Calculation Of Activity Coefficients Research Articles

The Examination of the Activity Coefficients of Cu(II) Complexes with OH− and Cl− in NaClO4 Using Pitzer Equations: Application to Other Divalent Cations

The stability constants for the hydrolysis of Cu(II) and formation of chloride complexes in NaClO4 solution, at 25 °C, have been examined using the Pitzer equations. The calculated activity coefficients of CuOH+, Cu(OH)2, Cu2(OH)3+, Cu2(OH) 2 2+ , CuCl+ and CuCl2 have been used to determine the Pitzer parameter (β (0) , β (1) , and C i ) for these complexes. These parameters yield values for the hydrolysis constants (log 10 β 1 * , log 10 β 2 * , log 10 β 2,1 * and log 10 β 2,2 * ) and the formation of the chloride complexes (log 10 β CuCl * and $\log_{10}\beta_{\mathrm{CuCl}_{2}}^{\mathrm{*}})$ that agree with the experimental measurements, respectively to ±0.01,±0.02,±0.03,±0.06,±0.03 and ±0.07. The stability constants for the hydrolysis and chloride complexes of Cu(II) were found to be related to those of other divalent metals over a wide range of ionic strength. This has allowed us to use the calculated Pitzer parameters for copper complexes to model the stability constants and activity coefficients of hydroxide and chloride complexes of other divalent metals. The applicability of the Pitzer Cu(II) model to the ionic strength dependence of hydrolysis of zinc and cadmium is presented. The resulting thermodynamic hydroxide and chloride constants for zinc are $\log_{10}\beta_{\mathrm{ZnOH}^{+}}=-9.04\pm0.04$ and $\log_{10}\beta_{\mathrm{Zn(OH)}_{2}}=-16.90\pm0.02$ . For cadmium the thermodynamic hydrolysis constants are $\log_{10}\beta_{\mathrm{CdOH}^{+}}=-10.24\pm0.05$ and $\log_{10}\beta_{\mathrm{Cd(OH)}_{2}}=-20.42\pm0.07$ . The Cu(II) model allows one to determine the stability of other divalent metal complexes over a wide range of concentration when little experimental data are available. More reliable stepwise stability constants for divalent metals are needed to test the linearity found for the chloro complexes.

Effects of uncertainties in the thermodynamic properties of aerosol components in an air quality model – Part 1: Treatment of inorganic electrolytes and organic compounds in the condensed phase

Abstract. Air quality models that generate the concentrations of semi-volatile and other condensable organic compounds using an explicit reaction mechanism require estimates of the physical and thermodynamic properties of the compounds that affect gas/aerosol partitioning: vapour pressure (as a subcooled liquid), and activity coefficients in the aerosol phase. The model of Griffin, Kleeman and co-workers (e.g., Griffin et al., 2003; Kleeman et al., 1999) assumes that aerosol particles consist of an aqueous phase, containing inorganic electrolytes and soluble organic compounds, and a hydrophobic phase containing mainly primary hydrocarbon material. Thirty eight semi-volatile reaction products are grouped into ten surrogate species which partition between the gas phase and both phases in the aerosol. Activity coefficients of the organic compounds are calculated using UNIFAC. In a companion paper (Clegg et al., 2008) we examine the likely uncertainties in the vapour pressures of the semi-volatile compounds and their effects on partitioning over a range of atmospheric relative humidities. In this work a simulation for the South Coast Air Basin surrounding Los Angeles, using lower vapour pressures of the semi-volatile surrogate compounds consistent with estimated uncertainties in the boiling points on which they are based, yields a doubling of the predicted 24-h average secondary organic aerosol concentrations. The dependency of organic compound partitioning on the treatment of inorganic electrolytes in the air quality model, and the performance of this component of the model, are determined by analysing the results of a trajectory calculation using an extended version of the Aerosol Inorganics Model of Wexler and Clegg (2002). Simplifications are identified where substantial efficiency gains can be made, principally: the omission of dissociation of the organic acid surrogates; restriction of aerosol organic compounds to one of the two phases (aqueous or hydrophobic) where equilibrium calculations suggest partitioning strongly in either direction; a single calculation of activity coefficients of the organic compounds for simulations where they are determined by the presence of one component at high concentration in either phase (i.e., water in the aqueous phase, or a hydrocarbon surrogate compound P8 in the hydrophobic phase) and are therefore almost invariant. The implications of the results for the development of aerosol models are discussed.

Water Activity in Aqueous Amino Acid Solutions, with and without KCl, at 298.15 K

Using a water activity instrument meter, water activity in aqueous solutions of dl-alanine, glycine, or l-serine, with potassium chloride, molality ranging from 0.0 to 3.0, has been measured at 298.15 K. The reliability of the method was checked comparing the experimental data with literature values. The method proved to be accurate, and the water activities measured for water + amino acid systems are reproducible when compared to the data reported using the isopiestic method. Additionally, a simple theoretical approach applied to those binary systems enabled the calculation of unsymmetric molal amino acid activity coefficients in high agreement with the values found using the isopiestic measurements. Finally, the usefulness of the ternary data to extend the capabilities of thermodynamic models to higher salt and amino acid concentrations was briefly discussed.

Properties of electrolyte solutions relevant to high concentration chloride leaching I. Mixed aqueous solutions of hydrochloric acid and magnesium chloride

A computer code based on a Pitzer model has been developed for the comprehensive prediction of the thermodynamic properties of MgCl2–HCl aqueous solutions over a wide range of conditions from 25 to 120 °C and from 0–350 g L−1 chloride. This code was applied to the calculation of (i) water activities and mean ionic activity coefficients for mixed aqueous solutions of hydrochloric acid and magnesium chloride over a wide range of concentrations and to 100 °C, (ii) solubilities of MgCl2·6H2O in MgCl2–HCl solutions to 80 °C, (iii) partial pressures of HCl(g) and H2O(g) over MgCl2–HCl aqueous solutions at various temperatures and (iv) partial pressures of HCl(g) at the normal boiling point of these mixed electrolyte solutions. The model predictions are in excellent agreement with available experimental data and confirm evidence from the literature that MgCl2(aq) and HCl(aq) mix almost ideally at constant water activity.

ISORROPIA II: a computationally efficient thermodynamic equilibrium model for K<sup>+</sup>–Ca<sup>2+</sup>–Mg<sup>2+</sup>–NH<sub>4</sub><sup>+</sup>–Na<sup>+</sup>–SO<sub>4</sub><sup>2−</sup>–NO<sub>3</sub><sup>−</sup>–Cl<sup>−</sup>–H<sub>2</sub>O aerosols

Abstract. This study presents ISORROPIA II, a thermodynamic equilibrium model for the K+–Ca2+–Mg2+–NH4+–Na+–SO42−–NO3−–Cl−–H2O aerosol system. A comprehensive evaluation of its performance is conducted against water uptake measurements for laboratory aerosol and predictions of the SCAPE2 thermodynamic module over a wide range of atmospherically relevant conditions. The two models agree well, to within 13% for aerosol water content and total PM mass, 16% for aerosol nitrate and 6% for aerosol chloride and ammonium. Largest discrepancies were found under conditions of low RH, primarily from differences in the treatment of water uptake and solid state composition. In terms of computational speed, ISORROPIA II was more than an order of magnitude faster than SCAPE2, with robust and rapid convergence under all conditions. The addition of crustal species does not slow down the thermodynamic calculations (compared to the older ISORROPIA code) because of optimizations in the activity coefficient calculation algorithm. Based on its computational rigor and performance, ISORROPIA II appears to be a highly attractive alternative for use in large scale air quality and atmospheric transport models.

The speciation equilibrium coupling with phase equilibrium in the H 2O–CO 2–NaCl system from 0 to 250 °C, from 0 to 1000 bar, and from 0 to 5 molality of NaCl

A model is developed for the calculation of the speciation equilibrium of H +, OH −, HCO 3 −, CO 3 2−, and CO 2(aq) in aqueous solutions coupled with liquid-vapor phase equilibrium in the CO 2–H 2O–NaCl system from 273 to 523 K, from 0 to 1000 bar (possibly to 2000 bar) , and from 0 to 5.0 molality ( m) of NaCl. The first and second ionization constants of carbonic acid, pH values in aqueous solutions, and their dependence on temperature ( T), pressure ( P) and NaCl concentrations can be calculated with accuracy close to those of experiments in the stated T– P– m range. Using the specific ion-interaction equations of Pitzer to calculate activity coefficients, the model is also able to accurately predict the difference between dissociation constants and apparent dissociation constants, and the concentrations of H +, OH −, HCO 3 −, CO 3 2−, and CO 2(aq) at a given T– P– m condition. Compared to previous models, this model covers larger T– P– m space and much more closely reproduces experimental results, especially in the elevated pressure region. Even without fitting to the experimental data for pH values, dissociation constants and apparent dissociation constants at pressures above the saturation pressures of pure water, this model can accurately predicted these properties up to high pressures. A computer program based on this model is developed and online calculation is made available at: www.geochem-model.org/models.htm.

Relationships between hydration number, water activity and density of aqueous sugar solutions

Hydration numbers ( n h) of simple sugars have been investigated for decades using thermodynamic, spectroscopic as well as molecular modelling techniques. Results were shown to depend on the technique employed. The most reliable values only concern the first hydration shell assuming a maximum oxygen–oxygen distance below 2.8 Å. As concentration increases, sugar–sugar interactions become preponderant and n h decreases. Assuming that no long range structuring effect is exerted by the solute on water, it is possible to estimate the volume occupied by each of hydration water (with nearly 9% volume contraction) and bulk water from density measurements. Likewise, the volume occupied by non-hydrated sugar molecules in the aqueous medium allows finding for sugar density in the aqueous medium a value comparable to that of solid crystalline form. On the other hand, using the literature values of aqueous sugar solution densities, it was possible to calculate the hydration numbers at different temperatures and concentrations. These values of n h show a noticeable decrease as temperature is raised and concentration increased. Decrease in n h can be explained assuming a partial occupation of potential hydration sites (OHs) because of differences in H-bonds lifetimes on the one hand and molecular folding around glycosidic bond on the other. Calculation of water activity coefficients ( f w) based on n h values was made for sucrose solutions. Results show the same trend as found previously [Starzak, M., & Mathlouthi, M. (2006). Temperature dependence of water activity in aqueous solutions of sucrose. Food Chemistry, 96, 346–370] for sucrose, i.e. a decrease of f w with increasing molar concentration. Temperature effect on water activity coefficients and hydration numbers is also determined. It shows a decrease in hydration number as temperature is increased. In this paper, empirical relations are proposed to calculate water activity coefficients and hydration numbers at different concentrations and temperatures by use of accurate density values. These models were first applied to sucrose, the most documented sugar and applied to disaccharides (maltose, trehalose) and monosaccharides (glucose, fructose).

Vapour–liquid equilibrium of cyclic ethers with 1-chlorohexane: Experimental results and UNIFAC predictions

A dynamic recirculating still was employed to measure isobaric vapour–liquid equilibrium at the pressures of 40.00 and 101.33 kPa for binary systems containing the cyclic ethers: tetrahydrofuran, tetrahydropyran, 1,3-dioxolane or 1,4-dioxane and 1-chlorohexane. The experimental data were tested for thermodynamic consistency, the calculated activity coefficients have been correlated using the Wilson, NRTL and UNIQUAC equations. Finally, predictions with the UNIFAC group contribution method were obtained and compared with the experimental results.

Thermodynamic Properties of Malonic, Succinic, and Glutaric Acids: Evaporation Rates and Saturation Vapor Pressures

This work provides thermodynamic data, in particular, liquid-state saturation vapor pressures of three common slightly water soluble secondary organic aerosol components, namely, malonic, succinic, and glutaric acids. A modified tandem differential mobility analyzer system was used to measure evaporation rates of nanometer sized aqueous malonic, succinic, and glutaric acid droplets at relative humidities and temperatures relevant in the lower troposphere. Liquid phase saturation vapor pressures and other thermodynamic properties were derived from the measurements using a binary condensation model. The obtained expressions for liquid phase saturation vapor pressures compare well with extrapolated literature data. The importance of the choice of method for calculating activity coefficients is discussed.

Excess enthalpies and (vapour + liquid) equilibrium data for the binary mixtures of dimethylsulphoxide with ketones

Excess enthalpies ( H E), at ambient pressure and T = 298.15 K, have been measured by using a solution calorimeter for the binary liquid mixtures of dimethyl sulphoxide (DMSO) with ketones, as a function of composition. The ketones chosen in the present investigation were methyl ethyl ketone (MEK), methyl isobutyl ketone (MIBK), and cyclohexanone (CH). The H E values are positive over the entire composition range for the three binary mixtures. Furthermore, the (vapour + liquid) equilibrium (VLE) was measured at 715 Torr for these mixtures, of different compositions, with the help of Swietoslawski-ebulliometer. The experimental temperature–mole fraction ( t– x) data were used to compute Wilson parameters and then used to calculate the equilibrium vapour-phase compositions as well as the theoretical points for these binary mixtures. These Wilson parameters are used to calculate activity coefficients ( γ) and these in turn to calculate excess Gibbs free energy ( G E). The intermolecular interactions and structural effects were analyzed on the basis of the measured and derived properties.

Solubility of NaF in NaF + NaX + H2O (X = ClO4 and NO3) Ternary Systems and Density and Refractive Index of the Saturated Solutions at 298.15 K

Data on the solubilities of NaF were determined in NaF + NaClO4 + H2O and NaF + NaNO3 + H2O ternary systems, and the density and refractive index of the saturated solutions were measured at 298.15 K. The solubility data and physical properties were determined for different molalities of the second electrolyte present in the ternary systems, within a range of 0 to 1.0 mol·kg-1. Experimental data on the density and refractive index were fit to polynomials as a function of NaClO4 or NaNO3 concentration. Finally, a thermodynamic description was made of the solid−liquid equilibrium for the two ternary systems, based on the Reilly, Wood, and Robinson (R−W−R) model for calculation of the mean ionic activity coefficient of NaF in the ternary saturated solution. The Bromley model was used for the calculation of the activity and osmotic coefficients of the binary systems because these are required for making calculations with the R−W−R model.

Vapor Pressures, Enthalpies of Vaporization, and Limiting Activity Coefficients in Water at 100 °C of 2-Furanaldehyde, Benzaldehyde, Phenylethanal, and 2-Phenylethanol

In our previous work, limiting separation factors of 2-furanaldehyde, benzaldehyde, phenylethanal, and 2-phenylethanol in water were determined using a recirculating still of the Gillespie type. Limiting activity coefficients were calculated out of the limiting separation factors in the previous work, too. As it was announced in those papers, a revised calculation of the limiting activity coefficient γ∞ data was performed in this work. For this work, vapor pressure measurements were performed with the transpiration method. The results of these measurements are given here together with the derived enthalpies of vaporization of the investigated components and their formerly calculated and recalculated limiting activity coefficients in water at 100 °C.

Re-evaluation of the Activity Coefficients of Aqueous Hydrochloric Acid Solutions up to a Molality of 16.0 mol·kg−1 Using the Hückel and Pitzer Equations at Temperatures from 0 to 50 °C

The simple three-parameter Pitzer and extended Huckel equations were used for calculation of activity coefficients of aqueous hydrochloric acid at various temperatures from 0 to 50 °C up to a molality of 5.0 mol·kg−1. A more complex Huckel equation was also used at these temperatures up to a HCl molality of 16 mol·kg−1. The literature data measured by Harned and Ehlers J. Am. Chem. Soc. 54, 1350–1357 (1932) and 55, 2179–2193 (1933) and by Akerlof and Teare [J. Am. Chem. Soc. 59, 1855–1868 (1937)] on galvanic cells without a liquid junction were used in the parameter estimations for these equations. The latter data consist of sets of measurements in the temperature range 0 to 50 °C at intervals of 10 °C, and data at these temperatures were used in all of these estimations. It was observed that the estimated parameters follow very simple equations with respect to temperature. They are either constant or depend linearly on the temperature. The values for the activity coefficient parameters calculated by using these simple equations are recommended here. The suggested new parameter values were tested with all reliable cell potential and vapor pressure data available in literature for concentrated HCl solutions. New Harned cell data at 5, 15, 25, 35, and 45 °C up to a molality of 6.5 mol·kg−1 are reported and were also used in the tests. The activity coefficients obtained from the new equations were compared to those calculated by using the Pitzer equations of Holmes et al. [J. Chem. Thermodyn. 19, 863–890 (1987)] and of Saluja et al. [Can. J. Chem. 64, 1328–1335 (1986)] at various temperatures, and by using the extended Huckel equation of Hamer and Wu [J. Phys. Chem. Ref. Data 1, 1047–1099 (1972)] at 25 °C.

BeerOz, a set of Matlab routines for the quantitative interpretation of spectrophotometric measurements of metal speciation in solution

The modelling of the speciation and mobility of metals under surface and hydrothermal conditions relies on the availability of accurate thermodynamic properties for all relevant minerals, aqueous species, gases and surface species. Spectroscopic techniques obeying the Beer–Lambert law can be used to obtain thermodynamic properties for reactions among aqueous species (e.g., ligand substitution; protonation). BeerOz is a set of Matlab routines designed to perform both qualitative and quantitative analysis of spectroscopic data following the Beer–Lambert law. BeerOz is modular and can be customised for particular experimental strategies or for simultaneous refinement of several datasets obtained using different techniques. Distribution of species calculations are performed using an implementation of the EQBRM code, which allows for customised activity coefficient calculations. BeerOz also contains routines to study the n-dimensional solution space, in order to provide realistic estimates of errors and test for the existence of multiple local minima and correlation between the different refined variables. The paper reviews the physical principles underlying the qualitative and quantitative analysis of spectroscopic data collected on aqueous speciation, in particular for studying successive ligand replacement reactions, and presents the non-linear least-squares algorithm implemented in BeerOz. The discussion is illustrated using UV–Vis spectra collected on acidic Fe(III) solutions containing varying LiCl concentrations, and showing the change from the hexaaquo Fe(H 2O) 6 3+ complex to the tetrahedral FeCl 4 − complex.

Prediction of phase equilibria for mixtures containing water, hydrocarbons and alcohols at high temperatures and pressures by cubic equation of state with GE type mixing rule based on COSMO-RS

The phase equilibria for the mixtures containing water, hydrocarbons and alcohols at high temperatures and pressures were predicted by a cubic equation of state with an excess Gibbs free energy ( G E) type mixing rule using information from the conductor—like screening model for real solvents (COSMO-RS) method. The systems predicted were water + hydrocarbon, water + alcohol, hydrocarbon + alcohol binary systems and water + hydrocarbon + hydrocarbon and water + hydrocarbon + alcohol ternary systems. COSMO-RS method was used to calculate activity coefficients of the mixtures. UNIQUAC equation was adopted as an activity coefficient model in G E type mixing rule. The interaction parameters in UNIQUAC equation were determined with the activity coefficients calculated by COSMO-RS method. The modified Huron–Vidal mixing rule (MHV1) and a modified Soave–Redlich–Kwong (MSRK) equation of state were chosen as the G E type mixing rule and the cubic equation of state, respectively. Further, the MHV1 mixing rule was modified to apply in high temperature regions. It was found that the prediction method proposed here can represent the phase behavior of the mixtures containing water, hydrocarbons and alcohols at high temperatures and pressures. The predicted results are compared with those calculated by a predictive Soave–Redlich–Kwong (PSRK) equation of state. The predicted results in this work are more accurate than those calculated by PSRK equation of state.

Modelling of a decarbonation reactor for geothermal waters: application to geothermal waters of Chott El Fejjej

The geothermal waters exploitation infrastructures in Chott El Fejjej (Tunisia) are constantly being damaged by a scaling phenomenon. Experimental work has been carried out on a new kind of decarbonation reactor: the open double partition reactor (RODP), which may be used in the construction of a new plant at the site of Chott El Fejjej. The objective of this paper is to propose a model, which may contribute to the design and optimisation of the process of decarbonation. Our model is based on the thermodynamic laws, which describe the physical and chemical phenomena occurring in an electrolytic three-phase continuous reactor. However, due to electrostatic forces, these kinds of systems cannot be successfully described by the concept of an ideal solution. Indeed, the calculation of activity coefficients is required. So, by using a group contribution method to calculate the activity coefficient, the model remains user-friendly. The non-linear system obtained is solved by using Newton–Raphson’s method. Confrontation with experimental results shows that the model correctly describes the ODP reactor.

Correlation of phase equilibria of amino acids and peptides in aqueous solution based on the perturbation theory of equation of state

In this paper, the activity coefficients of amino acids and simple peptides in aqueous solutions were correlated by using a three parameters model based on the perturbation theory. The adjustable parameters of this model were obtained from the experimental data and a relation for calculating the activity coefficient is derived. The calculated activity coefficients of amino acids and simple peptides obtained show that the equation of state based on the perturbation model can be used to correlate the activity coefficients of amino acids more accurately than the other models. A correlation for the solubility of amino acids in aqueous solutions is also derived. The results show that this correlation can accurately correlate the solubility of amino acids in aqueous solution over a wide range of temperatures (0–100 °C).

Geochemistry of unsaturated soil systems: Aqueous speciation and solubility of minerals and gases in capillary solutions

It has been shown that the capillary state of aqueous solutions in the unsaturated zone (UZ) modifies chemical equilibria due to the decreasing capillary pressure of the whole system (isobaric scenario) or of the aqueous phase only (anisobaric scenario). Meanwhile, the role of salinity in capillary solutions has not been explicitly taken into account up to now, at least not in a manner independent to capillarity. A consistent way to do so is considered in this paper by calculating activity coefficients as a function of capillary pressure through the extrapolation of the Davies model. The integrated approach thus defined is applied to the interpretation of some laboratory experiments (taken from the literature) carried out under different capillary conditions. Calculations and measurements of the boehmite ↔ bayerite hydration equilibrium in an atmosphere of varying humidity agree very satisfactorily if the anisobaric scenario is selected. The solubility of reactive gases O 2 and CO 2 is found to increase in the pore water when the relative humidity decreases. Consequently, and in agreement with experimental measurements, the extent of the pyrite oxidation depends on the relative humidity. The proposed model refines the manner with which chemical equilibria and mineralogical assemblies may be interpreted. In particular, the different scenarios that may be envisioned (isobaric, anisobaric and dual) are considered in the light of the possible modes of precipitation of the minerals (precipitation within the capillary solution, epitaxial type precipitation). Finally, it should be noted that the geochemical approach proposed here for the UZ fully ties in with and conforms to the methods used in the water-saturated zone (SZ).

Modeling Speciation in Highly Concentrated Alkaline Silicate Solutions

A model describing the distribution of silicate sites in concentrated alkali metal silicate solutions used in the synthesis of geopolymeric gels is presented. The model is based on a free energy minimization algorithm, using free energy of formation data obtained from the literature. A simplified form of the Pitzer method (Pitzer, K. S. J. Phys. Chem. 1973, 77, 26) is used to calculate activity coefficients, with interaction parameters calculated from known values for ions of similar size and charge type. Ion pairing effects are incorporated into the model formulation by a simple approximation, whereby larger cyclic or cagelike ions form strong ion pairs with dissolved alkali metal cations, and smaller monomeric (Q0) or end-group (Q1) sites pair less strongly with the cations. The model is able to describe accurately the speciation data obtained from a systematic 29Si NMR investigation of sodium, potassium, and mixed (1:1) sodium/potassium silicate solutions. The model is also tested against a large data set from the literature on sodium silicate solutions with a wide range of compositions. The model provides understanding of the speciation of silicate solutions and a basis for further understanding and modeling of the geopolymerization process.

Experimental and predicted vapour–liquid equilibrium of 1,4-dioxane with cycloalkanes and benzene

Isobaric vapour–liquid equilibrium measurements for mixtures containing 1,4-dioxane and cyclopentane, cyclohexane or benzene at the pressures of 40.0, 66.6 and 101.3 kPa are reported. The experimental data were tested for thermodynamic consistency, the calculated activity coefficients have been correlated using Wilson, NRTL and UNIQUAC equations. The system 1,4-dioxane with cyclohexane shows minimum temperature azeotropes. Using our VLE results UNIFAC and predictive Soave–Redlich–Kwong methods were tested.