Thermodynamics of concentrated electrolyte mixtures and the prediction of mineral solubilities to high temperatures for mixtures in the system Na-K-Mg-Cl-SO4-OH-H2O

Mineral solubilities in binary and ternary electrolyte mixtures in the system Na-K-Mg-Cl-SO₄-OH-H₂O are calculated to high temperatures using available thermodynamic data for solids and for aqueous electrolyte solutions. Activity and osmotic coefficients are derived from the ion-interaction model of Pitzer (1973, 1979) and co-workers, the parameters of which are evaluated from experimentally determined solution properties or from solubility data in binary and ternary mixtures. Excellent to good agreement with experimental solubilities for binary and ternary mixtures indicate that the model can be successfully used to predict mineral-solution equilibria to high temperatures. Although there are currently no theoretical forms for the temperature dependencies of the various model parameters, the solubility data in ternary mixtures can be adequately represented by constant values of the mixing term θ_ij and values of ψ_ijk which are either constant or have a simple temperature dependence. Since no additional parameters are needed to describe the thermodynamic properties of more complex electrolyte mixtures, the calculations can be extended to equilibrium studies relevant to natural systems. Examples of predicted solubilities are given for the quaternary system NaCl-KCl-MgCl₂-H₂O.     

A thermodynamic model for multicomponent melts,with application to the system CaO-Al2O3-SiO2

A thermodynamic model is proposed for calculation of liquidus relations in multicomponent systems of geologic interest. In this formulation of mineral-melt equilibria, reactions are written in terms of the liquid oxide components, and balanced on the stoichiometry of liquidus phases. In order to account for non-ideality in the liquid, a ‘Margules solution’ is derived in a generalized form which can be extended to systems of any number of components and for polynomials of any degree. Equations are presented for calculation of both the excess Gibbs free energy of a solution and the component activity coefficients.Application to the system CaO-Al₂O₃-SiO₂ at one atmosphere pressure is achieved using linear programming. Thermodynamic properties of liquidus minerals and the melt are determined which are consistent with adopted error brackets for available calorimetric and phase equilibrium data. Constraints are derived from liquidus relations, the CaO-SiO₂ binary liquid immiscibility gap, solid-solid $\text{P-T}$ reactions, and measured standard state entropies, enthalpies, and volumes of minerals in this system.Binary and ternary liquidus diagrams are recalculated by computer programs which trace cotectic boundaries and isothermal sections while checking each point on a curve for metastability. The maximum differences between calculated and experimentally determined invariant points involving stoichiometric minerals are 17°C and 1.5 oxide weight per cent. Because no solid solution models have been incorporated, deviations are larger for invariant points which involve non-stoichiometric minerals.Calculated heats of fusion, silica activities in the melt, and heats of mixing of liquids compare favorably with experimental data, and suggest that this model can be used to supplement the limited amount of available data on melt properties.     

Measurements and calculations of solid-liquid equilibria in the ternary systems NaBr–Na<Subscript>2</Subscript>SO<Subscript>4</Subscript>–H<Subscript>2</Subscript>O and KBr–K<Subscript>2</Subscript>SO<Subscript>4</Subscript>–H<Subscript>2</Subscript>O at 348 K

According to the compositions of the underground brine resources in the west of Sichuan Basin, solubilities of the ternary systems NaBr–Na₂SO₄–H₂O and KBr–K₂SO₄–H₂O were investigated by isothermal method at 348 K. The equilibrium solid phases, solubilities of salts, and densities of the solutions were determined. On the basis of the experimental data, the phase diagrams and the density-composition diagrams were plotted. In the two ternary systems, the phase diagrams consist of two univariant curves, one invariant point and two crystallization fields. Neither solid solution nor double salts were found. The equilibrium solid phases in the ternary system NaBr–Na₂SO₄–H₂O are NaBr and Na₂SO₄, and those in the ternary system KBr–K₂SO₄–H₂O are KBr and K₂SO₄. Using the solubilities data of the two ternary subsystems at 348 K, mixing ion-interaction parameters of Pitzer’s equation θxxx, Ψxxx and Ψxxx were fitted by multiple linear regression method. Based on the chemical model of Pitzer’s electrolyte solution theory, the solubilities of phase equilibria in the two ternary systems NaBr–Na₂SO₄–H₂O and KBr–K₂SO₄–H₂O were calculated with corresponding parameters. The calculation diagrams were plotted. The results showed that the calculated values have a good agreement with experimental data.     

Thermodynamic data for phases in the FeO-MgO-SiO2 system and phase relations in the mantle transition zone

Based on the available experimental data on phase equilibria in the FeO -MgO -SiO₂ system the mixing properties of the solid solutions (olivine, β- and γ-spinel, pyroxene, majorite, ilmenite and perovskite and magnesiowustite), the enthalpies of FeO and fictive FeSiO₃ phases with ilmenite and majorite structures have been assessed. The entropies, temperature dependance of heat capacities for fictive FeSiO₃ end-members were estimated from structural analogies. The calculated phase diagrams for Mg₂SiO₄-Fe₂SiO₄ and MgSiO₃ — FeSiO₃ systems at pressures up to 30 GPa and temperatures between 1000 and 2100 K are quite consistent with the available experimental determinations except for the fine features of the phase diagram at 2073 K.     

The Bayesian approach to an internally consistent thermodynamic database: theory, database, and generation of phase diagrams

An internally consistent thermodynamic dataset has been derived for 148 endmember phases (145 solids and 3 fluids) comprising the elements Li, Na, K, Be, Mg, Ca, Ti, Cr, Mn, Fe, Zn, Al, Si, C, H, and O. This has been achieved by simultaneous treatment of phase property (like standard enthalpy of formation, standard entropy, molar heat capacity, molar volume, thermal expansivity, bulk modulus etc.) and reaction reversal data by the Bayesian method. The theory underlying the approach, and the computational methods involved, are briefly outlined. (For the benefit of readers unfamiliar with inference statistics, the basic concepts of the Bayes method are also presented in such a way that they can be grasped intuitively.) Although not yet addressed, this method can be extended to refine the thermodynamic mixing properties of crystalline solutions. The sources of the input data, culled from the literature, are summarized in the Appendix. The resulting database is succinctly documented in this paper. It includes the enthalpies of formation and entropies, their uncertainties, and the correlation among them. The database allows calculation of P-T, T-X _CO2, P-X _CO2, and T-f _O2 sections, with error propagation into the computed phase diagrams on a routine basis. A user-friendly computer program has been written to generate such phase diagrams. It is public domain software. The software and the thermodynamic database (which includes a complete documentation of the thermodynamic data above and beyond those listed (Table 2, here) may be downloaded from the web site http://homepage.ruhr-uni-bochum.de/niranjan.chatterjee/Index.htm. Examples of computed phase diagrams are given to illustrate the quality of the data and the capabilities of the software. Received: 11 March 1998 / Accepted: 11 June 1998     

The stability of sapphirine + quartz: calculated phase equilibria in FeO–MgO–Al2O3–SiO2–TiO2–O

The presence in rocks of coexisting sapphirine + quartz has been widely used to diagnose conditions of ultra‐high‐temperature (UHT) metamorphism (>900 °C), an inference based on the restriction of this assemblage to temperatures >980 °C in the conventionally considered FeO–MgO–Al₂O₃–SiO₂ (FMAS) chemical system. With a new thermodynamic model for sapphirine that includes Fe₂O₃, phase equilibra modelling using thermocalc software has been undertaken in the FeO–MgO–Al₂O₃–SiO₂–O (FMASO) and FeO–MgO–Al₂O₃–SiO₂– TiO₂–O (FMASTO) chemical systems. Using a variety of calculated phase diagrams for quartz‐saturated systems, the effects of Fe₂O₃ and TiO₂ on FMAS phase relations are shown to be considerable. Importantly, the stability field of sapphirine + quartz assemblages extends down temperature to 850 °C in oxidized systems and thus out of the UHT range.     

Flory-Huggins' formulation,its validity and implications in retrieval of equilibrium temperature of binary and ternary systems at higher pressures,using liquid silicate data and 1 bar liquidus temperature

For a pure phase at equilibrium with a polycomponent melt, two sets of expressions can be derived; one expressing its activity as a function of enthalpy, entropy, heat capacity and temperature, and the other by coupling a Flory-Huggins' polymerisation model with the van Laar heat of mixing term. Interaction parameters for binary and ternary systems have been computed at 1 bar by equating these two expressions. Assuming the interaction parameter to be independent of temperature, equilibrium temperatures at higher pressures can be calculated by an iterative procedure. Such retrieval calculations were carried out in simple eutectic, volatile-free systems like CaAl₂Si₂O₈-CaMgSi₂O₆, Mg₂SiO₄-TiO₂, MgSiO₃-TiO₂, Mg₂SiO₄-CaMgSi₂O₆, NaAlSi₃O₈-SiO₂ and CaAl₂Si₂O₈-CaMgSi₂O₆-Mg₂SiO₄. The close agreement between the theoretically retrieved and the experimentally determined equilibrium temperatures testifies to the validity of the model at higher pressures. The successful application of the model to simple eutectic, binary and ternary systems involving vastly dissimilar phases without imposing added constraints implies that it can be possibly extended to hitherto unknown systems provided the thermodynamic parameters of the phases involved are known.     

Phase Equilibria in the Ternary Systems KBr–K2B4O7–H2O and KCl–K2B4O7–H2O at 373 K

According to the compositions of the underground gasfield brines in the west of Sichuan Basin,the phase equilibria in the ternary systems KBr-K2B4O7-H2O and KCl-K2B4O7-H2O at 373 K were studied using the isothermal dissolution equilibrium method.The solubilities of salts and the densities of saturated solutions in these ternary systems were determined.Using the experimental data,phase diagrams and density-composition diagrams were constructed.The two phase diagrams were simple co-saturation type,each having an invariant point,two univariant curves and two crystallization regions.The equilibrium solid phases in the ternary system KBr-K2B4O7-H2O are potassium bromide （KBr） and potassium tetraborate tetrahydrate （K2B4O7·4H2O）,and those in the ternary system KCl-K2B4O7-H2O are potassium chloride （KCl） and potassium tetraborate tetrahydrate （K2B4O7·4H2O）.Comparisons of the phase diagrams of the two systems at different temperatures show that there is no change in the crystallization phases,but there are changes in the size of the crystallization regions.As temperature increases,the solubility of K2B4O7·4H2O increases rapidly,so the crystallization field of K2B4O7·4H2O becomes smaller.     

The calculation of mineral compositions and modes of olivine-two pyroxene-spinel assemblages

Duhem's theorem states that the equilibrium position of a system of known bulk composition and size is completely determined by two independent variables. Given these data all other equilibrium parameters can be calculated. The method of calculation is set down in detail for olivine-two pyroxene-spinel assemblages which may serve as models for portions of the upper mantle. To make the calculations, equations for the Gibbs energy of the constituent minerals, olivine, spinel, orthopyroxene and clinopyroxene are formulated. The olivine is treated as an ideal, binary, cationic solid solution. The spinel is treated as a ternary, reciprocal solution between the four end members MgAl₂O₄, MgCr₂O₄, FeAl₂O₄ and FeCr₂O₄. The pyroxenes are treated as seven-component, simplified reciprocal systems. The eight oxides SiO₂, TiO₂, Al₂O₃, Cr₂O₃, FeO, MgO, CaO and Na₂O are thought sufficient for expressing the composition of the model mantle.The paucity and inconsistency of the required thermodynamic data preclude accurate calculations. To illustrate the calculation and to arrive at some highly tentative results, the required free energy data are estimated by combining analytical data from Iherzolite xenoliths with interpretations of experimental data.It seems likely that the Al content of pyroxenes cannot be used as a geobarometer without considering the bulk composition of the assemblage or, equivalently, the composition of the associated spinel phase. Temperature estimates based on the Ca to Ca+Mg ratio in Ca-rich pyroxenes may be approximately correct in the restricted composition range of mantle derived xenoliths.     

A thermodynamic model for Ca–Na clinoamphiboles in Na2O–CaO–FeO–MgO–Al2O3–SiO2–H2O–O for petrological calculations

A calibration is presented for an activity–composition model for amphiboles in the system Na₂O–CaO–FeO–MgO–Al₂O₃–SiO₂–H₂O–O (NCFMASHO), formulated in terms of an independent set of six end‐members: tremolite, tschermakite, pargasite, glaucophane, ferroactinolite and ferritschermakite. The model uses mixing‐on‐sites for the ideal‐mixing activities, and for the activity coefficients, a macroscopic multicomponent van Laar model. This formulation involves 15 pairwise interaction energies and six asymmetry parameters. Calibration of the model is based on the geometrical constraints imposed by the size and shape of amphibole solvi inherent in a data set of 71 coexisting amphibole pairs from rocks, formed over 400–600 °C and 2–18 kbar. The model parameters are calibrated by combining these geometric constraints with qualitative consideration of parameter relationships, given that the data are insufficient to allow all the model parameters to be determined from a regression of the data. Use of coexisting amphiboles means that amphibole activity–composition relationships are calibrated independently of the thermodynamic properties of the end‐members. For practical applications, in geothermobarometry and the calculation of phase diagrams, the amphibole activity–composition relationships are placed in the context of the stability of other minerals by evaluating the properties of the end‐members in the independent set that are in internally consistent data sets. This has been performed using an extended natural data set for hornblende–garnet–plagioclase–quartz, giving the small adjustments necessary to the enthalpies of formation of tschermakite, pargasite and glaucophane for working with the Holland and Powell data set.     

The prediction of mineral solubilities in natural waters: the NaKMgCaClSO4H2O system from zero to high concentration at 25° C

A chemical model of the seawater system, NaKMgCaClSO₄H₂O, is developed for predicting mineral solubilities in brines from zero to high ionic strengths. The calculated solubilities are shown to be in agreement with the experimental data from gypsum saturation (I < 0.06 m) to bischofite saturation (I > 20 m). The model utilizes activity coefficient expressions recently developed by Pitzer and co-workers and an algorithm for rapidly identifying the coexisting phases and their composition at equilibrium. The activity coefficient expressions are parameterized using binary and ternary system solubility and osmotic data. The results indicate that a free energy model defined by binary and ternary system data will accurately predict solubilities in more complex systems. The algorithm for solving the general chemical equilibrium problem is briefly discussed. The method can be used to model systems with an arbitrary number of possible non-ideal solution phases. The iterative procedure is guaranteed to converge and has been found to be efficient and easy to implement.Calculated phase diagrams associated with the seawater system are compared to experimental data. Our calculations are within experimental accuracy whereas the prediction of other seawater models are in substantial disagreement with the data even at low concentration. The calculation of evaporation sequences is also briefly discussed and qualitatively compared to field data. The mineral assemblages predicted by this method are in substantially better agreement with core samples than the sequences predicted by phase diagram methods (Braitsch, 1971), which do not explicitly include the Ca component.     

Equations of state for the NaCl-H2O-CH4 system and the NaCl-H2O-CO2-CH4 system: Phase equilibria and volumetric properties above 573 k

An equation of state (EOS) based on thermodynamic perturbation theory is presented for the NaCl-H₂O-CH₄ system. This equation consistently reproduces PvTX properties and phase equilibria with an accuracy close to that of data in the temperature, pressure and concentration ranges from 648 K to 873 K, 0 to 2500 bar and up to 2.37 mol % NaCl. Good agreement with recent ternary immiscibility data from 673 K to 873 K suggests that the EOS may provide accurate predictions for NaCl concentrations as high as 40 mol %. We could not find any experimental data above 873 K that can be used to validate the predictions of the EOS inside the ternary. However, parameters for the mixed ternary system were established from parameters evaluated for pure and binary systems and accurate combination rules. Therefore, predictions in the ternary should be reliable to the high temperatures and pressures where the EOS for the lower order systems are valid (about 1300 K and 5000 bar). Using the same combining approach, an EOS for the quaternary NaCl-H₂O-CO₂-CH₄ is constructed on the basis of parameters from our earlier model for the NaCl-H₂O-CO₂ system and the present NaCl-H₂O-CH₄ model. This suggests that predictions of the quaternary EOS are reliable also to about 1300 K and 5000 bar.     

The accuracy of standard enthalpies and entropies for phases of petrological interest derived from density-functional calculations

The internal energies and entropies of 21 well-known minerals were calculated using the density functional theory (DFT), viz. kyanite, sillimanite, andalusite, albite, microcline, forsterite, fayalite, diopside, jadeite, hedenbergite, pyrope, grossular, talc, pyrophyllite, phlogopite, annite, muscovite, brucite, portlandite, tremolite, and CaTiO₃–perovskite. These thermodynamic quantities were then transformed into standard enthalpies of formation from the elements and standard entropies enabling a direct comparison with tabulated values. The deviations from reference enthalpy and entropy values are in the order of several kJ/mol and several J/mol/K, respectively, from which the former is more relevant. In the case of phase transitions, the DFT-computed thermodynamic data of involved phases turned out to be accurate and using them in phase diagram calculations yields reasonable results. This is shown for the Al₂SiO₅ polymorphs. The DFT-based phase boundaries are comparable to those derived from internally consistent thermodynamic data sets. They even suggest an improvement, because they agree with petrological observations concerning the coexistence of kyanite?+?quartz?+?corundum in high-grade metamorphic rocks, which are not reproduced correctly using internally consistent data sets. The DFT-derived thermodynamic data are also accurate enough for computing the P–T positions of reactions that are characterized by relatively large reaction enthalpies (>?100 kJ/mol), i.e., dehydration reactions. For reactions with small reaction enthalpies (a few kJ/mol), the DFT errors are too large. They, however, are still far better than enthalpy and entropy values obtained from estimation methods.     

The SiO2 polymorphs: The equations of state and thermodynamic properties of phase transformations

Thermal equations of state have been derived for polymorphic forms of SiO₂ and values of \(\mathop \smallint \limits_0^P\) V _T dP have been tabulated. Available experimental data on the phase equilibria at high pressures and temperatures have been used with these equations to calculate the standard thermodynamic functions for the α-quartz-coesite and coesite-stishovite transformations. A study of sensitivity of calculated thermodynamical properties to uncertainties of phase equilibria data, initial data for elastic constants and equations of state has been carried out. The discrepancies between standard thermodynamic properties of these transformations calculated from phase equilibria data and solution calorimetry data still persist.     

CONTRIBUTIONS TO RESEARCH ON QUARTZ-FORMING SYSTEMS

Investigation of quartz inclusions has yielded data on concentrations, temperatures, and pressures of mineral-forming systems; important in application to problems of mineral genesis and prospecting criteria. An integrated method involving preliminary microscopic examination of inclusions, analyses of aqueous extracts, and spectrographic analyses of powdered material, both derived from inclusions. Solutions available only in small quantity or of high dilution were subjected to optical-spectrographic and microturbidimetric, qualitative and quantitative analyses. Chemically, the solutions showed alkaline reaction; . Na, K, Ca and Cl were prominent. In secondary inclusions cations Na⁺, K⁺, Ca²⁺, Mg²⁺, Al³⁺, Fe, and Mn with anions Cl^-SO₄ ^2- and B0₃ ^3- were present in dissolved state. The presence of boron was noted for the first time in analysis, although it is known to be characteristic in quartz forming solutions. The concentration calculated for dissolved salts ranged from 30 to 35 percent. Transfer of Si appears to occur through metasilicate anions, SiO^3- and Si₂0₅; OH^- is present along with Si0₂ resulting evidently from hydrolysis of SiO₃ ^2-.

PT diagrams of binary systems are used to gage the extent of mineral-formation possibilities. Three-component fields indicate: phase composition of saturated solutions, saturated solutions with gaseous phase, and saturated gaseous solutions. Recurrent metastability phenomena are discussed with respect to their modification of theoretical phase systems. Use of VT diagrams implements calculation of homogenization temperature for included phases, by examination of volume-ratio curves for liquid and gaseous phases during heat application. Multicomponent-system PT diagrams are derived from those describing simpler systems they are a resultant of the phase interrelationships under consideration. -- D. D. Fisher.     

Thermodynamic characterization of boltwoodite and uranophane: Enthalpy of formation and aqueous solubility

Boltwoodite and uranophane are uranyl silicates common in oxidized zones of uranium ore deposits. An understanding of processes that impact uranium transport in the environment, especially pertaining to the distribution of uranium between solid phases and aqueous solutions, ultimately requires determination of thermodynamic parameters for such crystalline materials. We measured formation enthalpies of synthetic boltwoodites, K(UO₂)(HSiO₄)·H₂O and Na(UO₂)(HSiO₄)·H₂O, and uranophane, Ca(UO₂)₂(HSiO₄)₂·5H₂O, by high temperature oxide melt solution calorimetry. We also studied the aqueous solubility of these phases from both saturated and undersaturated conditions at a variety of pH. The combined data permit the determination of standard enthalpies, entropies and Gibbs free energies of formation for each phase and analysis of its potential geological impact from a thermodynamic point of view.     

Effects of the degree of polymerization on the structure of sodium silicate and aluminosilicate glasses and melts: An O NMR study

Revealing the atomic structure and disorder in oxide glasses, including sodium silicates and aluminosilicates, with varying degrees of polymerization, is a challenging problem in high-temperature geochemistry as well as glass science. Here, we report ¹⁷O MAS and 3QMAS NMR spectra for binary sodium silicate and ternary sodium aluminosilicate glasses with varying degrees of polymerization (Na₂O/SiO₂ ratio and Na₂O/Al₂O₃ ratio), revealing in detail the extent of disorder (network connectivity and topological disorder) and variations of NMR parameters with the glass composition. In binary sodium silicate glasses [Na₂O-k(SiO₂)], the fraction of non-bridging oxygens (NBOs, Na-O-Si) increases with the Na₂O/SiO₂ ratio (k), as predicted from the composition. The ¹⁷O isotropic chemical shifts (¹⁷O δ_iso) for both bridging oxygen (BO) and NBO increase by about 10-15 ppm with the SiO₂ content (for k = 1-3). The quadrupolar coupling products of BOs and NBOs also increase with the SiO₂ content. These trends suggest that both NBOs and BOs strongly interact with Na; therefore, the Na distributions around BOs and NBOs are likely to be relatively homogenous for the glass compositions studied here, placing some qualitative limits on the extent of segregation of alkali channels from silica-enriched regions as suggested by modified random-network models. The peak width (in the isotropic dimension) and thus bond angle and length distributions of Si-O-Si and Na-O-Si increase with the SiO₂ content, indicating an increase in the topological disorder with the degree of polymerization. In the ternary aluminosilicate glasses [Na₂O]_x[Al₂O₃]_1−xSiO₂, the NBO fraction decreases while the Al-O-Si and Al-O-Al fractions apparently increase with increasing Al₂O₃ content. The variation of oxygen cluster populations suggests that deviation from “Al avoidance” is more apparent near the charge-balanced join (Na/Al = 1). The Si-O-Si fraction, which is closely related to the activity coefficient of silica, would decrease with increasing Al₂O₃ content at a constant mole fraction of SiO₂. Therefore, the activity of silica may decrease from depolymerized binary silicates to fully polymerized sodium aluminosilicate glasses at a constant mole fraction of SiO₂.