Individual Ion Activity Coefficients Research Articles

A semi-empirical equation modeling long-range contributions to the activity coefficients of individual ions at high ionic strengths

Molecular thermodynamics is well developed with models like UNIQUAC, UNIFAC and SAFT. These models include robust codes and extensive databases for chemical and petroleum industries through ASPEN, DORTMUND, and others. However, important systems in the pharmaceutical and biotech industries now exist where these codes and their databases are not always sufficient. Cell culture media used to grow mammalian cells are multicomponent aqueous solutions with 50, 100, or more compounds including amino acids, salts, acids or bases, as well as sugars, fatty acids, etc. These components can exhibit both complexation and incomplete dissociation. The thermodynamics of these systems have proven challenging to model, as current solubility descriptions frequently assume complete dissociation in solutions containing a single solute in water at relatively low concentrations. However, not all compounds that are generally considered to be strong electrolytes are completely dissociated even at low concentrations. Further, mixtures of such components can show dramatic changes in dissociation and complexation with concentration, pH, and temperature. In this paper, we present a semi-empirical model for individual ions in aqueous solutions. We use data from 317 compounds to isolate the ionic contribution to a compound’s activity coefficient in aqueous systems from confounding short-range effects. We compare eight robust regression M−estimators with a least-squares estimate and provide a two-parameter equation relating ionic strength, valence, and the activity coefficient not requiring arduous characterization of the solute of interest. We demonstrate that the Fair M−estimator generates an accurate model that can serve as an appropriate reference state for use with a model based on perturbation theory up to 29 molal.

Activity coefficient acquisition with thermodynamics‐informed active learning for phase diagram construction

AbstractThis work explores the use of thermodynamics‐informed Gaussian processes (GPs) and active learning (AL) to model activity coefficients and construct phase diagrams. Relying on synthetic data generated from an excess Gibbs energy model, GPs were found to accurately describe the activity coefficients of several binary mixtures across large composition and temperature ranges. Moreover, GPs could estimate their own uncertainty and identify composition/temperature regions where activity coefficient data provide the most information to the models. This was leveraged to build AL algorithms targeted at modeling phase equilibria. In many cases, a single active‐learning‐acquired data point was sufficient to describe the phase diagrams studied. Finally, the ability of AL to greatly reduce the amount of data needed to obtain accurate models was further verified on experimental case studies, namely individual ion activity coefficients, the solid–liquid and vapor–liquid equilibrium of deep eutectic solvents, and phase equilibria in ternary mixtures.

Partial osmotic pressures of ions in electrolyte solutions and the Gibbs-Guggenheim uncertainty principle.

The concept of the partial osmotic pressure of ions in an electrolyte solution is critically examined. In principle these can be defined by introducing a solvent-permeable wall and measuring the force per unit area which can certainly be attributed to individual ions. Here I demonstrate that although the total wall force balances the bulk osmotic pressure as required by mechanical equilibrium, the individual partial osmotic pressures are extrathermodynamic quantities dependent on the electrical structure at the wall, and as such they resemble attempts to define individual ion activity coefficients. The limiting case where the wall is a barrier to only one species of ion is also considered, and with ions on both sides the classic Gibbs-Donnan membrane equilibrium is recovered, thus providing a unifying treatment. The analysis can be extended to illustrate how the electrical state of the bulk is affected by the nature of the walls and the container handling history, thus supporting the "Gibbs-Guggenheim uncertainty principle," namely the notion that the electrical state is unmeasurable and usually accidentally determined. Since this uncertainty is conferred also onto individual ion activities, it has implications for the current (2002) IUPAC definition of pH.

Comparison of models for the relative static permittivity with the e-CPA equation of state

This study compares five different models of the relative static permittivity when they are used in the electrolyte Cubic Plus Association (e-CPA) equation of state. To get the best possible performance of the models, the parameters of e-CPA are readjusted for every model. Two different combinations of adjustable parameters are tested. The static permittivity models that are compared include both simple correlations and theoretically derived expressions. A new theoretically based model, that has not been applied to e-CPA before, is also investigated. The novel model describes the impact ions have on the relative static permittivity based on water–ion association. The model is parameterized in two ways: firstly, so that the model describes the reported experimental relative static permittivity data, and secondly to describe the permittivity when kinetic depolarization is not included. All the models are tested for their quantitative agreement with mean ionic activity coefficients (MIAC), osmotic coefficients and density. The model that describes the experimental data the best is the one based on ion association, when it is parameterized to describe the experimental relative static permittivity data. The prediction of the individual ion activity coefficients (IIAC) is also investigated. The only model that is capable of describing the qualitative trend of the IIAC data is the ion association model, but the quantitative agreement with the IIAC data is quite poor. Because of this, an additional parameterization of the ion association model is performed based on an altered parameter optimization strategy. It is shown that with the new parameterization it is possible to describe the IIAC data well, without significant loss of performance for any of the other properties.

Extension of the eSAFT-VR Mie equation of state from aqueous to non-aqueous electrolyte solutions

In this work, the eSAFT-VR Mie equation of state (EoS) is extended to low relative permittivity, non-aqueous solutions. The effect of using different relative permittivity relations for the electrolyte solutions is studied, ranging from experimentally measured values to a salt-composition independent relative permittivity. Furthermore, the effect of using different approaches for the characteristic diameters in the Debye-Hückel and Born terms is presented. The eSAFT-VR Mie EoS is reparametrized using aqueous mean ionic activity coefficients, individual ion activity coefficients and densities with different relations for the relative permittivity. Afterwards, the performance of these models on non-aqueous solutions is evaluated based on the Mean Ionic Activity Coefficients of salts in non-aqueous solutions. The conclusion is that a mole fraction based mixing rule for the relative permittivity yields the best extrapolation from aqueous to non-aqueous solutions, and achieves quantitative predictions for the mean ionic activity coefficients of monovalent salts in methanol and ethanol without additional adjustable parameters.

Activity Coefficients and Solubilities of NaCl in Water-Methanol Solutions from Molecular Dynamics Simulations.

We obtain activity coefficients and solubilities of NaCl in water-methanol solutions at 298.15 K and 1 bar from molecular dynamics (MD) simulations with the Joung-Cheatham, SPC/E, and OPLS-AA force fields for NaCl, water, and methanol, respectively. The Lorentz-Berthelot combining rules were adopted for the unlike-pair interactions of Na+, Cl-, and the oxygen site in SPC/E water, and geometric combining rules were utilized for the remainder of the cross interactions. We found that the selection of appropriate combining rules is important in obtaining physically realistic solubilities. The solvent compositions studied range from pure water to pure methanol. Several salt concentrations were investigated at each solvent composition, from the lowest concentrations permitted by the system size used up to the experimental solubilities. We first calculated individual ion activity coefficients (IIACs) for Na+ and Cl- from the free energy change due to the gradual insertion of a single cation or anion into the solution, accompanied by a neutralizing background. We obtained the salt solubilities by comparing the chemical potentials in solution with solid NaCl chemical potentials calculated previously using the Einstein crystal method. Mean ionic activity coefficients obtained from the IIACs are in reasonable agreement with experimental data, with deviations increasing for solutions of higher methanol content. Predictions for the salt solubility are in surprisingly good agreement with experimental data, despite well-known challenges in the simultaneous calculation of activity coefficients and solubilities with classical MD simulations. The present study demonstrates that good predictions for these two important phase equilibrium properties can be obtained for mixed-solvent electrolyte solutions using existing nonpolarizable models and further suggests that the previously proposed single ion insertion technique can be extended to complex mixed-solvent solutions as well.

Calculations of individual ionic activity coefficients of chloride salt in aqueous solutions

Individual ionic activity coefficients of several chloride salts in aqueous solutions are predicted using the II + IW theory. The two combinations of the II + IW theory proposed in this work are mean spherical approximation plus Born equation. Different calculated ion radii are used in calculations. The predicted results were presented and analyzed compared with the experimental data. The results show that the II + IW theory predicts reasonably well the individual ionic activity coefficients of the specific electrolytes in aqueous solutions. The results show that adjusting ion radius can adjust the ion-ion interaction and improve modeling performance for individual ionic activity coefficients. The electrostatic interaction terms and the effects of ion radius are further studied and discussed.

Activity coefficients of aqueous electrolytes from implicit-water molecular dynamics simulations.

We obtain activity coefficients in NaCl and KCl solutions from implicit-water molecular dynamics simulations, at 298.15K and 1bar, using two distinct approaches. In the first approach, we consider ions in a continuum with constant relative permittivity (ɛr) equal to that of pure water; in the other approach, we take into account the concentration-dependence of ɛr, as obtained from explicit-water simulations. Individual ion activity coefficients (IIACs) are calculated using gradual insertion of single ions with uniform neutralizing backgrounds to ensure electroneutrality. Mean ionic activity coefficients (MIACs) obtained from the corresponding IIACs in simulations with constant ɛr show reasonable agreement with experimental data for both salts. Surprisingly, large systematic negative deviations are observed for both IIACs and MIACs in simulations with concentration-dependent ɛr. Our results suggest that the absence of hydration structure in implicit-water simulations cannot be compensated by correcting for the concentration-dependence of the relative permittivity ɛr. Moreover, even in simulations with constant ɛr for which the calculated MIACs are reasonable, the relative positioning of IIACs of anions and cations is incorrect for NaCl. We conclude that there are severe inherent limitations associated with implicit-water simulations in providing accurate activities of aqueous electrolytes, a finding with direct relevance to the development of electrolyte theories and to the use and interpretation of implicit-solvent simulations.

Statistical theory of individual ionic activity coefficients of electrolytes with multiple – Charged ions including seawater

Based on methods of statistical mechanics for the calculation of individual activity coefficients for electrolytic solutions we present analytical nonlinear extensions of the standard Debye - Hückel and Mean-Spherical approximations. These extensions, so - called DHX and MSX approximations, respectively, generalize the exact results from cluster theory and are given for the model of charged hard spheres with non - additive contact distances. The flexibility of the fully analytical model which is able to reflect e.g. cationic solvation effects and allows the treatment of ionic activities in a 6-component seawater model. Denoting by e the elementary charge, DHX and MSX include in the third order, e6, the Poirier effects of asymmetry of ion charges, and in the fourth order, e8, as well as in higher orders, weak association effects which are relevant for electrolytes with multivalent ions. Following the Justice method, weak association effects are treated in a semi - chemical description avoiding mass - action laws. An extrapolation of the theory of individual activities to a description of individual conductivities is proposed.

Individual Ion Activity Coefficients in Aqueous Electrolytes from Explicit-Water Molecular Dynamics Simulations.

We compute individual ion activity coefficients (IIACs) in aqueous NaCl, KCl, NaF, and KF solutions from explicit-water molecular dynamics simulations. Free energy changes are obtained from insertion of single ions-accompanied by uniform neutralizing backgrounds-into solution by gradually turning on first Lennard-Jones interactions, followed by Coulombic interactions using Ewald electrostatics. Simulations are performed at multiple system sizes, and all results are extrapolated to the thermodynamic limit, thus eliminating any possible artifacts from the neutralizing backgrounds. Because of controversies associated with measurements of IIACs from electrochemical cells with ion-selective electrodes, the reported experimental data are not widely accepted; thus there remains a knowledge gap with respect to the contributions of individual ions to solution nonidealities. Our results are in good qualitative agreement with these reported measurements, though significantly larger in magnitude. In particular, the relative positioning for the activity coefficients of anions and cations matches the experimental ordering for all four systems. This work establishes a robust thermodynamic framework, without a need to invoke extra hypotheses, that sheds light on the behavior of individual ions and their contributions to nonidealities of aqueous electrolyte solutions.

Thermodynamic modeling of gas solubility in aqueous sodium chloride solution

Abstract An electrolyte Equation of State is presented by combining the Cubic Plus Association Equation of State, Mean Spherical Approximation and the Born equation. This new model uses experimental relative static permittivity, intend to predict well the activity coefficients of individual ions (ACI) and liquid densities of aqueous solutions. This new model is applied to model water + NaCl binary system and water + gas + NaCl ternary systems. The cation/anion-water interaction parameters of are obtained by fitting the experimental data of ACI, mean ionic activity coefficients (MIAC) and liquid densities of water + NaCl binary system. The cation/anion-gas interaction parameters are obtained by fitting the experimental data of gas solubilities in aqueous NaCl solutions. The modeling results show that this new model can correlate well with the phase equilibrium and volumetric properties. Without gas, predictions for ACI, MIAC, and liquid densities present relative average deviations of 1.3%, 3.6% and 1.4% compared to experimental reference values. For most gas-containing systems, predictions for gas solubilities present relative average deviations lower than 7.0%. Further, the contributions of ACI, and salting effects of NaCl on gases are analyzed and discussed.

Analysis of Some Electrolyte Models Including Their Ability to Predict the Activity Coefficients of Individual Ions

This work presents a modeling study, with selected models, of the mean ionic activity coefficients of electrolytes in aqueous solutions, where predictions of the activity coefficients of individual...

A Two-Parameter Theoretical Model for Predicting the Activity and Osmotic Coefficients of Aqueous Electrolyte Solutions

A thermodynamic model of electrolyte solutions is proposed. The model consists of a long-range term expressed by the Pitzer–Debye–Huckel equation (PDH) and a short-range term expressed by the molecular interaction volume model (MIVM). The new model was fitted with 39 different types of single electrolyte systems and was compared with the Pitzer equation, and the mean standard deviation (SD) and the mean average relative deviation (ARD%) are 0.0264, 0.0040 and 2.09%, 0.40%, respectively. Meanwhile, the physical meaning of the two electrolyte-specific interaction parameters ( $$B_{ca,s}$$ and $$B_{s,ca}$$ ) of the new model is also discussed. By further comparison with the Pitzer equation and a state-of-the-art model, eUNIQUAC-NRF, the new model exhibits relatively robust extrapolation capability, and also shows the potential ability to predict the activity coefficients of individual ions. In addition, only using binary parameters to predict 29 different types of ternary systems, the overall prediction results of the new model are slightly better than those of the Pitzer equation, and the mean SD and ARD% are 0.0288, 0.0396 and 2.88%, 3.81%, respectively. For some cases involving Rb and Cs, the Pitzer equation needs two ternary adjustable parameters ( $$\theta$$ and $$\psi$$ ) to achieve the prediction accuracy of the new model. Furthermore, we also compared the predictions of the new model with the eUNIQUAC-NRF model for several ternary systems; the new model also shows better performance, and its overall prediction accuracy was about twice that of the eUNIQUAC-NRF model, with the average SD and ARD% values being 0.0261, 0.0546 and 2.63%, 5.80%, respectively.

Thermodynamically Consistent Force Field for Coarse-Grained Modeling of Aqueous Electrolyte Solution.

We propose a thermodynamically consistent methodology to parameterize interactions between charged particles inside the dissipative particle dynamics (DPD) formalism. We used osmotic pressure experimental data as a function of the salinity in order to optimize the interaction parameters. Results for NaCl aqueous solution show that both mean osmotic and activity coefficients of individual ions allowed the determination of Na+-water, Cl--water, and Na+-Cl- DPD repulsion parameters. A simple linear relationship between the hydration-free energies of ions and the ion-water repulsion parameters that allows the parameterization of the complete series of halide and alkaline ions is proposed. Two strategies have been used to obtain the anion-cation interaction parameters for halide and alkaline ions. In the first one, the parameters are obtained based on the numerical optimization of the anion-cation repulsion parameter with respect to the experimental osmotic pressure data (with mean average deviations <4%). Second, we propose a predictive approach based on the free-energy difference of hydration energies of anions and cations in the spirit of the law of matching water affinities [with a mean absolute relative deviation of about 13%, better than 6% if small ions (Li+ and F-) are removed].

A Nonextensive Electrolyte UNIQUAC Model for Prediction of Mean Activity Coefficients of Binary Electrolyte Solutions

In this work, an electrolyte-UNIQUAC model was developed by replacement of Boltzmann weight binary interaction parameters by the nonextensive Tsallis weight. A summation of the long-range electrostatic term (Debye-Huckel equation) and a short-range interaction term were considered in the calculation of thermodynamic properties. A framework proposed by Chen and co-workers was employed for the derivation of the local mole fractions. Application of the nonextensive theory increased the degree of freedom of the present model (T-E-UNIQUAC). Furthermore, the strength of the model lies in its ability to calculate individual activity coefficients of ions. The applicability of the T-E-UNIQUAC model has been tested using aqueous electrolyte solutions and results have been compared with Messnaoui, Chen and Pitzer models.

Activity coefficients of individual ions in LaCl3 from the II+IW theory

ABSTRACTWe investigate the individual activity coefficients of ions in LaCl3 using our theory that is based on the competition of ion–ion (II) and ion–water (IW) interactions. The II term is computed from Grand Canonical Monte Carlo simulations on the basis of the implicit solvent model of electrolytes using hard sphere ions with Pauling radii. The IW term is computed on the basis of Born's treatment of solvation using experimental hydration-free energies. The results show good agreement with experimental data for La3+. This agreement is remarkable considering the facts that (i) the result is the balance of two terms that are large in absolute value (up to 20 kT) but opposite in sign, and (ii) that our model does not contain any adjustable parameter. All the parameters used in the model are taken from experiments: concentration-dependent dielectric constant, hydration free energies and Pauling radii.

Ionic molal conductivities, activity coefficients, and dissociation constants of HAsO42 − and H2AsO4− from 5 to 90 °C and ionic strengths from 0.001 up to 3 mol kg− 1 and applications in natural systems

Arsenic is known to be one of the most toxic inorganic elements, causing worldwide environmental contamination. However, many fundamental properties related to aqueous arsenic species are not well known which will inhibit our ability to understand the geochemical behavior of arsenic (e.g. speciation, transport, and solubility). Here, the electrical conductivity of Na2HAsO4 solutions has been measured over the concentration range of 0.001–1molkg−1 and the temperature range of 5–90°C. Ionic strength and temperature-dependent equations were derived for the molal conductivity of HAsO42− and H2AsO4− aqueous ions. Combined with speciation calculations and the approach used by McCleskey et al. (2012b), these equations can be used to calculate the electrical conductivities of arsenic-rich waters having a large range of effective ionic strengths (0.001–3molkg−1) and temperatures (5–90°C). Individual ion activity coefficients for HAsO42− and H2AsO4− in the form of the Hückel equation were also derived using the mean salt method and the mean activity coefficients of K2HAsO4 (0.001–1molkg−1) and KH2AsO4 (0.001–1.3molkg−1). A check on these activity coefficients was made by calculating mean activity coefficients for Na2HAsO4 and NaH2AsO4 solutions and comparing them to measured values. At the same time Na-arsenate complexes were evaluated. The NaH2AsO40 ion pair is negligible in NaH2AsO4 solutions up to 1.3molkg−1. The NaHAsO4− ion pair is important in NaHAsO4 solutions >0.1molkg−1 and the formation constant of 100.69 was confirmed. The enthalpy, entropy, free energy and heat capacity for the second and third arsenic acid dissociation reactions were calculated from pH measurements. These properties have been incorporated into a widely used geochemical calculation code WATEQ4F and applied to natural arsenic waters. For arsenic spiked water samples from Yellowstone National Park, the mean difference between the calculated and measured conductivities have been improved from −18% to −1.0% with a standard deviation of 2.4% and the mean charge balances have been improved from 28% to 0.6% with a standard deviation of 1.5%.

Individual ion activity and liquid junction potential—Two interrelated, interconnected electrochemical terms

Liquid junction potentials (LJPs) arise by force when two electrolyte solutions with different compositions are in contact. Provided that individual ion activity coefficients are precisely known, a statement can be made about LJPs in galvanic cells with transference. The LJP and the single-ion activity are two electrochemical terms that are mutually interrelated, interconnected and thus interdependent. Interdisciplinary research is necessary in order to obtain knowledge about individual ion activity coefficients. With mathematically obtained individual ion activity coefficients, it is possible to make exact statements of the magnitude of LJPs in galvanic cells with transference. The obtained LJPs are smaller than the values evaluated by the equation of Henderson. LJPs in galvanic cells with transference are calculated, and true conventional potentials of reference electrodes are given.

Assessment of H+ in complex aqueous solutions approaching seawater

pH measurements are used to assess both the free acidity of aqueous solutions and the concentrations, mi, of chemical species i related to hydrogen ion, H+, through chemical equilibrium, provided that measurements in reference solutions with well-known compositions are also carried out. Seawater is a highly complex electrolytic matrix of variable composition, rich in NaCl, where the determination of pH=−lg aH+ (aH+=mH+γH+ is the activity of the specified ion, H+ with the concentration mH+ and activity coefficient γH+≠1) is an issue of controversy.Electroneutrality prevents measurement of individual ion activity coefficients, which are affected and depend on the other ions in presence, namely the counter ions. Nevertheless, mean activity coefficients, γ±=γ+γ−, in this case γ±=γH+γCl−, can be experimentally assessed from the application of the Nernst equation to Harned cell (Pt|H2 vs. Ag|AgCl electrodes, without transference) in solutions of known mH+ and mCl−. In this work measurements of electrical potential, E/V, have first been taken in strategically planned solutions, ranging from 0.01molkg−1 HCl+(NaCl) to 0.01molkg−1 HCl+(NaCl+KCl+CaCl2+MgCl2), of ionic strength, I≈0.67molkg−1 at 15°C, 25°C and 35°C with experimental pressure corrected to reference pressure of 1atm. Mean activity coefficients were experimentally assessed with associated uncertainty. Further addition of sodium sulfate, Na2SO4, targeting artificial seawater (ASW), originates acid–base equilibria that affect the free hydrogen ion concentration, mH+. The changes introduced by the addition of sulfate in the measured Harned cell potential have been used to assess the free proton concentration and, based on the obtained results, to propose for the first time a procedure aiming to calculate values of bisulfate association constants with the corresponding uncertainty at 25°C for both studied media. Further, the effective ionic strength of artificial seawater (ASW) was calculated and compared with the value obtained using the salinity based equation largely conveyed in the literature.

Modeling for mean ion activity coefficient of strong electrolyte system with new boundary conditions and ion-size parameters

A rigorous approach is proposed to model the mean ion activity coefficient for strong electrolyte systems using the Poisson–Boltzmann equation. An effective screening radius similar to the Debye decay length is introduced to define the local composition and new boundary conditions for the central ion. The crystallographic ion size is also considered in the activity coefficient expressions derived and non-electrostatic contributions are neglected. The model is presented for aqueous strong electrolytes and compared with the classical Debye–Hückel (DH) limiting law for dilute solutions. The radial distribution function is compared with the DH and Monte Carlo studies. The mean ion activity coefficients are calculated for 1:1 aqueous solutions containing strong electrolytes composed of alkali halides. The individual ion activity coefficients and mean ion activity coefficients in mixed solvents are predicted with the new equations.