Standard State Chemical Potential Research Articles

Gibbs Free-Energy-Based Objective Function for Electrolyte Activity Coefficient Models

This paper proposes Gibbs free-energy-based objective functions in the parameter fitting of activity coefficient and specific heat capacity of ions. The activity coefficient parameters are fitted through the averaged squared error between the Gibbs free energy calculated by using the measured activity coefficient data and that by using the model equation. The standard-state heat capacity parameters of ions are fitted through the minimization of the average squared error between the Gibbs free energy of dissolution calculated through the saturation activity over a temperature range and that calculated through the standard-state chemical potential as a function of temperature via standard-state specific heat. This methodology is tested with Bromley and Pitzer models. The proposed methodology reduces the need for experiments and avoids the uncertainty of extrapolation to infinite dilution when determining standard-state specific heat of ions. The proposed methodology provides solubility estimates that are mo...

Estimating the solubility of 1:1 electrolyte aqueous solutions: the chemical potential difference rule

ABSTRACTA simple empirical rule to estimate the solubility of force fields of 1:1 aqueous electrolyte solutions at ambient conditions is proposed. The empirical prescription states that the logarithm of the solubility can be described by a second-order polynomial of the chemical potential difference of the salt in the solid and the salt in the standard state in solution. The rule will be denoted as the chemical potential difference rule . It is shown that the recipe is able to provide reasonable values of the solubility of 1:1 aqueous electrolytes (having the NaCl structure in the solid phase) for a number of different force fields for which the solubility has been computed in a rigorous way. This clearly indicates that reproducing only the experimental values of the free energy of hydration of ions at infinite dilution (which yield the standard state chemical potential of the salt in water) is not enough to foresee the experimental values of the solubility. The difference between the chemical potential of the salt in the solid phase and in the standard state seems to be the variable that controls the value of the solubility. This finding should be taken into account in the future when developing force fields for 1:1 electrolytes in water aimed at reproducing the experimental solubilities.

Thermodynamics of the MgO‐FeO‐SiO2system up to 140 GPa: Application to the crystallization of Earth's magma ocean

AbstractAt the end of Earth's accretion and after the core‐mantle segregation, the existence of a basal magma ocean at the top of the core‐mantle boundary (CMB) depends on the physical properties of mantle materials at relevant pressure and temperature. Present‐day deep mantle structures such as ultralow‐velocity zones and low‐shear velocity provinces might be directly linked to the still ongoing crystallization of a primordial magma ocean. We provide the first steps toward a self‐consistent thermodynamic model of magma ocean crystallization at high pressure. We build a solid‐liquid thermodynamic database for silicates in the MgO‐FeO‐SiO2system from 20 GPa to 140 GPa. We use already published chemical potentials for solids, liquid MgO, and SiO2. We derive standard state chemical potential for liquid FeO and mixing relations from various indirect observations. Using this database, we compute the ternary phase diagram in the MgO‐FeO‐SiO2system as a function of temperature and pressure. We confirm that the melt is lighter than the solid of same composition for all mantle conditions but at thermodynamic equilibrium, the iron‐rich liquid is denser than the solid in the deep mantle. We compute a whole fractional crystallization sequence of the mantle and show that an iron‐rich and fusible layer should be left above the CMB at the end of the crystallization.

Chemical Potentials, Activity Coefficients, and Solubility in Aqueous NaCl Solutions: Prediction by Polarizable Force Fields.

We describe a computationally efficient molecular simulation methodology for calculating the concentration dependence of the chemical potentials of both solute and solvent in aqueous electrolyte solutions, based on simulations of the salt chemical potential alone. We use our approach to study the predictions for aqueous NaCl solutions at ambient conditions of these properties by the recently developed polarizable force fields (FFs) AH/BK3 of Kiss and Baranyai (J. Chem. Phys. 2013, 138, 204507) and AH/SWM4-DP of Lamoureux and Roux (J. Phys. Chem. B 2006, 110, 3308 - 3322) and by the nonpolarizable JC FF of Joung and Cheatham tailored to SPC/E water (J. Phys. Chem. B 2008, 112, 9020 - 9041). We also consider their predictions of the concentration dependence of the electrolyte activity coefficient, the crystalline solid chemical potential, the electrolyte solubility, and the solution specific volume. We first highlight the disagreement in the literature concerning calculations of solubility by means of molecular simulation in the case of the JC FF and provide strong evidence of the correctness of our methodology based on recent independently obtained results for this important test case. We then compare the predictions of the three FFs with each other and with experiment and draw conclusions concerning their relative merits, with particular emphasis on the salt chemical potential and activity coefficient vs concentration curves and their derivatives. The latter curves have only previously been available from Kirkwood-Buff integrals, which require approximate numerical integrations over system pair correlation functions at each concentration. Unlike the case of the other FFs, the AH/BK3 curves are nearly parallel to the corresponding experimental curves at moderate and higher concentrations. This leads to an excellent prediction of the water chemical potential via the Gibbs-Duhem equation and enables the activity coefficient curve to be brought into excellent agreement with experiment by incorporating an appropriate value of the standard state chemical potential in the Henry Law convention.

Thermodynamic Modeling of Fluid-Rock Interaction at Mid-Crustal to Upper-Mantle Conditions

Fluid-rock interaction is important in a wide range of settings in the middle crust to the upper mantle. Fluids are liberated in the crust during prograde metamorphism in convergent-margin settings. At greater depth, devolatilization of subducting lithosphere plays a major role in global element cycling, metasomatic alteration of the mantle, and the genesis of arc magmas. Mafic magmas produced in these settings may stall in the lower crust and liberate volatiles that metasomatize surrounding rocks and trigger production of silicic magmas of crustal derivation. Mounting evidence points to an important role for metasomatic alteration by saline brines in the genesis of some lower-crustal granulite-facies metamorphic terranes. Unlike shallow geothermal systems or low-grade metamorphic environments, there has been only limited progress in modeling the fluid-rock interaction that characterizes the deeper geologic systems enumerated above. Such phase-equilibrium models require as input the chemical potential, μ i μ i ≡ ( ∂ G i ∂ n i ) P , T (1) of the participating phases or species i , where G i is the Gibbs free energy of i , n i is the number of moles of i , P is pressure, and T is temperature. At the P and T of interest, μ i = μ i ∘ + R T l n a i (2) Where μ° is standard state chemical potential, a is activity, R is the gas constant, and T is absolute temperature. It is challenging to quantify the right-hand terms in Equation (2) at P and T ranging from the middle crust to the upper mantle. This is chiefly because of (1) limitations in our quantitative knowledge of key properties of relevant solutes and H2O, and (2) uncertainty in how properly to model solute activities at the requisite conditions. As a consequence, aqueous geochemists have been discouraged from plying their trade in the numerous deep geologic systems in which fluid-rock interaction may play a major role in the Earth’s chemical and physical evolution. …

Modelo teórico para el cálculo de las solubilidades mutuas entre gases no-polares y agua con sales disueltas

We propose a thermodynamic model for the equilibrium of non-polar gases, water and aqueous sodium chloride solutions valid to high pressures, high temperatures, and to high ionic strength solutions. The model couples Henry's Law and an equation of state to reproduce ex- perimental data in the aqueous phase and the gas-rich fluid phases. The standard state chemical potential of the solute in brines is related to the standard state chemical potential of the solute in pure water through a salting-out or Setschenov coefficient. The salting-out coefficients are obtained from experimental data. The model reproduces all crucial phenomena, including three phase equilibrium, at temperatures below the critical temperature of water. The modeling approach cannot account for gas-gas equilibrium phenomena that occur in the vicinity of the water critical point, nor for the hydrate-vapor-liquid equilibrium near the melting point of water.

DES-code: A metacode to aid calculation of the chemical potential of aqueous solutions at elevated temperatures and pressures

The DH-ASF (Debye–Hückel-asymmetric formalism) model is a recently developed activity-composition model that can be used to calculate the thermodynamic effects of mixing in strong electrolyte and mixed solvent supercritical solutions at high pressures ( > 3 kbar ) and temperatures ( > 400 ∘ C ). The model uses a mole fraction concentration scale, and calculations are based on a pre-defined independent set of end-members that define speciation within the solution. This differs from the conventional use of molal concentration scales with sets of end-members that define the composition of the solution, but not the speciation (apparent end-members). This work presents DES (dual end-member sets)-code, a code that takes a conventional molal scale description of solution composition and implements the DH-ASF model for that solution. The code converts between apparent and independent end-member sets, and calculates standard state chemical potentials, ideal activities and activity coefficients using the DH-ASF model for molal and mole fraction concentration scales. The code runs in Mathematica TM 4.1 onwards, but it is written in a general meta-code form so that it can be implemented on a variety of platforms. Inputs to the code can be made manually, read from an auxiliary file, or presented to the input modules as passed variables. The code is provided with thermodynamic data from the Holland and Powell data set, but can be used with any data specified by the user. Outputs are designed to be modified by the user. Calculations on the systems NaCl– H 2 O , NaCl– CaCl 2 – H 2 O and NaCl– CO 2 – H 2 O are used to demonstrate the utility of the DES-code. Calculations predict that ion association increases with increasing temperature and concentration of salt and CO 2 , and with decreasing pressure. This is consistent with experimental observation and the results of molecular simulations. The DES-code is suitable for use as it stands, or for modification and incorporation into existing or new Gibbs energy minimisation or equilibrium solving thermodynamic codes.

The Ionization Constant of Water over Wide Ranges of Temperature and Density

A semitheoretical approach for the ionization constant of water, KW, is used to fit the available experimental data over wide ranges of density and temperature. Statistical thermodynamics is employed to formulate a number of contributions to the standard state chemical potential of the ionic hydration process. A sorption model is developed for calculating the inner-shell term, which accounts for the ion–water interactions in the immediate ion vicinity. A new analytical expression is derived using the Bragg–Williams approximation that reproduces the dependence of a mean ion solvation number on the solvent chemical potential. The proposed model was found to be correct at the zero-density limit. The final formulation has a simple analytical form, includes seven adjustable parameters, and provides good fitting of the collected KW data, within experimental uncertainties, for a temperature range of 0–800 °C and densities of 0–1.2 g cm−3.

Formation of Spherical Micelles in a supercritical Solvent: Lattice Monte Carlo Simulation and Multicomponent Solution Model

We modify Larson's lattice model [Larson, R.G., Scriven, L.E. and Davis, H.T. (1985). J. Chem. Phys. , 83 , 2411-2420] and use it to study formation of spherical micelles in a supercritical solvent by large-scale Monte Carlo (MC) simulations and by the multicomponent solution model. Carbon dioxide and perfluoroalkylpoly(ethylene oxide) serve as prototypes for the solvent and surfactant, respectively. Larson-model type parameters for carbon dioxide and perfluoroalkylpoly(ethylene oxide) are obtained using experimental values of critical parameters and solubility along with a modified Berthelot combining rule. We perform canonical MC simulations at a supercritical temperature and low surfactant concentrations, varying the number of surfactant head and tail segments and the solvent density. Various properties such as the critical micelle concentration, the aggregate size distribution and the size of the micelles is evaluated. The multicomponent solution model and the simulation results for the aggregate size distribution are then combined to determine the standard state chemical potential for the spherical micelles and the intermicellar interaction; we present a novel approach to model this standard state chemical potential. The implications of these results for the thermodynamics of the formation of the spherical micelles in supercritical solvents are explored.

Thermodynamic Description of Geochemical Important Aqueous Non Electrolytes in the Wide Range of State Parameters

The virial equation approach for description the thermodynamic properties of aqueous non electrolytes is proposed. It is based on the precise equation of state for the solvent (H20) given by Hill (1990) and requires only three empirical parameters that are independent on temperature and pressure. The standard state chemical potential of the dissolved species at given pressure P and temperature T can be expressed as: p~ = p~ + RT(-ln(55.51) + ln~l + 2p~ where g~ is the chemical potential of the associated pure gaseous component at standard pressure (1 bar), R is the gaseous constant, ~11 and

Structural interactions in the wetting and spreading of van der Waals fluids

The structural interactions for the submonolayer of a van der Waals (vdW) fluid have been modeled with the two-dimensional vdW equation of state (Hill-deBoer isotherm). The cohesion and co-area parameters can be estimated from the critical temperature and pressure of the bulk fluid. The difference in the standard state chemical potentials is estimated from the substrate/ fluid/vapor Hamaker constant calculated with the Lifshitz theory and an interaction distance of a single molecule with the substrate. For the systems studied, this interaction distance was approximately equal to the distance from a flat substrate to the middle of a molecule lying flat on the substrate. The resulting adsorption or disjoining pressure isotherm is assumed to describe the submonolayer part of the isotherm until it intersects the isotherm predicted from the Hamaker-Lifshitz theory. The composite isotherm is used to predict the equilibrium film thickness, film pressure, initial spreading coefficient, equilibrium spreading coefficient, and contact angle. Good agreement was observed for all systems with polytetrafluoroethylene (PTFE) as the substrate and for fluids which do not hydrogen-bond with water as the substrate. This agreement gives confidence that this model represents the structural interations of fluids that have only vdW interactions. It is possible to model the interaction of a monopolar liquid with a polar substrate. The acid-base component of the interaction energy from the van Oss, Chaudhury, Good theory was added to the model. With this addition, the model can predict the film pressure and spreading of benzene and chloroform on water.

Heat capacity and other thermodynamic properties of Na 2SO 4(aq) in hydrothermal solutions and the solubilities of sodium sulfate minerals in the system Na-Cl-SO 4-OH-H 2O to 300°C

Heat capacities of Na 2SO 4(aq) solutions have been measured from 140°–300°C at 200 bars using a flowcalorimeter over the molality range of 0.05–1.5 mol·kg −1. Using the ion-interaction or virial coefficient approach developed by Pitzer (1973, 1979, 1987) and coworkers, and approximating the pressure-dependencies of the various Na 2SO 4(aq) thermodynamic quantities with those of NaCl(aq) calculated from the equations of Rogers and Pitzer (1982), our measured heat capacities were combined with literature values on heat capacities, enthalpies, and osmotic coefficients at temperatures to 225°C and at pressures mostly at 1 bar or vapor-saturation pressure to yield a comprehensive set of equations for the thermodynamic properties of Na 2SO 4(aq) at temperatures 25°–300°C, pressure to at least 200 bars, and molalities to 3.0 mol·kg −1. Standard state chemical potentials and activity and osmotic coefficients of Na 2SO 4(aq) derived from our model, together with standard state properties of the solids available in the literature, permit the calculation of sodium sulfate solubilities in water and in electrolyte mixtures to 300°C. Good agreement between experimental and predicted solubilities in water indicate that the ion-interaction model can be used successfully to predict mineral-solution equilibria to 300°C without an explicit accounting for ion-pairs, and demonstrates that heat capacity measurements can be used to obtain reliable high-temperature and high-pressure activity properties of electrolyte solutions. The binary and ternary mixing parameters θ ü and Ψ ük are required by the ion-interaction model for calculations for multicomponent mixtures. It was found sufficient to adopt previously determined values for θ ü at 25°C without temperature dependence and, from the solubility data, to determine temperature-dependent Ψ ük functions.

Transfer chemical potentials for ions, solubilities of salts and kinetics of reactions involving inorganic complex ions at ambient pressure and 298.2 K in binary aqueous mixtures containing ethanol and propan-2-ol

Solubilities of several complex salts at ambient temperature and pressure are reported for aqueous solutions and for solutions in binary aqueous organic mixtures. The organic cosolvents are ethanol and propan-2-ol. These and earlier published solubilities are analysed using the TATB assumption which sets the transfer chemical potentials of Ph4As+ ions equal to those of Ph4B– ions. The calculations yield transfer parameters for various ions including complex metal ions. Derived transfer parameters for ions are compared for a range of solvent systems comprising binary aqueous mixtures, including those systems where the organic cosolvents are methanol, ethanol, propan-2-ol and acetone. These transfer parameters are used in an analysis of kinetic data describing chemical reaction between hydroxide ions and iron(II) complex cations, [Fe(phen)3]2+ in aqueous solutions. Definitions of standard state chemical potentials for solutes and solvents are considered with reference to descriptions of composition of these systems using the mole fraction scale.

The Polymer Model of Thermochemical Clay Mineral Stability

The Nriagu polymer model of 2:1 layer type clay minerals develops from the premise that clay minerals are condensation copolymers of solid hydroxides. In the Mattigod-Sposito formulation of the model, standard state chemical potentials (standard Gibbs energies of formation from the elements) of 2:1 clay minerals are predicted quantitatively with a linear correlation equation relating the standard Gibbs energy of the polymerization reaction (ΔGor) to the half-cell layer charge of the clay mineral and to the valence and ionic radius of the exchangeable cation. It is now shown that this correlation equation can be derived from two basic assumptions: (1) that the standard Gibbs energy change for the transfer of a cation in a pure hydroxide solid to a hydroxide component in the tetrahedral or octahedral sheet of a 2:1 clay mineral is independent of the nature of the cation and (2) that the difference between ΔGor for the polymerization reaction to form a 2:1 clay mineral and ΔGor for the same reaction to form the zero layer-charge analog of the clay mineral is proportional to the number of interlayer exchangeable cations per unit cell of the clay mineral and to the radius of its exchangeable cation. Both of these assumptions can be tested experimentally, independent of the polymer model.

A chlorite solid solution geothermometer the Los Azufres (Mexico) geothermal system

Chlorite constitutes a major hydrothermal alteration product of metamorphism of andesites, in the active geothermal system of Los Azufres (Mexico). Electron microprobe analyses performed on a set of crystals from each sample show wide variations in composition. Correlation coefficients among chemical constituents were calculated. It is shown that the tetrahedral charge is positively correlated with the octahedral vacancy and negatively with the iron content, and there is almost no correlation with the octahedral aluminium and magnesium content. A procedure is proposed to select end-members and substitution vectors, and to give a general formula for these chlorites. Their formation temperatures are estimated with great accuracy, combining results of microthermometric data on fluid inclusions from gangue minerals of chlorites (quartz, calcite), direct measurements in wells (Kuster equipment), and chemical geothermometers. Correlations between chlorite compositions, range and nature of site occupancy, and temperature are good. Formation temperatures of chlorites range from 130° C to 300° C. As no other thermodynamic parameter varies significantly in the studied field (composition of the host rocks, nature of the geothermal fluids, pressure, ...), these variations of site occupancy (mainly Al(IV) and the octahedral occupancy (6-Al(VI)-(Mg+Fe(2+)) = VAC) are considered mainly as temperature dependent. Molar fractions of each end-member show very different variations with increasing temperature: X-kaolinite decreases, and X-chamosite increases, while X-talc-3 brucite does not show significant change. From these data, activity coefficients and standard state chemical potential of major components, and molar free energy formation of chlorite have been calculated for each temperature of crystallisation.

Regular Solution Site-Mixing Model for Chlorites

AbstractActivity expressions are presented for a six end-member, regular solution, site-mixing model for chlorites. The end members are ideal 14-Å chlorites for which experimental stability data are lacking. Estimates were made of the standard state 25°C and 1 bar molar 3rd law entropies, volumes, and the Maier-Kelley heat capacity coefficients. Experimental stability data from the literature for 14-Å chlorites were used with different sets of exchange energies, for cations on adjacent sites, to compute estimates of the standard state chemical potentials of the end members at 25°C and 1 bar. More experimental data are needed for an adequate definition of the exchange energies and the standard state chemical potentials. The model was applied to diagenesis in clastic reservoirs. Aqueous activity ratios of Mg2+:Fe2+ were computed as an equilibrium function of the corresponding molar ratios in authigenic chlorites. The aqueous activity ratio was independent of the chlorite Al content at a constant molar ratio of Mg2+:Fe2+ in the chlorite. The model predicts a wide range of molar Mg2+:Fe2+ ratios in authigenic chlorites in equilibrium with reservoir fluids. Trends in these molar ratios should be independent of the Al content in the chlorites. The model can be applied directly to 7-Å chlorites when experimental data become available to estimate the various thermodynamic parameters of the 7-Å end members.

The Solubility of Electrolytes in Homogeneous Mixtures of Water-Miscible Organic Solvent (MOS)

The solubility of several electrolytes in homogeneous mixtures of water-miscible organic solvent (acetone, methanol, tetrahydrofuran, dioxane and acetonitrile) was found to obey the equation S = Sw exp (- λf) where Sw and S are the solubilities in water and the mixture respectively, and f is the volume fraction (or mole fraction) of the organic solvent. This dependence can be explained assuming the additivity of the standard state chemical potential of the electrolyte in the mixture, i.e. μ = f · μ + (1 - f) μ0.

A simple thermodynamic model for estimating the solubility of H2O in magmas

A symmetrical, strictly regular solution model is used to estimate H2O solubilities in silicate melts. The standard state chemical potential of dissolved H2O and the adjustable parameter in the activity coefficient are determined by least squares analyses of data on H2O solubility in silicate melts. The adjustable parameter in the expression for the activity coefficient (Inγ) is a function only of the anhydrous melt composition and eleven values are provided for melts ranging in composition from picrite to rhyolite. At the 95% confidence level, the model should estimate H2O contents to within 4.8% of the amount present if the amount present is less than 10 wt.%. This compares to the reproducibility of 2.25% of the amount present for experimental determinations. To apply the model to rocks and magmas estimates ofT, P, and the fugacity of H2O are required. Variation of the H2O content of the melt changes the activity of other components. Knowledge of this variation removes the requirement that the fugacity of H2O be estimated. Application of the properties of exact differentials to the Gibbs function for the hydrous melt provides an expression relating the chemical potential of a feldspar component to the H2O content of the melt. This expression contains a second adjustable parameter which depends on the anhydrous melt composition. Using this second expression, the H2O content can be estimated ifT, P, and feldspar composition are known. Data are too meagre to evaluate the quantitative success of the second method.