Ionic Activity Coefficients Research Articles

Investigation of the Limits of the Linearized Poisson–Boltzmann Equation

This work presents a comparison between a numerical solution of the Poisson-Boltzmann equation and the analytical solution of its linearized version through the Debye-Hückel equations considering both size-dissimilar and common ion diameters approaches. In order to verify the limits in which the linearized Poisson-Boltzmann equation is capable to satisfactorily reproduce the nonlinear version of Poisson-Boltzmann, we calculate mean ionic activity coefficients for different types of electrolytes as various temperatures. The divergence between the linearized and full Poisson-Boltzmann equations is higher for lower molalities, and both solutions tend to converge toward higher molalities. For electrolytes of lower valencies (1:1, 1:2, 2:1, and 1:3) and higher distances of closest approach, the full version of the Debye-Hückel equation is capable of representing the activity coefficients with a low divergence from the nonlinear Poisson-Boltzmann. The size-dissimilar full version of Debye-Hückel represents a clear improvement over the extended version that uses only common ion diameters when compared to the numerical solution of the Poisson-Boltzmann equation. We have derived a salt-specific index (Θ) to gradually classify electrolytes in order of increasing influence of nonlinear ion-ion interactions, which differentiate the Debye-Hückel equations from the nonlinear Poisson-Boltzmann equation.

On the derivations of the Debye–Hückel equations

This work presents the derivations of basic thermodynamic properties and activity coefficients equations from the linearised Poisson–Boltzmann equation. We consider two main approaches, the first one is based in classical thermodynamics, which has been used in the original work of Debye and Hückel, leading to the model which has been an important cornerstone of electrolyte thermodynamics since its original publication in 1923. The second approach relies on more modern derivations based on statistical mechanics, the so-called charging approaches. Both derivation routes have differences and shortcomings. We demonstrate the necessary steps to reach all original models derived from the Debye–Hückel model and further explore their capabilities and limitations concerning individual ion, and mean ionic activity coefficients for different size dissimilarities scenarios between ions. One immediate conclusion is that there is an unnecessary consideration in the Debye and Hückel derivation which is cancelled by another one they have made, leading to a correct expression for the activity coefficient. Also, the long-lasting consideration that both the Debye and Güntelberg charging processes lead to the same thermodynamic properties is demonstrated to be inaccurate, as it is rigorously true only when a common distance of closest approach is used.

Temperature dependence and short-range electrolytic interactions within the e-PPC-SAFT framework

Understanding and modelling electrolyte systems is of interest both in an industrial context and from a fundamental point of view. Their importance is related to their key role played in traditional and emerging industries, and to the ubiquitous presence of saline solutions in biological, chemical, and geological environments. The development of electrolyte equations of state has advanced greatly in recent years, with the aim to describe the properties of electrolyte systems using a limited number of parameters, in a way that allows describing accurately the equilibrium properties of these solutions. Here, we explore the temperature and salt concentration dependence of various PPC-SAFT-based models, using simple alkali halide salts (primarily NaCl). The dispersion and association free-energy terms characteristic of SAFT-like equations are compared for the description of the ion-solvent non-Coulombic interactions. Several formulations of the dielectric constant are also investigated. In all models the MSA term is used to account for the electrostatic interactions between the ions and the Born term is added to model the ion-solvent electrostatic interactions. The molecular model parameters are determined by comparison to experimental enthalpy of solution, activity coefficient, osmotic coefficient and apparent molar volume data.We find that equivalent results can be obtained with or without incorporating a explicit salt concentration dependence in the treatment of the dielectric constant; the most accurate results (smallest average absolute relative deviation) when comparing to experimental the data chosen are found in the case that salt concentration is not explicitly considered. For each approach, the most sensitive parameters are identified, which makes it possible to reduce the number of adjustable parameters without significantly affecting the overall quality of the model. We find that both the approach incorporating a dispersion term and that using an association term in the treatment of short-range interactions yield similar results. Unfortunately, none of the models can describe accurately and with physically meaningful parameters the correct low-temperature trend of the mean ionic activity coefficient observed experimentally.

Thermodynamic study of interfacial tension between oil and aqueous phases composed of ionic liquids and brine

The first fundamental step in determining the physicochemical properties of an equilibrium system is to determine the activity coefficient of electrolyte and non-electrolyte ions. Based on understanding the importance of activity coefficient in thermodynamic systems in this study, in order to predict interfacial tension between oil and aqueous phases composed of ionic liquids and brine, a modified thermodynamic equation based on concentration and coefficient of activity of ionic liquids is defined. For this study, the Extended UNIQUAC model is desired and its adjustable parameters are optimized with Genetic + PSO algorithm. The modified model has practical features such as investigating the effect of concentrations of salts in the water of oil fields formation on the interfacial tension of the system, investigating the effect of concentrations of various organic compounds such as ionic liquids on the interfacial tension of the system and investigating the interaction energy between organic and inorganic ions. In this study, the optimization of the modified thermodynamic equation to predict the interfacial tension of solutions containing [C8Py][Cl], [C18Py][Cl], [C12mim][Cl] and [C18mim][Cl] with the presence of brine and distilled water is investigated. Also, the effect of ionic strength of the solution in 32 equilibrium systems on interfacial tension is investigated. According to the optimization results of this study, the design of a computer program can be considered to predict the interfacial tension with the presence of ionic liquids and salts.

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.

Isopiestic determination of osmotic coefficients in the ionic strength range Im = (0.9670–2.2160) mol⋅kg−1 and activity coefficients determined by electromotive force measurements in the range Im = (0.0897–1.0054) mol⋅kg−1 of the {yKCl + (1 – y) K2HPO4}(aq) system at T = 298.15 K

The mean activity coefficients of KCl in the ternary system {y KCl + (1 – y) K2HPO4}(aq) were experimentally determined at T = (298.15 ± 0.01) K by the electromotive force measurement (EMF method) using the cell of the type, K–ISE∣KCl(mKCl), K2HPO4(mK2HPO4)∣Ag∣AgCl (ISE = ion-selective electrode), in the range of total ionic strength of the solution Im = (0.10 to 1.00) mol·kg−1 at different ionic strength fractions of KCl being y = (0.1011; 0.1997; 0.3016; 0.4027; 0.5006; 0.6018; 0.7018; 0.8051; 0.8988; 1). In addition, the osmotic coefficients of the system have been measured at the same temperature by the isopiestic vapor pressure method over the range of ionic strengths from (0.96698 to 2.21602) mol⋅kg−1 using KCl(aq) as the reference solution, at ionic strength fractions of KCl being y = (0.2063; 0.3849; 0.6099; 0.8011; 1). Values of the mean ionic activity coefficients of KCl and osmotic coefficients in the system {yKCl+(1–y)K2HPO4}(aq) were treated by an extended form of Pitzer’s ion-interaction model, Scatchard’s neutral-electrolyte model and Clegg–Pitzer–Brimblecombe’s equations based on the mole-fraction-composition scale. All three models gave similar quality representations of both sets of experimental results. There are two sets of previously published isopiestic data that were taken into consideration during treatment. The model parameters were fitted and optimized to all available thermodynamic data from EMF and isopiestic measurements to obtain an accurate and self-consistent description of properties and to form a set of parameters that can be used in a wide range of ionic strength of the system. The standard deviation of the fit for the ionic strength range from Im = (0.0897 to 10.4350) mol·kg−1 with two optimized mixing parameters of Pitzer’s model without higher–order electrostatic terms was for the mean ionic activity coefficients of KCl and osmotic coefficients, s.d. (γ±) = 3.5⋅10–3 and s.d. (ϕ) = 4.7⋅10–3 respectively, with two optimized mixing parameters of the Scatchard model standard deviations were s.d. (γ±) = 4.9⋅10–3 and s.d. (ϕ) = 5.0⋅10–3 and with Clegg–Pitzer–Brimblecombe’s model higher–order electrostatic terms not included, standard deviations were s.d. (γ±) = 4.2⋅10–3 and s.d. (ϕ) = 4.8⋅10–3.

Thermodynamic modeling of aqueous polyelectrolyte solutions with mixed-valent counterions

Thermodynamic modeling of aqueous polyelectrolyte solutions with salts is of significant interest for many industrial applications. This study applies the polyelectrolyte Nonrandom Two-liquid activity coefficient model to aqueous polyelectrolyte solutions with mixed-valent counterions. A modified Delocalized Binding Theory was proposed to determine the polyion condensation fractions of the mixed counterions. This modified theory accounts for the electrostatic binding of the counterions on the polyion, the dissociation entropy of the counterions, and the electrostatic interactions between the uncondensed ionic species. Given the polyion condensation fractions, the critical value of Manning’s parameter ξ and the amounts of uncondensed polyions and counterions can be calculated along with the activity coefficients of mobile ions. The model successfully correlates experimental data for various aqueous polyelectrolyte systems with mixed-valent counterions.

Influence of fixed charge concentration and water uptake on ion sorption in AMPS/PEGDA membranes

Water content and charge density strongly influence ion sorption in charged membranes. However, isolating the individual effect of each parameter on ion sorption can be difficult due to experimental challenges associated with independently controlling these two variables during membrane synthesis. This frustrates development and validation of fundamental models and structure/property relationships. In this study, three series of uncharged and negatively charged hydrogel membranes were prepared using poly(ethylene glycol) diacrylate (PEGDA) as a cross-linker and 2-acrylamido-2-methylpropanesulfonic acid (AMPS) as an ionic monomer. Water content and charge density were controlled by varying AMPS content and PEGDA chain length. Ion sorption in these materials was measured using aqueous NaCl solutions at concentrations ranging from 0.01 to 1.0 M. At constant charge density, salt sorption increased as water content increased. At fixed water content, salt sorption decreased as charge density increased. Salt sorption in membranes with the highest water content and charge density was predicted using the Donnan/Manning model, which accounts for solution and membrane phase thermodynamic non-idealities. However, this approach could not predict sorption in weakly charged and uncharged membranes, presumably due to additional membrane phase thermodynamic non-idealities associated with ion solvation and uncharged polymer segments. • High ion activity coefficients in uncharged membranes suggest ions have a much greater affinity for aqueous solution than swollen, uncharged membranes. • The Donnan/Manning model predicts experimental sorption well for highly hydrated and/or highly charged membranes. • Model predictions agree poorly with experimental data for membranes that are less hydrated, weakly charged, or uncharged due to phenomena not accounted for in Manning's theory (e.g., dielectric exclusion and non-idealities introduced by polymer segment-water-ion interactions).

Derivation of the solvation effect-incorporated Poisson-Boltzmann equation

Based on molecular kinetic consideration relating to dynamic properties for both the ionic atmosphere and ion solvation shell, solvation effect-incorporated PB (SEI-PB) equation was derived.Consequently, there appeared a solvated ionic screening thickness containing not only the ionic atmosphere thickness (Debye length), but also a new screening thickness of the solvation shell in both the linearized SEI-PB equation and its solution. The thickness of solvated ionic atmosphere has physical meaning of the screening thickness taking into account the volumetric effect of ions together with ionic solvation effect.As a result of the solution of linear SEI-PB equation, the total electric potential was divided into two terms: the first one was derived from the ions, which were affected by the solvation shells and the second one was derived from the solvation shells which were affected by the ions. An analytical equation for ion–solvent molecular dipole moment interaction (ISMDI) potential energy was established based on the solution of linear SEI-PB equation.Also by using the linear SEI-PB equation, analytical expressions of the activity coefficients for ions and electrolytes as well as solvent were derived and an equation for estimating the mean lifetimes of solvation shells according to ionic species was proposed and applied to estimate those of ionic hydration shells. Estimated mean lifetimes for ionic hydration shells according to ionic species were compatible with the previous XRD, NMR data and MD simulations.We confirmed that the SEI-PB equation has peculiarity containing not only solvation effect, but also dielectric saturation and that the dielectric saturation is connected implicitly with ionic solvation effect.

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.

Chemical speciation, thermodynamic properties, and salt solubility in aqueous hydrogen fluoride at various temperatures part I: Liquid chemical composition and thermodynamics properties of the HF-H2O system

A thermodynamic model of the aqueous hydrofluoric acid solutions is parameterized and applied to represent the osmotic coefficient, ionic mean activity coefficient, pH, and liquid chemical speciation of system HF-H2O. The developed model is suitable for temperature varying from 273.15 K up to 473.15 K and concentrations from infinite dilution to 20 mol.kg−1. Many obvious computational errors and a critical analysis of the measurement data of the aqueous solutions of the HF-H2O system in the literature are recorded and discussed.

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.

Importance of the Relative Static Permittivity in electrolyte SAFT-VR Mie Equations of State

The influence and importance of the relative static permittivity (RSP) in electrolyte equations of state is examined for the case of aqueous sodium chloride. Using the SAFT-VR Mie model, the Debye-Hückel (DH) or Mean-Spherical Approximation (MSA) terms, as well as the Born-solvation term, are used to formulate an electrolyte equation of state. The RSP is obtained from a variety of models, each differing in their dependencies; we consider constant, temperature-, density- and composition-dependent models. For a fair comparison between different combinations of electrostatic and RSP models, all ion-related parameters are obtained a priori. A novel combining rule is proposed to obtain the unlike parameters between solvents and ions; its reliability is examined for a variety of electrolyte systems. We also compare its performance relative to parameterised electrolyte models. Both the DH and MSA terms yield similar results for almost all properties and conditions. The RSP models used have the more-significant impact. Liquid densities and solvent saturation pressures showed limited changes between RSP models whereas osmotic coefficients, mean ionic activity coefficients and carbon dioxide solubilities observed drastically different behaviour. Analysing the contributions of the various terms to the activities of each species in an electrolyte mixture reveals an important balance between the Born-solvation and the DH or MSA terms where the RSP models have a significant influence over this balance, particularly when these carry a solvent- or ion-composition dependence.

Estimation of e-NRTL binary interaction parameters and its impact on the prediction of thermodynamic properties of multicomponent electrolyte systems

The main objective of the current work is to obtain and validate the binary interaction parameters of the e-NRTL model for different salts in water, using the data of mean ionic activity coefficients vs. molality compiled by Robinson and Stokes (Electrolyte Solutions, Dover Publications, 2012). This way, the interaction parameters τsalt,water and τwater,salt of the e-NRTL thermodynamic model for 171 binary (salt + water) electrolyte systems have been calculated by non-linear regression of the data available in the literature. The fitting was performed by the Marquardt-Levenberg algorithm which was computationally implemented in an Excel VBA homemade program. The e-NRTL equation was solved by using two different procedures: (i) included in the Excel VBA code or, (ii) implemented in Aspen Plus by linking the process simulator and the Excel VBA code via automation as the third-party software. The interaction parameters obtained by both approaches were able to adequately fit the experimental results. They were successfully validated by the thermodynamic prediction of multicomponent systems reported by other authors. Additionally, the current predictions were compared with those obtained with the parameters contained in the Aspen Properties database showing a significant improvement for several systems. Finally, this work demonstrates that it is possible enhancing the quality of the prediction of multicomponent systems with proper values of the binary electrolyte interaction parameters, without considering electrolyte-electrolyte interactions foreseen in the original e-NRTL model, which makes possible the simulation of processes containing a variety of chemical species regardless their nature.

Association‐based activity coefficient model for electrolyte solutions

AbstractThis work presents an association‐based activity coefficient model that explicitly considers the solution nonideality due to associations among ions and solvent species. Built upon the electrolyte nonrandom two‐liquid (eNRTL) model, the model greatly improves the accuracy of eNRTL model for strongly associating electrolyte solutions due to presence of ionic species with high surface charge density. The model successfully correlates mean ionic activity coefficients of 46 aqueous single‐salt systems from 10 cations and 5 anions at 298.15 K up to their solubility limits. With the ion‐specific association parameters identified, the model accurately predicts activity and osmotic coefficients for aqueous mixed‐salt systems at 298.15 K. The temperature dependence of the model results has also been examined at 273–373 K. With superior accuracy over a wide range of concentration and temperature, the model represents a major advancement over eNRTL model and has a great potential to be a next‐generation model for electrolyte solutions.

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.

Studies of thermodynamics of micellization and pattern of self-association of decyltrimethylammonium bromide in aqueous solutions at 298.15 K

Using recently reported density, thermal coefficient of expansion and internal pressure data along with the literature osmotic coefficient and enthalpy data and adopting Mass–action model, the properties of decyltrimethylammonium bromide (C10TAB) in water are studied at 298.15 K. The number density, aggregation number, equilibrium constant, standard free energy changes due to micellization, standard entropy changes due to micellization and mean ionic activity coefficient are calculated in pre–micellar and post–micellar regions of concentration assuming that the cationic surfactant C10TAB is fully dissociated in water. Characteristic pattern of association are examined through the models/concepts developed by Tanford, Mukerjee as well as by O. P. Ts’o and noted that hydrophobic interactions between the non–polar chains and water structural effects (in the form of ion–solvent and ion–ion interactions) govern the equilibrium properties of such solutions. The critical micelle concentration (cmc) value and aggregation number value agree with the literature data. It has been observed that aggregation number value decreases slightly, equilibrium constant value goes through a minimum, standard free energy changes due to micellization parameter go through a maximum while positive magnitude of entropy changes due to micellization are observed as a function of concentration of C10TAB. The variation of mean ionic activity coefficient of the salt ions with concentration indicates deviation from ideal–associative solution behaviour. All these results are interpreted on the basis of monomer–micelle equilibria existing in solution phase, affecting the solution properties through electrostatic, hydrophobic solvation and hydrophobic interaction effects.

Toward the understanding of water-in-salt electrolytes: Individual ion activities and liquid junction potentials in highly concentrated aqueous solutions.

Highly concentrated electrolytes were recently proposed to improve the performances of aqueous electrochemical systems by delaying the water splitting and increasing the operating voltage for battery applications. While advances were made regarding their implementation in practical devices, debate exists regarding the physical origin for the delayed water reduction occurring at the electrode/electrolyte interface. Evidently, one difficulty resides in our lack of knowledge regarding ion activity arising from this novel class of electrolytes, which is necessary to estimate the Nernst potential of associated redox reactions, such as Li+ intercalation or the hydrogen evolution reaction. In this work, we first measured the potential shift of electrodes selective to Li+, H+, or Zn2+ ions from diluted to highly concentrated regimes in LiCl or LiTFSI solutions. Observing similar shifts for these different cations and environments, we establish that shifts in redox potentials from diluted to highly concentrated regimes originate in large from an increased junction potential, which is dependent on the ion activity coefficients that increase with the concentration. While our study shows that single ion activity coefficients, unlike mean ion activity coefficients, cannot be captured by any electrochemical means, we demonstrate that the proton concentration increases by one to two orders of magnitude from 1 to 15-20 mol kg-1 solutions. Combined with the increased activity coefficients, this phenomenon increases the activity of protons and thus increases the pH of highly concentrated solutions which appears acidic.

Aqueous ion partitioning in Nafion: Applicability of Manning's counter-ion condensation theory

Nafion is a well-studied polymer of significant interest as a fuel cell membrane due to rapid transport of protons, which is often ascribed, at least in part, to its phase-separated morphology. The literature detailing ion sorption and transport in Nafion diverges considerably from that of water purification membranes, with contrasting analyses and models. To help bridge these gaps and facilitate comparisons of membranes with various structures, this study presents measurements of ion activity coefficients and sorption in Nafion 117 equilibrated with aqueous solutions of NaCl, MgCl2, and Na2SO4, which are relevant to water purification. The experimental data are compared to a theoretical framework based on Manning's counter-ion condensation theory and Donnan theory. Good agreement between the model and data is observed for NaCl and MgCl2, using no adjustable parameters. However, the model significantly underpredicts Na2SO4 solubility, which is attributed to ion pairing effects. The results highlight the general applicability of counter-ion condensation theory to highly charged polymers, including those exhibiting nanoscale ordering, while revealing fundamental limitations related to ion pairing.