, M.P., Altunin, V.V., & Timoshenko, N.I., 1962, Experimental investigation of the specific volume of carbon dioxide at temperatures from 200-750°C and pressures up to 600 kg/cm2 (in Russian). Teploenergetica, 9, 56-62. [105] Vukalovich, M.P., Altunin, V.V., & Timoshenko, N.I., 1963a, Experimental determination of the specific volume of CO2 at temperatures 40-150°C and pressures up to 600 kg/cm2 (in Russian). Teploenergetica, 10, 85-88. [106] Vukalovich, M.P., Altunin, V.V., & Timoshenko, N.I., 1963b, Specific volume of carbon dioxide at high pressure and temperature (in Russian). 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Chapter 2 Properties of C 0 2 - H 2 0 Mixtures at High Pressure and Temperature 2.1 Introduction The importance of knowing the thermophysical properties of aqueous fluid mixtures at elevated pressure and temperature is obvious in a number of earth science applications: i) the computation and prediction of phase equilibria in metamorphic and igneous petrol- ogy (Berman, 1988, Ferry &: Baumgartner, 1987, Ghiorso & Carmichael, 1987); ii) the interpretation of fluid inclusions (Roedder, 1984); iii) the modeling of mass transfer in aqueous geochemistry (Helgeson & Lichtner, 1987, Pitzer, 1987, Sverjensky, 1987); and iv) some high temperature-high pressure engineering applications, such as steam flood- ing to enhance oil recovery and technologies utilizing geothermal energy. In this chapter the system water-carbon dioxide is discussed which comprises the most abundant fluid species of geological interest. It will be seen that in spite of the importance of this system its properties are rather poorly known. In addition to the variables pressure (P), specific volume (V) and temperature (T) the compositional variable (X) must be considered. This makes graphical presentation of P-V-T-X data difficult without resorting to a large number of graphs. Because a comprehensive review of such data is not available elsewhere an extensive collection of graphs is presented in appendix E. This chapter introduces the system H 2 0 - C 0 2 followed by a presentation of existing 38 Chapter 2. Properties of C 0 2 - H 2 0 Mixtures at High Pressure and Temperature 39 equations of state in some detail. The theoretical and physical foundations of some equa- tions of state and mixing rules are evaluated. Fundamental thermodynamic relationships for fluid mixtures are presented with special emphasis on the treatment of phase equi- librium data that are used to constrain the mixing properties of H 2 0 - C 0 2 . Subsequent sections review and evaluate the available experimental data from both P-V-T-X and phase equilibrium studies. A strategy and methodology is proposed for the development of an equation of state for H 2 0 - C 0 2 mixtures that can take advantage of both P-V-T-X and phase equilibrium data. The chapter concludes with an assessement of where new data is most urgently required. 2.2 The System C 0 2 - H 2 0 The system C 0 2 - H 2 0 has three components, C, H, 0, and contains numerous species: C 0 2 , H 2 0 , CO, 0 2 , H 2 , CH.4, H 2 C03, complex hydrocarbons, and the condensed phases graphite and diamond. Charged species may be neglected on the basis of measurements of the electrical conductivity at elevated pressures and temperatures (Franck, 1956) and spectroscopic data (Franck, 1979). Hydrocarbons and other oil brine ingredients are not considered here even though some organic compounds in intermediate oxidation states may persist to metamorphic temperatures (Helgeson, 1990). Methane, carbon monoxide and in some cases hydrogen may be present in significant amounts under metamorphic conditions and in laboratory experiments at comparable conditions. Despite the existence of numerous species it is possible to take a stoichiometric ap- proach for the binary join H 2 0 - C 0 2 which considers only two species, C 0 2 and H 2 0 . The experimental data, however, on which the equation is calibrated, must be known to contain only those two species. The bulk composition of the fluid, whether experimental or natural, may be displaced from the binary join if the fugacity of one or more species Chapter 2. Properties of CO2 - H2O Mixtures at High Pressure and Temperature 40 is buffered (i.e. controlled by a solid phase assemblage), or sources or sinks for C-H-0 species exist. This is particularly significant in the case where graphite is stable. Care must be taken in phase equilibrium experiments involving CO2 and H 2 0 to ensure that the bulk composition of the fluid remains on the binary join during the experiment and that other species than H 2 0 and C 0 2 can be neglected if the results are to be used to con- strain mixing properties of C 0 2 and H 2 0 . Gain or loss of hydrogen is the most common problem encountered and may be solved by buffering the hydrogen fugacity at a very low partial pressure, removing likely sources or sinks for hydrogen, and keeping experimental run times as short as possible. Experimentally measured volumes of C 0 2 - H 2 0 mixtures may suffer from oxygen loss due to the oxidation of the steel pressure vessel or from hydrogen loss due to diffusion or formation of metal hydrides. Graphite must not be stable if a binary equation of state or binary phase equinbrium and P-V-T-X experiments are to be used as constraining data for the binary C 0 2 - H 2 0 system. Under reducing conditions in a C-O-H system with graphite present significant mole fractions of methane may persist to high temperatures (ca. 700 °C for a quartz- fayalite-magnetite [QFM] buffered system). Carbon monoxide becomes an important species in the presence of graphite at higher temperatures (Eugster & Skippen, 1967). In the three-component system C - H - 0 with graphite present the state of the sytem can be fully described by specifying only 3 parameters. These are commonly taken as P, T, and the potential of one of the species such as H 2 or 0 2 through the use of buffers (Eugster & Skippen, 1967, Ulmer & Luth, 1988). Under sufficiently oxidizing conditions methane, hydrogen and carbon monoxide are present only in trivial amounts and a binary C-O-H fluid consists chiefly of the species C 0 2 and H 2 0 (see for example Holloway, 1987, Eugster k Skippen, 1967). The two-phase region of the C 0 2 - H 2 0 binary system is encountered geologically under conditions of very low grade metamorphism, as well as under those applicable to Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 41 many hydrothermal processes. The critical curve which bounds the two-phase region extends from the critical point of pure H 20 (374.15 ° C , 221.3 bar) to a temperature minimum (266 ° C , 2450 bar, X Q Q 2 =0.415) and then probably to higher temperatures with increasing pressure and increasing X.QQ2. Experimental data on the two-phase region at high pressures extends to 3.5 kbar (Todheide &; Franck, 1963, Takenouchi & Kennedy, 1964). For C02-rich compositions the lower branch of the critical curve extends from the critical point of pure CO2 (31.05 ° C , 73.86 bar) to a lower critical endpoint at somewhat higher temperature and pressure which is experimentally poorly constrained. Equations of state designed for the computation of phase equilibria at temperatures below the critical point of water must therefore account for the two-phase region. Some geologically reasonable pressure-temperature-time paths like those responsible for blue schist metamorphism may intersect the two-phase region twice, once at low pressure and low temperature, and again at high pressure. Interestingly, this behaviour is predicted by the Kerrick & Jacobs (1981) equation where the two-phase region is calculated to occur at about 15 kbar and 400 °C. Another aspect of the CO2-H2O system that affects the physics of processes but not the thermodynamics is the density inversion at which the C02-rich vapour phase becomes more dense than the H 20-r ich coexisting liquid phase (Todheide &; Franck, 1963). This density inversion occurs at increasing pressure with increasing temperature and passes through the coordinates 50 °C , 800 bar and 250 °C , 2000 bar. At 2000 bar and 250 °C the 'vapour' contains 67.5 mol% C 0 2 and the 'liquid' dissolves 16.6 mol% C0 2. For computations that include fluid species other than CO2 and H 20 it is necessary to compute the abundance of all species from the relevant equilibrium constants using a 'distribution of species' calculation. This calculation must be performed at each pressure, temperature and bulk composition of interest. The calculation involves the simultaneous solution of all mass action and mass balance equations pertaining to the fluid species Chapter 2. Properties of C 0 2 - H20 Mixtures at High Pressure and Temperature 42 likely to be present. Some simple sample calculations are presented by Skippen & Eugster (1967), Holloway (1987) and Ferry & Baumgartner (1987). Alternately, a Gibbs potential minimization technique may be applied to achieve the same goal in a more general way (i.e. DeCapitani & Brown, 1989). 2.3 Constraints on Equations of State from Theory and Experimental Evidence It is fair to state that the problem of the prediction of P-V-T properties of fluids and their mixtures is not so much due to the lack of theories and models but due to the lack of our ability to assess their relative merits. Models based on first principles are not yet sufficiently developed to test equations of state, and the body of experimental data is not large enough to distinguish between the extrapolation properties of the numerous semi- empirical equations. As a consequence the computation of properties of fluid mixtures heavily relies on experimentally measured quantities, semi-empirical equations of state, and in many cases arbitrarily chosen mixing rules. A physical interpretation of the constants in an equation of state is usually offered in hindsight after some combination of parameters and mixing rules is found to achieve a satisfactory fit to experimental data. Measurements on mixtures above several kilobars pressure are few and the problem of extrapolation to conditions of interest to petrologists is obvious. There are no P-V- T data obtained with static experiments on gases above 10 kbar despite the fact that current technology would permit measurements in internally heated apparati up to 15-20 kbar. It is therefore necessary to examine closely some of the theories that underly the equations predicting P-V-T-X properties and thermodynamic functions in the CO2-H2O system. The exact shape of the intermolecular potential as a function of temperature and Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 43 pressure is not known, nor are the potentials resulting from their mutual interaction in mixtures. For this reason most equations of state applicable to real systems are based on macroscopic models with parameters related to the dominant intermolecular forces, namely attraction and repulsion. Examples of this class of equations are the virial- type equations, the van der Waals equation and the Redlich-Kwong (Redlich & Kwong, 1949) equation with its numerous modifications. The parameters in equations of state are either constants or some function of temperature and possibly pressure (or volume). Variations on the theme take other interactions into consideration, such as those arizing from polarizability, asymmetry, permanent dipole moments, induced dipole moments, multi-pole moments, formation of complexes and other forms of chemical association. The equation of state 'constants' may be derived from critical data (corresponding states sytematics), from low pressure data, from optimization to experimental data, or from a combination of the above. There is a spectrum from equations with a sound physical basis to those relying mainly on empiricism. Even inert gases, the simplest kind of gas systems, are difficult to treat on the basis of rigorous theory. These simple gases do not obey exactly the theory of corresponding states, making its application to more complex fluids theoretically unjustified. Corresponding-state systematics is, however, a useful tool for systems with little data and for systematic comparisons. The behaviour of mixtures of gases may in some cases be described by equations in which the parameters are formed by some combination of the parameters for the pure gases. However, the equations must be mathematically and physically compatible. These 'mixing rules' may be independent of the nature of the endmember gases in that they simply account for geometrical effects upon mixing. If one or more of the endmem- ber gases deviate substantially from ideal behaviour, mixing rules may be adjusted for non-geometric effects upon mixing, such as induced dipole effects, polarizability, strong electronic interactions, chemical association and others. In most cases only a two-body Chapter 2. Properties of CO? - H20 Mixtures at High Pressure and Temperature 44 interaction parameter is developed to account for these effects, and three-body and higher order interactions are neglected. The requirement that equations of the endmem- ber gases be physically and mathematically compatible puts severe constraints on the possibility of introducing modifications to either endmember to fit experimental data. Several equations introduced to geologists have been fit piecewise to data in order to achieve an acceptable fit with a relatively simple equation (i.e. Bottinga k Richet, 1981, for C 0 2 ; Holloway, 1977, 1981, for H 2 0 ; Saxena k Fei, 1987b, for H 2 0 and C 0 2 ; Powell k Holland, 1985, Holland k Powell, 1990, for the logarithm of the fugacity of H 2 0 and C0 2 ) . The tradeoff is the introduction of discontinuities to the thermodynamic functions derived from these equations of state. An important point that is easily underestimated is the dependency of the parameters in the equation of state on the mathematical form of the equation of state itself. Dif- ferent modifications of the Redlich-Kwong equation, for example, have different values for the repulsive and the attractive parameters, although they do have the same physical significance. The same mixing rules applied to a van der Waals equation and a Redlich- Kwong equation will result in different excess properties for a mixture, even though the physical justification and the underlying assumptions may be identical. This problem is particularly pronounced for semi-empirical equations. Increased flexibility to fit mea- sured volumes may be introduced by making the attractive term or the repulsive term more flexible, or a combination of both. Thus, an equation with a simple hard-sphere repulsive term can mimic increased compressibility at high pressure by introducing a vol- ume (or pressure) dependency on the attractive term (see for example Kerrick k Jacobs equation). The quality of fit to observed data alone implies absolutely nothing about the physical meaning of the variation of the parameters with P, V and T, or about the correctness of the mixing rules. Table 2.5 lists values for the b parameter, the 'excluded volume', for several equations Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 45 Equation b [cm3/mole] Reference C 0 2 H 2 0 van der Waals b - 42.8 30.4 Moelwyn-Hughes, 1964 Redlich-Kwong 6 = 29.7 21.1 Redlich k Kwong, 1949 de Santis MRK b = 29.7 14.6 de Santis et al., 1974 Holloway MRK b = 29.7 14.6 Holloway, 1977 Kerrick & Jacobs MRK b = 58 29 Kerrick & Jacobs, 1981 V°° = 14.5 7.25 Bottinga k Richet MRK b = b{V) - Bottinga k Richet, 1981 Bowers k Helgeson b = 29.7 14.6 Bowers k Helgeson, 1983 Mader k Berman b = b(V,T) - Equation 1.5, chapter 1 b° = 28.1 6°° = 22 - 23 Critical temperature 304.2 647.3 [K] Critical pressure 73.86 221.4 [bar] Critical volume 94.07 55.44 [cm3/mole] Table 2.5: Excluded volume for H 2 0 and C 0 2 for several equations of state. The van der Waals constants are derived from low pressure P-V-T data. The constants for the Redlich-Kwong equation are computed from the corresponding states relationship b — 0.0867 • RTC/PC. MRK: modified Redlich-Kwong equation. The b paramater of Kerrick k Jacobs is not a 'true' excluded volume. V°° is the limiting volume at infinite pressure. 6° is the value computed at 1 bar pressure and 298.15 K. 6°° is the temperature dependent limiting value at infinite pressure. Critical data for C 0 2 are from Angus et al. (1974) and for H 2 0 from Moelwyn-Hughes (1964). based on models with a repulsive and an attractive term. The physical interpretation in these 'billiard ball' (or hard-sphere) molecule models is that b represents the volume made inaccessible to other molecules by one molecule. The excluded volume is independent of temperature and pressure according to van der Waals theory. That this assumption breaks down even at moderate pressures (10-20 kbar) has been briefly shown in chapter 1 for C 0 2 . In the case of water, where the molecular interaction is dominated by hydrogen bonding rather than repulsion, the definition of an 'excluded volume' is particularly questionable. The number of parameters in an equation of state is not always correlated with the goodness of fit to data. In most equations one or more parameters, such as the Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 46 excluded volume, are fixed prior to regression to measured volumes (i.e. Kerrick &: Jacobs, 1981, Holloway, 1977). For some of the mathematically more complex equations a simultaneous fit of all parameters is not possible due to problems with numerical stability during optimization (i.e. Bottinga & Richet, 1981). An equation with few parameters fit simultaneously such as the 5-parameter equation presented in chapter 1 for CO2 may achieve a fit comparable to a 10-parameter equation such as that proposed by Kerrick & Jacobs (1981). A many-parameter equation may not extrapolate well, particularly if the parameters are combined empirically, for example in a power series expansion. The virial equation is frequently cited as being the best equation to adopt because of its basis on sound theory, i.e. the physical meaning of the virial coefficients which can be related to the intermolecular potential functions (see Prausnitz et al., 1986 for details). Application of the virial equation to mixtures requires no further assumptions and is therefore commonly preferred for mixtures of a large number of different fluid species. It is necessary to emphasize, however, that the virial equation itself is limited for practical purposes to moderate densities, not exeeding about one-half of the criti- cal density (Prausnitz et al., 1986). This is due to the lack of stringent experimental constraints on the third and higher virial coefficients which are therefore conveniently omitted by truncation or empirically estimated by some corresponding states scheme. The shape of the intermolecular potential function is not exactly known, particularly at high pressures, and therefore empirical functions are used. For most commonly used intermolecular potential models only mathematical expressions for the first and second virial coefficient are available (Prausnitz et al., 1986). The difficulty experienced in fit- ting the virial equation to P-V-T data for H 2 0 and C 0 2 is not surprising because of the polar nature of H 2 0 and the large quadrupole moment of C 0 2 . We cannot expect to predict the mixing properties with confidence using mixing rules that do not take proper account of the nature of H 2 0 , C 0 2 and their interactions. In the geologic literature most Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 47 virial equations fit to measured data (i.e. Saxena &; Fei, 1987a, 1987b) are virial-like equations in as much as the zeroth coefficient is not 1, but some value dictated by the fitting process. The sound theoretical basis is thus removed and the equations are semi- empirical in nature, offering no particular advantage over van der Waals-like equations except that mathematical manipulation may be simpler. A detailed account of the current state of theories of fluid mixtures and their physical justifications is provided by Prausnitz et al. (1986). For the purpose of this study it is sufficient to address a few problems specific to water and carbon dioxide. Carbon dioxide: One would not expect significant changes in the nature of the CO2 molecule in the range of pressure and temperature relevant to this study. Raman spectroscopic mea- surements to 490 °C and 2.5 kbar (Kruse, 1975) indicate no significant changes in the spectrum from data at low densities. Not much appears to be known about the fate of CO2 at very high temperatures and very high densities, i.e. the extent of dissociation, ionization, or dimerization. Water: While the peculiar nature and properties of water compared to other liquids are well known (i.e. Franks, 1972-1984, vols.I-VII), it is not well understood how these prop- erties change with increasing pressure and temperature. Insight is gained from experi- mental measurements of P-V-T properties, dielectric permittivity, Raman and infra red spectroscopy, thermodynamic properties, and more recently, from molecular dynamics simulations. Water at low pressures and temperatures is dominated by hydrogen bonding, which leads to a highly structured liquid dominated by tetrahedrally coordinated molecules with a very open structure similar to that in the corresponding solid, ice-I. The hydrogen Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 48 bond persists to high pressures in ice (i.e. ice-VII at pressures > 21 kbar) with an increase in density due to distortion of the tetrahedral framework. The relatively low compressibility of liquid water despite the open framework indicates that the structure dominated by the hydrogen bond may persist to high pressure in the fluid state as well. There is indirect evidence that water starts to behave like a 'normal' liquid between 4 and 8 kbar pressure. A normal liquid is composed of simple molecules with attractive and repulsive interactions that are not modified by polarization, dipole moments, multipole moments, or chemical association. A comparison of compressibilities for water, methanol and hexane shows some noteworthy trends (Franck, 1981): methanol, a polar solvent, hydrogen bonded like water but without forming a three dimensional open structure, deviates significantly from water at 1-4 kbar but shows almost identical compressibilities at 8 kbar. Hexane, a 'normal' liquid, shows compressibilities similar to methanol up to 4 kbar which is the upper limit of available measurements. The pressure limit of spectroscopic observation is limited to about 3-4 kbar which is not suffuciently high to observe the proposed collapse of the tetrahedral network at low temperatures. It is observed, however, that with increasing pressure the second- nearest and third-nearest neighbour distances decrease significantly, indicative of severe distortion of the network (Franck, 1981). The tetrahedrally coordinated network probably does not persist to high temperatures at low pressures. Normal liquids display increasing viscosity with rising pressure. Mea- surements of viscosity of water at low temperatures indicate a minimum at about 2 kbar. Above 50 °C this minimum no longer exists and water appears to behave like a normal liquid (Todheide, 1971). Spectroscopic data also indicate that at supercritical conditions H 2 0 has lost many of its structured characteristics which dominate at low temperatures and pressures. Supercritical water is, however, still polar in nature which is important for its ability to interact with other species. No data are available on structural changes Chapter 2. Properties of C02 - H2O Mixtures at High Pressure and Temperature 49 as a function of temperature at high pressures. At very high temperature and pressure (i.e. 100-200 kbar) water may be largely an ionic fluid due to the observed monotonic increase of the ion product with pressure and temperature (Holzapfel &; Franck, 1966, Mitchell & Nellis, 1979, and discussion in Franck, 1981). Recent progress in molecular dynamics simulations (Brodholt &; Wood, 1990) at a density of 1 g/cm 3 and temperatures up to 2100 °C (38 kbar) leads to reasonable agree- ment with measured volumes and transport properties. This approach may allow testing of intermolecular potential formulations and equations of state in the near future. Results of numerical simulations are generally not accurate enough for the purpose of thermody- namic computations. Designing equations of state for H2O applicable to a wide range of pressures and temperatures is therefore not an easy task. Models based on the dominant repulsive nature of compressed gas systems will be difficult to fit to observed data. It is not surprising that in work based on corresponding states systematics or van der Waals theory H2O is always a spoiler and requires special treatment such as more adjustable parameters or separate optimization for individual pressure or temperature intervals. The Haar et al. (1984) equation used for the thermodynamic properties of supercrit- ical water by Berman (1988) and in this study is reliable to at least 2500 K and 30 kbar. The equation is based on a formulation of the Helmholtz free energy as a function of tem- perature and density. The theoretical model and its empirical extensions obey the first and second laws of thermodynamics and thus allow for computation of any thermody- namic value by differentiation. The equation is consistent with virtually all experimental observations. Mixtures of water and carbon dioxide: As a first approximation inferences from P-V-T-X measurements about interactions Chapter 2. Properties of CO2 - H2O Mixtures at High Pressure and Temperature 50 between CO2 and H2O may be made and tested against what limited spectroscopic and other data exist. From the strong depression of the critical composition curve to lower temperatures (see previous section) compared to mixtures with an inert gas (i.e. argon) one might expect strong chemical association up to at least 400 °C and 4 kbar. The positive deviation of the volume from ideality on mixing is presumably due to repulsive interaction, i.e. dominated by dispersion forces. This effect may be reduced or even reversed by strong chemical association. Greenwood's (1969) P-V-T-X measurements indicate distinct negative volumes on mixing at temperatures above 600 °C and pressures below 300-400 bar, with the most negative values being at lowest measured pressures (50 bar) and small Xco2 • This would indicate that chemical association remains significant beyond 800 °C but is decreasing with increasing pressure. From theoretical consider- ations (Prausnitz et al., 1986) one would expect chemical association to become more pronounced with increasing pressure. Franck &z Todheide (1959) interpret their P-V- T-X measurements in terms of deviations from a van der Waals equation of state with 'geometric' mixing rules and estimate the maximum amount of chemical association in terms of carbonic acid (see also appendix E). De Santis et al. (1974) use these data to- gether with solubility measurements of carbonic acid at low pressure and temperature to formulate a temperature dependent equilibrium constant (C0 2 + H 2 0 ^ H 2 C O 3 ) that is incorporated into their modified Redlich-Kwong equation of state (see below). The P-V-T-X data by Franck & Todheide are considered suspect for reasons explained later, underestimating the volume effect on mixing significantly. Read's (1975) measurements on electrical conductance suggest at least some chemical association up to the limits of his experiments at 250 °C and 2 kbar. The volume effect on mixing is most pronounced in water-rich compositions (see figures in appendix E) over most of P-T-X space experimentally covered by Greenwood (1969) and Gehrig (1980). The partial molar excess volume of C 0 2 at high dilution is Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 51 therefore larger than that of H2O at the same dilution. Furthermore, the activity of water in the mixture deviates significantly from ideality over almost the entire range of fluid composition, whereas the activity of CO2 becomes nearly ideal at Xco2 larger than 0.6 at temperatures above 500 °C. From this one might conclude that H 2 0 simply dilutes C 0 2 up to some concentration above which interaction becomes significant. Conversely, the ideal behaviour at high Xco2 might be fortuitous in that the interaction based on dispersion forces is offset by effects of chemical association. Infra red and Raman spectroscopy on fluids at elevated temperature and pressure are rare due to obvious technical difficulties that put an upper pressure limit at 2-3 kbar dictated by the use of sapphire windows. The limited data available (Kruse, 1975, and summary by Franck, 1981) only show a weak band on a Raman spectrum at low temperatures which may be assigned to carbonic acid. Spectra at higher temperatures and pressures are dominated by features related to the CO2 and H 2 0 molecule without clear evidence for chemical association. While inferences on the nature of molecular interaction from P-V-T-X data are ham- pered by its macroscopic nature and by inconsistencies amongst data sets it is nevertheless concluded that interactions, physical or chemical, are complex and change with pressure, temperature and fluid composition. Spectroscopic data and measurements of electrical conductance do not put any quantitative constraints on chemical association at present. This makes justification for any type of equation of state and mixing rules ambiguous. Chapter 2. Properties of C02 - H2O Mixtures at High Pressure and Temperature 52 2.4 Existing Equations of State The most commonly used equations of state that predict P-V-T-X properties of H 2 0 - CO2 mixtures at conditions of interest to metamorphic and igneous petrologists in- clude the Kerrick k Jacobs (1981) equation and the Holloway-Flowers equation (Hol- loway, 1977, Flowers, 1979) which are both modifications of the Redlich-Kwong equation (Redlich k Kwong, 1949). Powell and Holland (1985) and Holland k Powell (1990) fit a polynomial function for the logarithm of the fugacity to data computed with other equations of state. The mixture is modeled by a subregular, asymmetric solution model for the activities which are fit to values computed with the Kerrick k Jacobs equation. Shmulovich et al. (1980) present a virial equation and use a Margules formulation to model the excess volume on mixing as a function of composition from which they derive the fugacities. In many applications to geologic problems water and carbon dioxide have been assumed to mix ideally (i.e. Eugster k Skippen, 1967, Greenwood, 1967b) due to the lack of constraints on non-ideality. This may result in distorted phase diagram topologies over a wide range of temperature, pressure and fluid composition (see appendix E and below). Spycher k Reed (1988) present a pressure-explicit virial equation applicable to calculations of hydrothermal boiling, with upper limits at 1 kbar and 1000 °C. The nu- merous equations and modifications proposed in the chemical literature (more than 100 differently modified Redlich-Kwong equations), many of them based on corresponding states systematics, fail at pressures above 500-1000 bars and temperatures as low as 500 °C. A notable exception was a paper by de Santis et al. (1974) which included higher pressures and formed the basis of Holloway's (1977) adaptation of the Redlich-Kwong equation to geological problems. Chapter 2. Properties of CO2 - H2O Mixtures at High Pressure and Temperature 53 Tabulations of P-V-T-X properties and/or fugacities or activity coefficients of CO2- H 2 0 mixtures are offered in several publications, most of them based on graphical in- tegration or some function fitting technique to experimental P-V-T-X data: Franck &; Todheide, 1959, Greenwood & Barnes, 1966, Greenwood, 1969, 1973, Ryzhenko &: Ma- linin, 1971, Wood & Fraser, 1977, Shmulovich et al., 1980. Kerrick & Jacobs equation: Kerrick & Jacobs (1981) adopt Carnahan & Starling's (1972) hard-sphere Redlich- Kwong equation with a modified attractive parameter, a, that is a function of temperature and volume (and therefore of pressure): RT(l + y + y2-y*) _ a(V,T) V(l-y)3 VTV(V + b) { ' with y = b/(4V) and a(V,T) = c + d/V + e/V2, where c, d and e are second order polynomials in temperature. This results in a total of 10 adjustable parameters, given in units of bar, K and cm3/mole: CH2O = [290.78 - (0.30276 • T) + (1.4774 • 10~4 • T2)] • 106 dH2o = [-8374 + (19.437 • T) - (8.148 • 10 - 3 • T2)] • 106 enao = [76600 - (133.9 • T) + (0.1071 • T2)} • 106 cCo7 = [28.31 + (0.10721 • T) - (8.81 • 10"6 • T2)] • 106 dCo2 = [9380 - (8.53 • T) + (1.189 • 10"3 • T2)] • 106 eco2 = [-368654 + (715.9 • T) + (0.1534 • T2)] • 106 b„2o = 29, bCo2 = 58 The parameters are not fit simultaneously to experimental data because of convergence problems using non-linear least squares regression techniques. A b parameter is arbi- trarily chosen in such a way as to yield a(V) curves for various isotherms having a Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 54 monotonic form in order to permit fitting of the simple power expansions of c, d and e to experimental data. The equation of state is thus semi-empirical and a rigorous physical interpretation of the parameters is not possible. Simple geometric mixing rules are employed to obtain parameters for an equation for mixtures: fcmix = T.btXi (2.7) t Omix = E E ^ ^ ' a ' i (2-8) «' J where the binary interaction parameters are obtained from a,j = y/ajaj. Note that these mixing rules are of a 'geometric' nature and do not explicitly account for interactions arising from the polar nature of water, from the very different size and shape of the molecules, or from chemical association. Thermodynamic properties of the gas mixture may be computed analytically from the equation. The expression for the fugacity coefficient of a species in the mixture is too long to be reproduced (equation 27 in Kerrick & Jacobs, 1981). The volume at spec- ified pressure, temperature and composition has to be computed iteratively. APL and FORTRAN codes are published in a separate paper (Jacobs & Kerrick, 1981a). Jacobs & Kerrick (1981b) expand the two-species binary system to include methane. In a com- panion paper Jacobs & Kerrick (1981c) use their equation to compute numerous phase diagrams. The authors present results for the pure gases from about 300-1000 °C, and from 400-800 °C for mixtures. Jacobs & Kerrick (1981a) recommend the equation over the range 300-1050 °C and 1-20000 bar. Because of the functional form of the a(V,T) term there is no positive volume defined for pure carbon dioxide at low temperatures. This lower temperature limit increases with rising pressure and is located at about 480 °C at 50 kbar. At 400 °C difficulties were encountered in evaluating the volumes of mixtures in a small pressure interval at about 200 bars using their published computer code (see Chapter 2. Properties of CO2 - H2O Mixtures at High Pressure and Temperature 55 figures E.36 and E.37 in appendix E). This may be a convergence problem of the root finding algorithm used by the authors, or more likely the mathematical behaviour of the equation close to the two-phase region. This problem is 'invisible' for the computation of thermodynamic properties at higher pressures which require integration of the partial molar volume with respect to pressure. In contrast to the Holloway-Flowers-de Santis equation (see below) the Kerrick & Jacobs model does not allow for chemical association and thus no negative volumetric contribution to the effect on mixing is permissible. Thus all deviations from ideality are constrained to be positive except very close to critical conditions. Holloway-Flowers-de Santis equation: Holloway (1976, 1977) introduced the Redlich-Kwong equation of state (Redlich & Kwong, 1949) to geologists, adapted and modified from de Santis et al. (1974): , V-b VfV(V + b) 1 ' For simple, non-polar molecules the a parameter is approximately constant and may be derived from critical data (Redlich & Kwong, 1949). De Santis et al. (1974) introduced a temperature dependence, a(T) = a0 + a'(T), which takes into account large dipole or quadrupole moments. The temperature dependence of a'(T) was refit by Holloway to P-V-T data extending to higher pressures and temperatures (H 2 0 extrapolated to 1800 °C from previous extrapolations by Holloway et al. (1971)). The functions chosen are second or third order polynomials in temperature. The b and a° parameters are adopted from de Santis et al. (1974). The mixing rules are identical to those adopted by Kerrick h Jacobs (equations 2.7 and 2.8). The binary interaction parameter is expanded by de Santis et al. (1974) to account for complications due to the formation of a complex by a reaction of the form C0 2 + H2OT=i complex (i.e. H 2 C0 3 ) which is supported by experimental evidence at low Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 56 temperatures (Coan k King, 1971). The equilibrium constant K of this reaction is then entered into the binary interaction parameter a,j for H2O-CO2 mixtures in the form: dj = y/oyj+il^R'T^K (2.10) &mix = Y^biXi (2.11) amix = (2.12) where R is the gas constant. The temperature dependence of the logarithm of the equi- librium constant K is expanded in a third order series of inverse powers of temperature. Numerical values of the equilibrium constant are derived from low temperature solubility data (Coan k King, 1971) and P-V-T-X data of Franck k Todheide (1959) at 400-750 °C up to 2 kbar. Franck k Todheide (1959) discuss the maximum possible amount of formation of carbonic acid by comparison of their measured volumes of the mixtures with those computed by a van der Waals equation with geometric mixing rules. However, the application of standard mixing rules to H2O-CO2 mixtures is questionable, as has been discussed above. The data of Franck k Todheide are suspect (see discussion in section 'P-V-T-X Data on H2O-CO2 Mixtures' below) and the term for chemical association derived by de Santis et al. (1974) introduced some additional uncertainty into the equa- tions of state of Holloway (1977), Flowers k Helgeson (1983) and Bowers k Helgeson (1983) who adapted the above treatment. Note also that equation 2.10 is derived for non-polar gases that have no significant dipole or quadrupole moments. More recent data on electrical conductance and inferences on the first ionization potential of carbonic acid by Read (1975) suggest that at least some carbonic acid is present up to 250 °C and 2 kbar which is the upper limit of the measurements. The approach by Kerrick k Jacobs is hindered because it excludes chemical association (see above). An evaluation of the de Santis et al. approach versus the Kerrick k Jacobs approach is not yet possible due to insufficient or contradictory experimental data. Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 57 The resulting equation for H2O-CO2 mixtures of Holloway (1977) has 15 parameters which are all fit to various sets of experimental data, but not simultaneously. The tem- perature is in °C for a(T) and in Kelvin for In K(T), pressure is in atmospheres, and the volume in cm3/mole: a°co2 = 46 • 106, bCo7 = 29.7 a%0 = 35 -10 6, 6tf20 = 14.6 a'co2(T) = 73.03 • 106 - 71400 • T + 21.57 • T2 aH2o(T) = 166.8 • 106 - 193080 • T + 186.4 • T2 - 0.071288 • T3 , ~ , < 5953 2746 • 103 464.6 • 106 In A = -11.07 + — — + R = 82.05 Holloway (1981a, 1981b) refit the a'(T) term for H 2 0 to an extended data base with separate polynomials for three different temperature intervals. The expression for temper- atures below 400 °C was derived by J. Connolly (J.R. Holloway, written communication, June, 1990). T < 600°C : a'H20(T) = 4.221-10 9-3.1227-10 7-T + 87485-T 2 -107.295-T 3 + 4.86111-10- 2-T 4 600 < T < 1200 : a'Hi0{T) = 166.8 • 106 - 193080 • T + 186.4 • T2 - 7.1288 • 10 - 2 • T 3 T > 1200°C : a'H20(T) = 140 • 106 - 50000 • T T < 400°C : a'H20(T) = 0.987(7.4174 • 107 + 2.132 • 105 • T - 442.1 • T2 + 0.28623 • T 3 ) Flowers (1979) pointed out that Holloway's (1977) treatement of the fluid mixture for H2O-CO2 violated the Gibbs-Duhem relationship (i.e. X{d In a,; = 0 ) which led to an erroneous expression for the fugacity coefficient (7;) that resulted in excessive deviations from ideality. The correct expression for the fugacity coefficient relative to ideal mixing Chapter 2. Properties of CO2 - H20 Mixtures at High Pressure and Temperature 58 at P and T is given by Flowers (1979) and Prausnitz et al. (1986): ln7< = layv^J + v^-- w o * * -H— ^mix^i b2 • RTZI2 In V where z'=C02 or H2O. The same equation presented in the review article by Ferry & Baumgartner (1987, eq. 108, p. 342) contains a typographical error. Powell & Holland equation: Powell & Holland (1985) fit the expression RTlnf for endmember gases to a sec- ond order polynomial in temperature with coefficients that expand into power series of pressure with up to four terms. Their calibration is based on fugacity data for H2O by Burnham et al. (1968) and the high pressure extrapolation of Delaney & Helgeson (1978). CO2 fugacities are fit to data derived by Shmonov & Shmulovich (1974) up to 10 kbar and the high pressure extrapolation predicted by the Bottinga & Richet (1981) equation of state. Activities in mixtures are modeled by a subregular, asymmetric solution model with Margules parameters that have a linear dependence on temperature and pressure. The Margules parameters are fit to activities of mixtures computed with the Kerrick & Jacobs (1981) modified Redlich-Kwong equation of state. Powell & Holland's equa- tions are stated to be applicable in the range 2-10 kbar and 400-800 °C, with separate Margules parameters to extend the pressure range below 2 kbar. In an extended version of their thermodynamic database, Holland & Powell (1990) expand the power series of the expressions for the logarithm of the endmember fugacities to a total of 15 and 13 parameters for C 0 2 and H2O, respectively. The improved range of applicability is said to be 0.1-40 kbar and 300-1200 °C for H 2 0 and 300-1400 °C for CO2. The mixtures are treated with the subregular solution model presented in 1985. The limits of appliction of the mixing model are not stated but probably coincide with Chapter 2. Properties of C 0 2 - H 2 0 Mixtures at High Pressure and Temperature 59 those of Kerrick & Jacobs (1981). Bowers-Helgeson equation: Bowers & Helgeson (1983) present a modified Redlich-Kwong equation of state for H 2 0 - C 0 2 - N a C l mixtures for the range 350-600 °C and pressures greater than 500 bars (Bowers &: Helgeson, 1985). The treatement of H 2 0 and C 0 2 is based on the work of de Santis et al. (1974), Holloway (1977, 1981a) and Flowers (1979). The a(T) and b parameters for C 0 2 are identical to those of Holloway (1977). Sodium chloride is not treated as a separate species but rather incorporated into the a(T) and b terms of H 2 0 by making them a function of NaCl content. The authors achieve a better fit by making a0 for H 2 0 a function of temperature. The mixing rules are identical to those of de Santis et al. (1974), including the formulation for the effect of complex formation (see discussion above). Spycher & Reed equation: Spycher & Reed (1988) present quadratic, pressure-explicit virial-like equations for H 2 , C 0 2 , C H 4 , H 2 0 and H 2 0 - C 0 2 - C H 4 mixtures applicable to problems within the hydrothermal regime, including boiling processes. Their equations for H 2 0 and C 0 2 and their mixtures, calibrated to P-V-T data and solubility data, are applicable up to 500 bar at temperatures below 350 °C, and up to 1000 bar for temperatures between 450 and 1000 °C. Saxena & Fei equations: Saxena & Fei (1987a, 1987b) present pressure-explicit or volume-explicit virial-like equations of state for many pure gas species, including H 2 0 and C 0 2 , based on shock wave data and P-V-T measurements. Efforts are under way to extend this work to fluid mixtures (Fei & Saxena, 1987). A detailed discussion and comparison of their equations must await full publication of their research. Their equation for pure carbon dioxide is Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 60 discussed in chapter 1. 2.5 Thermodynamic Relationships for Fluid Mixtures In this section the thermodynamic relationships for fluids and fluid mixtures are devel- oped. While most standard texts cover the subject adequately, they do not provide the necessary details relevant to our treatment of the phase equilibrium experiments. I found the following references particularly useful: Prausnitz et al. (1986) as a comprehensive treatise and the geologically oriented summaries of Wood &: Fraser (1977), Anderson (1977) and Skippen & Carmichael (1977). The notation is tabulated for clarity in table 2.6 and also explained in the text. Pure fluid: The fugacity / (or 'thermodynamic pressure') is best introduced by considering the change in chemical potential of the pure fluid from one state to another state at constant temperature T: For a perfect gas (PV = RT) equation 2.14 is easily integrated to obtain iiP2'T - fip"T = RT\n(P2/Pl). (2.15) Lewis (1907) defined the fugacity in such a way as to preserve the mathematical form of equation 2.15 ^state 2 _ ^state 1 _ RT\n (/state 2//state = RT In a (2.16) where states 1 and 2 are at the same temperature but may differ in fluid pressure or fluid composition. The activity a describes how 'active' a species (gas, solid, liquid, pure or in a mixture) is with respect to a standard state that needs to be specified in order to assign a numerical value to the activity. Chapter 2. Properties of C 0 2 - H20 Mixtures at High Pressure and Temperature 61 P pressure p, partial pressure of component i T absolute temperature in Kelvin V molar volume v? molar volume of pure component i Vi partial molar volume of component i in a mixture vm molar volume of a mixture V volume of a mechanical (ideal) fluid mixture ve molar excess volume of a mixture partial molar excess volume of component i in a mixture Xi mole fraction of component i in a mixture chemical potential chemical potential of component i in a mixture chemical potential of pure component i f fugacity fi fugacity of component i in a mixture ft fugacity of pure component i 7 fugacity coefficient relative to a perfect gas (PV = RT) 7i fugacity coefficient of component i in a mixture relative to perfect gases that mix ideally 7,? fugacity coefficient of component i relative to ideal mixing at pressure and temperature of interest a activity a,- activity of component i in a mixture A; activity coefficient of component i in a mixture stoichiometric coefficient of component i in a reaction normalized stoichiometric coefficient of component i R gas constant Superscript * refers to a variable at some standard state Superscript l d refers to a variable in a state of ideal mixing Superscript e x p refers to an experimentally measured variable Superscripts ^.v,*.- r e f e r to a variable at state P, T, V or Xi Table 2.6: Notation for thermodynamic equations. Chapter 2. Properties of C 0 2 - H20 Mixtures at High Pressure and Temperature 62 If / i P l , T in equation 2.14 is chosen as a reference state at a pressure Pi sufficiently low in order for the perfect gas law to hold true, i.e. Pi = 1 bar, it follows that fPl'T ~ Pi = 1 bar, and equation 2.16 will reduce to: _ ^tbar . r = RTh.^T/i bar) = RTln fp'T. (2.17) Note that fugacity has the unit of pressure but the expression In / is a ratio and therefore has no unit. For this particular standard state the activity a is numerically equal to the fugacity. The fugacity coefficient 7 may be introduced to describe the deviation from a perfect gas at P and T: / p ' T = P . 7 p ' r . (2.18) Given an equation of state that relates pressure, volume and temperature for a fluid the fugacity may be readily computed by integration from equation 2.14 and 2.17 RTlnfp'T= fP [V(P)]TdP (2.19) Jl bar Fluid Mixtures: The compositional variable Xi denotes the mole fraction of component i in the mixture. Equation 2.16 may be used to introduce the fugacity /, for component i in a mixture, where now state 1 and state 2 may differ not only in fluid pressure but also in composition Xi. ^state 2 _ ^state 1 = JJJ,^ ^-state 2/^state = R T ^ fl. ^ . 2 0 ) For the following derivations differences of the chemical potential with respect to various standard states are examined in order to obtain fugacities, activities and volume integrals. First, operating at constant composition: (2.21) Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 63 Denning unit fugacity at Px = 1 bar and using the perfect gas approximation and ideal P T X' mixing one can obtain /, l ' ' ' = X{PX = X\ • 1 bar and write: /zf 2 ' T ' X i - ^ l b a r - T '*« = Xi [V,(P)}TX. dP = RT\nff2'T'Xi - RTlnXi. (2.22) J l b a r ' 1 The fugacity coefficient 7 ; of component i that describes the deviation from an ideal mixture of perfect gases becomes: fp T' Xi =Xi-P- i?'T'Xi. (2.23) In an ideal mixture of perfect gases the fugacity coefficient becomes unity and the fugacity is equal to the partial pressure P, = XiP. In an ideal fluid mixture there is no excess volume on mixing and the fugacity is related to the fugacity of the component in its pure P T X ~ oP T state by the mole fraction: / , ' ' ' = X , • / , ' . Because the partial pressures must sum to the total pressure equation 2.23 leads to the relationship: P = Pi — Yji filli- For many practical purposes 7 * may be used relative to a state of ideal mixing of real gases at pressure and temperature of interest and equation 2.23 becomes: fP,T,Xt = ^ . f o P , T , X i . 7.F,r,X,. ( 2 2 4 ) This fugacity coefficient 7 * relates directly to the excess properties of a mixture. Note that the volume Vi in equation 2.21 is the partial molar volume which may be obtained at constant P and T through differentiation of the volume of the mixture with respect to the compostional variable X , : "̂'-O,/ (2-25) It is possible to measure the volume Vm of a mixture experimentally as a function of P, T and X , (see section below). In order to compute the fugacity / , of component i the volume of the mixture at constant temperature as a function of both pressure and Chapter 2. Properties of C02 - H2O Mixtures at High Pressure and Temperature 64 composition must be known. The computation by graphical means without an equation of state requires therefore an enormous number of experimental measurements. For a binary mixture with components A and B , and mole fractions XA and XB = 1 — XA one can write (see figure 2.8 for a geometrical interpretation): yP,T,XA = yP,T _ x ( 9V£T\ V P T , X B = z V P , T + x f d V ^ B - = v™ ( 2 ' 2 6 ) It is practical to introduce excess volume as the difference between the real volume of a mixture, Vm, and the volume of an hypothetical mechanical mixture V = J2i X{V°. The molar excess volume of a mixture is then defined by VE — VM — V, and the partial molar excess volume of a component on mixing becomes Vei = VJ- — V°. Equations 2.26 may now be rewritten in terms of excess volumes: VP,T,XA = yP,T _ % ( 9VeP'T\ {dxB ) P T V™. = vr + XA[°?g) (2,7) A simple geometric interpretation of equation 2.27 is provided in figure 2.8. Using equation 2.22 and 2.17 the integral of the excess volume with respect to pressure is related to the fugacities: .p .p p I VeT'XidP = I V?'XidP - I V°TdP = Y (RT In fpT'Xi - RT In Xi - RT In / ° F ' T ) fP,T,Xi • RT'In y^oP,T J ~ RTlnX, ± (RTln a-^-\ = l-RTlm?™ (2.28) Chapter 2. Properties of C0 2 - H20 Mixtures at High Pressure and Temperature 65 Figure 2.8: Geometric interpretation of the partial molar excess volume in a binary fluid mixture with components A and B. Note that the standard state for the activity is the pure component at pressure and temperature of interest. Similarly the volume integral of the excess volume of the mixture may be expressed as a linear combination of the partial molar excess volumes (see figure 2.8): f V ^ d P ^ ^ f V^dP. (2.29) J l b a r • J l b a r For a mixture we obtain: cP,T,Xi (P Vep'TdP = ^ i j r i n ^ ^ - ^ P T l n X , - J1 bar .• r 1 = E ^ i n ^ = E ^ i n 7 ; ™ . (2.30) Returning to equation 2.16 one can examine the change of chemical potential as a function of the compositional variable Xi at constant P and T compared to the pure state of component i: ^ ' - tfP'T = Win {f?<T'Xi I ff'T) = * T log a ™ . (2.31) This choice of standard state of the pure component at the pressure of interest is most Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 66 useful to define the activity of a component in a mixture. A pure component has thus an activity equal to unity at any pressure and temperature and the value of the activity for a component in a mixture is a number between zero and unity. The activity coefficient A, may be defined to describe the deviation of a mixture from ideality (a, = Xi) at constant P and T: ap'T'Xi = Xi • XP'T'Xi. (2.32) In an ideal solution the activity coefficient is unity and the activity of a component in a mixture is equal to its mole fraction. The fugacity coefficient is related to the activity coefficient through the relationship between the fugacity ratio and the activity, with the pure state at P and T as a standard state: fP,T,Xi ^P<T,Xi « f ' ™ = W or A f ^ = ̂ = 7 r ^ . (2.33) fV'- J ' 7,° Returning for a third time to equation 2.16 pressure and composition are changed simultaneously to obtain the relationship lP2,T,Xi _ oPuT _ (..P2.TJCi _ ..Pi,TJCi\ , (llPi,T,Xi _ ..oPi,T\ ri ri ~ ^ri ri J 1 \ri ri ) = RT In (fP2'T'Xi/fPl 'T'Xi) + RT In ( / f 1 'T'Xl Iff1 'T) (2.34) which simplifies to the following expression choosing Pi = 1 bar as a reference state and assuming ideal mixing of perfect gases at 1 bar: HP*'T>Xi - n°Pl'T = RTln(fP'T'Xi/Xi • lbar) + RTln{Xi • lbar/1 bar) = RT\nfP'T'Xi. (2.35) From the above relationships it is also obvious that the activity and the fugacity are not independent and only one of them needs to be specified. Equations for the treatment of phase equilibrium data: Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 67 T P = constant experimental datum 0 ideal H 2 0 JVfexp i d C 0 2 Figure 2.9: Schematic diagram of a single fluid species equilibrium. See text for expla- nation. In this section expressions are derived that relate the displacement from experimen- tally measured phase equilibrium positions to a hypothetical position where the fluid components mix ideally. In order to use this information to constrain volumetric prop- erties of the fluid, the observed deviation from ideality is expressed in terms of fugacities (or activities) together with the corresponding volume integrals over pressure. These relationships are used in a subsequent section to compare experimentally measured devi- ations from ideality of a H 2 0 - C 0 2 fluid to non-idealities predicted by equations of state and mixing rules. Single fluid species equilibria: First, phase equibbria are considered in which only one species in a mixture takes part in the reaction. Figure 2.9 illustrates such an equilibrium in a mole fraction versus temperature diagram of a binary fluid mixture at constant pres- sure. The solid line depicts an experimentally determined equilibrium boundary whereas the dashed boundary is that of the same equilibrium computed with the assumption of an ideally mixing fluid phase. The deviation from ideality is expressed at constant pressure and temperature as a difference in composition between Xexp and Xld. At both Chapter 2. Properties of C02 - H2O Mixtures at High Pressure and Temperature 68 locations, Xexp and X,d, the change of the Gibbs potential of a reaction is formulated: ,P,T,A7XP ARG = 0 = gp'T[^ y ; , 5 ; , ( c P ) , - (T) , ttfc(r), & ( P ) ] + ^ JET in - (2.36) A R G = 0 = Qp*\vi% Vk\ 5*, (CPUT), a*(T), &(P)] + ^ .PPln ^ — — , (2.37) where subscript j denotes all phases that take part in the reaction, k relates to solid phases only and i is the subscript for the single gas species i. The function Q contains all contributions to the Gibbs potential from the solid phases and the contributions from the gas species with the exception of the volume integral (RTIn f term). Because the two states under consideration are at the same pressure and temperature the Q terms cancel and one can equate the fugacity ratios directly: ,P,T,.Y?XP rP,T,X\d l n ^ 7 r - = l n ^ _ _ . (2.38) Note that the computed Q carries the sum of the uncertainties of the thermodynamic data of all phases involved which propagate into relationship 2.38. Remembering that for an ideal mixture /; = X,/° and that the fugacity ratio is equal to the activity relative to the standard state defined by the fugacity (i.e. f°P'T) one obtains: rP,T,Xfxp = a w r p = X ( l d ( 2 3 9 ) Ji Equally well the deviation from ideality may be related to the fugacity coefficient (see equation 2.23): 7 , W i = ^ • ±p- or 7 : P ' T ' X< (2.40) Equation 2.38 is used to relate the volume integral of the partial molar excess volume Ve; with respect to pressure given in equation 2.28: RT\nXld = RT \n fP'T'XrP - RT In f°p'T = XJxp [P VeidP + RTIn A 7 x p , (2.41) Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 69 or: Xrp (P VeidP = RT In = RT In 7 ; P - T ' * e x p . (2.42) J l b a r - A , Equilibria in which several fluid species take part: The equations that describe de- viations from an ideally mixing fluid are derived in a similar fashion as in the previous case of a single fluid species equilibrium. Only expressions for some combination of the fugacities (or activities) of the species that take part in a reaction may be obtained. Unlike the first case, the stoichiometric coefficients of the fluid species will remain in the equations. From equations 2.36 and 2.37 that relate the change of the Gibbs potential of a reaction for the two states at X**9 and X;d it follows: J^ViRTln1' =J2^RTln^-—-. (2.43) i Ji i Ji Using the relationship for ideal mixing / , = X{f° one can simplify and obtain: P T X*xp 5 *i In f-^—pT- = £ In a r X J P " P - £ i/,- In X'f. (2.44) « Ji i i The activity is defined relative to a standard state of the pure species at P and T of interest. In order to introduce the volume integrals of the fluid species into equation 2.43 equation 2.38 is used or equation 2.42 is expanded directly to obtain: rp yid ZviXT" L VeidP = Y,»>RTIn (2.45) For a mixture in which all fluid species take part in a reaction, and which all have identical stoichiometric coefficients, the left-hand-side of equation 2.45 becomes equal to the molar excess volume of the mixture (see equation 2.29): l b a r K r ^ = £ ^ l n | t (2-46) Chapter 2. Properties of CO2 - H20 Mixtures at High Pressure and Temperature 70 For convenience equations 2.44 and 2.45 are made independent of the magnitude of the stoichiometric coefficients by normalizing them to unity by introducing a normalized stoichiometric coefficient m: V, = (2-47) E,- For computer manipulation of a large number of phase equilibrium experiemnts it is convenient to generalize the above expressions to deviations from ideality at arbitrarily chosen pressure, temperature and fluid composition. Relationships analogous to equa- tions 2.36 and 2.37 lead to: GPCXP>T"P - Gpid'T>d = E »>R(T',d ~ T e x p ) ^ , i d - In ^ ' I ^ p j • (2.48) The reference state is that of a pure fluid at P and T of interest. Note that this rela- tionship is not independent of the thermophysical properties of the minerals involved in the equilibrium. The computation of Q is described by Berman (1988) and in chapter 1 (equation 1.1) for stoichiometric phases. For numerical optimizations where fluid prop- erties are to be constrained and where thermophysical parameters of selected minerals are to be adjusted simultaneously, the computing time required for the algorithm may become substantial. 2.6 P - V - T — X Properties Constrained by Phase Equilibrium Data Similar to the method developed in chapter 1 for pure C 0 2 phase equilibrium experiments involving a H20-C0 2 mixture may be used to put bounds on the integrated partial molar excess volumes (fugacity) of CO2 and/or H2O. The combined bounds from many exper- imental half-brackets, which each put one inequality constraint of the form of equation 2.48, will confine any equation of state within feasible limits with respect to measured phase equilibria. This may be used to test existing equations of state or to constrain pa- rameters of a new equation using numerical optimization procedures analogous to those Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 71 outlined in chapter 1. This approach is valid if the following requirements are fulfilled: i) the thermophysical properties of the minerals involved are accurately known, ii) the constraining phase equi- librium experiments are trustworthy, iii) the standard state thermodynamic properties of the fluid (i.e. the pure species at P and T of interst) to which the excess volumes on mixing are related to is accurately known, and iv) the presence of significant species other than CO2 and H2O in the experimental charges can be ruled out. Some mathematical form of an equation of state must be adopted. More details and a possible approach are discussed in a later section after evaluation of the available experimental data. This thesis proceeds only as far as a thorough review of existing data and equations of state, for reasons discussed below, and does not offer a new equation of state for H2O-CO2 mixtures. The approach outlined above leads to a focussed review of existing work with respect to suspect data and directing one to future experimental work that constrains thermophysical properties of fluids most efficiently. 2.7 P - V - T - X Data on H 2 0 - C 0 2 Mixtures Appendix E contains a series of pressure versus volume and fluid composition versus volume diagrams comparing measured volumes of H2O-CO2 mixtures to several equations of state already introduced in a previous section. A brief account of the experimental methods for all relevant studies is provided in the same appendix partly because several publications are written in German and in Russian. The most striking feature apparent from inspection of the figures in appendix E are the substantial inconsistencies between data sets of Franck &: Todheide (1959), Greenwood Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 72 (1973), Gehrig (1980) and Shmulovich et al. (1979, 1980). There is reasonable agree- ment, although not within stated experimental uncertainties, between data by Green- wood (1973) and Gehrig (1980). The Kerrick k Jacobs (1981) equation of state which reproduces experimentally measured phase equilibria quite well (Berman, 1988, Jacobs k Kerrick, 1981c, and discussion below) loosely approximates the data by Greenwood and Gehrig between 400-600 °C. It is concluded that the volumes measured by Franck k Todheide are most likely in error, particularly at pressures below 1 kbar, and should not be used. Franck k Todheide's data were used by de Santis et al. (1974) for a modification of the Redlich k Kwong equation of state which was subsequently adopted by Holloway (1977), Flowers k Helgeson (1983) and Bowers k Helgeson (1983) (see discussion of equations of state above). Franck k Todheide's data above 1.4 kbar are not included in the figures of this appendix due to large uncertainties compared to the small volume effect on mixing at high pressures. These uncertainties are a result of not reporting P-V-T data on the pure fluids. For the computation of the excess volume below 1.4 kbar the pure gas data of Kennedy (1954) and Holser k Kennedy (1958, 1959). were used (figures E.36 to E.45 in appendix E) which Franck k Todheide state to agree within 1 % of their measurements. The same procedure had to be adopted for the data of Gehrig (1980) who reported data of the mixtures between Xco2 = 0.1-0.9 but none for the pure fluids. Shmulovich et al. (1980) measured volumes at 400 and 500 °C and selected fluid compositions between 1 and 5 kbar. The data are plotted up to 2 kbar in figures E.37 and E.40. Shmulovich's measurements are probably of good quality and extrapolate reasonably from data at lower pressures. Measurements by Shmulovich et al. (1979) at 400 and 500 °C from 150- 1000 bar and 400-5700 bar show excess volumes significantly larger than all of the other data over much of the range measured. It is not possible to relate the apparently erroneous measurements of Franck k Chapter 2. Properties of C 0 2 - H 2 0 Mixtures at High Pressure and Temperature 73 Todheide (1959) to any particular cause in their experimental method (appendix E). A combination of several causes seems possible: i) the large volume of connections rela- tive to the volume of the bomb which may introduce larger uncertainties due to thermal gradients than are suggested by their cakbration; ii) the measurement of the volume of the bomb with the mercury method tends to underestimate the volume (Greenwood, 1969); iii) the copper packing of the Bridgman seal can creep, leading to an increase of the bomb volume with time and thus to underestimated volumes; and iv) the possibility of an incremental change in fluid composition each time the valve to the pressure gauge is opened. An interesting and important feature of the measured P-V-T-X properties is the negative volumes on mixing observed by Greenwood (1969, 1973) at temperatures above 600 °C and pressures below 300-400 bar. If these measurements are correct, significant chemical association must occur in order to reverse the positive volume effect resulting from dispersion forces alone. Fugacity coefficients (the integrated partial molar excess volume weighted by its mole fraction) may be substantially smaller in this range com- pared to predictions from equations of state (i.e. Kerrick &; Jacobs, 1981) that do not account explicitly for some mode of chemical association. The Holloway-Flowers equa- tion (Holloway, 1977, Flowers, 1979) which accounts for formation of carbonic acid (de Santis et al., 1974) does not represent Greenwood's (1973) data better (see appendix E). In summary it is concluded that a set of P-V-T-X data on C 0 2 - H 2 0 mixtures is much needed to resolve the inconsistencies and extend the range of pressure beyond that of Greenwood (1973) and Gehrig (1980) who measured up to 500 and 600 bar. Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 74 2.8 Phase Equilibrium Data Involving H 2 0 - C 0 2 Mixtures Berman (1988) presented a thermodynamic analysis of all equilibria relevant to this study with extensive graphical comparison of computed equilibria with experimental measure- ments. Some more recent experimental data on mixed-volatile equilibria involving mag- nesite and cbnochlore were presented by Chernosky k Berman (1989). New data on the equilibrium calcite+andalusite+quartz anorthite+C02 in the presence of H 2 0 were reported in a preliminary form by Chernosky k Berman (1990). The system CaO-MgO- S i 0 2 - C 0 2 - H 2 0 was most recently addressed by Trommsdorff k Connolly (1990) with respect to some inconsistencies between phase diagram topologies computed with the data base of Berman (1988) and those derived from field observations (see discussion of magnesite in chapter 1). The following discussion of data on selected phase equilibria builds on the analysis and figures of Berman (1988), who used the Kerrick k Jacobs (1981) equation of state for the H 2 0 - C 0 2 mixtures. The abbreviation 'B88' refers to Berman (1988). Calcite+quartz ^ wollastonite+C0 2 (figure 8a, B88): Data by Ziegenbein &; Johannes (1974, 2-6 kbar) are inconsistent with experiments by Greenwood (1967, 1-2 kbar) and Jacobs & Kerrick (1981c, 6 kbar). The latter two data sets are in accord with the thermodynamic data base of B88. The constraints on the equilibrium with pure C 0 2 (Harker k Tuttle, 1956, 0.5-3 kbar) are only compatible with Greenwood's data, indicating that the experiments by Ziegenbein k Johannes (1974) are most likely in error. Some uncertainty in the properties of calcite exists due to the fact that B88 used constraints from mixed volatile equilibria and the equation of Kerrick k Jacobs (1981) which may deviate somewhat from reality. Wollastonite-(-calcite-|-quartz T=* grossular+C0 2 (figure 9a-c, B88): Only data by Gordon k Greenwood (1971, 1-2 kbar) could be reconciled by B88. Experimental Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 75 460 4 4 0 - y 4 2 0 - LU cn CE cn LU 400 - 3 8 0 - 3 6 0 - 3 4 0 - X (C02) at P = 2000, 1000 (bar) Figure 2.10: Equilibrium calcite+andalusite+quartz ^ anorthite+C02 at 2 kbar. Solid curves are computed with the assumption of ideal mixing in the fluid phase. Dotted curves were computed with the Kerrick & Jacobs (1981) equation of state. See text for discussion of inconsistencies. Filled symbols refer to brackets that were used to constrain RTlnj in figure 2.13. The diagram was produced with GEO-Calc (Brown et al., 1988). Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 76 brackets by Shmulovich (1978, 1-6 kbar) and Hoschek (1974, 1 & 4 kbar) deviate too much from the position of the equilibrium assuming ideal mixing to be accounted for by positive deviations from ideality within the range of those computed with the Kerrick & Jacobs (1981) equation. Calcite+andalusite+quartz ^ anorthi te+C0 2 (figure 2.10; figure 8c, B88): New data (Chernosky &z Berman, 1990, 1-2 kbar) are reproduced by the B88 data base within the uncertainties of the thermodynamic properties of the minerals and render previous experiments by Jacobs Sz Kerrick (1981c, 2 kbar) and Kerrick & Ghent (1979, 2 kbar) distinctly inconsistent. An important implication is that the large observed de- viations from ideal mixing in the fluid phase at these low tempeartures of Chernosky's experiments (350-410 °C) are reasonably reproduced by the Kerrick & Jacobs (1981) equation of state. Calcite+kyanite+quartz ^ anorthi te+C0 2 (figure 8b, B88): Experiments by Jacobs & Kerrick (1981c, 6 kbar) and Kerrick Sz Ghent (1979, 6 kbar) between 450 and 580 °C are in reasonable agreement with the data base of B88, but do not put tight constraints on C 0 2 fugacities. System C a O - A l 2 0 3 - S i 0 2 - C 0 2 at high T (figure lOa-f, B88): Experiments on equilibria in this system at temperatures of 670-850 °C amongst anorthite, calcite, grossu- lar, gehlenite, corundum, wollastonite, and C 0 2 diluted with H 2 0 have one trend in common: brackets by Hoschek (1974, 1 Hz 4 kbar) would demand negative deviations from ideal mixing in the fluid phase compared to the data base of B88. While negative deviations from ideality at high temperatures may be supported by P-V-T-X data of Greenwood (1973, c.f. appendix E) this discrepancy might equally well be due to errors in the data base or problems with the experimental study. One would expect that H 2 0 and C 0 2 would mix almost ideally at these high temperatures and at an Xco 2°f 0.5-0.8 where this discrepancy is observed. Chapter 2. Properties of C 0 2 - H20 Mixtures at High Pressure and Temperature 77 System C a O - M g O - S i 0 2 - C 0 2 at high T (figure 24a-d, B88): Data by Zharikov et a/.(1977, 1 kbar) on equilibria involving calcite, merwinite, monticellite, akermanite, diopside, forsterite, periclase, C 0 2 diluted with H 2 0 at temperatures of 700-950 °C demand slightly negative deviations from an ideally mixing fluid at Xco2 between 0.15- 0.3 which is the upper limit of the range examined. Such negative deviations from ideality at these conditions are reasonable in the light of Greenwood's (1973) P-V-T-X data which allow for negative deviations from ideality. Note that the Kerrick k Jacobs' (1981) model is virtually indistinguishable from ideality at these conditions. System C a O - M g O - S i 0 2 - C 0 2 - H 2 0 (figure 27a-d k 28, B88): Experiments in- cluding tremolite-C0 2-H 20±(diopside, forsterite, calcite, dolomite, quartz) at 430-700 °C (Chernosky k Berman, 1986, 1 kbar; Metz, 1967, 1 kbar; Metz, 1976, 5 kbar; Eg- gert k Kerrick, 1981, 6 kbar; Slaughter et al., 1975, 2 kbar) are consistent with the data base of B88 with two exceptions: brackets on the reaction dolomite+tremolite T=* forsterite+ calcite-f-C02+H20 at 0.5 kbar (Metz, 1967), and experiments on the equi- librium calcite+quartz+tremofite T=* diopside+H20+C02 at 1 and 5 kbar (Metz, 1970). This discrepancy is not fully explained by the effects of composition of Metz' natural tremolite (Campolungo, Switzerland) according to B88. Data by Slaughter et al. (1975, 1 & 5 kbar) on the latter reaction is consistent with the analysis of B88. The equilibrium quar tz+dolomite+H 2 0^ calci te+talc+C0 2 (figure 2.11) stud- ied at 400-550 °C by Gordon k Greenwood (1970,1-2 kbar), Metz k Puhan (1970,1971, 1-5 kbar), Skippen (1970, 1-2 kbar) and Eggert k Kerrick (1981, 6 kbar) is consistent with the data base of B88 at pressures above 2 kbar. Data at 2 kbar at 425-450 °C and particularly at 1 kbar and 400 °C suggest more positive deviations from ideal mixing in the fluid phase than computed with the Kerrick k Jacobs (1981) equation of state. Note that there is some uncertainty as to the ordering state of dolomite at higher temperatures (see discussion in B88). Chapter 2. Properties of CO2 - H2O Mixtures at High Pressure and Temperature 78 X (C02) Figure 2.11: Equilibrium quartz+dolomite+H20 ^ calcite + talc+C0 2 at 1, 2 and 3 kbar. Curves were computed with the Kerrick & Jacobs (1981) equation of state. Experimental data at 5 and 6 kbar are not shown for clarity. See text for discussion of inconsistencies. The diagram was produced with GEO-Calc (Brown et al, 1988). Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 79 System A l 2 0 3-Si02 - H 20 - C 0 2 (figure 33a-b, B88): Experiments on equilibria involving clinochlore-calcite-diopside-forsterite-spinel-C02-H20± dolomite at 550-700 °C by Chernosky k Berman (1986, 1-2 kbar) and Widmark (1980, 1-4 kbar) are in reasonable agreement with the data base of B88. See discussion in B88 about the tem- perature dependence of the aluminum content of clinochlore which makes careful char- acterization of experimental charges nesessary. Clinochlore+magnesite ^ forsterite-f spinel-f-H20-|-C02 (figure 34a, B88): - Brackets on this equilibrium by Chernosky Sz Berman (1986, 1-4 kbar, 530-670 °C) are in good agreement with the data base of B88, but are displaced to slightly higher temperatures at 1-2 kbar using the new enthalpy for magnesite of chapter 1 which renders magnesite less stable. This discrepancy might be explained by more nonideal mixing in the fluid phase, although the effects of uncertainties in the order-disorder state of spinel could be significant as well. System M g O-Si02-C0 2 - H 20 (figure 34b-c k 35a-c, B88): Unreversed exper- iments that constrain only the high-temperature side of the equilibrium magnesite-H quartz+H 20^ talc+C0 2 by Johannes (1969, 1-7 kbar, 330-610 °C) agree with B88. Brackets on the reaction magnesite+talc T = ± forsterite-f-H20+C02 (figure 2.12) by Johannes (1969, 1-7 kbar, 480-660 °C) are at odds at 2 kbar with the analysis of B88 and expermemnts by Greenwood (1967, 1-2 kbar, 470-550 °C). Greenwood's data no longer agree with the computed equilibrium using the new enthalpy for magnesite derived in chapter 1. If thermodynamic properties of magnesite, talc, forsterite and the pure fluids are accurate this latter discrepancy must be due to too little positive deviation from ideality (less stable fluid phase) compared to the Kerrick k Jacobs (1981) equation, or due to problems with both Greenwood's and Johannes' experiments. There is some indication that the thermophysical properties of talc are in need of some revision (see discussion in section 'Conclusions' below). Chapter 2. Properties of C02 - H 2 0 Mixtures at High Pressure and Temperature 80 X (C02) at P = 0.5, 1, 2, 4, 7 kbar Figure 2.12: Equilibrium magnesite-ftalc ^ forsterite+H20+C02 at 0.5, 1, 2, 4 and 7 kbar. Solid symbols depict experimental charges with the reactant-stable phase assem- blage. Solid curves were computed with the Kerrick & Jacobs (1981) equation of state and magnesite properties as revised in chapter 1. The dotted curves at 0.5, 1, 4 and 7 kbar were computed with the assumption of ideal mixing in the fluid phase. The dashed curves at 1 and 2 kbar were computed with the magnesite properties of B88 and the Kerrick k Jacobs (1981) equation of state. See text for discussion of inconsistencies. The diagram was produced with GEO-calc (Brown et al, 1988). Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 81 Brackets on three equilibria with magnesite and chrysotile (350-480 °C), talc, or anthophyllite (520-550 °C) by Johannes (1969, 1-4 kbar) at XCo20.03-0.08 do not put stringent constraints on the properties of the fluid mixture. Systems with muscovite, phlogopite, paragonite, potassium feldspar, al- bite, H2O, CO2 (figure 40a-c k 41a-c, B88): Experiments on the breakdown of parag- onite by Shvedenkov et al. (1983, 1 kbar, 460-510 °C) and that of muscovite+calcite+ quartz by Hewitt (1973, 2-7 kbar, 450-620 °C) are in good agreement with the data base of B88. Brackets on the breakdown of muscovite+quartz in the presence of an H2O-CO2 fluid by Shvedenkov et al. (1973, 1 kbar, 450-520 °C) are displaced to lower tempera- tures relative to computed equilibria by 20-40 °C for reasons unknown (see discussion in B88). This latter equilibrium and that involving paragonite are the only reactions studied experimentally that involve H2O diluted by CO2. Brackets on equilibria involving phlogopite by Bohlen (1983, 5 kbar, 775-800 °C), Puhan (1978, 4-6 kbar, 460-650 °C), Puhan k Johannes (1974, 2 kbar, 420-530 °C), Hoschek (1973, 2-6 kbar, 500-650 °C) and Hewitt (1975, 2-4 kbar, 460-520 °C) agree reasonably with the data base of B88. Several inconsistencies remain with experiments in the presence of pure H2O. The problem of non-stoichiometry of phlogopite is discussed by Berman (1988) in this context. Rutile+quartz-f-calcite ^ sphene + C 0 2 (figure 45, B88): Experiments at 6 kbar by Jacobs k Kerrick (1981c, 500-620 °C) are in good agreement with the data base of B88, but are shifted to higher temperatures at 2 kbar, 480-530 °C and Xco 2at 0.4-0.8. This discrepancy of 25-45 °C is too large to be attributed solely to more nonideal mixing in the fluid phase than computed with the Kerrick k Jacobs (1981) equation of state. Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 82 While the body of experimental data including H2O-CO2 mixtures is substantial, in- compatibilities amongst experimentalists studying the same equilibrium are disturbingly numerous. One reason for this is that such experiments are technically difficult to control with ample possibilities for errors that remain undetected. This places obvious difficul- ties on selecting experiments that put reasonable constraints on the deviation from ideal mixing of the fluid phase. Suggestions about which inconsistencies should be resolved experimentally are outlined in the concluding section along with proposed additional studies. 2.9 Results, Discussion and a Possible Approach P-V-T-X data and existing equations of state: Substantial inconsistencies exist between P-V-T-X data measured by different au- thors. The rough agreement between data of Greenwood (1969,1973) and Gehrig (1980) and that of the Kerrick & Jacobs (1981) equation of state points towards errors in the measurements by Franck & Todheide (1959). Data by Shmulovich et al. (1980) are difficult to evaluate at present but extrapolate reasonably from measurements ar lower pressures. Of particular interest is the negative volume effect on mixing observed by Greenwood at temperatures above 600 °C and pressures below 3 0 0-400 bar. Negative volumes would imply some form of chemical association which has consequences for the theoretical basis on which mixing rules are formulated. While the Holloway-Flowers-de Santis equation of state (Holloway, 1977, Flowers, 1979, de Santis et al, 1974) does ac- count for chemical association explicitly, it does not represent P-V-T-X data adequately. The Kerrick & Jacobs (1981) model does not allow for negative volumes on mixing but is in reasonable agreement with P-V-T-X data up to 600 °C and with much of the phase equilibria measured in a H2O-CO2 fluid. Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 83 Phase equilibrium data and constraints on fugacities: A disturbing number of inconsistencies amongst different data sets exist which need to be resolved before all equilibria become useful as constraints on the fugacities of the fluid species in the mixture. The number of feasible experimental brackets is further decreased by those that are not very constraining due to large experimental uncertainties. Three examples of how a particular experimental datum puts a constraint on the fugacity are provided in figure 2.13 and discussed below. Theory: There is as yet no theory that could be adopted successfully to yield an equation of state for H2O-CO2 mixtures that covers the range of pressure, temperature and com- position of interest to petrologists. Empirical extensions, mostly on van der Waals-like theory, are restricted by mixing rules that demand compatible mathematical forms for equations of the endmember fluids. Furthermore, analytic expressions for the fugacity of a species in the mixture are required for efficient numerical calculations, i.e. the equation of state must be integrable. A possible approach: Let us first consider improvements on existing equations of state. The obvious advantage of the Holloway-Flowers-de Santis equation (Holloway, 1977, Flowers, 1979, de Santis et al., 1974) is that parameters for many fluid species are available, particularly those relevant to the system C-O-H. Improvements to an equation for one particular species are limited by the requirement of mathematical compatibility which dictates that the parameters a and b can only be functions of temperature. The derivation of equation 2.13, for example, includes integration at constant temperature and thus only modifications of the temperature dependence of a and b are permitted. In chapter 1 significant improvements on the equation of state for CO2 could only be obtained by Chapter 2. Properties of C 0 2 - H2O Mixtures at High Pressure and Temperature 84 introducing a volume (pressure) dependence of the b parameter. From P-V-T-X data and spectroscopic observations one would expect that the equilibrium constant for chemical association is a function of both temperature and pressure, and perhaps fluid composition. Again, an equilibrium constant dependent on pressure is not permissible if we must maintain mathematical compatibility. While minor improvements are possible based on an extended data base and more robust optimization techniques, major improvements would certainly demand more complex expressions for the parameters than permissible by the existing model. A possible new approach would follow an alternate route in that the mixing properties are formulated independently of the pure fluids. This has the advantage that any equation of state for the pure fluid may be used. The disadvantage is that the extension of such a model to additional fluid species such as methane, carbon monoxide, hydrogen and oxygen is more complex. Two approaches may be taken: i) a mathematical model that describes the excess volume of the mixture as a function of pressure, temperature and fluid composition; ii) a mathematical model that directly describes the logarithm of the fugacitiy of a fluid species as a function of pressure, temperature and fluid composition. Both approaches would require reliable P-V-T-X data over a much larger range in pressure than available at present. It may be advantagous to modify option i) such that the partial molar excess volume is expressed as a function of P, T and Xco2 which would facilitate easy computation of fugacities by integration with respect to pressure. The measured excess volumes of the mixture would then constrain the combination of the partial molar excess volumes of H2O and CO2 which requires integration with respect to Xco2. Using option ii) care must be taken in choosing meaningful P, T and Xco2 dependencies of the function describing the logarithm of the fugacity, i.e. the integrated partial molar volume. Powell Sz Holland (1985), for example, choose a linear dependence Chapter 2. Properties of C02 - H2O Mixtures at High Pressure and Temperature 85 on pressure and temperature which cannot be justified and must lead to significant de- viations and uncertain extrapolation properties. The dependence on fluid composition could be treated with a Margules parameter formulation which has the advantage of providing mathematically tractable expressions on integration and differentiation. One could follow traditional avenues for formulating the excess Gibbs potential on mixing, G ^ x (i.e. Brown, 1977, Berman &: Brown, 1987): G w = WGL2XLX* + WG2lXlX2 = RT{XX l n 7 l + X2 l n 7 2 ) (2.49) This two-parameter Margules expansion may easily be expanded to allow for more flex- ibility in the dependence of the excess properties on composition. If one were to follow the approach commonly adopted for solid solutions, the pressure and temperature de- pendence of the excess Gibbs potential could be incorporated by suitable expansion: WG = WH- TWs + {P- 1)WV (2.50) The last term including Wy stems from the assumption of constant volume and is not adequate for a solution of fluids where the volume term will form the largest contribution to the excess Gibbs potential. The first two terms are neglected in the subsequent treatement for simplicity but could easily be incorporated if required. Some pressure and temperature dependence of Wy is chosen arbitrarily: WG= jF Wv(P,T)dP (2.51) Jlbar The volume of the mixture can then be obtained by differentiation of the excess Gibbs function with respect to pressure: = Wv,2(P,T)X1X* + Wv2i(P,T)X:2X2 (2.52) Chapter 2. Properties of C02 - H 2 0 Mixtures at High Pressure and Temperature 86 Although the approximate shape is known from figures E.36 to E.54, the exact mathe- matical form of the functions describing the Wy parameters or the partial molar excess volume has not been established. Perhaps one could borrow from some formulations used in profile fitting of diffraction patterns (Howard k Preston, 1989). Constraints from phase equilibrium experiments may be incorporated as outlined in a previous section and discussed in detail in chapter 1. Each half-bracket puts an inequality constraint derived from equation 2.48 on the integrated partial molar excess volume. All constraints from valid experimental data combined define a feasible region for adjustable parameters in an equation of state that are consistent with all constraining equilibria. The formulation of the inequality constraints require integration and possibly differentiation of the equation of state. For all but the simplest equations of state the problem becomes non-linear. Figure 2.13 illustrates how experimental half-brackets put constraints on the log- arithm of the fugacity of C 0 2 . These examples are taken from an equilibrium which involve C 0 2 diluted by H 2 0 : calcite + andalusite + quartz ^ anorthite + C 0 2 (figure 2.10). The constraints are obtained from the displacement of the experimental datum from a hypothetical position assuming ideal mixing in isobaric-isothermal sections. See section 'Thermodynamic Relationships' and figure 2.9 for details. Results are compared to several equations of state. Mathematical programming offers an appropriate formalism for such an approach and computer codes are readily available (see chapter 1, section 'Method' for details). Parameters for the model describing the excess properties on mixing would be optimized to P-V-T-X data and simultaneously subject to constraints from phase equilibrium ex- periments. If standard state properties for selected solid phases are to be derived simul- taneously the optimization may become computationally time consuming. Walter (1963) and Ziegenbein k Johannes (1982) also tried to constrain the fugacity Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 87 - S i n o II C5 i n o 8 - -4 h > a> i l So o o (J) <-> cn + c n c o e L 1 N <D 1 ° 03 1 + «tf i -o D> | C - X 141 (0 + o 1 o c 1 <-> L 10 0 0 IWftftfl) HI a o ao o II © II 8-, 3 -stS UJ - Q. - B J O O mo- OQ'O- IMOWQ) N~l Figure 2.13: Graphs of In 7 versus pressure for CO2 comparing phase equilibrium brackets to equations of state by Kerrick k Jacobs (1981) and Holloway, 1977 (see section 2.4 for equation of state parameters). Each vertical bar represents the range in In 7 permissible by one particular exparimental half-bracket on the equilibrium calcite + andalusite -f quartz anorthite + CO2. The figures on the left show two half-brackets by Jacobs k Kerrick (1981c) at 363 and 377 °C, including uncertainties. The figures on the right show half-brackets by Chernosky k Berman (1990) at 405 and 412 °C. See figure 2.10 for location of the experiemntal data. Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 88 of CO2 in a H2O-CO2 mixture from experimentally measured phase equilibria. Walter (1963) used data on the equilibrium magnesite periclase + C 0 2 in the presence of H 2 0 (Walter et al., 1962), and Ziegenbein & Johannes (1982) used measurements on the reaction calcite + quartz ^ wollastonite + CO2 diluted with H2O (Ziegenbein h Johannes, 1974). The results are limited by the consideration of a single equilibrium which leads to curves of activity versus fluid composition at constant pressure that are polythermal, i.e. the temperature is rising with increasing Xco2 • Ziegenbein & Johannes (1982) derive deviations from ideal mixing that are much larger than those computed from the Holloway-Flowers (Holloway, 1977, Flowers, 1979) equation of state or those derived by Ryzhenko &; Malinin (1971) from P-V-T-X data. This is because experimental data by Ziegenbein &: Johannes (1974), on which the analysis in their 1984 paper is based, are shifted distinctly to lower temperatures compared to brackets by Greenwood (1967) and the data base of Berman (1988) and also are not compatible with experiments in pure C 0 2 by Harker k Tuttle (1956). 2.10 Conclusions and Future Work From the discussions in the previous sections and the data presented throughout chapter 2 and appendix E, it is concluded that at present insufficient experimental data and inadequate theoretical foundations are available on which to build a constrained empirical model that allows one to compute fugacities of H2O-CO2 mixtures over the range of pressure and temperature of interest to petrologists. However, the existing body of data is extensive enough that the effort needed to provide additional information is not insurmountable. The following problems or lack of data deserve particular attention: 1. Inconsistensies amongst existing sets of P-V-T-X measurements must be resolved and the range of pressure must be increased to at least 8 kbar by a complete series Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 89 of P-V-T-X measuruements. 2. Several inconsistencies amongst studies on phase equilibria involving H2O-CO2 mixtures need to be resolved in order to permit refinement of thermophysical prop- erties of solid phases which will then provide better constraints on the fluid mixture. Numerous inconsistencies are addressed by Berman (1988) which may serve as a guide for designing the most useful experiments. Several examples are presented below. 3. Phase equilibria involving H2O-CO2 mixtures can be utilized to constrain devi- ations from ideal mixing where P-V-T-X data are not available. Deviations are pronounced at low temperatures where resolution would be best for experimental work. P-V-T-X measurements are not yet possible above 15 kbar which render phase equilibrium studies the only means with which to examine deviations from ideality at high pressures. Several equilibria that could be studied in a piston- cylinder apparatus are proposed below. 4. At present there are hardly any constraints from phase equilibrium data on the activity of H 2 0 in a H2O-CO2 mixture at any pressure and temperature. Several possible studies to supply such information are discussed below. The last few sections below are devoted to specific experimental projects that are feasible with the technology and equipment available in many petrologic laboratories. Some of the projects may be realized in a one-month time frame while others would require two years of concentrated effort. Proposed P-V-T-X experiments: To resolve inconsistencies amongst existing studies (appendix E) and to form a firm basis for an equation of state, an extensive P-V-T-X study is required that covers at least the range 200-8000 bar and 200-800 °C and the entire compositional space. Such Chapter 2. Properties of C 0 2 - H20 Mixtures at High Pressure and Temperature 90 experiments would best be done in an internally heated gas apparatus furnished with an expandable noble metal sample container ('accordion type') and an attached linear voltage differential transducer (LVDT) to measure linear expansion of the sample 'bag'. This linear expansion can be calibrated with well known gases (i.e. argon, water) to yield volumes of the 'bag' at measured displacements. Such a pressure vessel with a large enough volume can be purchased 'off the shelf with modifications necessary to install the LVDT mechanism and temperature probes to control thermal gradients. Furnace assemblies can be machined from pyrophyllite to any shape desired. 'Accordion bags' are commercially available, commonly made of gold. This apparatus could be designed to hold pressures up to 12-15 kbar, but vessels with a large volume are difficult to seal at these pressures and store a dangerous amount of energy in the form of compressed hot fluids. In order to obtain excess volumes of sufficient precision from the difference of two large numbers, the volume of the mechanical mixture and the measured volume of the mixture, the volumes of the pure fluids must be measured as precisely as possible. A precision of at least ±0.5 % is required to yield an excess volume with a precision of ±1 % of the volume of the mixture. At pressures below 300 bar the precision has to be much better. Because of the limited expansion of the sample bags the proposed pressure range is only accessible with repeat runs that contain different amounts of sample. The appropriate pressure intervals are approximately 200-500 bar, 500-2000 bar, and 2000-8000 bar. While not the entire range of composition is of importance to geologists it is required to constrain the partial molar volumes which are used to compute fugacities (see section 'Thermodynamic Relationships'). In this respect it is important to undertake measure- ments at very small and very large mole fractions of C 0 2 . Ideally, one would like to obtain data at XCo2 = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95. Furthermore, any empirical equation of state gains strength from a large number of measurements. Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 91 One problem that needs attention is hydrogen diffusion through gold at elevated temperatures. Container materials different from gold could be used and run times should be kept as short as possible. Alternatively, one could impose externally a hydrogen fugacity close to the one predicted to occur in the mixture and thus remove gradients. Some monitor of speciation should be incorporated to assertain the presence of H2O and CO2 as the dominant species. Perhaps the hydrogen or oxygen fugacity could be measured by some means that does not affect the sample mass in an uncontrollable way. A P-V-T-X project of the proposed extent is a major undertaking that probably requires 2 years of concentrated effort. The proposed phase equilibrium studies require less effort and are experimentally not as difficult as P-V-T-X measurements. Proposed phase equilibrium studies: 5 magnesite + 1 talc ^ 4 forsterite + 1 H 2 0 + 5 C 0 2 : New brackets on this equilibrium (see figure 2.12) would resolve the inconsistencies amongst data by Johannes (1969) and Greenwood (1967). It would put further constraints on the enthalpy of magnesite as well as put some limits on the deviation from ideal mixing of the fluid phase. This is an easy experiment to do which would require only one tight bracket at 1, 2 and possibly 5 kbar, placed at the culmination of the T-X curve at a fluid composition of X c o 2 = 5 / 6 . An experiment with this starting fluid composition and both reactant and product assemblages present will remain at constant fluid composition during the experiment. Four to six experiments spaced 5 K apart are probably all that is required to obtain a very tight reversal at a given pressure. All minerals involved can readily be synthesized and are relatively reactive. From Greenwood's (1967b) experience it appears that run times of 15-20 days are sufficient. Equil ibria involving margarite: The proposed studies would aim at providing constraints on the activity of H2O in a H 20-C02 mixture in a temperature range where Chapter 2. Properties of C 0 2 - H20 Mixtures at High Pressure and Temperature 92 deviations from ideality are expected to be significant. Equilibria with margarite in pure H 2 0 have been studied previously by Jenkins (1984), Storre & Nitsch (1974), Nitsch et al. (1981), Chatterjee et al. (1984), and Chatterjee (1974). Al l data are consistent with the analysis of Berman (1988, figures 13a-e) with one obvious exception of the brackets by Storre & Nitsch on the equilibrium margarite + quartz kyanite + zoisite + H 2 0 that cannot be reconciled with any other data. The two equilibria suitable for study in a mixed fluid are: 1 margarite + 1 quartz 1 andalusite + 1 anorthite + 1 H 2 0 and 1 margarite ^ 1 anorthite + 1 corundum + 1 H 2 0 (figure 2.14). The range accessible to experimentation at 2 kbar fluid pressure spans 400-500 °C and XQO2 = 0.0-0.7. The range in temperature may be increased by studying the equilibria at pressures below and above 2 kbar. At 1 kbar the isobarically invariant point margarite-quartz-andalusite-anorthite-calciteis predicted to be at 375 °C and XCo2 =0.78 with the data base of Berman (1988). Margarite becomes metastable with respect to calcite + andalusite at fluid composi- tions larger than Xco 2 = 0.7. The equilibrium, 1 margarite + 1 C 0 2 1 calcite + 2 andalusite + 1 H 2 0 , has a very large dT/dX at constant pressure such that it could form tight constraints on the position of invariant points and thus thermodynamic properties of phases involved. The ratio of the stoichiometric coefficients of H 2 0 and C 0 2 is 1:1 which allows one to relate deviations from ideality directly to the molar excess volume of the mixture (equation 2.46). Constraints on the temperature at the culmination of the equilibrium 1 calcite + 1 margarite + 2 quartz ^ 2 anorthite + 1 H 2 0 + 1 C 0 2 at -Xco2 = 0.5 would pose constraints similar to those of the previous equilibrium. This reaction could be investigated with the strategy already discussed for the magnesite-talc-forsterite equilibrium. Chapter 2. Properties of C 0 2 - H2O Mixtures at High Pressure and Temperature 93 0.0 0.2 0.4 0.6 0.8 1.0 X (C02) at P = 2000 bar Figure 2.14: Phase diagram with equilibria involving margarite (Mrg), andalusite (And), corundum (Co), quartz (aQz), anorthite (An), H 2 0 and C 0 2 at 2 kbar pressure. The solid curves were computed with the Kerrick & Jacobs (1981) equation of state. Dotted lines were calculated assuming ideal mixing in the fluid phase. Reactions with zoisite or grossular that modify the diagram at very H 2 0-rich compositions are not shown. The diagram was computed with GEO-Calc (Brown et al, 1988). Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 94 While many other hydrous phases exist, most of them are not suitable for our purposes because their stabilities are limited to water-rich fluid compositions. At temperatures above approximately 500 °C anthophyllite would be accessible to experimentation over a large range of fluid compositions. There is however some debate at present with respect to thermophysical properties of anthophyllite (the volume in particular) and possibly talc (B.W. Evans and R.G. Berman, pers. comm., June 1990). Experiments involving an- thophyllite would therefore be appropriate in the context of resolving these uncertainties rather than deriving constraints on the fluid mixture. Equilibria at pressures greater than 15 kbar: Such studies would aim at con- straining deviations from ideal mixing in the fluid phase at pressures not accessible by P-V-T-X mesurements. Experiments at pressures beyond that attainable in gas appa- rati have to be performed in solid-media pressure vessels such as the piston-cylinder apparatus used for work presented in appendix C. The obvious disadvantages are the small sample volume and the difficulty in controlling fluid composition, speciation in the fluid phase and diffusion of hydrogen at high temperatures. Sample charges are best externally buffered at some oxidation potential high enough to ensure the absence of significant amounts of species other than H 2 0 and C 0 2 . Alternately, if the fugacities of other species are known corrections may be applied. Suitable equilibria should include only solid phases that are thermodynamically well known and which are constrained experimentally in the presence of pure C 0 2 or pure H 2 0 . Furthermore, only H 2 0 or C 0 2 should take part in the reaction and the slope of these dehydration/decarbonation reactions should have a large dP/dT in order to im- prove resolution. Because only the relative displacement from the equilibrium in the presence of the pure fluid is sought some of the uncertainties arising from extrapolation of thermodynamic properties cancel. The possibilities are too numerous to be discussed Chapter 2. Properties of C02 - H20 Mixtures at High Pressure and Temperature 95 in detail but preference should be given to equilibria that may be studied at high pres- sures and relatively low temperatures in order to decrease experimental uncertainties and errors arising from extrapolated thermodynamic properties. Feasible studies include the reactions: 1. 2 diaspore ^ corundum + 1 H 2 0 (580 °C at 20 kbar) 2. 1 forsterite + 1 tremolite ^ 5 enstatite + 2 diopside + 1 H 2 0 (650 °C at 20 kbar) 3. 1 talc + 1 forsterite ^ 5 enstatite + 1 water (700 °C at 20 kbar) 4. 1 talc ^ 3 enstatite + 1 quartz 4- 1 H 2 0 (800 °C at 20 kbar) 5. 1 brucite ^ 1 periclase -f 1 H 2 0 (950 °C at 20 kbar) 6. 1 magnesite + 1 coesite ^ 1 forsterite + 1 C 0 2 (1100 °C at 35 kbar) 7. 1 calcite + 1 quartz ^ 1 wollastonite + 1 C 0 2 (1350 °C at 20 kbar) In order to resolve deviations from ideal mixing in the fluid phase the best precision possible has to be achieved, which is about ±5 K for piston-cylinder work. This author is not convinced at present whether piston-cylinder work can be successfully applied as proposed to yield results of the desired quality. Only a trial will demonstrate feasibility of such experiments. Bibliography [1] Anderson, G.M. 1977, Fugacity, activity, and the equilibrium constant. 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Appendix A Integration of the Equation of State The quantity Jpo VQQ2 dP is most easily obtained from the relationship J^VdP = £° PdV + Vp(P-P0)-Po(Vo-VP) fV° R T jr, fV° A l Jir fV° A1 v*+c • + VP{P - P0) - Po(V0 - VP) (A.53) with B = Bx + B2T and C = B3/B. The attractive terms (Ax and A2 terms) become upon integration: Aifl 1 \ A2 ( 1 1 T The repulsive term (RT term) requires integration of a rational function of the form rv0 mV3 + n dV (A.55) / Ivp V(aV3 + bV2 + cV + d) with m = RT,n = RTC, a = l,b= -B, c = 0, d = C = B3/B and B = Bx + B2T. Integral (A.55) may be solved after splitting into partial fractions provided the real roots of the polynomial aV3 + bV2 -f- cV + d are known. Case I: polynomial aV3 + bV2 + cV + d yields 3 real roots Xi, X2, X3 Integral (A.55) may be split into partial fractions by solving mV3 + n _ £ J K L V{aV3 + bV2 + cV + d) ~ V + V - X1 + V - X2 + V - X3 ^ ' ' Cross multiplication and collecting terms in powers of V yields four equations in four unknowns (I, J, K, L) by comparing coefficients: n = I(-XXX2X3) (A.57) 0 = I(X1X2+X1X3 + X2X3) + J(X2X3) + K(X1X3) + L(XlX2) (A.58) 0 = I(X1+X2 + X3) + J(X2 + X3) + K(X1+X3) + L(X1+X2) (A.59) m = I + J + K + L (A.60) 106 Appendix A. Integration of the Equation of State 107 Equations (A.57) through (A.60) are solved simultaneously to obtain: I = (A.61) XXX2X3 j = mXi + n (A62) Xl{-Xl2 + XlX2-X2X3 + XlX3) y ' ; K = 1 7 1 X 2 + n (A.63) X2(X2 — X\X2 — X2X3 + X\X3) i = mXi + n (A64) X3(X$ + X\X2 — X2X3 — X\X3) Integral (A.55) may now be integrated by parts to obtain: Case II: polynomial aV3 + bV2 + cV + d yields 1 real root X\ Integral (A.55) may be split into partial fractions by solving: mV3 + n I J KV + L = - + TZ 77" 4- 7 7 7 — 7 7 — , (A.66) V(aV3 + bV2 + cV + d) V V - Xx V2 + aV + 8 with a = b/a + X\ and B = c/a + Xxa which yields four equations in four unknowns (7, J, K, L): n = -I0Xi (A.67) 0 = IiP-aXJ + JB-LXi (A.68) 0 = I(a-X1) + Ja-KX1 + L (A.69) m = I + J + K (A.70) Equations (A.67) through (A.70) are solved simultaneously to obtain: 1 = -w, x 3 ( A J 1 ) 3 = ~ Xx{aXx\ ^ X2) ( A - ? 2 ) _ a/3mX1 + an + f32m + nXx . K ~ 8(aX1 + 8 + X2) ( A > 7 3 ) _ a2n + anXx + 82mXx - fin . . . 1 ~ 8(aX1+(3 + X2) ( A J 4 ) Appendix A. Integration of the Equation of State 108 z2 + t2 dz (A.75) The last term of equation (A.66) is expanded to standard integrals by appropriate sub- stitutions: f ^ V + J ^ r_K^ rL-Ks J V2 + aV + f3 J z2 + t2 J with z = V + s, s = a/2, t = j3 — s2 and dV = dz. Integration of (A.55) yields: + p L Ks VQ-XX\ K ny0 + s)2 + t2' + 2 n\(vP + sy + t2i (A.76) with a = bja + Xx, (3 = c/a + aXi, s = a/2 and t — y//3 — s2. Appendix B F O R T R A N - 7 7 Subroutine This appendix contains the listing of a FORTRAN-77 subroutine that computes the volume and the fugacity, / , of carbon dioxide at a specified pressure and temperature. The compressibility factor z = PV/RT, density, fugacity coefficient 7 = f/P and the volume integral / VdP = RT In / may also be retrieved. Quantities that are not required may be removed from the program with the appropriate variables blanked out in the calling sequence of the subroutine and its corresponding argument list. The computing efficiency gained by this procedure is not significant in the case where the fugacity is required. If the volume is the only quantity sought then the parts and subroutines designed for the fugacity computations should be removed to make the code more efficient. The comment statements provided with the code are sufficient for this purpose. C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C # MADER & BERMAN EQUATION OF STATE FOR C02 # C # # C # Ref: - Mader et a l . , 1988, # C # Geol. Soc. Amer., Abstr . w. P r o g r . , p.A190 # C # - Mader & Berman, 1991 # C # In prep. ( J . Pet.) # C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C These subroutines compute the volume and fugacity of C02 C at spec i f i ed pressure and temperature: C C P = RT / (V-B) - A l / T*V**2 + A2 / V**4 C B = B l + B2*T - B3/(V**3+C); C = B3/(B1+B2*T) C C Program writ ten by Urs Mader, June 1987 C modified: Sept. 1988, by Urs (better volume i t e r a t i o n routine) C modified: July 1989, by Urs (clean up integrat ion code) C modified: March 1990, by Urs ( fool proof volume i terator) C C Program i s written i n standard FORTRAN 77 C C Note: Rootfinder i s not designed to handle s u b c r i t i c a l pressure and C temperature simultaneously ( i . e . 89.2 bar , 335.6 K ) . To do so, the C root f inder must be capable of recognizing s u b c r i t i c a l conditions and C f i n d the saturated volumes by equating f(vap) = f ( l i q ) . C C L i s t of subroutines: C MABE90: main subroutine, returns volume, fugac i ty , etc. at P,T C C02V0L: computes volume at P.T using C02BIS,C02R00.C02FUN C C02INT: computes int(V)dP using C02R00,C02IN3,C02IN1,C02TIR C C02BIS: converges to V at P J by i n t e r v a l ha l f ing 109 Appendix B. FORTRAN-77 Subroutine C C C C C C C C C C C C C C C C C c c c c c c c c c c c c c c c c c c c c C02FUN: returns the function value of the equation of state C02R00: f inds root(s) of 3rd order polynomial C02IN3: integrates a t tract ive term (1st case, 3 r e a l roots) C02IN1: integrates a t tract ive term (2nd case, 1 r e a l root) C02TIR: a u x i l i a r y routine for C02IN1 L i s t of var iab les : PBAR.P TK.T VO VP.V VJ R RTLNF FUG FUGLN FUGCOE Z RHO RHOCGS A1,A2,B1,B2,B3 B C MAXIT V1.V2 NROOT X1.X2.X3 pressure [bar] temperature [K] volume at 1 bar and T [cm**3/mole] volume at P and T [cm**3/mole] volume at P and T [Joule/bar] 10*gas constant (10*8.3147 [Joule/mole/K]) R/10*T*ln(f)C02 [Joule] fugacity [bar] In(FUG) fugacity coef f ic ient compress ibi l i ty {=R*T/(P*V)> density [kg/m**3] density [g/cm**3] constants for equation of state = B l + B2*T = B3/B i t e r a t i o n l i m i t for routine C02BIS volumes used during i t e r a t i o n no. of roots of 3rd order polynomial roots of 3rd order polynomial example of a main program PROGRAM MAIN IMPLICIT REAL*8(A-H,0-Z) PBAR=1000.0D00 TK=1073.0D00 CALL MABE90(PBAR.TK,VP.VJ,FUG,FUGCOE,FUGLN,RTLNF,Z,RHO,RHOCGS) STOP END C main subroutine SUBROUTINE MABE90(PBAR,TK,VP,VJ,FUG,FUGCOE,FUGLN,RTLNF,Z,RHO,RHOCGS) IMPLICIT REAL*8(A-H,0-Z) COMMON /PAR/ R,B,B3,C,A1,A2 COMMON /LIMIT/ MAXIT LOGICAL FIRST DATA FIRST/T/ C IF(.NOT.FIRST) GOTO 11 FIRST = .FALSE. C f i t parameters: March 26, 1990 C 121 phase equ i l . Bl = 28.064740D00 B2 = 1.7287123D-4 B3 = 8.3653408D04 Al = 1.0948021D09 A2 = 3.3747488D09 R = 83.147D00 MAXIT = 50 11 B = B1+B2*TK constraints , 440 PVT Appendix B. FORTRAN-77 Subroutine C = B3/B C test for s u b c r i t i c a l P and T IF((PBAR.LT.89.2D0).AND.(TK.LT.335.6D0)) WRITE(6,*) & ' s u b c r i t i c a l P or T i n C02 rout ine' C compute VP,VJ,Z,RHO at PBAR,TK CALL C02V0L(VP,VO,PBAR,TK) VJ = VP/10.0 Z = PBAR*VP/R/TK RHOCGS = 44.010/VP RHO = RHOCGS*1000.0 C compute RTLNF,FUGLN,FUGCOE,FUG at PBAR.TK CALL C02INT(RTLNF,PBAR,TK,VP,VO) FUGLN = RTLNF/R/TK*10.0 FUG = DEXP(FUGLN) FUGCOE = FUG/PBAR RETURN END C C subroutine to compute VP of C02 at P and T SUBROUTINE C02V0L(VP,VO,P,T) IMPLICIT REAL*8(A-H,0-Z) COMMON /PAR/ R,B,B3,C,A1,A2 COMMON /LIMIT/ MAXIT C approx. VO at 1 bar and set i n i t i a l guess (VI,V2) to f i n d VP VO = R*T+B V2 = (R*T/P+B)*1.2 C compute minimum volume VMIN at T for P = i n f . C (solve V**3 - V**2*B + C = 0; from V = B - B3/(V**3 + C)) WP = -B WR = C CALL C02R00(WP,WR,XI,X2,X3,NROOT) VI = XI IF(NROOT.EQ.3) THEN IF(X2.GT.V1) VI = X2 IF(X3.GT.V1) VI = X3 END IF CALL C02BIS(V1,V2,VP,P,T) RETURN END c C subroutine to compute int(V)dP of C02 SUBROUTINE C02INT(RTLNF,P,T,VP,V0) IMPLICIT REAL*8(A-H,0-Z) COMMON /PAR/ R,B,B3,C,A1,A2 C integrate equation of state from 1 bar to P; C note: int(V)dP = int(P)dV + V*(P-1) - (V-V0)*1 WP=-B WR=C C compute roots X1.X2.X3 of 3rd order polynominal i n V C (denominator of repuls ive term of equation of state) C and integrate repuls ive term (TERM1.TERM2) CALL C02R00(WP.WR,XI,X2.X3,NROOT) IF(NR00T.EQ.3) CALL C02IN3(X1.X2,X3,V0,VP,T1,T2) IF(NROOT.Eq.l) CALL C02IN1(X1,WP,VO,VP,T1,T2) TERM1 = R*T*T1 TERM2 = R*T*C*T2 C integrate a t t rac t ive terms of equation of state TERM3 = A1/T*(1.0/V0-1.0/VP) Appendix B. FORTRAN-77 Subroutine TERM4 = -A2/3.0*(1.0/V0/V0/V0-1.0/VP/VP/VP) TERM5 = VO-VP TERM6 = (P-1.0)*VP RTLNF = (TERM1+TERM2+TERM3+TERM4-TERM5+TERM6)/10.0 RETURN END C subroutine for rootf inding for C02 equation of state SUBROUTINE C02BIS(X1,X2,XMID,P,T) IMPLICIT REAL*8(A-H,0-Z) COMMON /PAR/ R,B,B3,C,A1,A2 COMMON /LIMIT/ MAXIT ACC = Xl*1.0D-6 X = X2 DX = X1-X2 DO 11 J=l,MAXIT DX = DX*0.5 XMID = X+DX CALL C02FUN(XMID,FMID,P,T) IF(FMID.LT.O.ODOO) X=XMID IF((DABS(DX).LT.ACC).OR.(DABS(FMID).LT.1.OD-08)) RETURN 11 CONTINUE WRITE(6,*) 'MAXIT (C02BIS) EXEEDED' RETURN END C subroutine to evaluate equation of state SUBROUTINE C02FUN(V,FV,P,T) IMPLICIT REAL*8(A-H,0-Z) COMMON /PAR/ R ) B,B3,C,A1,A2 VSQ = V*V FV = R*T/(V-B+B3/(VSQ*V+C))-A1/T/VSQ+A2/VSQ/VSQ-P RETURN END C subroutines for integrat ion of repuls ive term SUBROUTINE C02R00(WP,WR,X1,X2,X3,NROOT) IMPLICIT REAL*8(A-H,0-Z) C f i n d root(s) of polynominal X**3 + WP*X**2 + WR*X qA = (-WP*WP)/3.0 QB = (2.0*WP*WP*WP+27.0*WR)/27.0 DET = QB*QB/4.0+QA*QA*QA/27.0 IF(DET.GT.O.O) THEN ARGF = DSQRT(DET)-QB/2.0 ARGE = -DSQRT(DET)-QB/2.0 SIGF = DABS(ARGF)/ARGF SIGE = DABS(ARGE)/ARGE qF = DEXP(DLOG(DABS(ARGF))/3.0)*SIGF qE = DEXP(DLOG(DABS(ARGE))/3.0)*SIGE XI = qF+qE-WP/3.0 NROOT = 1 ELSE PI = 3.1415926536D00 PHI = DARCOS(-qB/2.0/DSqRT(-qA*qA*qA/27.0)) RM = 2.0*DSqRT(-qA/3.0) XI = RM*DC0S(PHI/3.0)-WP/3.0 X2 = RM*DCOS(PHI/3.0+2.0*PI/3.0)-WP/3.0 X3 = RM*DC0S(PHI/3.0+4.0*Pl/3.0)-WP/3.0 NROOT = 3 Appendix B. FORTRAN-77 Subroutine END IF RETURN END C integrat ion for case with 3 r e a l roots X1.X2.X3 SUBROUTINE C02IN3(X1,X2,X3,VO,V,T1,T2) IMPLICIT REAL*8(A-H,0-Z) C term 1 : V**2 / (V**3 - V**2 * B + C) DIV = = X1*X1*(X2-X3)+X2*X2*(X3-X1)+X3*X3*(X1-X2) SA = X1*X1/(X1*(X1-X2-X3)+X2*X3) SB = X2*X2*(X3-X1)/DIV SC = X3*X3*(X1-X2)/DIV TI = & SA*(DL0G(VO-Xl)-DL0G(V-Xl))+SB*(DL0G(VO-X2)-DL0G(V-X2))+ SC*(DL0G(V0-X3)-DL0G(V-X3)) C term 2 : 1 / V / (V**3 - V**2 * B + C) SA = -1.0/X1/X2/X3 SB = 1.0/X1/(X1*(X1-X2-X3)+X2*X3) SC = -1.0/X2/(X2*(Xl+X3-X2)-Xl*X3) SD = 1.0/X3/(X3*(X3-X2-X1)+X1*X2) T2 = ft SA*(DL0G(VO)-DL0G(V))+SB*(DL0G(VO-Xl)-DL0G(V-Xl))+ SC*(DL0G(VO-X2)-DL0G(V-X2))+SD*(DL0G(VO-X3)-DL0G(V-X3)) RETURN END C integrat ion for case with 1 r e a l root XI SUBROUTINE C02IN1(X1.WP.VO.V.Tl,T2) IMPLICIT REAL*8(A-H,0-Z) ALP = Xl+WP BET = XI*(Xl+WP) C term 1 DIV = X1*(X1+ALP)+BET SA = X1*X1/DIV SB = (ALP*X1+BET)/DIV SC = X1*BET/DIV CALL C02TIR(SB.SC,ALP,BET,VO,V,TR) TI = SA*(DLOG(VO-X1)-DLOG(V-X1))+TR C term 2 SA = -1 .0 /BET/X1 SB = 1.0/X1/DIV SC = (ALP+XD/BET/DIV SD = (ALP*(ALP+X1)-BET)/BET/DIV CALL C02TIR(SC,SD,ALP,BET,VO,V,TR) T2 = SA*(DLOG(VO)-DL0G(V))+SB*(DL0G(VO-Xl)-DL0G(V-Xl))+TR RETURN END C a u x i l i a r routine for C02IN1 SUBROUTINE C02TIR(S1,S2,ALP,BET,VO,V,TR) IMPLICIT REAL*8(A-H,0-Z) RDET = DSQRT(4.0*BET-ALP*ALP) IF(RDET.LT.1.0D-15) RDET = 1.0D-15 TR = S1/2.0*(DL0G(V0*V0+ALP*V0+BET)-DL0G(V*V+ALP*V+BET))+ ft (2.0*S2-ALP*S1)/RDET*(DATAN((2.0*VO+ALP)/RDET)- ft DATAN((2.0*V+ALP)/RDET)) RETURN END Appendix C Piston-Cylinder Experiments C . l Piston-Cylinder Apparatus The piston-cylinder apparatus used for this study was developed and built in the depart- ment of geological sciences at UBC with the expert help of Dr. Don Lindsley (Stony Brook). It differs from the original Boyd &; England (1960) design by a concentric ar- rangement of the rams, modifications of the bridge, and an improved design of the upper electrode and the water cooling cycle. Two different bridges allow easy interchange between a 3/4-inch and a 1/2-inch bore pressure, vessel. The ram diameter to piston diameter ratio is such that 10 kbar piston pressure require 362 bar oil pressure for the 3/4-inch assembly and 160 bar for the 1/2-inch assembly respectively. This design works reliably to 25 kbar with the 3/4-inch assembly and 40 kbar with the 1/2-inch pressure vessel. Carbides under these conditions last more than 100 runs, most likely limited by careless handling of carbide parts. The apparatus is end-loaded with a force of 1000 kNewton (310 bar oil pressure) dur- ing operation. The 3/4-inch piston excerts a force of 431 kNewton (548 bar oil pressure) to achieve a nominal sample pressure of 15 kbar. Oil pressure is generated by a pump driven with compressed air and a manually operated piston pump to provide backup pressure for ram retraction when required. The power supply for the graphite resistance furnace is designed to provide up to 50 AC Amperes at about 25 AC Volts which is sufficient for temperatures up to 1600 °C with most furnace designs. Thermocouple output is measured with a Leeds k Northrup 7554 Type K-4 poten- tiometer furnished with a constant voltage supply (Leeds k Northrup, cat. 9879) and a 1-Volt standard cell (Eppley Laboratory, Inc., cat. no. 100). Temperature is con- troled by a voltage balanced against the thermocouple signal with a Leeds k Northrup 9828 DC Null Detector and a KRISEL solid-state SCR controler (Hadidiacos, 1969, with modifications). C.2 Sample Assembly Each application demands its own optimized design of a sample assembly and choice of pressure transmitting media. The assembly developed for this study is suitable for high temperature applications (> 900 °C) and consists of cheap, non-toxic and easily 114 Appendix C. Piston-Cylinder Experiments 115 sample cavity alumina disk crushable ceramic alumina ceramic AlSiMag ceramic 304 stainless steel graphite pyrophyllite pyrex glass talc 0.004" lead foil Figure C.15: Pyrex sample assembly for piston-cylinder apparatus. machinable materials. The assembly proved to perform reliably with no failures due to thermocouple breakage and only few failures due to excessive deformation of the furnace. The 3/4-inch assembly (figure C.15a) consists of a talc cylinder (0.743 in. OD, 0.500 in. ID) which contains the pyrex assembly made from standard 1/2 in. medium-wall and 8 mm medium-wall pyrex glass tubing embedding a 0.375 in. OD graphite cylinder furnace with a 0.06 in. wall thickness. A 0.2 in. thick graphite base disc and a 0.2 in. thick graphite plug at the top ensure good electrical contacts. The chamber within the inner pyrex tube is 35 mm long and 5 mm in diameter which allows the accommodation of relatively large sample capsules. The inner diameter can be further reduced by a 5 mm OD pyrex tube to 3 mm (1/8 in.). Machinable AlSiMag-222 ceramic rods serve as filler below the sample capsule. The two-hole thermocouple ceramic tube (McDanel MV20 mullite, 1/16 in. OD, AWG 26) is held in place by a piece of mullite ceramic tubing (McDanel MV20, 1/8 in. OD, 1/16 in. ID) sitting next to the sample capsule, and about 4 to 8 mm of crushable alumina ceramic tubing (1/8 in. OD, 1/16 in. ID) positioned below the steel plug which firmly grips the thermocouple tube during runup to prevent extrusion. The tip of the thermocouple is separated from the sample capsule by an alumina ceramic disc (0.02 in. thick). The sample capsule is packed at the top and Appendix C. Piston-Cylinder Experiments 116 1 inch I st T alumina ceramic AlSiMag ceramic 304 stainless steel graphite crushable ceramic pyrex glass Pt capsule Figure C.16: Sample assembly for thermal gradient calibration. Only center part is shown (compare figure C.15). base with pyrex powder to fill all voids due to welding seams and other irregularities in the outside of the capsule. Reduction of friction against the carbide core is achieved with a 0.004 in. thick lead foil lubricated with dry molybdenum disulfide (Molylube spray, Bel-Ray Company, Inc.). The steel plug, 0.5 in. long, is machined from 304 stainless steel with a 1/16 in. bore and an 0.03 in. thick insulator made from unfired pyrophyllite. The 1/2-inch sample assembly (figure C.15b) consists of an outer talc cylinder (0.493 in. OD, 0.375 in. ID) and a standard 5/8 in. diameter thick-wall pyrex tube which contains the graphite furnace (0.212 in. OD, 1/8 in. ID). Sample capsules of 0.093 in. OD are contained within a thin-walled ceramic insulator (McDanel MV20, 1/8 in. OD) and packed with pyrex powder against the ceramic filler rod at the base and the ceramic tube containing the two-hole thermocouple ceramic insulator (1/16 in. OD, AWG 26) at the top. The thermocouple junction is separated from the sample by a 0.02 in. thick alumina ceramic disk. Steel plug design and friction reduction are the same as in the 3/4-inch assembly. C.3 Temperature Calibration Knowing the temperature of the sample at pressure within a piston-cylinder apparatus is a difficult task. Uncertainties of ± 30 °C or more are not uncommon if no special care is taken. Precision and accuracy of a temperature measurement depend on: 1. thermal gradients along the axis of the sample assembly, 2. effect of pressure on the electromotive force of the thermocouple, Appendix C. Piston-Cylinder Experiments 117 3. calibration of the thermocouple, 4. calibration of the device measuring the electromotive force, 5. calibration of the ice point compensation, 6. additional voltages due to extension wires, junctions, induced emf, 7. ability to control the temperature during an experiment, 8. changes in geometry of the sample assembly during an experiment, 9. contamination of the thermocouple during an experiment. Items 1 to 7 need to be calibrated, item 8 can be examined after the run, item 9 is neg- ligible for short experiemnts and can be checked by recalibration after long experiments with some care. P [kb] T r [°C] T 3 [°C] T 9 [»C] T s [°C] 10 .900 908 893 350 ±30 1100 1110 1096 420 ±30 1300 1309 1300 495 ±40 1500 1509 1507 535 ±50 15 900 908 886 1100 1109 1087 1300 1310 1292 1500 1508 1499 Table C.7: Temperature measurements within the piston-cylinder apparatus made for calibra- tion of thermal gradients. Position T r is the location of the reference thermocouple junction (cf. figure C.16). Positions T3 and T9 are located 3 mm and 9 mm below T r . Position Ta is located at the top of the graphite furnace adjacent to the steel plug (cf. figure C.16). Tem- peratures at Ta above T r = 1200 °C are extrapolated. Uncertainties in T g are estimated from measurements obtained from two independent experiemnts. Thermal gradients: Gradients within the 3/4-inch assembly were measured at 10 and 15 kbar and 900-1500 °C. All calibration measurments were done with Pt-Ptl0%Rh thermocouples (Johnson &; Matthey Canada, Inc., 0.25 mm wire diameter) from the same spool. The gradients were determined with repeat runs each with two independent thermocouples contained within a four-hole ceramic tube (McDanel 998 Alumina, 1/16 in. OD, AWG 26) with one junction placed at the measuring position, T r , for reference above the position of the sample capsule. The second junction was placed at depths of 3 and 9 mm, T 3 and T 9 , within a dummy platinum sample capsule (3 mm OD, 10 mm long) lined with an insulator cup machined from crushable alumina ceramic tubing (figure C.16). Appendix C. Piston-Cylinder Experiments 118 T 3 , 9 - T r [K] + 15 : + 10 + 5 0 - 5 - 10 - 15 H 3/4—inch assembly ~i—i—i—i—i—i—r~ 0 3 6 9 Thermocouple position [mm] Legend 10 15 kbar 0 * 1100 °( o • 1300 °( <? * 1500 °( Figure C.17: Thermal gradients within the piston-cylinder apparatus. The data points at 10 kbar are offset from the Ti position for clarity. Curve shows the inferred distribution of temperature at 1300 °C along the axis of the furnace assembly. Temperatures were read from two 4^—digit thermometers (Analog Devices, type AD 2051-A) which were calibrated against each other between 1000 and 1315 °C to agree within instrument precision of ± 0.5 °C. Temperatures at the top of the furnace were also determined in order to estimate the pressure seal temperature of the thermocouple wires (position T« in figure C.16). The results are tabulated in table C.7 and plotted in figure C.17. The uncertainty has two major components: the uncertainty associated with the thermal gradient over the length of the sample capsule and the uncertainty associated with the precision with which the measuring thermocuple position can be repeated for all experiments. The position of the thermocouple was carefully monitored before and after each run and was observed to be precise to ±0.5 mm. This contribution accounts for about ±4 K and is thus the largest contribution to the overall temperature uncertainty. The total uncertainty due to thermal gradients amounts to ± 6.5 K together with the contribution of ± 2.5 K from the thermal gradient over the length of the sample capsule. This uncertainty puts an upper limit to the precision with which runs can be duplicated. This precision also indicates that the temperature interval between runs should be at least 10 K for practical purposes. The temperature correction due to thermal gradients applied to the nominal temperatures of the experiments amounts to +9 K ± 6.5 K (cf. figure C.17). Appendix C. Piston-Cylinder Experiments 119 Effect of pressure on the emf: The electromotive force (emf) of a thermocouple is affected by pressure whenever thermal gradients occur under pressure. The effect on commonly used thermocouple materials has been measured by single wire experiments within a piston-cylinder apparatus with controlled hot and cold pressure seal tempera- tures and approximate isobaric conditions over the length of the thermal gradient (Get- ting k Kennedy, 1970). Measurements on platinum and platinum-10% rhodium wire were made up to 33 kbar and 1000 °C. Extrapolation of the correction volatge to 1500 °C seems to be permissable, based on approximate linear trends of single wire voltage ver- sus pressurized temperature interval and an approximately constant Seebeck coefficient at temperatures between 1000-1500 °C. Corrections are positive (low apparent temper- ature) at all pressure-temperature conditions. Uncertainties of the correction voltage are reported by Getting k Kennedy (1970) to be ±(10% + 10/xV) below 1000 °C and esti- mated to ±(15% + 10/xV) above 1000 °C. The corrections to be applied to experiments depend on the temperature at which the thermocouple is subject to the run pressure (pressure seal temperature). The position of the pressure seal in our experiments at 15 kbar is assumed to be at the point where the thermocouple ceramic tube is contained within a crushable alumina tube to enable a firm grip (position T s in figure C.16 and table C.7). This approximation is based on the observation that thermocouple wires slide easily out of the part of the ceramic tube enclosed by the steel plug but do not separate from ceramic parts contained by pyrex and the furnace. The uncertainty associated with the correction is larger at 25 kbar because the steel plug appears to grip the thermo- couple' tube and wires at least to some extent. Table C.8 is a summary of temperature corrections and their estimated uncertainties at pressures and temperatures relevant to this study. The table entries were derived graphically from Getting k Kennedy's (1970) computed curves of correction voltage versus temperature. P [kb] p x cor T [°C] Correction [K] Uncertainty [K] 10 7.5 1000 1200 1400 +2.0 +1.8 +1.5 ±1.0 ±1.0 ±1.5 15 12.1 1000 1200 1400 +4.2 +4.0 +3.7 ±1.5 ±1.5 ±2.0 25 21.5 1000 1200 1400 +6.0 +5.8 +5.5 ±2.0 ±2.0 ±2.5 Table C.8: Effect of pressure on the emf of the thermocouple calculated for pressure seal temperatures of 380 °C, 450 °C and 540 °C for run temperatures of 1000 °C, 1200 °C and 1400 °C respectively. P c o r denotes pressures corrected for the effect of friction. Calibration of thermocouples: Platinum-platinum-10%Rh thermocouples (0.25 mm wire diameter) supplied by Johnson k Matthey Canada, Inc., are reliable and precise with an accuracy of better than ±0.25 % at temperatures between 538 and 1482 °C (ASTM Standard E230-72, Powell et al., 1974). This will introduce an uncertainty of ±3 K at the temperatures relevant to this study. No further calibration against melting points of standard reference materials was felt necessary. Appendix C. Piston-Cylinder Experiments 120 Temperature control: Temperature in all experiments was controlled automatically to better than ±20 /iV which corresponds to ±2 °C at temperatures above 1000 °C. This uncertainty is small compared to effects of thermal gradients. Electromotive force measurement: The emf of the thermocouple was measured with a Leeds &; Northrop 7554 type K-4 potentiometer with a sensitivity of better than 1 fj,V. The certified 1-volt standard cell (Eppley Laboratory, Inc., cat. no. 100) was checked and calibrated against a high precision digital volt meter to an accuracy of better than ±6/iV. Direct comparison under run conditions at 1280 °C, including the ice point compensation, against a F L U K E 8840A voltmeter showed agreement within 2 fiV. The F L U K E 8840A voltmeter was calibrated against a F L U K E 8800A meter which was recently certified by the US National Bureau of Standards to an accuracy of better than ±5.8/xV. The combined uncertainty of emf measurement, ice point compensation and all extensions and connections is therefore better than ±8^JV which is equivalent to ±0.8 K at temperatures above 1000 °C. Ice point compensation: A battery powered flMEGA ice point compensator (type CJ) for type S and R thermocouples was used initially with an accuracy of ±0.25 K set by factory calibration. The calibration was checked against the temperature of a well stirred ice bath and agreed to within the specifications. This unit did not perform properly at elevated temperatures creating an excessive correction voltage and thus simulating high apparent temperatures. The unit was replaced by the well-proven ice bath in a dewer flask. For the thermal analysis experiments the ice point compensation was performed at the input terminals to the KEITHLEY (Series 500) data collection system with a tem- perature sensor attached to the thermal mass of the terminal block. This configuration is accurate to ±0.5 K from combined calibration uncertainties and thermal gradients amongst the input terminals. Changes in geometry: The sample assembly was found to perform very stably under pressure and temperature. Thermocouple junction and sample capsule remained in po- sition during all runs reported. The graphite furnace was observed to contract about 0.2 mm in diameter at mid-height, and a slightly elliptical crossection was observed in most runs at 15 kbar and 45-240 minutes duration. At 25 kbar all runs displayed a distinct elliptical cross section without any visible cracks in the furnace wall. Two 120-minute runs at 25 kbar pressure and > 1400 °C were discarded due to excessive asymmetrical deformation of the furnace. Run durations of less than 60 minutes seem to avoid the problem of excessive deformation. Any effect from this change in geometry is accounted for by the thermal gradient calibration with an identical sample configuration and three independent runs. Each sample assembly was carefully examined after the run in order to exclude erroneous experiments due to changes in sample geometry not included in the calibrations. Appendix C. Piston-Cylinder Experiments 121 Contamination of the thermocouple: Contamination is negligible for runs of short duration (1-4 hours) and at temperatures below approximately 1500 °C. Most runs of this study were shorter than two hours, with a few runs of 4 hours. Al l thermocouple ceramics made of mullite were boiled in hydrochloric acid as a precaution against iron contamination. McDanel MV20 two-bore and four-bore mullite ceramic tubing has an Fe203 content of less than 0.8 %, four-bore McDanel 998 alumina ceramic of less than 0.025 %. The superior mechanical properties and smaller manufacturing tolerances of the mullite tube outweigh the disadvantage of higher iron content for short runs. The high precision of the melting point determinations for pressure calibration (cf. below) preclude significant drift of the thermocouples- used. Thermocouple contamination can be neglected for the run durations of the experiments of this study. C .4 Pressure Calibration The difference between the pressure acting on the carbide piston (i.e. oil pressure) and the pressure experienced by the sample is called 'friction' despite the fact that the actual processes leading to such a difference are not known in detail. This 'friction' can be determined by calibration against well known phase equilibria at high pressure which were determined in gas pressure apparati or with sample assemblies with negligible 'friction'. Several useful equilibria are discussed by Bohlen (1984) and Mirwald et al. (1975). Calibration should be performed at pressures and temperatures close to the experiment with comparable run times and procedures and identical sample assemblies. This is in most applications not possible and the uncertainty associated with the differences between calibration and experiment remains unknown. Friction corrections for talc-pyrex assemblies similar to the design developed in this study are discussed below. The cells differ in diameter, in the type of pressure medium used within the graphite furnace (listed separately if different from pyrex) and in many details concerning the design and run procedures. Direct comparisons are therefore not possible and the actual cause of 'friction' remains speculative. Akella (1979) uses a 1/2- inch talc-pyrex-AlSiMag (a machinable, porous Mg-ceramic material) cell and reports a double value of friction of 2.5-2.9 kbar at 30 kbar and 1000-800 °C calibrated on the quartz T=* coesite transition. The same cell, but with a boron nitride core, was used by Akella & Kennedy (1971) with a friction of about 2.2 kbar at 30 kbar derived from data on the melting curve of silver compared to Mirwald's et al. (1975) measurements. Chipman & Hays (in Johannes et al., 1971) use a 1/2-inch talc-pyrex-pyrophyllite cell with a porcelain sample capsule sleeve and employ a —8% correction on piston-in runs calibrated on the equilibrium quartz ^ coesite at 40 kbar and 1400 °C. The interlaboratory comparison study (Johannes et al., 1971) suggests a correction of —1.5 kbar in piston-in runs and no correction for piston-out runs at 17 kbar and 600 °C based on the equilibrium albite Appendix C. Piston-Cylinder Experiments 122 ^ jadeite + quartz. Hariya k Kennedy (1968) use a 1/2-inch talc-pyrex cell with lead foil and apply a correction of —10% but offer no calibration. It is not clear whether they employed piston-in or piston-out procedures. Irving k Wyllie (1975) use a 1/2-inch talc- pyrex-boron nitride cell with lead foil and grease-base molybdenum disulfide lubricant. They use piston-out procedures, apply no correction and offer no calibration. Huang k Wyllie (1975) use a 1/2-inch talc-pyrex-boron nitride assembly, similar or identical to Irving k Wyllie (1975), and probably use lead foil and molybdenum disulfide lubrification. They use the piston-out technique and apply a correction of —3 kbar based on calibrations against the equilibria quartz T=* coesite (30-35 kbar, 800-1100 °C) and albite ^ jadeite -f quartz (21-32 kbar, 800-1100 °C). Newton k Sharp (1975) use 3/4-inch talc-soda-lime glass and 1/2-inch talc-pyrex assembbes similar to Hariya & Kennedy (1968) but offer no calibration and assume a correction of —10% for the 1/2-inch cell and —5% for the 3/4-inch cell using piston-in procedures. Haselton, Sharp & Newton (1978) use the same assemblies and procedures as Newton k Sharp (1975) and quote a correction of —8% for the 1/2-inch cell based on a relative comparison to 3/4-inch NaCl-cell data (T.J.B. Holland, unpublished) at 1100-1200 °C on the albite ^ jadeite 4- quartz equilibrium. The 3/4-inch talc-soft-glass cell is compared to NaCl-cell data at 14 kbar and 950 °C (magnesite + rutile ^ geikielite + C0 2 ) indicating negligible friction. We may conclude that the lack of consensus about what 'friction' correction to apply is a result of the variety of cells and run procedures used and, importantly, also the result of the lack of calibrations in several studies. The few calibrations available for 1/2- inch cells using piston-in procedures indicate a correction of —1.5 to —3 kbar, whereas no calibrations seem to be published for 3/4-inch talc-pyrex cells. The uncertainty of knowing the pressure may be as large as —3 kbar (± precision) if no calibration is offered. For this work the melting curves of gold and silver were chosen as determined by Mirwald et al. (1975). Melting point calibration was preferred over bracketing a well studied mineral equilibrium for practical reasons: it is much faster and allows one to collect a complete calibration with one experiment. For a pressure calibration of ±0.5 kbar accuracy the melting temperature of gold or silver needs to be determined to an accuracy of ±3 K. Although the temperature dependence of the melting pressure of silver and gold is rather large there are practical reasons for chosing them: metals display a large heat effect on melting and the high densities of gold and silver permit the use of small sample beads. Furthermore, graphite containers can be used conveniently and both metals are readily available in sufficient purity. Only few laboratories seem to routinely employ differential thermal analysis in piston- cylinder equipment, notably Mirwald (i.e. 1975) in Bochum. The writer is not aware of any direct thermal analysis experiments having been published, i.e. using a single ther- mocouple rather than two measuring junctions to record differences. As it will be shown below, one thermocouple combined with a near-linear heating rate is sufficient to define Appendix C. Piston-Cylinder Experiments 123 alumina ceramic AlSiMag ceramic 304 stainless steel graphite crushable ceramic pyrex glass gold bead I • i • i • ' I . I . I Figure C.18: Sample assembly for melting point calibration of gold and silver. equilibria with fast reaction rates and large heats of transformation. Our piston-cylinder apparatus was adapted to thermal analysis experiments in the following manner: Tem- perature was recorded with a personal computer with a Kiethley Series 500 interface that performs the zero-point correction of the thermocouple signal, amplification and analog to digital conversion. Calibration of the temperature measuring system against an ice bath, including the zero-point correction, indicated a correction of +2 K. This correction was verified at run temperatures against a calibrated F L U K E 8840A voltmeter. The raw data (figure C.19 and C.20) were adjusted by +2 K (table C.9 and C.10, figure C.21). The sensitivity of the triggering system for the phase-controled (SCR) AC furnace supply was increased with an additional potentiometer placed in series with the existing set-point potentiometer of the KRISEL controller (Hadidiakos, 1969). The potentiometer was now able to control a range of 80-90 K in ten turns. A motor with variable speed attached to the potentiometer permitted smooth variation of the temperature. A heating/cooling rate of about 20 K per minute turned out to be optimal. Rates of 10 K per minute proved to be unsatisfactory due to the relatively large thermal fluctuations inherent in piston- cylinder apparati. These fluctuations produce apparent thermal arrests at slow heating rates. The biggest problem posed was the signal noise due to inductive pickup from a noisy AC environment. After proper guarding (active driven guard) and grounding, 20-30 % of the emf signals collected at the fastest sampling rate (ca. 10 per second) were unuseable due to noise. On-line software filtering effectively removed the noise with a three-stage filter that sucsessively eliminated the largest spikes and then the smaller ones down to a level of ± 20 ftV (± 2 K for a Pt-Rh thermocouple above 1000 °C). The filter had to be simple and efficient in order not to slow down the sampling program written in Appendix C. Piston-Cylinder Experiments 124 BASIC. Satisfactory results were obtained with collecting 10 readings, subsequent filter- ing, averaging, and output to screen and floppy-disk. This procedure allowed recording a temperature measurement about every 2 seconds. Backward computed temperature differences were diplayed on the screen as a simple means of monitoring thermal arrests. Gold: The melting of gold under pressure was determined by Mirwald et al. (1975), Akella & Kennedy (1971) and Cohen et al. (1966) in piston-cylinder apparati. Mirwald's data was obtained in a virtually frictionless salt cell (Mirwald et al., 1975, Mirwald & Massonne, 1980, Johannes, 1978, Bohlen, 1984). This is as accurate a data set as one can possibly expect from piston-cyblnder work and little error is introduced by calibrating against it. Figure C.18 depicts the sample assembly used for the melting point determinations of gold and silver by thermal analysis. The graphite container is 14 mm long with a sample cavity of 4 mm length and 3.5 mm diameter to accommodate a gold bead of 0.6 grams. The gold bead was cast in graphite from acid-cleaned scrap gold tubing with a fineness of better than 99.95 and no detectable impurities measured by electron microprobe analyses (J. Knight, written communication, March, 1990). The well for the thermocouple tube (1/16 in. OD) extends through the graphite lid into the gold bead to insure measuring true sample temperature. The thermocouple junction was recessed below the tip of the mullite (MacDanel MV20 two-bore) insulater in a thin groove cut with a diamond blade and covered with Sauereisen cement for electrical insulation. The cement was sucked partway into the tube with the wires in place to ensure mechanical strength of the insulation. The sample assembly surrounding the graphite container is identical to the one depicted in figure C.15. The assembly performed very stably and survived at least 14 heating and cooling cycles. One or two tiny metal beads were detected embedded in pyrex adjacent to a hairline fissure in the wall of the graphite container in both the gold and silver experiments. One attempt for each the gold and silver calibration was successful to obtain the entire series of measurements required. Run procedures were identical to the run procedures described for the phase equilib- rium experiments including hot compaction for 20 minutes below the melting temperature of gold. The temperature was steadily increased from about 20 K below the expected melting point to about 20 K above melting and then reversed. At least four determina- tions were recorded at each pressure. Pressure was always increased and never decreased and time for relaxation of stresses was allowed before starting measurements at a new pressure. The measurements are remarkably precise. Only determinations at three dif- ferent pressures were required to establish a relatively straight melting curve (compare figure C.21). Data are presented in figure C.19 and C.21 and in table C.9. Silver: The melting of silver under pressure was determined by Mirwald et al. (1975) and Akella & Kennedy (1971) in piston-cylinder apparati. Mirwald's data was obtained in a virtually frictionless salt cell (Mirwald et al., 1975, Mirwald & Massonne, 1980, Appendix C. Piston-Cylinder Experiments 3 1103.0(1.0) X fci- W - G o l d a t 10 k b a r —r— 180. 60. 200. 3 S J «> IN 3 - • K 2 a. 4) 1128.0(1.41 V G o l d a t j t A k b a r \ — I — 60. I 120. 1 180. 3 0 0 . O =4 3 • 1152.0(1.0) C o l d a t 20 k b a r 1 r- 60. 180. 120. Time (see) 240. 300. Figure C.19: Thermal analysis traces of melting of gold under pressure. Appendix C. Piston-Cylinder Experiments 126 T em pe ra tu re (C ) S O . 98 8 . 9 9 6 . 1 0 0 4 . 1 0 1 2 . 1 0 2 1 '•• 1 > 1 1 •• V 1 1 1 v ' * 8 \ T em pe ra tu re (C ) S O . 98 8 . 9 9 6 . 1 0 0 4 . 1 0 1 2 . 1 0 2 Stiver a t 10 k b a r *» <P • * * n • • cn 0 ? 1 1 1 1 1 1 1 1 1 60. 120. 180. 2<U). 3 l l 1 l 1 1 1 l l DO T em pe ra tu re (C ) 0 0 8 . 1 0 1 6 . 1 0 2 4 . 1 0 3 2 . 1 0 4 0 . II • \ f y A o • « . 1029.2(0.8> o 0 N «P° - T em pe ra tu re (C ) 0 0 8 . 1 0 1 6 . 1 0 2 4 . 1 0 3 2 . 1 0 4 0 . II 7"* S r = 3 x 1 S i l v e r a t 15 *. \ 07 6. 1 1 I I ' I I I I I I i • 60. 120. 180. 240 3 l 1 l l l 1 l l l 00 T em pe ra tu re (C ) 10 36 . 1 0 4 4 . 1 0 5 2 . 1 0 6 0 . 1 0 6 8 . 1 1 % / \ / X / _ T em pe ra tu re (C ) 10 36 . 1 0 4 4 . 1 0 5 2 . 1 0 6 0 . 1 0 6 8 . 1 1 — ' ^ - v — ^»» t —a* 2 3 - *v— * y / ° S i l v e r a t 2 Q n r ^ a r / • ' i i - T •- r i r •• i • " i i - Time (sec) Figure C.20: Thermal analysis traces of melting of silver under pressure. Appendix C. Piston-Cylinder Experiments 127 P [kb] Mirwald (1975) [°C] Mader [°C] A(Ma-Mi) 'friction' T m T corr unc. N m T corr prec. unc. A T [°C] A P [kb] 10.0 1118 1120 ±4.0 4 1105 1107 ±1.0 ±4.0 -13.0±3 -2.5±0.5 15.0 1146 1150 ±4.5 6 1130 1134 ±1.5 ±5.0 -15.0±3 -2.9±0.5 20.0 1173 1178 ±5.0 4 1154.6 1159.6 ±1.0 ±5.0 -18.4±3 -3.5±0.5 Table C.9: Melting point determination of gold under pressure. T m denotes the uncorrected melting temperature, T c o r r the melting temperature corrected for the effect of pressure on the emf, 'prec' is short for precision and 'unc' for uncertainty. Mirwald's data is interpolated from experimental measurements at 28-56 kbar. Data for T m (Mader) has been corrected by +2 K from figure C.19 based on calibration (see text). P[kb] Mirwald (1975) [°C] Mader [°C] A (Ma-Mi) 'friction' T m T •'-corr unc. N T T corr prec. unc. A T [°C] A P [kb] 10.0 1018 1020 ±4.0 4 1003.0 1005 ± 1 ±4.0 -15±3 -2.9±0.5 15.0 1047 1051 ±4.5 4 1031.2 1035 ± 1 ±4.5 -16±3 -3.1±0.5 20.0 1075 1080 ±5.0 6 1057.0 1062 ± 1 ±5.0 -18±3 -3.3±0.5 Table C.10: Melting point determination of silver under pressure. T m denotes the uncorrected melting temperature, T c o r r the melting temperature corrected for the effect of pressure on the emf, 'prec' is short for precision and 'unc' for uncertainty. Mirwald's data is interpolated from experimental measurements at 28-56 kbar. Data for T m (Mader) has been corrected by +2 K from figure C.20 based on calibration (see text). Johannes, 1978, Bohlen, 1984) and in a pyrophyllite cell for comparsion of friction. The salt cell data are demonstrated to be accurate and little error is introduced by calibrating against it. Akella and Kennedy (1971) used a talc-pyrex-boron nitride cell and observed melting at distinctly higher pressures compared to Mirwald et al. (1975) due to effects of friction or differential loading. The sample assembly for this study is identical to the one described for the gold caUbration. The silver bead (0.4 g) was machined from high purity silver. The smaller mass of the silver bead compared to the gold sample leads to a slightly less pronounced thermal arrest on melting or freezing. The measurments are presented in figure C.20 and figure C.21 and tabulated in table C.10. Cesium Chloride: The melting curve of cesium chloride was determined in an in- ternally heated gas pressure apparatus to 15 kbar and 1170 °C by Clark (1959) using differential thermal analysis and up to 5 kbar by Bohlen (1984) using the 'falling sphere' method. Melting was also determined by Bohlen (1984) in a piston-cylinder apparatus with a frictionless one-inch salt cell between 5 and 15 kbar employing the 'falling sphere' method. Clark's reported temperatures need to be corrected for a much exagerated pres- sure correction on the electromotive force of the Pt-Ptl0%Rh thermocouple based on Appendix C. Piston-Cylinder Experiments 128 Temperature [°C] 0 10. 20 30 Nominal pressure [kbar] Figure C.21: Melting of gold and silver under pressure. The calibration curves labeld 'Mirwald' are interpolated from data by Mirwald et al. (1975) at 28-56 kbar. Dotted lines trace the measured difference in melting temperature between Mirwald and this study and the resultant pressure difference ('friction'). The size of the symbols for the data points represents the precision of the measurements. Appendix C. Piston-Cylinder Experiments 129 wrongly assumed linear extrapolation (Getting & Kennedy, 1970) from Birch's (1939) measurements. Clark's temperatures need to be converted to the IPTS-68 temperature scale (Powell et al, 1974) which adds about 3 K between 1000 and 1200 °C. Taking these corrections into account Clark's melting curves for CsCl, LiCl and NaCl are sys- tematically underestimated by 10-15 °C compared to Bohlen's measurements in both solid-medium and gas-medium apparati. The reasons for this discrepancy are unclear but are partially attributed by Bohlen (1984) to reported impurities in Clark's starting materials. The temperature discrepancy translates into a difference in pressure of 0.3- 0.7 kbar. Cesium chloride is nevertheless a suitable material for pressure calibration up to 20 kbar having the largest temperature gradient with respect to pressure on melting compared to other alkali chlorides and halogenides (Clark, 1959). I prefer Bohlen's data because there are more possibilities to accidentally measure too low a melting tempera- ture than the converse. Thermal analysis experiments with gold and silver indicate that the ideal sample mass should not be much less than 0.3 grams spread over as short a vertical dimension as possible to reduce thermal gradients. This was achieved with a 4.8 mm OD platinum capsule (0.2 mm wall) with a well (2.0 mm ID) for the thermocouple. For easy fabrication the large and small diameter tubes of 15 mm length were secured concentrically by a stainless steel cylinder, crimped together in a drill chuck and fused with a welding torch. The annular sample container was then filled with 0.28 grams of CsCl, compacted with the steel cylinder to an effective sample length of 7 mm and crimped around the inner tube with a drill chuck resulting in a bottle shaped container with a 5 mm long neck. The ring-shaped welding seam at the top with three flanges resulting from the triangular crimp required some welding experience with a plasma arc welder. An aluminum block with an appropriate bore diameter served as heat sink during welding to prevent excessive heating of the sample. Spec pure cesium chloride (SPEX Industries, Inc., lot 03781) was dried at 400 °C for 10 hours and stored at 120 °C. The capsule was placed into the 3/4- inch assembly (figure C.15) in the same fashion as the graphite container in figure C.18. Al l irregular spaces around the platinum capsule were filled with pyrex powder. The thermocouple junction was positioned in the platinum well with a ceramic spacer. The sample was allowed to compact at 8 kbar gauge pressure initially at a temperature below the melting of CsCl (ca. 640 °C at 1 bar) and then at slowly increasing temperature. Furnace controle and temperature recording were done with the same set-up as the gold calibration. Heating rates of about 20 °C per minute were adequate. The recorded thermal analysis traces between 10 and 20 kbar are unfortunately useless for pressure calibration purposes. The melting signal is distinct although less clearly defined than with gold or silver with an arrest defined in most runs within i 2 K. The pressure within the sample assembly, however, is inferred to vary with temperature and perhaps also during melting without a change in gauge pressure. It appears that the Appendix C. Piston-Cylinder Experiments 130 P[kb] Correction [kb] Uncertainty [kb] 10.0 -2.5 (25 %) ± 1.0 15.0 -3.0 (20 %) ± 1.0 20.0 -3.3 (16 %) ± 1.0 25.0 -3.5 (14 %) ± 1.0 Table C . l l : Pressure correction due to friction for the 3/4-inch talc-pyrex assembly. large sample size and the large increase of the volume during melting of CsCl (ca. 5.6 cm3/mole, Clark, 1959) are enough to increase sample pressure during heating within the 2-3 kbar friction interval calibrated with gold and silver where the gauge pressure is not expected to rise or fall upon heating or cooling. At 10 kbar gauge pressure 8 heating-cooling cycles were recorded yielding thermal arrests spread over a 70 K interval from 935-1005 °C which suggests a pressure interval of 2 kbar. The melting temperature ' at 10 kbar obtained by Clark (1959) is at 1005 °C and that obtained by Bohlen (1984) is at approximately 1117 °C. Single heating runs at 1 kbar intaervals from 11 to 17 kbar indicate a 2 kbar friction defined by the lowest measured melting signals. The scatter of the data towards higher temperatures is substantial due to the variable amount of temperature increase and therefore also pressure increase before the melting curve is intersected. Several runs also indicate 'gliding' along the melting curve for as much as 40 K or 1.3 kbar. Friction correction: The term 'friction' applied to piston cylinder experiments lumps together contributions from friction between the sample assembly and the carbide wall, friction between the carbide piston and the carbide pressure vessel, stresses stored in the assembly materials not transmitted to the sample and various other contributions such as differential loading (negative and positive anvil effects). The magnitude of friction is a function of pressure, temperature, assembly geometry, assembly materials and run procedures. Details, such as the type of lubricant used, may play a significant role in the amount of friction observed. It is imperative that each pressure cell design be calibrated separately using identical run procedures as in the experiments to be performed. Friction correction applied by analogy with the experiences of other researchers in a different laboratory is not sufficient. Friction corrections relevant to this study are summarized in table C . l l with uncer- tainties derived from calibration uncertainties of the melting point determinations. The uncertainty is the sum of the effect of temperature uncertainty (± 3 K, ± 0.5 kbar) and the uncertainty of knowing the run pressure (± 0.5 kbar). The uncertainty associated with Mirwald's et al. (1975) calibration is negligible compared to the above contributions. Discussion: The relatively large friction in the talc-pyrex cell of this study is largely due to 'true friction' between the sample assembly and the carbide bore. It seems un- reasonable that the talc sleeve could support a differential load of several kilobars. The Appendix C. Piston-Cylinder Experiments 131 cold lower end of the sample assembly with talc, graphite and solid glass may form a plug responsible for the observed friction. In fact, such an effect is observed at the cold upper end during extrusion of the sample assembly after each run: the friction decays suddenly to about 20-30 % of the initial value after the steel plug with the pyrophyllite sleeve are pushed beyond the carbide bore. Friction may therefore be reduced substan- tially by introducing a pressure medium that is able to flow at low temperatures. It is hypothesized that replacing the talc sleeve with a NaCl sleeve may result in an almost negligible friction. C .5 Run Procedures for Phase Equilibrium Experiments The sample assembly was compacted slowly to about 8 kbar at room temperature and then softened by increasing the temperature. The softening point of pyrex under these conditions is at about 540 °C. Reactant starting materials were held at 50-100 K below the equilibrium temperature and product starting materials were quickly heated to 50- 100 K above the approximate equilibrium temperature. Pressure and temperature were then increased simultaneously up to final run pressure keeping the temperature either below or above the final run temperature. The samples were held at these conditions for approximately 15 minutes in order to allow stresses within the sample assembly to relax while holding pressure constant by advancing the piston. This stress relaxation was equivalent to pumping about 10-15% of the run pressure. After pressure stabilization the final run temperature was approached and first controlled manually and after a few minutes automatically. During the subsequent 45 to 140 minutes additional relaxation occured requiring further pumping of not more than a total of 2-3% of the run pressure. Al l runs and calibrations were performed in the same pressure vessel with the same carbide parts which developed only insignificant damage from wear and tear. The quench from run conditions was a temperature quench for reactant starting ma- terials and a combined pressure quench followed by a temperature quench for product starting materials. The temperature quench proceeded at an initial rate in excess of 100 K per second. C.6 Equilibrium magnesite ^ periclase 4- C 0 2 Starting materials: Synthetic periclase and magnesite were used as starting materials with silver-oxalate as the source for carbon dioxide. Reagent grade periclase was fired at 1200 °C for 6 hours and stored in a drying oven at 120 °C. Magnesite was synthesized from reagent grade basic magnesium carbonate (approx. (MgC03)4-Mg(OH)2-5 H2O) sealed in silver or gold capsules and kept for 100-110 houres at 2 kbar gas pressure and 690-700 °C in cold-seal pressure vessels at mole fractions of water in the CO2 fluid of Appendix C. Piston-Cylinder Experiments 132 25 kbar 73 77 78 79 68 • • • • ® ® ® ® ® ® © 72 75 74 81 56 61 T , 1360 1380 i 1400 1 1420 1440 15 kbar 65 69 64 63 62 • • • • • ® ® ® ® ® ® 51 52 28 54 55 27 1 1 1140 1160 1 1180 1 1200 1220 Nominal temperature [°C] Legend O periclase starting material • magnesite starting material # 9 complete reaction ®@ partial reaction OCH no reaction © no reaction, quench magnesite Figure C.22: Experimental brackets on the equilibrium magnesite ^ periclase + carbon diox- ide. Arrow marks define the limits of each bracket. Pressures and temperatures are nominal values and not corrected. See table C.12 and text for discussion of calibrations and corrections. Numbers refer to run numbers in table C.12. 0.3-0.4. Pure basic magnesium carbonate would produce a mole fraction of water in the fluid of approximately 0.6 on devolatilization which was diluted by adding silver oxalate as an additional source of CCv The synthesized material was checked by X-ray powder diffractometry and consisted of a very fine grained (5-20 /<m), fluffy, euhedral magnesite with no other phases detectable (see also figure C.25b). Carbon dioxide was added to the experimental charges in the form of silver-oxalate (Ag 2C20 4, Dixon Fine Chemichals, lot 02708), stored light-tight in a desiccator. Silver oxalate releases its carbon dioxide explosively at about 140 °C under ambient pressure. Silver is reduced in this process to the metallic state and does not participate in the reaction at run conditions. The explosive character of this reaction at ambient pressure requires cooling of the sample charges during the welding of capsule seams. Sample preparation: Starting materials for the 3/4 inch pyrex assembly were sealed into 3 mm OD platinum capsules (2.6 mm ID), 8-9 mm long including a triangular crimp of 3-4 mm length for easy welding. The capsules contained 7-10 mg magnesite or periclase. 7-10 mg of silver-oxalate were added to magnesite and 15-20 mg were added to periclase. The triangular crimp done in a drill chuck maintains a circular cross section of the platinum capsule and provides a very strong welding seam. Welding was Appendix C. Piston-Cylinder Experiments 133 done with a plasma needle-arc welder (LINDE PWM-6) or equally well with a simple graphite electrode electric welder. In either case the capsule was cooled by a few wraps of water soaked tissue contained in a bored aluminum block. Additional electrical grounding proved helpful by wrapping copper wire around the sample capsule about 2 mm below the welding seam. The capsules were placed into the assembly with the crimped end pointing downward which results in an effective sample length of no more than 4 mm. A relatively long crimped end also provides for extra volume if required by the amount of fluid enclosed in the capsule. Any void spaces around the capsule were well packed with pyrex powder. Results: Two brackets were reversed on the equilibrium MgCOa^ MgO-f- CO2, one at 15 kbar and one at 25 kbar nominal pressure, both with the 3/4-inch assembly. All runs are listed in table C.12 with uncertainties and corrections. Most capsules opened with a hiss from escaping carbon dioxide proving the presence of a CO2 fluid at run conditions. Critical runs that did not lose gas on opening were discarded and repeated. Run products were examined under the light microscope and by X-ray powder diffrac- tion. Some textures were further investigated under the scanning electron microscope using secondary electron imaging on gold-coated powder mounts. Reaction progress was established by three independent methods: estimated from grain mounts under the mi- croscope, by X-ray intensities, and by the weight of CO2 produced or consumed during the run calculated from the amount of CO2 released on opening the capsule after the run. The accuracy of the latter method is not satisfactory with an uncertainty of approx- imately ±4 mg on the amount of CO2 produced or consumed which translates into an uncertainty of about 10 percentage points in the amount of reaction progress computed in table C.13. Details of run products are summarized in table C.13. The equilibrium at 12.1±1.0 kbar is located between 1173±10 °C and 1183±10 °C with a temperature interval of 10±10 K where no reaction progress is observed at run times of 45-140 minutes. The bracket at 21.5±1.0 kbar spans an interval of 60±10 K and is located between 1375±10 °C and 1435±10 °C. Textures: Textures of run products were studied under the scanning electron micro- scope (SEMCO Nanolab 7) in order to assist with the interpretation of the extent of reaction deduced from light microscopy and X-ray diffraction. Specifically, the study aimed towards narrowing the 60 K-interval at 21.5 kbar where no nucleation and growth of new phases were observed without ambiguity. It was anticipated that textural obser- vations would facilitate the distinction between stably grown magnesite and magnesite formed during the temperature quench. Loose powder or gently crushed aggregates from sample capsules were mounted on aluminum stubs. Nail polish served as binder with its stickiness controled by placing the sample spread on a dry nail polish film under acetone vapour for an appropriate amount of time. The samples were coated with gold to a thickness of about 30-40 nm. The Appendix C. Piston-Cylinder Experiments 134 Run Pressure [kbar] Temperature [°C] Time [min] Starting materials Run products nominal corrected nominal corrected UP-24 15 (0.5) 12.1 (1) 1285 (7) 1298 (10) 55 Mc,AgOx a(Pe),qs(Mc) UP-25 15 (0.5) 12.1 (1) 1250 (7) 1263 (10) 65 Pe,AgOx e(Pe),qt(Mc) UP-26 15 (0.5) 12.1 (1) 1230 (7) 1243 (10) 50 Pe,AgOx e(Pe),qt(Mc) UP-27 15 (0.5) 12.1 (1) 1200 (7) 1213 (10) 65 Pe,AgOx e(Pe),qt(Mc) UP-28 15 (0.5) 12.1 (1) 1170 (7) 1183 (10) 85 Pe,AgOx e(Pe),qt(Mc) UP-50 15 1150 65 Pe,AgOx C 0 2 leak UP-51 15 (0.5) 12.1 (1) 1145 (7) 1158 (10) 155 Pe,AgOx e(Pe),am(Mc) UP-52 15 (0.5) 12.1 (1) 1160 (7) 1173 (10) 100 Pe.AgOx a(Mc),em(Pe) UP-53 15 (0.5) 12.1(1) 1170 (7) 1183 (10) 115 Pe,AgOx e(Pe),qt(Mc) UP-54 15 (0.5) 12.1 (1) 1180 (7) 1193 (10) 95 Pe,AgOx e(Pe),qtr(Mc) UP-55 15 (0.5) 12.1 (1) 1190 (7) 1203 (10) 105 Pe,AgOx e(Pe),qt(Mc) UP-56 25 (0.5) 12.1 (1) 1420 (7) 1435 (10) 115 Pe.AgOx e(Pe),qs(Mc) UP-57 15 (0.5) 12.1 (1) 1230 (7) 1243 (10) 110 Mc,AgOx a(Pe),qs(Mc) UP-58 15 (0.5) 12.1 (1) 1240 (7) 1253 (10) 105 Mc,AgOx a(Pe),qs(Mc) UP-59 15 (0.5) 12.1 (1) 1260 (7) 1273 (10) 115 Mc,AgOx a(Pe),qs(Mc) UP-60 15 (0.5) 12.1 (1) 1300 (7) 1313 (10) 130 Mc,AgOx a(Pe),qs(Mc) UP-61 25 (0.5) 21.1(1) 1440 (7) 1455 (10) 115 Pe,AgOx e(Pe),qs(Mc) UP-62 15 (0.5) 12.1 (1) 1210 (7) 1223 (10) 120 Mc,AgOx a(Pe),qs(Mc),t(Mc) UP-63 15 (0.5) 12.1 (1) 1190 (7) 1203 (10) 160 Mc,AgOx a(Pe),qs(Mc),q(Mc) UP-64 15 (0.5) 12.1 (1) 1170 (7) 1183 (10) 140 Mc,AgOx a(Pe),qs(Mc),q(Mc) UP-65 15 (0.5) 12.1 (1) 1150 (7) 1163 (10) 125 Mc,AgOx e(Mc), furnace def. UP-66 15 1160 115 Mc,AgOx furnace deformed UP-67 25 1420 110 Mc,AgOx furnace deformed UP-68 25 (0.5) 21.5 (1) 1420 (7) 1435 (10) 55 Mc,AgOx a(Pe),a(Mc),qs(Mc) UP-69 15 (0.5) 12.1 (1) 1160 (7) 1173 (10) 120 Mc,AgOx e(Mc) UP-70 25 (0.5) 21.5 (1) 1400 (7) 1415 (10) 60 Pe.AgOx e(Pe),qs(Mc) UP-71 25 (0.5) 21.5 (1) 1380 (7) 1395 (10) 60 Pe,AgOx e(Pe),qs(Mc),q(Mc) UP-72 25 (0.5) 21.5 (1) 1360 (7) 1375 (10) 60 Pe.AgOx a(Mc),e(Pe) UP-73 25 (0.5) 21.5 (1) 1370 (7) 1385 (10) 45 Mc,AgOx e(Mc) UP-74 25 (0.5) 21.5 (1) 1380 (7) 1395 (10) 60 Pe,AgOx e(Pe),qs(Mc),q(Mc) UP-75 25 (0.5) 21.5 (1) 1370 (7) 1385 (10) 60 Pe.AgOx e(Pe),qs(Mc),q(Mc) UP-76 25 1380 Mc,AgOx furnace failure UP-77 25 (0.5) 21.5 (1) 1380 (7) 1395 (10) 55 Mc,AgOx e(Ms) UP-78 25 (0.5) 21.5 (1) 1400 (7) 1415 (10) 55 Mc,AgOx e(Ms) UP-79 25 (0.5) 21.5 (1) 1410 (7) 1425 (10) 55 Mc,AgOx e(Ms) UP-80 25 Pe,AgOx furnace failure UP-81 25 (0.5) 21.5 (1) 1400 (7) 1415 (10) 105 Pe,AgOx e(Pe),qs(Mc),q(Mc) Table C.12: List of experiments on magnesite decarbonation. Numbers in parentheses refer to precision (columns 'nominal P, T') and uncertainty (columns 'corrected P, T'). Abreviations: Pe: periclase, Mc: magnesite, AgOx: silver oxalate, m: minor, t: trace, q: quench product, s: skins, a: anhedral, e: euhedral. Appendix C. Piston-Cylinder Experiments 135 electron microscope was operated with an acceleration potential of 15 kV. Secondary electron images were recorded on Polaroid 52 film. Figures C.23, C.24, and C.25 depict the range of textures observed in a study of 15 samples. At 15 kbar the nucleation and growth of periclase appear to proceed rapidly: the decarbonation reaction went to completion in all runs above the narrowly defined equilibrium temperature (figure C.22). The growth of magnesite from periclase proceeded at a much slower rate documented by the mixtures of periclase and well crystallized magnesite quenched from run conditions. In figure C.23a thin quench skins of magnesite are visible on periclase formed from magnesite where the skin is broken at several places (i.e. left of center, near top). The peculiar shape of the magnesite crystals presumably formed from Unking at random arrays of globular magnesite starting material (grain size of 5-20 fim) during reaction progress. At 25 kbar nominal pressure difficulties in nucleating either magnesite or periclase were encountered resulting in a temperature interval of 60 K where no reaction progress was observed. The low-temperature side of the reaction was defined by growth of large magnesite crystals (figure C.23b) with relict spherical periclase inclusions clearly visible. At higher temperatures periclase starting material displayed thick skins of magnesite (figure C.24a) which were interpreted as quench products based on the lack of mag- nesite crystal morphologies. Rare sponge-like aggregates of probable magnesite were observed in runs 71, 74, 75 and are illustrated in figure C.24b. These aggregates are bire- fringent and show domains of coherent optical orientation. 'Very fine-grained spongy aggregates' of magnesite were observed by Harker & Tuttle (1955) in cold-seal experi- ments at 1.3 kbar and 800-850 °C employing the 'slow-quench' method using compressed air. These aggregates were not observed in runs quenched more rapidly by plunging the pressure vessel into a bucket of water. The authors therefore conclude that these mag- nesite aggregates were quench products. Considering the extremely fast quench rates of a piston-cylinder apparatus compared to cold-seal equipment it is unlikely that the two 'sponge-like' aggregates represent the same feature. The lack of any crystal morphology may perhaps be indicative of a quench product. The size of the aggregates is similar to the periclase grains and may therfore be a replacement of former periclase grains. Con- densation from the vapour phase might also be considered a mechanism of formation, although one would expect nucleation during condensation to occur on surfaces rather than forming separate particles. While the significance of these aggregates remains enig- matic one cannot make a strong case for these features being stably formed magnesite. Runs with magnesite starting material, on the other hand, do not show any complex textures. Magnesite grains become more euhedral at 25 kbar (figure C.25a) than the hydrothermally produced starting material (figure C.25b). At a nominal temperature of 1440 °C partial decarbonation is observed marking the high-temperature side of the experimental bracket. Appendix C. Piston-Cylinder Experiments 136 While this analysis of textures is interesting and may contribute towards the analysis of kinetics of reactions, it does not put tighter limits on the location of the equilibrium position at 25 kbar. It is, however, concluded that reaction rates need not be as fast as portrayed in some studies (Irving Sz Wyllie, 1975) where run times of only minutes were considered sufficient to let reactions proceed to completion. Appendix C. Piston-Cylinder Experiments 137 Run P [kb] T [°C] Starting materials Products optical examination X-ray % R SEM UP-24 15 1285 Mc,AgOx p(Pe),qs(Mc) - - - UP-25 15 1250 Pe.AgOx e(Pe),qt(Mc) Pe,w(Mc) - - UP-26 15 1230 Pe.AgOx e(Pe),qt(Mc) - - - UP-27 15 1200 Pe,AgOx e(Pe),qt(Mc) - - e(Pe),qs(Mc) UP-28 15 1170 Pe.AgOx e(Pe),qt(Mc) - 100 Pe e(Pe),qs(Mc) UP-51 15 1145 Pe,AgOx e(Pe),am(Mc) Pe,m(Ms) - - UP-52 15 1160 Pe,AgOx a(Mc),em(Pe) Mc,m(Pe) 67 Pe - UP-53 15 1170 Pe.AgOx e(Pe),qt(Mc) Pe,w(Mc) 100 Pe - UP-54 15 1180 Pe,AgOx e(Pe),qt(Mc) Pe,w(Mc) 100 Pe - UP-55 15 1190 Pe,AgOx e(Pe),qt(Mc) Pe,w(Mc) 100 Pe - UP-56 25 1420 Pe,AgOx e(Pe),qs(Mc) Pe,d(Mc) 96 Pe e(Pe), Qs(Mc) UP-57 15 1230 Mc,AgOx p(Pe),qs(Mc) Pe 81 Mc - UP-58 15 1240 Mc,AgOx p(Pe),qs(Mc) - 0 Mc - UP-59 15 1260 Mc,AgOx p(Pe),qs(Mc) - 0 Mc - UP-60 15 1300 Mc,AgOx p(Pe),qs(Mc) - 0 Mc p(Pe), qs(Mc) UP-61 25 1440 Pe,AgOx e(Pe),qs(Mc) Pe 0 Pe e(Pe), Qs(Mc) UP-62 15 1210 Mc,AgOx p(Pe),qs(Mc),t(Mc) Pe,w(Mc) 46 Mc g+p(Pe), qs(Mc) UP-63 15 1190 Mc,AgOx p(Pe),qs(Mc),q(Mc) Pe,a(Mc) 100 Mc - UP-64 15 1170 Mc,AgOx p(Pe),qs(Mc),q(Mc) Pe,d(Mc) 28 Mc p(Pe), qs(Mc) UP-68 25 1420 Mc,AgOx a(Pe),a(Mc),qs(Mc) - 50 Mc - UP-69 15 1160 Mc,AgOx e(Mc) - 90 Mc f(Mc) UP-70 25 1400 Pe,AgOx e(Pe),qs(Mc),q(Mc) - q leak - UP-71 25 1380 Pe,AgOx e(Pe),qs(Mc),q(Mc), S - q leak - UP-72 25 1360 Pe,AgOx a(Mc),e(Pe) - 58 Pe e(Mc), r(Pe) UP-73 25 1370 Mc,AgOx e(Mc) - 100 Mc e+f(Mc) UP-74 25 1380 Pe,AgOx e(Pe),qs(Ms),q(Ms) Pe q leak e(Pe), Qs(Mc), S UP-75 25 1370 Pe,AgOx e(Pe),qs(Ms),q(Ms) Pe 72 Pe e(Pe), Qs(Mc), S UP-77 25 1380 Mc,AgOx e(Mc) - 86 Mc - UP-78 25 1400 Mc,AgOx e(Mc) - 90 Mc e(Mc) UP-79 25 1410 Mc,AgOx e(Mc) - 100 Mc e(Mc) UP-81 25 1400 Pe,AgOx e(Pe),qs(Mc),q(Mc) - 100 Pe e(Pe), Qqs(Mc) Table C.13: Summary of run products and textures of magnesite decarbonation experiments. Optical examination was carried out on powder mounts immersed in oil (n = 1.73). X-ray examination was performed on powder mounts on quartz slides ((001) surface) with Cu-K„ radiation. The '% Reaction' column indicates the amount of reaction progress expressed as a percentage of starting material not reacted. The amount of CO2 produced or consumed was determined from the measured amount of C 0 2 loss on opening of the capsule after each run and the total amount in the capsule introduced with silver oxalate and magnesite. The accuracy of this method is about 10 percentage points. Runs marked 'q leak' have experienced CO2 leakage during the quench. See text for discussion of the textures observed under the scanning electron microscope. Abreviations: Pe: periclase, Mc: magnesite, AgOx: silver oxalate, m: minor, t: trace, q: quench product, s: skins, w: weak, d: distinct, a: anhedral, e: euhedral, Qs: thick quench skins, S: spongy aggregates (see text), g: globular, f: facetted grains with rounded corners and edges, p: periclase with 'peculiar' shape (see figure C.23a). Appendix C. Piston-Cylinder Experiments 138 Figure C.23: Plate I of textures of experimental run products viewed under the scanning electron microscope using secondary electron imaging: a) run UP-60 , 15 kbar, 1300 °C, 130 minutes, b) run UP-72, 25 kbar, 1360 °C, 60 minutes. See text for discussion of textures. Appendix C. Piston-Cylinder Experiments 139 Figure C.24: Plate II of textures of experimental run products viewed under the scanning electron microscope using secondary electron imaging: a) run UP-75 , 25 kbar, 1370 °C, 60 minutes, b) run UP-75 , 25 kbar, 1370 °C, 60 minutes. See text for discussion of textures. Appendix C. Piston-Cylinder Experiments 140 Figure C.25: Plate III of textures of experimental run products viewed under the scanning electron microscope using secondary electron imaging: a) run UP-79 , 25 kbar, 1410 °C, 55 minutes, b) hydrothermal run, 2 kbar, 700 °C, 110 hours. See text for discussion of textures. Appendix D Comparison of Calculated Volumes with Measured Volumes This appendix contains a collection of pressure versus percent-deviation plots (figure D.26 to figure D.35) that compare experimentally measured volumes with calculated ones using the equation of state of this study. This type of plot is a part of the output from the home-grown peripherals of the non-linear programming code used for fitting parameters of the equation of state. The figures are discussed in some detail and expand the discussion provided in chapter 1. A brief account of the experimental methods is included. Shmonov & Shmulovich, 1974, fi9ure D.26, 1-8 kbar, 400-700 ° C: This data set covers a very important part oiP — T space and is the only study extending to high temperatures above 1.4 kbar. The experimental method is discribed in Shmonov (1977). The authors use a displacement method whereby the volume is extracted from the difference of two measurement series at the same temperature with sample volumes that differ by a known amount. This method has the advantage that corrections due to the deformation of the sample volume and due to thermal gradients along connections are not required. Smoothed volumes extrapolated to 10 kbar and 100-1000 °C are tabulated in Shmonov & Shmulovich (1974) and are quoted by several authors as primary measurements. Experimental data at 1 kbar are 2.5 % too large compared to data by Kennedy (1954) (cf. figure 1.5). Data by Tsiklis et al. (1971) at lower temperatures and 4-7 kbar are only 1 % smaller than measurements by Shmonov & Shmulovich (1974) at 673 K, but trends in isobaric sections (cf. figure 1.3) would extrapolate volumes by Tsiklis et al. (1971) to volumes smaller by about 2-3 %. Isobaric lines on a T — V diagram are almost straight but should be curved slightly downward based on van der Waals theory which would even enhance the discrepancy discussed above. Any smooth equation of state based on van der Waals-like theory constrained by data of Kennedy (1954), Tsiklis et al. (1971) and Shmonov &, Shmulovich (1974) will therefore predict volumes which are too small relative to Shmonov & Shmulovich. Phase equilibrium data at moderate pressures (1-5 kbar) appear to demand smaller volumes (more stable CO2) as well (cf. figure 1.5). In figure D.26 we also observe a marked nonlinearity of the 980 K data in isobaric sections with the lower temperature data in the pressure range of 2.5-4 kbar, perhaps suggesting some experimental problem. While this evaluation can only be relative it points towards uncertainties for P-V-T measurements at high pressure that are significantly larger than those estimated by the 141 Appendix D. Comparison of Calculated Volumes with Measured Volumes 142 5 to 00 eg Q3 Q0 MO<Q - CM (Mcorsio OJCMT I _ II || - o * > > * v 5 a a < a a J L o . o CD [ X K S p tXKXDi M O < D M O < D M O OG M O <G M <X30 1X1 O d M 0 < D M 0 < D M 0 < ] G M O O Q o .CO .tO CO n — H N ' V N I — i — O'S 0'£ 0"! ( 3 1 U 3 - S 9 0 ) o in o f X < n t n C D Q : o Q . ' CO . 9 C\J T O'l- 0"£- N 0 I 1 U I A 3 Q X O'S- Figure D.26: Comparison of computed volumes with experimental data of Shmonov & Shmulovich (1974). Appendix D. Comparison of Calculated Volumes with Measured Volumes 143 experimentalists. Kennedy, 1954, figure D.27, 25-1400 bar, 0-1000 °C: The experiments were per- formed at Harvard University. The apparatus (Kennedy, 1950, 1954) consists of an externally heated pressure vessel connected to a calibrated volumeter. Two methods are used, one with known amounts of sample gas in a known volume, and one with constant but unknown amount of sample in a constant volume. The first method is tedious but yields P-V-T data directly whereas the latter method has to be referenced to an isotherm along which the volume is known. Most data were obtained by the latter method and a reference isotherm at 150 °C was chosen based on data by Michels et al. (1935) and Michels & Michels (1935) and those obtained by the first method for gas densities below the range of Michels' data. Little primary data is published. Smoothed volumes based on several thousand measurements are tabulated. Precision of the data is reported to be ±0.2 % and smoothed values do not deviate from measured ones by more than 0.4 %. The accuracy is estimated at ±0.2 % (or ±0.0001 g/cm3) where data overlap with existing work and not stated at higher temperatures. The possibility of reaction of CO2 with the alloy of the pressure vessel wall is mentioned but not examined. Discrepancies with data by Shmonov & Shmulovich (1974) are discussed above. The data set is considered of inferior quality by Angus et al. (1973) compared to other avail- able data below 1 kbar. Kennedy's data, however, covers an extremely wide range of temperature (373-1273 K) which is crucial to constrain the temperature dependence of paramaters in an equation of state in such a way as to permit meaningful extrapolation to higher temperatures. The equation of state of this study does not reproduce volumes at 373-473 K ade- quately, particularly near the critical region (70-200 bar). Data below 473 K were not used to constrain the equation of state in order to permit a better fit to measurements at high temperatures. This trade-off demonstrates that a realistic temperature dependence of the parameters needs to be complex in order to fit the near-critical region. Juza et al., 1965, figure D.28, 700-4000 bar, 50-475 °C: These experiments were performed in Prague at the Czechoslovak Academy of Sciencies using an externally heated pressure vessel. The authors measure pressure and temperature of a known amount of sample in a known volume. The pressure measuring unit had to be maintained at 25 °C making necessary thermal gradient corrections for connection tubes. Uncertainties are estimated at ±0.3 %. Only smoothed volumes are reported by Juza et al. (1965) but the original measure- ments are tabulated in Angus et al. (1973). The overall fit of the equation of state to the data is reasonable with the exception of some measurements at low temperatures (323-423 K) which were not used as constraints. Tsiklis et al., 1971, figure D.29, 2-7 kbar, 50-400 °C: The experiments were per- formed at the State Institute for Nitrogen Research (GIAP) in Moscow. The authors o in' K E N N E D Y , 1954 ( T A B ) Mean % Deviation = Average % Deviation = Umimum % Deviation Mkumum % Deviation = J 1 I I i i -0.13 0.97 6.30 -4.25 J L X 1273 K * 1173 K X 1073 K X 9 7 3 K 8 7 3 K 7 7 3 K X 6 7 3 K + 5 7 3 K A 4 7 3 K 0 3 7 3 K J I <->o cr > o LU • , CD A o +. + A A CD O in. i i i i — i — i — i — r 150 300 450 — I 1 1 1 1 - 600 750 900 PRESSURE (BAR) n — i — i — r 1050 1200 J U Z A E T AL., 1965 ( S M ) Mean % Deviation = -0.45 Average % Deviation = 1.43 Maximum % Deviation = 2.24 Minimum % Deviation = -4.14 o 7 2 3 - 7 4 8 K X 6 2 3 - 6 9 8 K + 5 2 3 - 5 9 8 K A 4 2 3 - 4 9 8 K CD 3 2 3 - 3 9 8 K T 155 190 PRESSURE 400 Appendix D. Comparison of Calculated Volumes with Measured Volumes 146 S3 CO Ex. a? CO CO 0505030505 <O«0Tr05O5 OX + <Q - en ocogco cocn̂ r • • . . Q a o ( \ i 1 II „ - II C .2§ o (0 > £ _ , a) 5 ̂ «J a ) « c v > <d .S 2<32 t i - < ] + 1 x 0 1 < + <1 4 3 * ! I < < K I I I O E) f I a . a 00 o . oo CO a • CM, CO .Sx oCC f ee ZD to CO . LU on .ao cn o . CM cn a .co CM n 1—1—1— o's o'e o ' i o* i - o*e- o'g- ( 3 1 d 3 - S g O ) N0I1UIA3Q t CM Figure D.29: Comparison of computed volumes with experimental data of Tsiklis et al. (1971). Appendix D. Comparison of Calculated Volumes with Measured Volumes 147 use a displacement method whereby the volume is extracted from the difference of two measurement series at the same temperature with sample volumes that differ by a known amount. A measurement series consists of withrawing and weighing successive samples of fluid and monitoring pressure at constant temperature. This method has the advan- tage that corrections due to the deformation of the sample volume and due to thermal gradients along connections are not required. Only smoothed P-V-T data is presented with a claimed accuracy of ±0.5 %. Devia- tions from volumes measured by Juza et al. (1965) are as large as 1 %. The equation of state of this study represents the data well with the exception of some measurements at 323-373 K. This is an important data set which expands the range of temperature ex- perimentally examined at high pressure from those reported by Shmonov &: Shmulovich (1974). Volumes of Tsiklis et al. (1971) extrapolate to smaller values at higher temper- atures than those measured by Shmonov &; Shmulovich (cf. figure 1.3 and discussion above). Michels et al., 1935, figure D.30, 70-3000 atm, 25-150 °C: These experiments were performed at the van der Waals laboratory in Amsterdam using the classical method of mercury manometry. The measurements are exceedingly precise (±0.05 %) and likely of excellent accuracy estimated to better than ±0.2 % by Angus et al. (1973). This data set therefore portrays best the inaccuracies of the equation of state in the near-critical region. Data between 373 and 398 K were not used to constrain the equation of state. Vukalovich et al., 1962, figure D.31, 10-600 bar, 200-750 °C: The experiments were performed at the Moscow Energetics Institute (MEI). The method is to remove and measure the amount of fluid in the pressure vessel at known temperature, pressure and volume. The authors stopped the experiemntal project at 1023 K above which they observed graphite coatings on the pressure vessel walls indicating the presence of signifi- cant carbon monoxide. The measurements agree well with those of Kennedy (1954) and other data where some overlap exists. There is some uncertainty as to the speciation of CO2 at temperatures above 900-1000 K in the measuremnets of Kennedy (1954) which extend to 1273 K. No estimate of uncertainty is provided. Angus et al. (1973) assigned an uncertainty of approximately ±0.3 % based on his analysis. The equation of state does not represent the data within their accuracy towards higher pressures. The difficulty of fitting data well at 300-600 bar indicates too little flexibility of the equation of state to fit data to higher pressures simultaneously. Amagat, 1891, figure D.32, 50-1000 atm, 100-260 °C: This classical experimental study was performed using mercury manometry. The apparatus is described in Amagat (1880). The measurements are of good quality and agree well with measurements of Michels et al. (1935) and Kennedy (1954) at 1 kbar. Data at 373 K which show the largest deviations from the equation of state were not used as constraints. Michels & Michels, 1935, figure D.33, 16-250 atm, 0-150 °C: The experiments were Appendix D. Comparison of Calculated Volumes with Measured Volumes 148 CO CO i 5 * co s OX-K1G -\ CD r~ om—-CM •TOO • — • -\r I CM CO I ii c-2.2 o <o > £ *&§! 4> > «3.S g a 1 1 1 1 1 1 k i *k i ^ 1 ^ ! <• i x ! | - ) Q 9 3 i i — r 0"8 S'P 9'I ( 3 1 b 3 - S 9 0 ) 9* I- B'P- NQI1UIA3Q i o .in cn in " cn o . 0 0 CM CM ,2x CC m d -CD cn L U inCL . o o cn 0*8- Figure D.30: Comparison of computed volumes with experimental data of Michels et al. (1935). Appendix D. Comparison of Calculated Volumes with Measured Volumes 149 ©3 ©j ©j oj ©} oo e»j oj o>j ©j ©j QOjCJ5Q0P3f>.r̂ «5CO*O»O'̂ K>-N*XX+<l-MO<Q - CO o o oootooo •moo • o • csi i o — I II || - • gs-2 a3.22 o « •*£ «J a> x n 0) > <0.S J L J L o . o ID in D t S J M X f f - N O < G I M < # N O < l G O'E i — i — r i — r . CM O . — • ca . o U J ence . cn cn gLU l ~c% o . 0 0 a . CM _ a (D 8"1 9*0 9'0- 8"l- O'e- (31U3-S90) N0I1UIA3Q t Figure D.31: Comparison of computed volumes with experimental data of Vukalovich et al. (1962). AMAGAT, 1891 Mean % Deviation = Av ŝage % Deviation = Maximum % Deviation = Mmtrhjum % Deviation = - 1 . 5 2 2 .68 10.69 - 4 . 3 1 X + A O 5 3 1 K 4 7 1 K 410 K 3 7 3 K T — r 440 550 660 PRESSURE (BAR) uoo M I C H E L S A M I C H E L S , 1 9 3 5 a o c\i' O 4 2 3 K Mean % Deviation = -0.57 X 418 K Average % Deviation = 0.58 + 4 1 3 K Maximum % Deviation = -0.34 A 3 9 8 K Minimum % Deviation -0.74 O 3 7 3 K J I I I I J I I I I I I I I I I I O C M 1 • 1 (_J 1 in CO^ ZD*. I—'O _ CC t—1 — > C M I I I • L U o CM © A * > 0 A * D A - R > ^ AVA 4 ^ A ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20 25 30 35 40 45 50 55 60 65 70 PRESSURE (BAR) Appendix D. Comparison of Calculated Volumes with Measured Volumes 152 performed at the van der Waals laboratory in Amsterdam (see notes on Michels et al., 1935). The agreement of the measured volumes at these subcritical pressures and super- critical temperatures with the equation of state is adequate. Vukalovich etal., 1963b, figure D.34, 20-200 bar, 650-800°C: These high temperature- low pressure experiments were performed at MEI (see notes on Vukalovich et al., 1962). At these conditions the equation of state approximates the perfect gas law (minus the covolume b) which reproduces the measured volumes well. Vukalovich et al., 1963a, figure D.35, 10-600 bar, 40-150 °C: This data is an ex- tension of the study by Vukalovich et al. (1962) using the same method (see above). Similar to data discussed previously (Amagat, 1891, Michels et al, 1935, Kennedy, 1954) the equation of state does not represent measurements well at pressures just above the critical pressure at temperatures below 400 K. This data set was not used to constrain the equation. Appendix D. Comparison of Calculated Volumes with Measured Volumes 153 J L o . a (M a . co .o CO OS CO 45 s 3 X + <Q cnorsia o a a i II •I II - | ii c - .2 o <0 >*C V jr. la -I K O OJ 3 2 a .co .3 a.—. .CNJQ; a: CO g t u - c c ZD . CO cn L U Q_ . a co . a CN n i i—i—r— O'Z Z\ PO t?"0- Z'\- (31b3-SaO) N 0 U 0 I A 3 Q O'Z- Figure D.34: Comparison of computed volumes with experimental data of Vukalovich et al. (1963b). Appendix D. Comparison of Calculated Volumes with Measured Volumes 154 8 01 I CO o 25 4J 030003 +<G H LO gcocn— V Q O ) • . . .-^ ocvjcn 1 11 -4 1 1 A c 11 c ° | o <o >C1 (0 Qjt-JQ < + - K O "4 G ° <J + j < + 1 < -H to G G G 0'8 T 8> (3~ld3- 9*1 •S90) a . o co in a '9 a . CM CO g i u cnce CM U C D a .CM . a co CO CO U J 1—1—r 9 ' T - B'P- N Q I i d I A 3 0 X 0'8- Figure D.35: Comparison of computed volumes with experimental data of al. (1963a). Vukalovich et Appendix E P—V—T-X Measurements: Data and Experimental Procedures of Existing Work Existing P-V-T-X measurements on H2O-CO2 mixtures at supercritical conditions are recast in this appendix in terms of the molar excess volume on mixing. Figures E.36 to E.47 present the data in isothermal, isocompositional, pressure vesrsus excess volume plots. Figures E.48 to E.54 display the same data in graphs of fluid composition versus excess volume at constant pressure and temperature. Data are discussed in chapter 2. A summary of the experimental methods used by the authors of P-V-T-X data is provided below including a discussion of uncertainties and some other aspects where appropriate. Two studies of the systems B^O-Ar, B^O-Xe and H2O-CH4 are included for comparative purposes. Several of the original publications or Ph.D. theses are written in German and in Russian. Franck & Todheide, 1959: H20-C02, 300-2000 bar, 400-750 °C: The authors use a cylindrical autoclave with 80 cm 3 volume and 66 mm OD and a Morey-type conical plug with a Bridgman seal made of copper. The OD to ID ratio is 1:3 with a wall thickness designed for 1000 hrs at 2000 bar and 750 °C. Pressure is measured with Bourdon gauges (0.6 and 1) which are filled with pure water and separated from the experiment through a valve. The furnace design combines two independent heating coils. Temperature is measured with 5 Pt-PtRh thermocouples (1 for absolute T, 3 along bomb, 1 in lid) located in 25 mm deep wells. Starting materials are double-distilled water (conductivity < 2.0 - 5 Ohm - 1 cm - 1 ) and C 0 2 of 99.7-99.8% purity (impurities: 15% 0 2 , 1% CO, 84% N 2 ) . Moisture in C 0 2 is reporetd to be 0.2 mg per liter. The volume of the autoclave is 76.90±0.05 cm 3 determined with the mercury method. The volume of connections is 5.7 ± 0.1 cm 3, some of which is added to the bomb volume. The authors assign 79.1 ± 0.25 to the measuring volume and 3.5 ± 0.2 to unheated connections and reproduce pure gas data (Kennedy, 1954, Holser & Kennedy, 1958,1959) to within 1% accuracy. This procedure of determining the actual volume by calibration will eliminate some of the thermal gradient effects. Note that the data of Kennedy only extend to 1400 bar and measurements on pure gases are not reported. The bomb is filled with liquid water and CO2, with an uncertainty of 20 mg for water and 30 mg for C 0 2 . Connections are heated to empty its contents into the cold bomb. Measurements are taken during heating and cooling starting at 200 °C, with 50 K 155 Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 156 intervals. The anticipated pressure in the bomb is preadjusted in the manometer capillary with a screw press to minimize exchange of gas during pressure measurement with the valve opened. The authors observe higher pressures on cooling compared to heating below 450 °C, with a maximum deviation of 50 kg/cm2 at about 370 °C. They hypothesize on the extension of the two-phase region beyond the critical temperature of water. A similar observation was also made by Khitarov &; Malinin (1956) in their experiments. Note that the critical region does not extend beyond the critical temperature of water (Todheide Sz Franck, 1963). This must be either a metastable subcritical behaviour or the effect of metastably persisting species other than H2O and CO2. Measurements are smoothed graphically with a mean deviation of all data of 0.39%. Corrections are applied due to the volume expansion of the autoclave and unheated con- nections but no details are reported. No data on volumes of pure fluids are reported. The computation of excess volumes for this appendix is based on the tabulated volumes of Kennedy (1954) and Holser & Kennedy (1958, 1959) for the pure fluids. This introduces the large uncertainties depicted in figurs E.36-E.45 and E.48-E.54. Compressibility fac- tors (2) are reported with an uncertainty of 2.5% at total bomb concentrations of 10 mol/lit (1% from P, 0.2% from T, 1.3% from X C o 2 ) - A concentration of 10 mol/lit cor- responds to about 1000 bar with pure CO2 and 550 bar with pure H2O. Uncertainties are larger at lower total concentrations, but are not specified. Extrapolated pressures from data by Khitarov & Malinin (1956) are up to 8 % higher than Franck & Todheide's data. Ideal mixing yields compressibility factors that are only up to 5% too small compared to the measurements of Franck & Todheide (1959). Formation of H2CO3 is discussed: about 0.08 mol CO2 are soluble in water at ambient pressure of which 0.1% is H2CO3. The authors partition the excess volume into a non- chemical part and a chemical part comparing their measurements to a van der Waal's equation with the following mixing rules that accont for the non-chemical contributions: flmix = - ^ H 2 O G H 2 O + -Xco 2 aco 2 + 2Xn2oXco2 \/aH 2oaco 2 "mix — 2 O & H 2 O + ^co 2&co 2 + 2 X H 2 O ^ C O 2 Note that these rules are for simple molecules with no or weak dipole moments, probably not a good approximation for H 2 0 and C 0 2 . The authors conclude that at 600 °C and X = 0.5 no more than 10% carbonic acid is present, probably much less. Measurements of the electrical conductivity (Franck, 1956) indicate no formation of ionic species above 400 °C. Note that this treatment of the problem of chemical association forms part of the basis on which de Santis et al. (1974) built their modification of the Redlich-Kwong equation of state which was adopted by Holloway (1977) and others (see chapter 2 for more deatils). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 157 Todheide & Franck, 1963, two-phase region to 3500 bar: The pressure vessel is made from ATS 113 with two conical Morey caps with copper Bridgman seals. The top cap contains 4 capillaries, 1 for filling and 3 for extracting samples for analyses. The capillaries for extraction tap at the base and at the top, and a third capillary is adjustable in its position. The base plug connects to a second vessel containing an electro-magnetically activated stirring mechanism. No volumes of any parts are reported. The bomb is of approximately 80 mm OD and 450 mm length. The sample chamber is about 25 mm by 300 mm (ca. 147 cm3). Eight thermocouples are used for gradient control, 1 main, 5 along the side of the bomb, 1 in each cap, all in 18 mm deep wells. The furnace has 3 independent windings which are used to remove thermal gradients to better than 0.5 K. No volumes are measured in this study. Water is doubly distilled and C 0 2 is 99.7-99.8 % pure, with N 2 , 0 2 and minor CO. The high and low density phases are extracted (200-300 mg) into a series of evacuated containers and adsorption tubes (Mg-perchlorate and natron asbestos/Mg-perchlorate). The sample is filled into the autoclave at the temperature of interest until the desired pressure is achieved using calibrated injection pistons. Three samples of each phase are extracted at each pressure and temperature. The deviation from the mean is ± 0.15 mol- % in the liquid phase and ± 0.75 mol-% in the vapour phase. Equilibrium conditions are attained after 1 hour while stirring. Uncertainties are reported to be ± 1 % in pressure, ± 1 % in temperature, ± 0.5 mol-% in the liquid phase, and ± 1.0 mol-% in the gaseous phase. No details are provided on estimation of uncertainties. In particular, information about the treatment of volumes not at pressure and temperature of the experiemnt is not given. Measurements are reported to agree with Wiebe k Gaddy (1939, 1940, 1941) and Wiebe (1941) up to 120 °C and 700 bar. The composition of the liquid phase agrees with Khitarov k Malinin (1958) and Malinin (1959) (330 °C, 500 bar), but deviations of the gas phase composition are as large as 4 mol-%. The minimum of the critical curve is determined to be at 2.4-2.5 kbar, 266 °C and a mole fraction of C 0 2 of 43 mol-%. Gehrig, 1980: H20-C02-NaCl, 150-600 bar, 400-500 °C (to 3 kbar with NaCl): This author uses an apparatus that permits visual observation of the appearance of a new phase or its determination from the break in slopes of P-T curves at constant volume or V-T curves at constant pressure. This method provides precise measurements near phase boundaries or where densities of the two phases are nearly identical and cannot be determined reliably by the analytical method (i.e. Todheide k Franck, 1963). The autoclave is made from non-magnetic Ni-Cr-alloy (ATS 340) with dimensions of 400 mm length, 50 mm OD and 15 mm ID. A moveable piston allows the measur- ing volume to be adjusted. A VITON O-ring separates the pressure medium (water) chamber from the chamber containing the mixture. To protect the O-ring from excessive Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 158 temperatures a moveable cooling jacket is mounted externally at the position of the pis- ton. This probably leads to temperature gradients within the cell. A sapphire window (10 mm thick) is mounted in the cap closing off the measuring volume. The other end contains a capillary for pressure medium access and a guide for a rod to determine the position of the piston with an externally mounted induction coil. Stirring is achieved by means of a magnetic stick agitated by an external magnet. Light is beamed through the window onto a platinum mirror mounted on the piston face which iluminates the sample volume. The furnace is wound directly onto the autoclave in four independent sections to allow thermal gradient control. Heating rates were less than 3 K/minute and less than 1 K/minute near phase transitions. Furnace sector temperatures are controlled with ther- mocouples mounted in wells in the pressure vessel wall. Temperature is measured with a Chromel-Alumel thermocouple mounted internally. Various modifications of the sapphire window were tested to minimize problems with salt leakage. Many measurements were performed in a cell without a window and with a fixed volume. Temperature is measured with a resolution of 0.25 K. Pressure is measured on 3000 bar Heise-Bourdon gauge with 1 bar resolution. Volumes are measured via piston dis- placement which can be resolved inductively to 0.05 mm. The cell volume is calibrated with argon and carbon dioxide along isotherms. The volume correction term is expressed as 2nd-degree polynomial in temperature. The effect of pressure on the volume of the autoclave is assumed to be negligible (ca. 0.07 % per 1000 bar). Tri-distilled water was used and C 0 2 of 99.8 % purity. Errors are reported to be 0.5 % in pressure, 0.5 % in temperature (estimated), < 0.05 % in sample mass, 0.2-0.6 % in sample volume. The effect of the cooling jacket for the O-ring on the thermal field is not known, but might be substantial and will lead to an excessive apparent volume. There are some gradient measurements reported in an identical apparatus in the dissertation by Welsch (Karlsruhe, 1973), which show gradients increasing to 8 K at 500 °C. This effect translates into an overestimate of the volume of about 1 %. Data on H2O-CO2 mixtures are reported in form of tabulated smoothed values from XCo2 = 0.1-0.9, 150-600 bar and at 400, 450 and 500 °C. Volumes of pure fluids were measured but not reported. Data smoothing is carried out in V-Xco2 sections con- strained to obey the Gibbs-Duhem relationship. Smoothing was not constrained in V-P sections leading to a ragged apparance in several graphs of this appendix. Volumes for the pure gases were taken from the literature (Kennedy, 1954, Holser k Kennedy, 1958, 1959) in order to compute excess volumes on mixing from Gehrig's data. Activity coef- ficients are also tabulated based on the integrated excess volumes on mixing but details are not provided. Franck & Todheide's volumes are up to 10 % smaller and the excess volumes up to 60 % smaller than Gehrig's measurements. Greenwood's (1973) data is in fair agreement but does not agree within stated uncertainties at many P-T-X conditions Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 159 (see graphs in this appendix). Greenwood, 1969, 1973: H20-C02, 50-500 bar, 450-800 °C: The method and appa- ratus is described in great detail in Greenwood (1969). Measurement series were made in externally heated Morey bombs at constant temperature beginning with a C02-rich starting composition. Pressure is incrementally increased by injecting known amounts of H2O, thus producing more H 2 0-rich compositions along a given isotherm. The compo- sition of the mixture at the end of a measurement series is determined as described in Greenwood (1961, 1967b) from which all compositions along the P-Xco2 traverse can be determined by back calculation. The volume was determined by various methods to ensure an acceptable accuracy of about 0.3-0.4 %. Temperatures are reported to be precise to ±0.3 K and accurate to ±1 K. Pressure is measured by calibrated Heise-Bourdon gauges with an estimated accuracy of 0.2 %. The calibration of the screw press used to inject H2O shows maximum deviations of 0.085 % from the regression equation used to process the raw data. Data is smoothed to a polynomial in pressure and mole fraction for each isotherm that represent the observed compressibility factors to within 0.5 %. Tables of the compressibility z are presented in 50 K intervals. In a subsequent publication Greenwood (1973) recasts his experiments into con- strained polynomials with 12 coefficients for temperatures between 450 and 800 °C with 50 K intervals. Activity coefficients are tabulated at 50 K intervals between 450 and 800 °C at pressures of 100-500 bar. The polynomial equations obey the Gibbs-Duhem relationship and are used to represent Greenwood's data in the graphs of this appendix. Shmulovich et al., 1980: H20-C02, 1-5 kbar, 400 and 500 °C: These experiments were carried out at the Institute of Experimental Mineralogy, Academy of Sciences of the USSR, in Chernogolovska. A volume displacement method is used with an externally heated pressure vessel. The sample mixture is contained in a sealed gold capsule and the pressure medium is pure carbon dioxide. The volume of the sample is determined from the difference between two series of measurements, one with the filled gold capsule and one with the empty capsule. Measurements are performed at constant temperature starting from high pressure by incremental release of pressure medium. Pressure is measured by calibrated Bourdon gauges (type SV 6000) and reported to be accurate to ±15 bar. The amount of gas in the capsule is determined after each run by weighing with an accuracy of ±0.05 %. Temperature gradients are controlled to better than 1 K. The volumes measured are dependent on the P-V-T properties of CO2 used as pres- sure medium. These properties were taken from Shmonov & Shmulovich (1974) obtained by a displacement method which might lead to systematic errors of the same sign in both studies. P-V-T measurements on pure H2O show deviations from tabulations by Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 160 Burnham et al. (1969) of up to 0.5 %. The overall accuracy is estimated at ±1.6 % at pressures 1.5-2 kbar and ±0.7 % at 4.5-5 kbar. Enough raw data is provided that one could revise the volumes of the mixtures given more accurate data for the pressure medium CO2. Excess volumes derived by the authors are tabulated at 400 °C (-Xco2 =0.623, 0.316, 0.087) and at 500 °C (Xco2 =0.398, 0.227), at various pressures between 1 and 5 kbar. Shmulovich et al., 1979: H20-C02, 0.4-5.7 kbar at 500 °C and 150-1000 bar at 500 °C: The method is identical to that used by Shmulovich et al. (1980). The accuracy of the measured volumes is stated to be 0.2 %. The authors report excess volumes on mixing at Xco 2=0.2, 0.4 at 500 °C, and at ^co 2=0.2, 0.4, 0.6, 0.8 at 400 °C. Some data is smoothed from measurements at XCo2 =0.227, 0.398 (500 °C) and X Co 2=0.2, 0.38, 0.6, 0.8 (400 °C). The volume of the mixture is not reported. Volumes of the pure fluids were adopted from Burnham et al. (1969) and Shmonov & Shmulovich (1974). Most excess volumes are distinctly larger compared with other studies. Data ob- tained by Shmulovich et al. (1980) by a similar or identical method do not agree within uncertainties with this study where the two sets join together. Khitarov & Malinin, 1956: H20-C02, 1-1800 kg/cm2, 100-500 °C: The authors use an externally heated pressure vessel and measured the pressure generated at temperature by a known amount of mixture of known composition in a calibrated volume. The mea- surements are not precise with the largest uncertainties introdueced by the inadequate method of filling and mixing, and the rather large temperature uncertainty of ±2.5 K. The data are therefore not used in this analysis. Welsch, 1973, H20-Xe, H20-CH4, 300-2500 bar, 375-500 °Cand 375-425°C: These systems behave similarly to H2O-CO2, i.e. they all display temperature minima in the critical curve of the mixtures extending from the critical point of water to higher pres- sures. Methane is quite similar to CO2 also with respect to its size and polarizibility. Methane, however, has no quadrupole moment. The temperature minimum of the C H 4 - H 2 0 critical curve is at 353 °C, 990 bar, 38.3 cm3/mole and 25 mol-% methane. The temperature minimum of the C 0 2 - H 2 0 critical curve is at 266 °C, 2450 bar and 41.5 mol-% C 0 2 . H 2 0 - C H 4 : data is presented at 375, 400 and 425 °C between 400 and 2500 bar, plotted as a series of Vexcesa-X diagrams. At 400-500 bar all K x c e s s are positive, skewed towards the water-rich side with a maximum of about 33 cm3/mole at 400 °C and 400 bar. At higher pressures Kxcess becomes negative on the water-rich and methane-rich side. At pressures above 1500 bar all Vexce8a are negative, not more than —5 cm3/mole, decreasing again at higher T (425 °C). H 2 0-Xe: Vexcesa-X plots are presented for 375, 400, 450, 500 °C at 300-2500 bar. Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 161 Largest K . x c e S s are observed at 300 bar and 400 °C of 52 cm3/mole, with curves skewed towards the water-rich side. Small negative Vexces3 are depicted at 375-400 °C on the water-rich side at pressures above 1000 bar. Welsch also discusses the van der Waals equation of state and how it predicts the qualitative behaviour of the critical curve correctly in the systems Ar, Xe, CH4 mixed with H 2 0 . Greenwood, 1961, H20-Ar: 500-2000 bar, 500 °C; The system H 2 0 - A r shows a different behaviour than mixtures of H2O with Xe, CH4 or CO2 in that the critical curve proceeds to higher temperatures with rising pressure without the presence of a temperature minimum. The method used by Greenwood (1961) consists of measuring pressure and tempera- ture of a mixture contained in a known volume using an externally heated Morey-type pressure vessel. The composition of the mixture is determined after each experiemnt using a cold-trap to measure the amount of water and a manometer to determin argon. The uncertainties are estimated at 2 % for the composition of the mixture and 2 % for the molar volume. Measurements at 500 °C on pure H 2 0 agree well within uncertainties with measurements by Kennedy (1950). Date on pure argon at 150 °C is in good agreement with measurements by Michels et al. (1949). Original measurements are reported and tabulated fugacities are determined from graphically smoothed data. The molar excess volume on mixing is displayed as a function of mole fraction and pressure. The largest excess volume (32 cm3/mole) is observed at 500 bar. Appendix E. S P-V-T-X Measurements: Data and Experimental Procedures 162 I I I 1 I L. J I I I I l_ 400 C X(COZ) = 0 . 1 Kerrick * Jacoos von der Waals eg — — Holloway t. CBH(1980) X SUM (1980) —V 30 - r - 50 i 1 r 70 1 1 1 1 1 1 90 111 130 , ISO PRESSURE (BAR) (X10 1 ) i , i i ' ' i i i i i i i_ 400 C X(COZ) = 0 . 2 Kerrick * Jacobs van der Waals eg — H olloway A CEH(1980) O F*r(t959) X SSZ(1979) — I — 170 PRESSURE U $ R ) (XH) 1 ) 1 5 0 j ' ' ' I L. X(COZ) = 0 . 4 Kerrick * Jacobs van der Waals eg — — Holloway A CEH(I980) O FicT(19S9) X SSZ(1979) 90 110 130 , PRESSURE (BflR) (X10 1 l l I I I I I T" 0 ISO 170 190 210 Figure E.36: P - Ve graphs: 400 °C, X C O j =0.1-0.4. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 163 J i i i i ' i ' i ' ' ' i ' i i i i i 450 C X(COZ) =0.6 Kerrick Jt Jacobs van der Coals eq — — Holloway CD CRE( 1973) A CEH(tsao) O F*T(f959) T 1 1 1 1 1 1 I 1 1 1 1 1 1 r 70 90 IJQ I30 . ISO I70 ]90 PRESSURE I BAR) 1XI0 1 ) 210 ' i i ' ' ' 450 C X(C02) = 0.8 Kerrick 4c Jacobs van der Waals eq — • H ollarway O GRE(1973) A GEH(isao) O F*T(1959) 7 - I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 10 30 W 70 90 110 130 . 150 1 70 1 93 210 PRESSURE l ^ R ) IXH)1 1 450 C X(C02) = 0.9 Kerrick * Jacob* van der Waals eg — Holloway CO GRE(1973) A CEH(fSSO) ~\ 1 1 1 r 30 SO I I I I I I I I I I I I I I 70 , 9 0 . HQ 130 , 150 1 70 190 PRESSURE IBRR) IX10 1 ) 10 210 Figure E.37: P - Ve graphs: 400 °C, XCo2 =0.6-0.9. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 164 - I 1 1 1 I I I L J I I I I I I L. 4 5 0 C X(C02) = 0 . 1 Kerrick * Jacobs van der Waals eq — — Holloway O GRE(1973) A GEH(1980) - i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 30 50 70 90 no 130 . 150 170 PRESSURE I BRR) 1X10" ) 190 210 J I ' i _! I I I U 4 5 0 C X(C02) = 0 . 2 Kerrick * Jacobs van der Waals eq O — Holloway O CRE(t973) • F*T(1959) A GEH(1980) i ' i i i i 1 1 1 1 1 1 1 r 10 30 50 70 „ „ 9 0 _ MO 130 , 150 170 190 210 PRESSURE (BRR) 1X10' ) 70 90 110 130 . 150 170 PRESSURE (BRR) IX10 1 ) Figure E.38: P - Ve graphs: 450 °C, XCo2 =0.1-0.4. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 165 (_) X UJ _L: i 1 1 1 1 i i ' i i _L I I 1_ 400 C X(COZ) = 0 . 6 Kerrick £ Jacobs van der iTaata eg — — Holiouxxy A CEH(1980) O FAT(I9S3) X SUM (1980) X SSZ(19?9) - T ~ 30 - J — 50 -i 1 1 1 r—i 1 1 1 1 1— 70 90 HO 130 , 150 1 70 PRESSURE (BRR) (X10 1 ) 190 210 J I I I I I I I I I I I I I I I I 9 - 400 C X(C02) = 0 . 8 Kerrick A Jacob* van der iTaala eg — — Holloway A CEH(1980) O FJtT(l959) X SSZ(I979) n r 30 1 1 1 1 1—i 1 1 1 1 1 1 1 1—i r 50 70 90. 110 130 , 150 170 190 210 PRESSURE I BRR) IX10 1 ) 1 kD i i i i i i i i i i ' i i i 400 C X(C02) = 0 . 9 O 9 - Kerrick * Jacobs van der Waalm eg — Holloway :H /M OL E 3 0 .0 A GEH(1980) _ a 58B- : UJ ui~ i i N ^ o i 1 1 1 1 1 1 1 r SO 70 90 _ 110 130 , 150 PRESSURE (BfiR) (X10 1 ) 170 190 Figure E.39: P - Ve graphs: 450 °C, XCo2 =0.6-0.9. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 166 500 C X(C02) = 0.1 Kerrick * Jacobs van der Waals eq — — Holloway O CRE(1973) A CEH(1980) T 70 90 110 130 , ISO PRESSURE I BAR) (X10 1 ) 170 1 I l i 1__J !_ 500 C X(C02) =0.2 Kerrick * Jacobs van der Waals eq — Holloway CD CRE(I973) A CEH(I980) <> r*T(t9S9) X SUM (1980) _ X SSZf 1979; T 1 1 1 1—i 1 r 50 70 30 110 130 , 150 PRESSURE (BRR) IX10 1 - | r 170 i r 190 210 500 C X(C02) =0.4 Kerrick * Jacobs van der Waals eq — —Holloway O CRB(1973) A CEH(1980) O F*T(19S9) X SHU (1980) I X SSZ( 1979) i I- - - - - r r t 10 / 70 30 110 130 , 150 PRESSURE IBHR) IX10 1 ) Figure E.40: P - Ve graphs: 500 °C, XCo2 =0.1-0.4. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 167 PRESSURE (BRR) m o 1 ) .i i i i i i i ' ' i i i 500 C X(C02) = 0.8 Kerrick * Jacobs van der Waals eq — — Holloway (D CRE(t973) A GEH (1980) O F*T(19S9) § 9 \ o _ (_> _ l I I l _ 500 C X(C02) = 0.9 Kerrick A Jacobs van der ITaat* eq — Holloway O CRE(t973) A GEH(1980) </T=> UJ > = —J— 30 —r- SO - I 130 7 0 PRESSURE ('iffa) (ifo 1 ) ISO 170 210 Figure E.41: P - Ve graphs: 500 °C, XCo2 =0.6-0.9. Data of Gehrig (GEH), Franck k Todheide (F&T)', Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 168 J I I I I I I L . ' ' I I I I I I 600 C X(C02) = 0.1 Kerriek tc Jacobs — — von der Waal* eq — — Holloway O GRE(1973) — I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 10 30 50 70 90 110 130 , 150 1 70 1 90 210 PRESSURE (BAR! (X10 1 ) _i i : i i ' ' ' ' ' ' J L I l _ l L 600 C X(C02) = 0.2 Kerrick A Jacobs van der Waals eq — - Holloway CD GRE(1973) O F*T(1959) - i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 10 30 50 70 „ 90 110 130 . 150 170 190 210 PRESSURE BAR IX10 1 ) - i 1 1 1 1 1 1 1 i i i i i i i ' i i i j 600 C X(C02) = 0.4 Kerrick A Jacobs van der Waals eq — - Holloway O CRE(1973) O F*T(t9S9) ~i 1 1 1 r — i r — i 1 1 — i 1 1 1 1 1 1 1 r 3 0 5 0 7 0 „„2<L „ ,UJU I * , "SO 170 190 PRESSURE (BAR) IX10 1 ) 210 Figure E.42: P - Ve graphs: 600 °C, XCo2 =0.1-0.4. Data of Gehrig (GEH), Franck & Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data, and Experimental Procedures 169 - i — i — i i i i i i i i i i i i i i : i_ 600 C X(C02) = 0.8 Kerrick * Jacobs van der Waals eq — Holloway O GRE(t973) 0 F*T(1959) —I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 v 10 30 50 70 90 110 1 30 , 150 1 70 190 210 PRESSURE (BRR) IX10 1 ) - i 1 1 i i i i i i i_ o o <_> 1- 600 C X(C02) = 0.9 Kerrick * Jacobs van der Waals eq - — Holloway CD CRE(1973) ~i i i i i i i i i 1 1 1 1 — i 1 1 1 i r- 30 50 70 „ „ 9 0 110 130 , 150 170 190 210 PRESSURE (BRR) ( X 1 0 ! ) Figure E.43: P - Ve graphs: 600 °C, X C o 2 =0.6-0.9. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 170 ' ' ' i i i i i i i 7 0 0 C X(COZ) = 0 . 1 Kerrick 4c Jacobs van der Waals eq — — Holloway <D CRK(1973) I 30 n 1 1 1 1 1 1 1 1 1 r 50 TO 90_ 1JQ_ 130 . 150 170 —i r 130 7 0 PRESSURE I^RR) I X U J 1 ) 210 7 0 0 C X(C02) = 0 . 2 Kerrick A Jacobs van der Waals eq — // olloway O CRE(1973) —1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — 30 50 70 90 110 130 , ISO 170 190 PRESSURE (BRR) (X10 1 ) 2)0 Figure E.44: P - Ve graphs: 700 °C, XCo2 =0.1-0.4. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 171 • i I 1 1 1 ( — " 1 1 1 1 1 r—i 1 1 1 i 1 1 r 10 30 50 70 90 UP. 130 . 150 1 70 190 210 PRESSURE (BHR) IX10 1 ) -J 1 1 1 1 i i i ' i i i i i i i i i i _ 700 C X(C02) =0.8 derrick * Jacobs van der ITaais eq — - Holloway <D CK8(I973) • r*T(1959) - I 1 1 1— 10 30 50 ~i 1 1 1 1 1 1 1 1 1 1 1 1 r 150 170 190 210 7 0 PRESSURE I'iflR) ixfo' ) —i 1 i i i i i i i i i i_ 700 C X(C02) = 0.9 Kerrick * Jacobs van drr Waals eq — Holloway O GRE(1973) Xrtf- -I — - i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 10 30 50 70 , 9 0 110 130 , 150 170 190 210 PRESSURE (BAR) ( X 1 0 1 ) Figure E.45: P - Ve graphs: 700 °C, XCo2 =0.6-0.9. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 172 - i i i i i i i i i i ; i i — i — i — i — 1 _ 8 0 0 C X(C02) = 0 . 1 Kerrick tc Jacobs van der Waals eq — — Holloway O GRE(1973) ~i 1 — i 1 1 1 1 1 — i — r — i 1 1 1 1 r~—i r 30 50 70 90 110 130 , 150 170 190 PRESSURE (BAR) 1X10 1 ) 800 C X(C02) = 0 . 2 Kerrick * Jacobs van der Waals eq - — Holloway T 1 1 1 1 1 1 1 1 — i — i — i 1 : 1 1 1 1 r 30 50 70 _ 90 110 130 , 150 1 70 190 PRESSURE (BAR) (X10 1 ) 800 C X(C02) = 0.4 Kerrick * Jacobs van der Waals eq — Holloway C[GRE(1973) 70 90 HQ 130 , 150 PRESSURE (BAR) (X10 1 ) - i — 190 Figure E.46: P - Ve graphs: 800 °C, XCo2 =0.1-0.4. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 173 J i i i_ _] I I I I I 1 I I I I L_ 800 C X(C02) = 0.6 Kerrick * Jacobs -van der Waals eq — — Holloway CD CRE(19?3) 50 - r TO - -30 110 130 , 150 - 1 r no -1 r 193 30 PRESSURE (BAR) 1X10' ) 210 800 C X(C02) = 0.8 Kerrick * Jacobs '— van der Waals eq — - Holloway CD CRE(1973) —T— 30 —r~ 50 — I — 190 7 0 PRESSURE (tib) t i f fa i . , 5 ° no 210 _ i : i ' ' i o u 800 C X(C02) = 0.9 Kerrick * Jacobs van der Waals eq - — Holloway CD GKE(I973) - i 1 1 r 30 so 1 1 1 r 1 0 PRESSURE (Ufa <X?0> ) , 5 ° 1 7 0 210 Figure E.47: P - Ve graphs: 800 °C, XCo2 =0.6-0.9. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 174 _L Kerrick A Jacobs van der Waals eq Holloway ID . in T(C) = 4 0 0 . P(b) = 2 0 0 . A GEH (1980) X SSZ(1979) U _ l _ J a o C D . 0 . 0 0 . 2 0 . 4 ' 0 . 6 X(CG2J 0 . 8 1 .0 Kerrick A Jacobs van der Waal* eq H olloway T(C) = 4 0 0 . P(b) = 3 0 0 . A GEH(1980) <S> FAT(19S9) X SSZ(I979) 0 . 0 0 . 2 0 . 4 0 . 6 XIC02) 0 . 8 1 . 0 Figure E.48: X - Ve graphs: 400 °C, 200 k 300 bar. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 175 r - co. in J I I Kerrick A Jacobs van der Waals eq JfollouHty T(C) - 4 0 0 . P(b) = 4 0 0 . A GEH(1980) O FAT(1959) X SSZ(1979) 1 .0 Kerrick «fr Jacobs van der Waals eq Holloway in T(C) = 4 0 0 . P(b) = 5 0 0 . A GEH(1980) O FAT(f959) X SSZ(1979) c_) I I I — I 1 1 1 1 — ° 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 XIC02) Figure E.49: X - Ve graphs: 400 °C, 400 k 500 bar. Data of Gehrig (GEH), Franck k TSdheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 176 o CM . C_) Kmrick Jt Jacobs van der Waals eq Holloway T(C) - 500. P(b) = 200. O CRE(1973) A CEH(1980) 0.0 0.2 o ID . 0.4 0.6 X(C02) _ ] 1 1 0.8 Kerrick A Jacobs van der Waals eq Holloway T(C) = 500. P(b) = 400. O GRE(1973) O F*T(1959) X SSZ(1979) 1 . i r I l I l l r 0.0 0.2 0.4 0.6 X(C02) 0.8 1 .0 Figure E.50: X - Ve graphs: 500 °C, 200 k 400 bar. Data of Gehrig (GEH), Franck k' Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures o. I I I Kerrick 4c Jacobs van tier Waals eq Holloway -r ? T(C) = 500. P(b) = 500. O GRE(1973) A GEH(1980) O FScT(19S9) X SSZ( 1979) 0 . 4 0.6 XICO'2) O UJ _ J - § ° \ < \ i . C_> CJ CT)C5 UJ CJ X - UJ o o ' Kerrick 4c Jacobs van dsr s*aais eq Holloway ~r rcc; = 5oo. P(b) = 1000. X SHU (1980) X SSZ(1979) O FAT(1959) 0.0 C02) 1.0 Figure E.51: X - Ve graphs: 500 °C, 500 k 1000 bar. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 178 o o o _ Kerrick * Jacobs van dsr Waals e g Holloway T(C) = 6 0 0 . P(b) = 2 0 0 . O CRE(1973) (_) i i i r 0 .4 0 . 6 X(C02) Figure E.52: X - Ve graphs: 600 °C, 200 k 400 bar. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 179 o I D . I I I L Kerrick * Jacobs van der Vaals eg Holloway T(C) = 6 0 0 . P(b) = 5 0 0 . 0 GRE(1973) O F*T(1959) 0 . 0 o •<3 . o o i d . 0 . 2 0 . 6 X(CQ2) 0 . 8 1.0 Kerrick A Jacobs van der Waals eg Holloway T(C) = 6 0 0 . P(b) = 1000. <> F&T(19S9) 1 I I 1 I I 0 . 0 0 . 2 0 . 4 0 . 6 X(C02) 0 . 8 1 . 0 Figure E.53: X —Ve graphs: 600 °C, 500 k 1000 bar. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ). Appendix E. P-V-T-X Measurements: Data and Experimental Procedures 180 0 .0 Kerrick A Jacobs van der Waals eq Holloway T(C) ~ 7 0 0 . P(b) = 2 0 0 . O GRE(1973) 0 . 2 0 . 4 - 0 . 6 X(C02) i r 0 . 8 l . 0 Kerrick: A Jacobs van der Waals eq Holloway T(C) = 7 0 0 . P(b) = 5 0 0 . Figure E.54: X - Ve graphs: 700 °C, 200 k 500 bar. Data of Gehrig (GEH), Franck k Todheide (F&T), Greenwood (GRE), Shmulovich et al. (SHM), and Shmulovich et al. (SSZ).
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