Mean Activity Coefficients Research Articles

Mean Activity Coefficients of NaCl in the Ternary System NaCl–SrCl2–H2O at 288.15 K Determined by the Cell Potential Method and Their Application to the Prediction of Phase Equilibria

Mean Activity Coefficients of NaCl in the Ternary System NaCl–SrCl₂–H₂O at 288.15 K Determined by the Cell Potential Method and Their Application to the Prediction of Phase Equilibria

Real-Time Estimation of CO2 Absorption Capacity Using Ionic Conductivity of Protonated Di-Methyl-Ethanolamine (DMEA) and Electrical Conductivity in Low-Concentration DMEA Aqueous Solutions

The present study investigates the real-time estimation of CO2 absorption capacity (CAC) based on the electrical conductivity (EC) of low-concentration di-methyl-ethanolamine (DMEA) solutions (0.1–0.5 M). CO2 absorption experiments are conducted to measure the variation in CAC and EC during CO2 absorption, revealing a strong correlation between the two properties. The ionic conductivity of DMEAH+ formed during absorption is calculated to be 53.1 S·cm2/(mol·z), which is found to be larger than that of TEAH+ and MDEAH+. This can be attributed to the smaller molar mass and higher ionic mobility of DMEAH+. A significant finding is that the measured EC (ECM) of the DMEA solutions consistently demonstrates a lower value than the theoretically predicted value. This discrepancy is due to the larger ionic size of DMEAH+, which results in a reduction in the real mean ionic activity coefficient. This effect becomes more pronounced with increasing DMEA concentration. Consequently, a higher CAC is required to produce the same change in EC at higher amine concentrations. Based on these findings, an empirical equation is devised to estimate CAC from ECM in solutions of constant DMEA concentration. This equation will be employed as a practical approach for the in situ monitoring of CO2 absorption using DMEA aqueous solution.

Calculations of mean ionic activity coefficients of copper and manganese salts in aqueous solutions

Accurately predicting the activity coefficients of copper and manganese salts in aqueous solutions is crucial for treating industrial wastewater, and it is of practical interest to calculate the activity coefficients of heavy metals in aqueous solutions using the Equation of State because of its high flexibility and wide range of applicability. This work conducts a modeling study on the mean ionic activity coefficients of copper and manganese salts in aqueous solutions. The thermodynamic framework used in this work is an electrolyte version of the Cubic Plus Association Equation of State (e-CPA). The model's ion-water binary interaction parameters are obtained by regressing the osmotic coefficients and mean ionic activity coefficients in aqueous solutions. The modeling results indicate that the electrolyte Equation of State can satisfactorily correlate the water activity, osmotic coefficients, and mean ionic activity coefficients of copper and manganese salts in aqueous solutions over a wide range of salt concentrations. Among all the systems at 298.15 K, the biggest average relative deviation between calculated and experimental values is within 6.7 % and 2.3 % for the mean ionic activity coefficients and water activity, respectively. Furthermore, this work discusses the impact of ion pairs on the phase equilibrium of aqueous systems from a microscopic mechanism perspective and analyzes and discusses the model's adaptability. The model used in this work can accurately predict the activity coefficients of a variety of copper and manganese salts in aqueous solutions over wide concentration ranges and provides an improved interface for improving the calculation accuracy of the model under high concentration or other complex conditions.

Predictive potentialities of the Quasi-Random Lattice model for electrolyte solutions, discussion and improvement strategies

This article focuses on the predictive potentialities of the Quasi-Random Lattice (QRL) model, developed for describing the activity behaviour of electrolytic solutions, and elaborates strategies for their improvement.First, the study critically discusses the computational-experimental procedure (previously published) for determining the QRL parameterization, whose convergence within few iterations is counterbalanced by known experimental issues concerning, in particular, the mean activity coefficient. An alternative procedure is proposed, that makes use of osmotic data at medium-high concentrations, so as to make QRL more interesting from a practical point of view.Second, the study explores the applicability of the model beyond the concentration ranges earlier considered. To this purpose, the solution density is evaluated in detail. Its thermodynamic relationship with the mean activity coefficient yields a parametric Abel Equation of the Second Kind valid in the medium-high range of concentrations. A further density equation is formulated, useful in the low-medium range, based on a classical power-series combined with appropriate analytical constraints to improve estimation and prediction methods.QRL theory, methods and procedures are applied to binary aqueous solutions at 25°C.

Theoretical and practical investigation of ion-ion association in electrolyte solutions.

In this study, we present a new equation of state for electrolyte solutions, integrating the statistical associating fluid theory for variable range interactions utilizing the generic Mie form and binding Debye-Hückel theories. This equation of state underscores the pivotal role of ion-ion association in determining the properties of electrolyte solutions. We propose a unified framework that simultaneously examines the thermodynamic properties of electrolyte solutions and their electrical conductivity, given the profound impact of ion pairing on this transport property. Using this equation of state, we predict the liquid density, mean ionic activity coefficient, and osmotic coefficient for binary NaCl, Na2SO4, and MgSO4 aqueous solutions at 298.15K. Additionally, we evaluate the molar conductivity of these systems by considering the fraction of free ions derived from our equation of state in conjunction with two advanced electrical conductivity models. Our results reveal that, while ion-ion association has a minimal influence on the modification of the predicted properties of sodium chloride solutions, their impact on sodium and magnesium sulfate solutions is considerably more noticeable.

Thermodynamic modeling using extended UNIQUAC and COSMO-RS-ES models: Case study of the cesium nitrate-water system over a large range of temperatures

A comparison of two thermodynamic models is presented using the water-cesium nitrate system as case study. Both models were able to model the thermodynamic properties such as the osmotic coefficient, vapor pressure, mean activity coefficient and solubility with good accuracy. We show that it is possible to reproduce the temperature dependency of the properties using a simple set of parameters in the case of Extended UNIQUAC. Furthermore, COSMO-RS-ES is a completely predictive model adjusted to data at 298.15 K, which is applied for the first time to other temperatures.

Thermodynamic models for solution behavior and solid-liquid equilibrium in acetate binary systems from low to very high concentration at 25oC

In this study we developed new thermodynamic models for solution behavior and solid-liquid equilibrium in eight acetate binary systems from low to very high concentration at 25oC: six systems of the type 1-1 (X(CH3COO)-H2O with X= (Li, Na, K, Rb, Cs, and Tl)) and two systems of the type 2-1 (Y(CH3COO)2-H2O with Y= (Mg, and Ba)). Models were developed on the basis of Pitzer ion interactions approach. To parameterize models for binary systems we used all available experimental osmotic coefficients data (φ) for the whole concentration range of solutions. To develop models, we used different versions of the standard molality-based Pitzer approach. It was established that for all acetate systems under study application of extended approach with 4 parameters (β0 , β1 , β2 and Cφ ) and variation of 1 and 2 terms in fundamental Pitzer equations leads to the lowest values of standard model-experiment deviation. The predictions of new developed here models are in excellent agreement with experimental φ data, and with recommendations on the mean activity coefficients (±) from low to very high concentration: up to 23.8 m in K(CH3COO)-H2O, up to 39.2 m in Rb(CH3COO)-H2O, and up to 52.2 m in Cs(CH3COO)-H2O. On the basis of evaluated binary parameters important thermodynamic characteristics (Deliquescence Relative Humidity DRH, thermodynamic solubility products Ko sp; standard molar Gibbs free energy of formation, fGo m) for 7 acetate solids were determined at 25oC. The calculated DRH and fGo m values were compared with those reported in the literature. The models described in this study are of high importance, especially in development and improvement of technology for production and purification of acetate solutions and solid phases.

Modifying Kohlrausch's Law to Describe Nonaqueous Electrolytes for Lithium-Ion Batteries.

To develop next-generation lithium-ion batteries with enhanced stability and safety, it is crucial to understand the physicochemical principles of nonaqueous electrolytes. Kohlrausch's law describes a linear decrease in the molar conductivity (Λ) with respect to the square root of the molarity of strong electrolytes at lower concentrations. This empirical law explains the impeded ionic mobility at higher concentrations due to ionic interactions, i.e., relaxation and asymmetric effects. However, this law does not hold at higher concentrations due to the ionic association that alleviates the ionic interactions and retards the decrease in the Λ. Especially, the anomalously stagnant decrease in the Λ near the solubility limit has not been clearly explained, calling for the consideration of other concentration-dependent factors such as the mean activity coefficient (γ±), viscosity (η), and dielectric constant (ε). Herein, we develop a systematic method to modify Kohlrausch's law. First, we install the ionic association constant, and the theoretical estimation is compared with the experimental results to induce the correction function that is related with the previously neglected concentration-dependent factors. Thus, the induced correction function was close to the rectified linear unit (ReLU) function, which has been widely used in the field of artificial intelligence. The modified Kohlrausch's law with the ReLU-type correction function provides a highly precise and reliable data fitting, and the fitted parameters showed clear concentration dependency and straightforward interpretability. As a result, this method effectively generalized Kohlrausch's law for nonaqueous electrolytes at higher concentrations up to the solubility limit of 3.0-3.5 M. Moreover, the modified Kohlrausch's law inspired us to discover the physical origins of the anomalously stagnant Λ profiles near the solubility limit; and the most relevant physical origin of the anomaly was the concentration dependency of the γ± and η, which grow exponentially above a critical concentration.

Modification of e-CPA for estimating phase equilibria and development of predictive models for electrical conductivity in aqueous electrolyte solutions

We modify the electrolyte cubic plus association equation of state (e-CPA EoS) and inetgrate it with two electrical conductivity models to estimate the electrical conductivity of 11 monovalent electrolytes solutions in water. The e-CPA EoS is modified by including the ion-ion and ion-solvent association interactions and tuned using experimental data of mean ionic activity coefficient and density of the electrolyte solutions. The modified e-CPA model shows a better performance in estimating the bubble pressure data. The hybridization of two electrical conductivity models with e-CPA results in two predictive models for estimating the electrical conduction of dilute and concentrated electrolyte solutions. Finally, we generalize these predictive models by correlating their model fitting parameters to the electrolyte physical parameters such as mean hard-sphere diameter or to association volume. The two predictive electrical conductivity models estimate electrical conductivity with AARD of 11.15 % and 13.87 % over wide ranges of temperature and electrolyte concentration.

Binding Debye-Hückel theory for associative electrolyte solutions.

This study presents a new equation of state (EOS) for charged hard sphere fluids that incorporates ion-ion association. The EOS is developed using the Debye-Hückel (DH) theory, reference cavity approximation, and Wertheim's theory. Predictive accuracy is evaluated by comparing the model's predictions with Monte Carlo simulations for various charged hard-sphere fluids. The assessment focuses on mean ionic activity coefficient, individual ionic activity coefficient, and osmotic coefficients. The results demonstrate good agreement between the model and simulations, indicating its success for different electrolyte systems. Incorporating ion-ion association improves accuracy compared to the DH theory. The importance of the cavity function and ion-dipole interactions is emphasized in accurately representing structural properties. Overall, the developed EOS shows promising predictive capabilities for charged hard sphere fluids, providing validation and highlighting the significance of ion-ion association in thermodynamic predictions of electrolyte solutions.

Systematic evaluation of parameterization approaches for the ePPC-SAFT model for aqueous alkali halide solutions. II. Alkali bromides, iodides, fluorides, and lithium halides

In this work, state-of-the-art ion-specific parameter sets for the ePPC-SAFT model are obtained for 15 aqueous alkali halide solutions based on extensively collected and critically evaluated databases. The association approach, which has been proven the best for accounting for the short-range interactions, is here extended to all the alkali halides. Efforts are made to improve modeling accuracy. The impact of anion association site number on modeling accuracy is investigated. Recommended parameter sets are presented. Only 7 adjustable parameters altogether are needed for the 4 salts with the same anion, e.g., NaBr, KBr, RbBr, and CsBr. It is found that Li+-water dispersion needs to be accounted for to reach the same accuracy for Li+ salts as for the other salts. Thus, 7 parameters altogether are needed for LiCl, LiBr, and LiI. The obtained model and parameter sets are very accurate for correlating the properties included in the regression, mean-ionic activity coefficient, vapor-liquid equilibria, and density, and for predicting the osmotic coefficient. In all cases, maximum deviations are within a few percent. The contributions of the model terms are analyzed.

Determination of the Ionic Association Constants of Na+ with CO32- and HCO3- Ions, in NaCl-NaHCO3-H2O Ternary Systems, at 25 °C.

The ionic association constants of sodium with carbonate ion (K1C') and acidic carbonate ions (K2C') were measured in NaCl-NaHCO3-H2O ternary systems to determine the distribution of sodium among the chemical species present in the growth medium of Chlorella homosphaera 424 algae. The mean activity coefficients of sodium chloride (in pure sodium chloride and in a mixture of electrolytes) were determined experimentally using two electrochemical cells, namely Ag, AgCl| KCl (3 M)|| NH4NO3 (1 M)| NaCl (mNaCl)| Na+-ISE and Ag, AgCl|KCl (3 M)|| NH4NO3 (1 M)| NaCl (mNaCl)| Cl--ISE. The studies carried out show that the values of the association constants of K1C' and K2C' do not depend on the composition of the medium, but only on the effective ionic strength. The experimentally obtained γNaCl0 values in the binary system are comparable to the mean activity coefficients values for NaCl, calculated using data from the literature, with -0.9 to 0.1% relative standard deviation. The obtained results show that the experimentally determined mean activity coefficient in the ternary system, γNaCl, is smaller than γNaCl0 in the binary system over the entire field of ionic strengths studied. The ternary system NaCl-NaHCO3-H2O obeys Harned's rule.

Modeling the effects of salt concentration on aqueous and organic electrolytes

Understanding the thermodynamic properties of electrolyte solutions is of vital importance for a myriad of physiological and technological applications. The mean activity coefficient γ± is associated with the deviation of an electrolyte solution from its ideal behavior and may be obtained by combining the Debye-Hückel (DH) and Born (B) equations. However, the DH and B equations depend on the concentration and temperature-dependent static permittivity of the solution εr(c, T) and the size of the solvated ions ri, whose experimental data is often not available. Here, we use a combination of molecular dynamics and density functional theory to predict εr(c, T) and ri, which enables us to apply the DH and B equations to any technologically relevant aqueous and nonaqueous electrolyte at any concentration and temperature of interest.

Mixed Solvent Electrolyte Solutions: A Review and Calculations with the eSAFT-VR Mie Equation of State.

In this work, mixed-solvent mean ionic activity coefficients (MIAC), vapor-liquid equilibrium (VLE), and liquid-liquid equilibrium (LLE) of electrolyte solutions have been addressed. An extended literature review of existing electrolyte activity coefficient models (eGE) and electrolyte equations of state (eEoS) for modeling mixed solvent electrolyte systems is first presented, focusing on the details of the models in terms of physical and electrolyte terms, relative static permittivity, and parameterization. The analysis of this literature reveals that the property predictions can be ranked, from the easiest to the most difficult, in the following order: VLE, MIAC, and LLE. We have then used our previously developed eSAFT-VR Mie model to predict MIAC, VLE, and LLE in mixed solvents without fitting any new adjustable parameters. The model was parameterized on MIAC of aqueous electrolyte solutions and successfully extended to nonaqueous, single solvent electrolyte solutions without any new adjustable parameters by using a salt-dependent expression for the relative static permittivity. Our approach yields excellent results for MIAC and VLE of mixed solvent electrolyte solutions, while being fully predictive. LLE is significantly more challenging, and an accurate model for the salt-free solution is crucial for accurate calculations. When the compositions of the two phases in the binary salt-free system are accurately captured, then the electrolyte extension of our model shows a lot of potential and is currently among the best eEoS for LLE prediction in the literature.

Activity coefficients, phase diagram and separation process design for NaNO3 and Pb(NO3)2 of waste liquor in the production of lead–chromium pigments

Lead contamination is one of the common soil heavy metal pollution elements, and the waste solution discharged from chromium–lead pigment plants is one of the potential sources of lead contamination. In order to separate and recover sodium nitrate and lead nitrate from waste liquid of the production of lead–chromium pigments, it is necessary to make systematical studies on thermodynamic properties, and thermodynamic phase diagram of the salt–water system containing NaNO3 and Pb(NO3)2. Accordingly, in this work, the cell potential method were chosen to measure the thermodynamic activity coefficients of mixed solutions NaNO3– Pb(NO3)2–H2O. The thermodynamic properties (mean activity coefficient, osmotic coefficient, water activity, excess Gibbs free energy) of NaNO3–Pb(NO3)2–H2O mixed solutions at 308.15 K were also determined by Pitzer model. Furthermore, the phase equilibria of the NaNO3–Pb(NO3)2–H2O system was investigated by the isothermal dissolution equilibrium method, and its isothermal equilibrium phase diagram was drawn in detail. The solubility data were used to obtain the Pitzer mixing ion parameters, and the solubilities of salts in the ternary system were predicted by Pitzer model. Based on the plotted phase diagrams, as a result, the process to separate NaNO3 and enrich the recycling of Pb(NO3)2 was designed based on the calculated phase diagram. Among them, the recovery of NaNO3 reached 70.82% and the enrichment of Pb(NO3)2 reached about 15 times. The designed separation process in this work has the advantages of green, low energy consumption and sustainability, which is of positive practical significance for the full utilization of resources and the prevention of environmental pollution.

Self-Consistent Implementation of a Solvation Free Energy Framework to Predict the Salt Solubilities of Six Alkali Halides.

To assess the salt solubilities of six alkali halides in aqueous systems, we proposed a thermodynamic cycle and an efficient molecular modeling methodology. The Gibbs free energy changes for vaporization, dissociation, and dissolution were calculated using the experimental data of ionic thermodynamic properties obtained from the NBS tables. Additionally, the Marcus' and Tissandier's solvation free energy data for Li+, Na+, K+, Cl-, and Br- ions were compared with the conventional solvation free energies by substituting into our self-consistent thermodynamic cycle. Furthermore, Tissandier's absolute solvation free energy data were used as the training set to refit the Lennard-Jones parameters of OPLS-AA force field for ions. To predict salt solubilities, an assumption of a pseudo-solvent was proposed to characterize the coupling work of a solute with its environment from infinitely diluted to saturated solutions, indicating that the Gibbs energy change of solvation process is a function of ionic strength. Following the self-consistency of the cycle, the newly derived formulas were used to determine the salt solubilities by interpolating the intersection of Gibbs free energy of dissolution and the zero free energy line. The refined ion parameters can also predict the structure and thermodynamic properties of aqueous electrolyte solutions, such as densities, pair correlation functions, hydration numbers, mean activity coefficients, vapor pressures, and the radial dependences of the net charge at 298.15 K and 1 bar. Our method can be used to characterize the solid-liquid equilibria of ions or charged particles in aqueous systems. Furthermore, for highly concentrated strong electrolyte systems, it is essential to introduce accurate water models and polarizable force fields.

Mean ionic activity coefficient of associative electrolyte solutions: A comparison study

The performance of four implicit solvent models for associative and two models for non-associative electrolyte solutions has been compared and analyzed for the mean ionic activity coefficient (MIAC) of electrolytes in the solution. Two of these models for associative electrolyte solutions are based on a chemical approach (Fisher-Levin-Guillot-Guissani (FLGG) and Ebeling-Grigo (EG)), and one of them is based on the reference cavity approximation (Zhou-Yeh-Stell (ZYS)). The last one is based on the thermodynamic perturbation theory (Binding mean spherical approximation, BiMSA). Models without ion-pairing also consist of hard sphere (HS) contribution in addition to the electrostatic contribution from the Debye-Hückel (HS + DH) and the mean spherical approximation (HS + MSA) theories. To this aim, the models’ predictions are compared with the numerical solution of the Poisson-Boltzmann equation, the Monte Carlo simulations, and the experimental data. We have shown that considering the ion pairing results in a better agreement of the predicted MIAC against both simulations and experimental data of 2:2 electrolytes. We have also adjusted the ionic diameter and the distance between ion pairs to the MIAC experimental data and validated the estimated fractions of free ions with the electrical conductivity measurements. We have shown that the FLGG model captures the physics of the system more accurately compared to other models.

Thermodynamic properties of the ternary system (KBr + PEG 4000 + water) by potentiometric method at T = (288.2, 298.2, and 308.2) K

The goal of this research is to evaluate the mean ion activity coefficients (γ±) of the mixtures containing KBr + PEG 4000 + H2O (PEG stands for poly(ethylene glycol)) as well as some other thermodynamic quantities including solvent osmotic coefficients (φ), excess Gibbs energies (GE), standard Gibbs energies of transference (ΔGt0), and primary hydration number (nhydr). The experiment was employed by potentiometric measurement using the cell with two ion-selective electrodes (abbreviated as ISE) at (288.2, 298.2, and 308.2) K, and the mass fraction of PEG 4000 ranged from w = (0 to 0.20) with an increment of 0.05. The type of the cell is as follows:K–ISE | KBr (m), PEG 4000 (w), H2O (1-w) | Br–ISE.Nernst's equation was used to combine the Debye-Hückel and Pitzer models with potentiometric values. Pitzer ion interaction parameters (β0, β1, and Cφ) and Debye-Hückel model parameter (a) were evaluated. Moreover, the Bjerrum parameter (q) was calculated.

Thermodynamics of K2HPO4/d-Sucrose/Water System at 298.15 K: Experiment and Modeling

Thermodynamic properties of K2HPO4/d-sucrose/water ternary solutions were investigated at 298.15 K. Water activity, osmotic coefficient, and electrolyte saturation points are determined by measuring relative humidities in the wide range of K2HPO4 molality (ranging from 0.1 to approximately 5 mol·kg–1) and for various d-sucrose contents (0.1, 0.3, 0.5, 1, 2, 3, 4, 5, and 5.5 mol·kg–1) in the ternary K2HPO4/d-sucrose/water system. The solid state was characterized using powder X-ray diffraction (XRD) and attenuated total reflection Fourier-transform infrared (ATR-FTIR) techniques. The obtained data were then analyzed using the Pitzer-Simonson-Clegg (PSC) model to determine several thermodynamic properties. These properties encompass the stoichiometric ionic mean activity coefficient of K2HPO4, excess Gibbs energy, solubility in aqueous solutions, and the Gibbs energy of transfer of K2HPO4 between water (W) and d-sucrose/water (W+S) mixtures. Furthermore, the interactions among water, d-sucrose, and K2HPO4 are also examined and discussed.

Thermodynamic properties of aqueous lanthanum sulfate solutions taking into account the association

Thermodynamic properties (the mean activity coefficients, γ±, and the relative apparent molar enthalpies, φL) of lanthanum sulfate aqueous solutions, strongly deviating from the Debye-Hückel limiting law even at lowest experimentally accessible concentrations, can easily and quantitatively be reproduced assuming the formation of two sulfate complexes, LaSO4+ and LaSO42-, whose existence was established in numerous complex formation studies. The analysis was carried out using a simple SIT model for the activity coefficients of ions with only the electrostatic Debye-Hückel terms. Excellent agreement between measured values of γ± and φL and the values of equilibrium constants was obtained by using the following values of equilibrium constants and enthalpy changes of reactions of formation of LaSO4+ (La3++SO42-⇄ LaSO4+) and La(SO4)2- (La3++2SO42-⇄ La(SO4)2-): log10K1o = 3.65 ± 0.05; log10K2o = 5.63 ± 0.2, ΔrH1o = 17.6 ± 1.0 kJ mol−1; ΔrH20 = 28.1 ± 4.0 kJ mol−1. The agreement with literature data is excellent for log10K1o and ΔrH1o and satisfactory for log10K2o and ΔrH2o. With these log10K1o, log10K2o, ΔrH1o and ΔrH2o results the experimental γ± and φL values are reproduced quantitatively.