Single-ion Activity Coefficients Research Articles

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.

Investigation of the performance of e-CPA for a wide range of properties for aqueous NaCl solutions

In the past years a number of parametrizations of the e-CPA model have been published. The e-CPA model has shown promising results for electrolyte solution properties. Several of these parametrizations have been investigated in this work for a wide range of properties for aqueous NaCl solutions. Previously published e-CPA parameter sets, which made use of a volume translation to be able to correlate the density, were found to show some inconsistencies, including a suboptimal temperature dependency of the density. A new parametrization is proposed, that is able to reproduce the density without the need of a volume translation, and without loss of accuracy of other properties. This parameter set, together with the previously published parameter sets were tested for their predictability of single ion activity coefficients, and it was found that all of them are incapable of getting the correct trend regarding which ion that is above the mean and which is below. An alternative parametrization has, however been shown to be able to change this without much loss of accuracy with respect to the other properties.

Single ion activity coefficients of chloride ions in aqueous sodium chloride and magnesium chloride estimated potentiometrically based on ionic liquid salt bridge at 298 K

The single ion activity coefficients of Cl-,γCl-, in aqueous NaCl and MgCl2 solutions have been estimated at 25 °C potentiometrically by use of galvanic cells with ionic liquid salt bridge consisting of tributyl(2-methoxyethyl) phosphonium bis(pentafluoroethanesulfonyl) amide. With increasing the ionic strength, Im, the γCl- decreases monotonously up to ca. 1 and 1.5 mol kg−1 for NaCl and MgCl2 solutions, respectively.The observed monotonous decrease is distinctively different from the leveling-off of the values greater than 0.5, previously reported, and is in agreement with the prediction by the Franekel’s Debye-Hückel-Smaller-ion Shell (DH-SiS) model that shows no minimum in γCl- versus Im curves in both NaCl and MgCl2 solutions. Quantitatively, the measured γCl- values are consistently smaller than those predicted by the DH-SiS model.

Activity Coefficients of Concentrated Salt Solutions: A Monte Carlo Investigation

Monte Carlo (MC) simulations were used to calculate single ion and mean ionic activity coefficients and water activity in concentrated electrolytes and at elevated temperatures. By using a concentration dependent dielectric constant, the applicability range of the MC method was extended to 3 mol·L−1 or beyond, depending on the salt. The calculated activity coefficients were fitted to experimental data by adjusting only one parameter, i.e., the cation radius. Fitted ionic radii obtained by such a procedure indicate the extent of cation–anion interaction in a salt solution. For example, the fitted radii of Li+ and Na+ in LiClO3 and NaClO3 indicate that Li+ is strongly hydrated and has a weak interaction with the ClO3− ion whereas Na+ forms ion pairs and loses its hydration. The single ion activity coefficients for protons and chloride ions in HCl were calculated by MC simulations and compared with experimental values obtained by ion selective electrodes. The calculated single ion activity coefficients for protons and chloride ions are much lower and higher, respectively, than the experimental values. However, the mean activity coefficients of HCl obtained by the MC simulations, ion selective electrodes and vapor pressure measurements are in good agreement. In the case of NaCl and KCl the calculated single ion activity coefficients of Na+, K+, and Cl− are much closer to the values obtained by ion selective electrodes. The results in HCl indicate that the hydrated proton is large and includes the chloride ion within the hydration shell, i.e., the apparent size of the chloride ion is negligible.

Modeling Acid-Base by Minimizing Charge-Balance.

In this study, we show that equilibrium pH can be obtained for any specified fluid with any number of buffers and dissociations. This is done by root finding in the equation for charge balance. We demonstrate that this equation is monotonic in proton concentration for conceivable buffers. We show that the total charge on any buffer is a function of only the total buffer concentration and pH, given the thermodynamic dissociation constants. Using the Davies’ equation as a placeholder for single-ion activity coefficients as a function of charge and ionic strength, we develop an iterative algorithm, whereby the apparent dissociation constants are updated from the thermodynamic dissociation constants, and from this, the equilibrium is also identified in the nonideal state. We show how this algebra leads to guaranteed conservation of both thermodynamic dissociation constants and total buffer concentrations because the distribution of buffer species is fixed by the updated dissociation constants, actual pH, and total buffer concentration. Strong ions are assumed to contribute fixed charges. In order to concentrate on the process of modeling the equilibrium pH alone, this algorithm is examined against a series of theoretical results in which the Davies’ equation was given the same status. However, a large sample of clinical pH measurements is also examined. To enhance the practical utility, CO2 and albumin are present as the default condition. We developed “ABCharge”, a package in R, an open source language. The main function returns pH, activity coefficients, buffer species distribution, ionic strength, and charge balance for both the ideal and nonideal cases, for any mixture of any buffers with any number of known thermodynamic dissociation constants. Our algorithm can be updated if a more reliable and practical assessment of single-ion activities becomes available. Can Stock Photo/miceking

An improved theory of the electric conductance of ionic solutions based on the concept of the ion-atmosphere's smaller-ion shell.

Although the electric conductivity of ionic solutions is of great importance in science and technology, complete understanding of the physical factors and principles underlying ion and charge transport in solution has so far been missing. The Debye-Hückel-Onsager (DHO) theory of the equivalent ionic conductivity (λi for single ion i, Λ for a binary electrolyte) is a limiting law of only a few electrolyte valence families, and its extension to solution of finite concentration has not been entirely successful; at most, it has led to semiempirical expressions of λi or Λ vs. concentration (c), incorporating adjustable parameters of no clear physical meaning. Improving the description of the ion atmosphere and including in the conductivity model the smaller-ion shell (SiS) of this atmosphere (D. Fraenkel, Mol. Phys., 2010, 108, 1435) allow now an extension of the DHO theory to a "DHO-SiS" theory that offers analytic expressions for λi and Λ as a function of c without adjustable parameters. The fit with experiment is very good, often excellent, for many electrolytes of the various valence families. λi values can be converted to single-ion activity coefficients, γis, suggesting that the latter ionic factors do have physical validity, contrary to what many literature sources claim. Together with the recently proposed theory of the limiting conductivities, λ0i and Λ0 (D. Fraenkel, Mol. Phys., 2018, 116, 2271), the new theory offers a simple and comprehensive ion conductivity treatment that improves upon existing theoretical treatments, with λi and Λ expressed by ionic sizes instead of λ0i and Λ0.

Traceability of pH to the Mole

Free acidity of aqueous solutions was initially defined in 1909 by Søren Peter Lauritz Sørensen as pH = −lgcH+ (c/mol·dm−3 or m/mol·kg−1 of the free hydrogen ions in solution, H+) soon (1910) was changed to pH = paH+ = −lgaH+, integrating the new concepts of activity, ai and activity coefficient γi, for the ionic species i under concern, H+ in this case; it is ai = −lg(miγi). Since individual ions do not exist alone in solution, primary pH values cannot be assigned solely by experimental measurements, requiring extra thermodynamic model assumptions for the activity coefficient, γH+, which has put pH in a unique situation of not being fully traceable to the International System of Units (SI). Also the concept of activity is often not felt to be as perceptible as that of concentration which may present difficulties, namely with the interpretation of data. pH measurements on unknown samples rely on calibration of the measuring setup with adequate reference pH buffers. In this work, the assignment of pH values to buffers closely matching the samples, e.g., seawater, is revisited. An approach is presented to assess the quantity pmH+ = −lgmH+ profiting from the fact that, contrary to single ion activity coefficients, mean activity coefficients, can be assessed based on experimentally assessed quantities alone, γExp ±, thus ensuring traceability to the mole, the SI base unit for amount of substance. Compatibility between γExp ± and mean activity coefficient calculated by means of Pitzer model equations, γPtz ±, validates the model for its intended use.

Poisson–Fermi model of single ion activities in aqueous solutions

A Poisson–Fermi model is proposed for calculating activity coefficients of single ions in strong electrolyte solutions based on the experimental Born radii and hydration shells of ions in aqueous solutions. The steric effect of water molecules and interstitial voids in the first and second hydration shells play an important role in our model. The screening and polarization effects of water are also included in the model that can thus describe spatial variations of dielectric permittivity, water density, void volume, and ionic concentration. The activity coefficients obtained by the Poisson–Fermi model with only one adjustable parameter are shown to agree with experimental data, which vary nonmonotonically with salt concentrations.

Meaning and Measurability of Single-Ion Activities, the Thermodynamic Foundations of pH, and the Gibbs Free Energy for the Transfer of Ions between Dissimilar Materials.

Considering the relationship between concentration and vapor pressure (or the relationship between concentration and fugacity) single-ion activity coefficients are definable in purely thermodynamic terms. The measurement process involves measuring a contact potential between a solution and an external electrode. Contact potentials are measurable by using thermodynamically reversible processes. Extrapolation of an equation to zero concentration and ionic strength enables determination of single-ion activity coefficients. Single-ion activities can be defined and measured without using any extra-thermodynamic assumptions, concepts, or measurements. This method could serve as a gold standard for the validation of extra-thermodynamic methods for determining single-ion activities. Furthermore, it places the concept of pH on a thermodynamically solid foundation. Contact potential measurements can also be used to determine the Gibbs free energy for the transfer of ions between dissimilar materials.

Computing excess functions of ionic solutions: the smaller-ion shell model versus the primitive model. 1. Activity coefficients.

The present study compares the Monte Carlo (MC) simulation of the primitive model (PM) ( Abbas, Z. et al. J. Phys. Chem. B 2009 , 113 , 5905 ), by which activity coefficients of many binary ionic solutions have been computed through adjusting ion-size parameters (ISPs) for achieving best fit with experiment, with a parallel fit and ISP adjustment, employing the Smaller-ion Shell (SiS) treatment ( Fraenkel, D. Mol. Phys. 2010 , 108 , 1435 ), a Debye-Hückel type theory ("DH-SiS") considering counterions of unequal size. DH-SiS is analogous to the unrestricted PM (UPM), so the comparison is with the MC simulation of the UPM, "MC-UPM". Among the representative electrolytes NaCl, KCl, NaClO4, CaCl2, Ca(ClO4)2, and LaCl3, in water at 25 °C, the 1-1 electrolytes exhibit a far better fit quality for DH-SiS than for MC-UPM, and the fit extends to higher concentration. Moreover, theoretical single-ion activity coefficients derived from DH-SiS agree with experimental estimation for solutions of NaCl, CaCl2, and other electrolytes ( Fraenkel, D. J. Phys. Chem. B 2012 , 116 , 3603 ), whereas parallel MC-UPM predictions are at odds with experiment. Additional advantages of DH-SiS over MC-UPM are in (a) employing co-ion ISPs that are usually equal to the crystallographic ion diameters and (b) easily applying ISP nonadditivity in adjusting counterion ISPs.

Theoretical analysis of aqueous solutions of mixed strong electrolytes by a smaller-ion shell electrostatic model

In spite of the great importance of mixed electrolytes in science and technology, no compelling theoretical explanation has been offered yet for the thermodynamic behavior of such systems, such as their deviation from ideality and the variation of their excess functions with ionic composition and concentration. Using the newly introduced Smaller-ion Shell treatment - an extension of the Debye-Hückel theory to ions of dissimilar size (hence DH-SiS) - simple analytic mathematical expressions can be derived for the mean and single-ion activity coefficients of binary electrolyte components of ternary ionic systems. Such expressions are based on modifying the parallel DH-SiS equations for pure binary ionic systems, by adding to the three ion-size parameters - a (of counterions), b+ (of positive coions), and b- (of negative coions) - a fourth parameter. For the (+ + -) system, this is "b++," the contact distance between non-coion cations. b++ is derived from fits with experiment and, like the other b's, is constant at varying ion concentration and combination. Four case studies are presented: (1) HCl-NaCl-H2O, (2) HCl-NH4Cl-H2O, (3) (0.01 M HX)-MX-H2O with X = Cl, Br, and with M = Li, Na, K, Cs, and (4) HCl-MCln-H2O with n = 2, M = Sr, Ba; and n = 3, M = Al, Ce. In all cases, theory is fully consistent with experiment when using a of the measured binary electrolyte as the sole fitting parameter. DH-SiS is thus shown to explain known "mysteries" in the behavior of ternary electrolytes, including Harned rule, and to adequately predict the pH of acid solutions in which ionized salts are present at different concentrations.

Concentration dependence of single-ion activity coefficients: An analysis

Single-ion activity coefficients reported in literature for 10 ternary common-ion liquid-junction systems with KCl(aq) in the reference solutions and MX(aq) electrolytes in the test solutions at 25°C are reproduced in the present work numerically over wide MX molality ranges by functions of mean activity coefficients and transport numbers of ions in the liquid junction, the MX being BaCl2, CaCl2, CsCl, HCl, KF, K2SO4, LiCl, MgCl2, NaCl and SrCl2. The activity coefficients of solutes in the ternary systems are represented in the calculations by Scatchard's Neutral Electrolyte Model, and the transport numbers by a molality-based version of Van Rysselberghe's rule. An analysis is given of factors affecting the magnitudes of the functions, among them composition profiles in the liquid junction, formulation of Van Rysselberghe's rule, concentration dependence of ion mobilities, and non-ideality of electrolyte mixing. Most importantly, it is found that the exact functions can be approximated reasonably closely by a product of the mean activity coefficients in binary MX and KCl solutions, both at molality of the test solution, raised to powers that can be understood as related to effective transport numbers of the respective non-common ions in the liquid junction. Advantage is taken of the simple formulas to interpret the observed concentration dependences of single-ion activity coefficients in terms of the mean activity coefficients and effective transport numbers.

Determination of single-ion activity coefficients of hydrogen and bromide ions in aqueous hydrobromic acid solutions based on an ionic liquid salt bridge

Single ion activity coefficients of H+ and Br− in aqueous hydrobromic acid solutions, γH+ and γBr−, have been estimated potentiometrically at 25°C with a hydrogen electrode and a silver–silver bromide electrode, respectively, based on the assumption that the change in the liquid junction potential at the interface between a sample solution and an ionic liquid salt bridge is negligibly small. The mean activity coefficients of HBr calculated from experimentally obtained values of γH+ and γBr− well agree with the literature values over the entire concentration ranges studied, 1.04 × 10−3 to 0.528molkg−1. Overall dependencies of γH+ and γBr− on ionic strength are similar to those found previously in aqueous HCl solutions, but the degree of the dependence is distinctively stronger in both γH+ and γBr−. Plots of γH+ and γBr− as a function of the ionic strength are nearly congruent with the predictions by Fraenkel's smaller-ion shell (SiS) model up to 0.05molkg−1. However, with a further increase in HBr concentration experimental γH+ and γBr− more rapidly increases and decreases, respectively, in comparison with the SiS predictions.

Numerical values of individual activity coefficients of single-ion species in concentrated aqueous electrolyte solutions and the attempt of a qualitative interpretation on a model of electrostatic interaction

Individual activity coefficients of single-ion species can be achieved by the factorizing of a new concentration function for the mean activity coefficient to the required power applying a purely mathematical method. These single-ion activity coefficients, calculated in this manner, are listed for some aqueous strong electrolytes. The reasons for the magnitude and variation of the activity coefficients as a function of the concentration are, without a doubt, of complex nature. Activity coefficients have their meaning as practical values. In relation to the analytical concentration, the individual activity coefficients represent the macroscopic effectiveness of the single-ion species in solution an easy manner. However, with increasing deviations from Debye–Huckel conditions of an infinitely diluted electrolyte solution, a physically correct interpretation of the macroscopically visible activity coefficient is becoming more and more difficult, if not impossible to find. On the basis of a model of electrostatic interaction, an attempt has been made to create a qualitative interpretation of the individual ion activity coefficients in concentrated aqueous electrolyte solutions which were calculated applying the purely mathematical method by Ferse.

New Data on Activity Coefficients of Potassium, Nitrate, and Chloride Ions in Aqueous Solutions of KNO3 and KCl by Ion Selective Electrodes

Ion selective electrodes (ISEs) are used to measure the single-ion activity coefficients in aqueous solutions of KNO3 and KCl at 298.15 K against a double-junction reference electrode. The EMF responses of ISEs up to 0.01 m are plotted to obtain the slope and intercept values. The obtained slopes and intercepts are used in Nernst equation for higher concentrated solutions for calculation of individual ion activity coefficient. The mean ionic activity coefficients are estimated from single ion activity coefficient, and the obtained results are compared with the literature values.

Self-diffusion and activity coefficients of ions in charged disordered media

Self-diffusion and single ion activity coefficients of ions of size symmetric electrolytes were studied in the presence of a collection of charged obstacles (called matrix) within a "soft" version of the primitive model of electrolyte solutions. The matrix subsystem possesses a net charge, depending on the concentration and charge of obstacles. The brownian dynamics method was used to calculate the self-diffusion coefficients of mobile species. The replica Ornstein-Zernike theory for the partly quenched systems was used to calculate the individual activity coefficients of mobile ionic species. The results reflect the competition between attractive (obstacle-counterion, co-ion-counterion), and repulsive (obstacle-co-ion) interactions in these model systems. For the simplest possible system of symmetric monovalent ions the latter effect wins: Co-ions are excluded from the area around obstacles, and this slows down their diffusion compared to that of counterions. Introduction of divalent charges into the system changes this result when the concentration of obstacles is low. We compare these results to those obtained for the corresponding fully annealed systems, i.e., where all the species are mobile. In most cases the self-diffusion and activity coefficients of counterions and co-ions in the presence of charged obstacles follow the trends of the fully annealed solution, which are dictated by the composition of the mixture. In few situations, however, the presence of charged obstacles modifies these trends. Our study allows us to clearly identify the effects due to obstacles, and to separate them from those arising from the composition of the solution. In the case of charge and size symmetric systems, the results for the individual activity coefficients fully support the hypothesis of the "electrostatic excluded volume". Thermodynamic and dynamic results are consistent in explaining the behavior of the systems studied.

Comment on “The nature of single-ion activity coefficients calculated from potentiometric measurements on cells with liquid junctions” by Dmitri P. Zarubin, J. Chem. Thermodyn. 43 (2011) 1135–1152

Comment on “The nature of single-ion activity coefficients calculated from potentiometric measurements on cells with liquid junctions” by Dmitri P. Zarubin, J. Chem. Thermodyn. 43 (2011) 1135–1152

Reply to comment by J.H. Vera and G. Wilczek-Vera on ‘The nature of single-ion activity coefficients calculated from potentiometric measurements on cells with liquid junctions’

Reply to comment by J.H. Vera and G. Wilczek-Vera on ‘The nature of single-ion activity coefficients calculated from potentiometric measurements on cells with liquid junctions’

Short answer to the reply from D.P. Zarubin to our comment on “The nature of single-ion activity coefficients calculated from potentiometric measurements on cell with liquid-junction”

Short answer to the reply from D.P. Zarubin to our comment on “The nature of single-ion activity coefficients calculated from potentiometric measurements on cell with liquid-junction”

Determination of Single-Ion Activities of H+ and Cl– in Aqueous Hydrochloric Acid Solutions by Use of an Ionic Liquid Salt Bridge

Single-ion activities of H(+) and Cl(-) in aqueous hydrochloric acid solutions up to 500 mmol·dm(-3) HCl have been determined, independently of each other, with reasonable accuracy, 95% confidence interval of log(10)γ being ±0.012 at 25 °C, by use of an ionic liquid salt bridge inserted in a Harned cell. The geometric averages of those activities agree well with the literature values of the mean ionic activity of HCl determined with a Harned cell up to an ionic strength of 500 mmol·dm(-3). The agreement supports the assumption used in estimating the single-ion activities, that is, the liquid junction potentials on both sides of the ionic liquid salt bridge stay constant, within the confidence interval of the cell voltage, ± 1 mV, over the concentration range of HCl between 10 and 500 mmol·dm(-3). The single-ion activity coefficient values of H(+) and Cl(-) over the concentration range studied show fairly good agreement with the recent theoretical predictions by the "smaller-ion-shell" model proposed by Fraenkel (Mol. Phys. 2010, 108, 1435; J. Phys. Chem. B 2011, 115, 557), which fact is significant and implicative in physical and analytical chemistry of electrolyte solutions.