Individual Activity Coefficients Research Articles

Uncertainty Analysis of Time-Integrated Activity Coefficient in Single-Time-Point Dosimetry Using Bayesian Fitting Method.

Calculation of the uncertainty of the individual time-integrated activity coefficient (TIACs) is desirable in molecular radiotherapy. However, the calculation of TIAC's uncertainty in single-time-point (STP) method has never been reported in the literature. This study presents a method based on the Bayesian fitting (BF) to calculate the standard deviation (SD) of individual TIACs in the STP dosimetry. Biokinetic data of 177Lu-DOTATATE in kidneys were obtained from PMID33443063. BF methods with extended objective function, which optimize the fitting using prior knowledge of the function's parameters, were used. Reference TIACs (rTIACs) were calculated by fitting a mono-exponential function to the all-time-point data. The goodness of fit was checked based on the visual inspection and the coefficient of variations (CV) of the fitted parameters < 0.5. BF with relative (BFr) and absolute-based (BFa) variance methods were used to obtain the calculated TIACs (cTIACs) from the STP dosimetry. Performance of the STP method was obtained by calculating the relative deviation (RD) between cTIACs and rTIACs. Visual inspection showed a good fit for all patients with CV of fitted parameters less than 50%. The mean ± SD of cTIAC's %RD were 7.0 ± 25.2 for BFr and 2.6 ± 8.9 for BFa. The range of %CV of the individual cTIAC's SD for BFr and BFa methods was 36-78% and 22-33%, respectively, while the %CV of the rTIAC SD was 0.8-49%. We introduce the BF method to calculate the SD of individual TIACs in STP dosimetry. The presented method might be used as an alternative method for uncertainty analysis in STP dosimetry. The online version contains supplementary material available at 10.1007/s13139-024-00851-8.

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.

Application of the symmetric Poisson-Boltzmann theory to equilibrium thermodynamic properties of primitive model electrolyte mixtures

A symmetric Poisson-Boltzmann theory is applied to a primitive model electrolyte solution containing 7 different ion species. Only 7 coupled equations need to be solved for the mean electrostatic potential around the ions once the uncharged hard sphere radial distribution function is specified. Comparison is made with the individual activity and osmotic coefficients of a simulation study of a sea water mixture. Good overall agreement is found with the simulation results.

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.

Statistical theory of individual activity coefficients of electrolytes including multiple ionic charges

In previous work we developed a new statistical method for calculating the individual activities of ions including the association of ions. Here we study multi-particle electrostatic interactions connected within higher cluster integrals and identify the ionization constants of the mass action law of associating ion clusters. In contrast to Bjerrum and Fuoss, our concept of association is not based on spatial criteria, but instead on the strength of interaction measured in powers of the Bjerrum parameter (e2/D0kBTa; a is contact) and defined by asymptotic properties of the cluster integrals. For ion pair formation our mass action constant is the classical counterpart of Planck's famous hydrogenic partition function. As a rule, the new association constants are smaller than traditional expressions, e.g., by Fuoss and Kraus, in the interesting regions of interaction parameters about fifty percent. Several examples including CaCl2, MgCl2, Na2SO4, K2SO4, LaCl3 and a model of seawater are studied. For several associating electrolytes and seawater, reasonable agreement with experiments and Monte Carlo results is achieved.

Influence of concentration and temperature dependent dielectric constants on the thermodynamics of electrolytes

The symmetric Poisson–Boltzmann theory, the modified Poisson–Boltzmann theory, and the mean spherical approximation are utilized to calculate the osmotic coefficient, and the individual and mean activity coefficients of model KCl, LiCl, NaCl, and NaClO4 electrolytes all dissociated in water where the dielectric constant depends on concentration. The theories are also employed to compute the same thermodynamic quantities for HCl and NaCl solutions where the dielectric constant is now temperature dependent. In each situation the theoretical predictions are compared with the corresponding exact benchmark Monte Carlo simulation results. The mean spherical approximation and the modified Poisson–Boltzmann theory reproduce the simulation data for osmotic coefficient quantitatively and that for the mean activity almost quantitatively. The individual activities are accurately represented by the modified Poisson–Boltzmann theory but only qualitatively by the symmetric Poisson–Boltzmann and the mean spherical approximation. In general, the symmetric Poisson–Boltzmann theory shows good agreement with the simulations at low concentrations but deviates at higher concentrations.

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.

Activity coefficient of multi-ions and Gibbs free energy calculation during electrochemical nitrate reduction in synthetic and actual wastewater

This study reports the calculation of individual ion's activity coefficient and total minimum Gibbs energy for electrochemical denitrification of nitrate ions in synthetic and actual wastewater. The concentration of nitrate ion continuously decreased during treatment via the electrochemical method. During the treatment, nitrate ions got reduced into nitrite and ammonium ion. The individual ion activity coefficient was examined by applying Pitzer, Lin and Lee modified Debye-Huckel theory which is based on the interaction of ion-ion (long-range, i.e., LR interaction-electrostatic effect) and ion-water (short-range, i.e., SR interaction-solvation effect) in the aqueous solution. The equilibrium composition of these ions was calculated by minimizing the error between total minimum Gibbs energy calculated by using experimental and predicted compositions. The individual ion activity coefficient, which was calculated through this model, was used to measure the total minimum Gibbs energy (Gtotal). Variation of individual ion activity coefficient and Gtotal with initial nitrate ion concentration, applied current density, initial pH, and concentration of NaCl as electrolyte has been studied. It was found that the experimental and calculated composition of nitrate, nitrite, and ammonium ions were in good agreement for electrochemical denitrification of synthetic and actual wastewater.

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.

Robust Spatial Modeling of Thermodynamic Parameters in a Full-Scale Reverse Osmosis Membrane Channel.

Full-scale reverse osmosis (RO) units usually consist of a set of pressure vessels holding up to six (1 m long) membrane modules in series. Since process parameters and water composition change substantially along the filtration channel in full-scale RO units, relevant thermodynamic parameters such as the ion activities and the osmotic coefficient change as well. Understanding these changes will lead to more accurate fouling prediction and to improvement in process and equipment designs. In this article, a rigorous thermodynamic model for RO concentrates in a full-scale module is developed and presented, which is capable of accounting for such changes. The change in concentrate composition due to permeation of water and ions is predicted locally in the membrane filtration channel. The local ionic composition is used to calculate the local activity coefficient and osmotic coefficient along the membrane channel through the Pitzer model for each modeled anion and cation. The approach developed was validated against related literature data, showing that Pitzer coefficient predictions were satisfactory. The spatial variation model was verified experimentally. It was found under the modeled conditions of high recovery that individual solute activity coefficients could be diminished up to 65%, in our case for sulfate, from their initial value from the membrane inlet to the outlet, and the water osmotic coefficient increased 3% as concentrate salinity increased from the membrane inlet to the outlet. Modeled at moderate recovery, the sulfate still achieved a statistically significant drop of 34% and an opposing trend of a decrease of 0.5% for the osmotic coefficient. These variations in internal water chemistry along the channel can significantly impact predicted recovery, fouling propensity, and permeate quality. Fouling prediction with our approach was also assessed through a theoretical fouling index to demonstrate the significance of ion activity over concentration-based calculations. Additionally, data from a pilot plant RO filtration channel was used to carry out a sensitivity analysis to show the capability of the developed model.

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.

Thermodynamics of primitive model electrolytes in the symmetric and modified Poisson-Boltzmann theories. A comparative study with Monte Carlo simulations

Osmotic coefficients, individual and mean activity coefficients of primitive model electrolyte solutions are computed at different molar concentrations using the symmetric Poisson-Boltzmann and modified Poisson-Boltzmann theories. The theoretical results are compared with an extensive series of Monte Carlo simulation data obtained by Abbas et al. [Fluid Phase Equilib., 2007, 260, 233; J. Phys. Chem. B, 2009, 113, 5905]. The agreement between modified Poisson-Boltzmann predictions with the "exact" simulation results is almost quantitative for monovalent salts, while being semi-quantitative or better for higher and multivalent salts. The symmetric Poisson-Boltzmann results, on the other hand, are very good for monovalent systems but tend to deviate at higher concentrations and/or for multi-valent systems. Some recent experimental values for activity coefficients of HCl solution (individual and mean activities) and NaCl solution (mean activity only) have also been compared with the symmetric and modified Poisson-Boltzmann theories, and with the Monte Carlo simulations.

Poisson-Fermi modeling of ion activities in aqueous single and mixed electrolyte solutions at variable temperature.

The combinatorial explosion of empirical parameters in tens of thousands presents a tremendous challenge for extended Debye-Hückel models to calculate activity coefficients of aqueous mixtures of the most important salts in chemistry. The explosion of parameters originates from the phenomenological extension of the Debye-Hückel theory that does not take steric and correlation effects of ions and water into account. By contrast, the Poisson-Fermi theory developed in recent years treats ions and water molecules as nonuniform hard spheres of any size with interstitial voids and includes ion-water and ion-ion correlations. We present a Poisson-Fermi model and numerical methods for calculating the individual or mean activity coefficient of electrolyte solutions with any arbitrary number of ionic species in a large range of salt concentrations and temperatures. For each activity-concentration curve, we show that the Poisson-Fermi model requires only three unchanging parameters at most to well fit the corresponding experimental data. The three parameters are associated with the Born radius of the solvation energy of an ion in electrolyte solution that changes with salt concentrations in a highly nonlinear manner.

CALCULATION METHOD OF REAL ACTIVITY COEFFICIENTS OF IONS BY EXAMPLE OF HYDROGEN AND CHLORINE IONS IN AQUEOUS HCl SOLUTIONS

In this paper, we proposed a calculating method of the real individual activity coefficients. To calculate the energy of the ion, transfer through the interface we offered to use the value of the electrolyte solution surface potential, and to use electronegativity elements values for accounting contribution of the charge transfer energy between the surface and the bulk solution. To verify the proposed equation, the activity coefficients of hydrogen and chlorine ions in an aqueous HCl solutions were calculated over a wide range of electrolyte concentrations. The Debye-Hückel equation within 2nd approximation describes satisfactorily the dependence of the activity coefficient on the electrolyte concentration for anions, but not reproduce the minimum on the curve for cations. This dependence for the cation H⁺ is most extremal, and namely it is the reason of the choice of HCl for the model calculation. The calculated values were compared with the data obtained from the results of determining the activity coefficients by experimental method of "vertical jet" [1]. The comparison showed that our proposed equation reproduces the character of the concentration dependence of real activity coefficients not for anion only but for cation as well. Previously we found, that at low concentrations of solutes up to 0.1 mol/kg the values of some thermodynamic characteristics are strongly determined by not the volume concentration of the electrolyte, but the restructuring of the solvent structure in the surface layer of the solution, and therefore, the changing the energy state of the molecules at the interphase surface. Hence, the number of cations and anions at the surface in the areas of high and low bulk concentrations of electrolyte also differ greatly. Obviously, the small variation between calculated and experimental values of the real activity coefficients may be connected with necessity of using at calculation the bulk and surface concentration of ions. For citation:Fedorovа A.A., Sharonov N.Yu., Filippov D.V. Calculation method of real activity coefficients of ions by example of hydrogen and chlorine ions in aqueous HCl solutions. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2017. V. 60. N 3. P. 29-35.