Mean Activity Coefficients Of Electrolytes Research Articles

Generalized Debye–Hückel model for activity coefficients of electrolytes in water–methanol mixtures

We propose a generalized Debye–Hückel model from Poisson–Fermi theory to calculate mean activity coefficients of electrolytes in water–methanol mixtures with arbitrary percentage of methanol from 0 to 100%. The model accounts for both short and long ion–ion, ion–water, ion–methanol, and water–methanol interactions, the size effect of all particles, and the dielectric effect of mixed-solvent solutions. We also present a numerical algorithm with mathematical and physical details for using the model to fit experimental data. The model has only 3 empirical parameters to fit the experimental data of NaF, NaCl, and NaBr, for example, in pure-water solutions. It then uses another 3 parameters to calculate the activities of these salts in mixed-solvent solutions for any percentage of methanol. Values of these parameters show mathematical or physical meaning of ionic activities under variable mixing condition and salt concentration. The algorithm can automatically determine optimal values for the 3 fitting parameters without any manual adjustments.

Mean activity coefficient of electrolytes: A critical evaluation of four physical models

Accurate prediction of phase equilibria in the presence of electrolytes is important for many applications. For hydrate phase equilibria in the presence of salts, the fluid phase equation of state needs to be coupled with a reliable electrolyte model. In this work, various electrolyte models: Debye-Hückel, truncated Debye-Hückel, Pitzer theory and Bromley activity models have been critically evaluated for predicting the mean activity coefficient for various aqueous salt solutions. The Debye-Hückel and truncated Debye-Hückel models give accurate predictions at lower salt concentrations, but a large deviation was observed at molal concentrations higher than about 1 mol/kg. The Bromley activity model and Pitzer theory were found to be good alternatives to the Debye-Hückel models and its modifications. For 1:1 electrolytes, the Pitzer theory and Bromley activity models give accurate predictions up to the saturation limit of salt solutions. Conversely, mean activity coefficient calculations for 1:2 electrolytes (CaCl2, MgCl2 and BaCl2) using the Pitzer theory, Bromley activity model, or Debye-Hückel model and its modification were not able to capture the electrolyte contribution of these salts.

The effect of concentration- and temperature-dependent dielectric constant on the activity coefficient of NaCl electrolyte solutions.

Our implicit-solvent model for the estimation of the excess chemical potential (or, equivalently, the activity coefficient) of electrolytes is based on using a dielectric constant that depends on the thermodynamic state, namely, the temperature and concentration of the electrolyte, ε(c, T). As a consequence, the excess chemical potential is split into two terms corresponding to ion-ion (II) and ion-water (IW) interactions. The II term is obtained from computer simulation using the Primitive Model of electrolytes, while the IW term is estimated from the Born treatment. In our previous work [J. Vincze, M. Valiskó, and D. Boda, "The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion-ion correlations," J. Chem. Phys. 133, 154507 (2010)], we showed that the nonmonotonic concentration dependence of the activity coefficient can be reproduced qualitatively with this II+IW model without using any adjustable parameter. The Pauling radii were used in the calculation of the II term, while experimental solvation free energies were used in the calculation of the IW term. In this work, we analyze the effect of the parameters (dielectric constant, ionic radii, solvation free energy) on the concentration and temperature dependence of the mean activity coefficient of NaCl. We conclude that the II+IW model can explain the experimental behavior using a concentration-dependent dielectric constant and that we do not need the artificial concept of "solvated ionic radius" assumed by earlier studies.

Expressions for the Activity Coefficients and Osmotic Coefficients of Solutions of Electrolytes and Nonelectrolytes Based on the Hydration Model of Zavitsas

In two papers Zavitsas described a model for the thermodynamic properties of aqueous solutions of a single electrolyte or nonelectrolyte (Zavitsas, J Phys Chem B 105:7805–7817, 2001; J Solution Chem 39:301–317, 2010) in which he assumed that part of the water is so strongly bound to the solute that it can be considered as part of it, and thus only the remaining unbound water is considered to be the solvent. He showed that when the usual water mole fraction was replaced by the resulting mole fraction of unbound water, obtained by optimizing an effective hydration number, basically linear relations were obtained to fairly high molalities for the freezing temperature lowering, boiling temperature elevation, and the water activity/vapor pressure of water. However, Zavitsas only considered the properties of the solvent, not the solute. In this paper we derive the corresponding expressions for the activity coefficient of the solute for the usual molality scale based on 1 kg of water, for the modified molality scale based on 1 kg of unbound water, for the mole fraction scale based on the total number of moles of water, and for the modified mole fraction scale based on the number of moles of unbound water. These equations show that if the hydration number is larger than the stoichiometric ionization number of the electrolyte, then all four types of mean activity coefficients are predicted to always be >1 (nearly all hydration numbers reported by Zavitsas for electrolyte solutions are greater than the corresponding ionization numbers), which directly conflicts with extensive experimental and theoretical evidence that the mean activity coefficients of electrolytes in aqueous solutions always initially decrease below unity. In contrast, for nonelectrolyte solutions, the hydration model of Zavitsas gives more realistic values of the activity coefficients.

Comment on “Factorizing of a concentration function for the mean activity coefficients of aqueous strong electrolytes into individual functions for the ionic species” by A. Ferse and H.O. Müller

A. Ferse and H.O. Muller have recently presented a mathematical method aimed at subdividing the activity coefficients of electrolytes into functions of individual ionic species; these functions are suggested to be the ionic activity coefficients. By examining the method, it is possible to verify that the peculiar mathematical structure of the functions in question really guarantees a unique result, unlike the usual subdivisions of electrolyte activity coefficients, which admit infinite possibilities for the ionic activity coefficients. But the subsequent step of the reasoning, i.e., that these functions have to be the activity coefficients of the ionic species, is an illation. And indeed, another kind of subdivision in individual functions can be exemplified, whose mathematical structure also guarantees results that are unique and perfectly compatible with all theoretical properties of the ionic activity coefficients. It is concluded that it is impossible to rely on mathematical method to pull the activity coefficients of ions out of the mean activity coefficients of the electrolytes. And hence, the individual functions for the ionic species determined by Ferse and Muller do not represent the ionic activity coefficients and do not have any particular utility.

Response to “Comment on ‘The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion–ion correlations’” [J. Chem. Phys. 134, 157101 (2011)

Fraenkel commented1 on our paper2 in which we offered a different view of the activity coefficient (or, equivalently, the excess chemical potential) of electrolytes. Fraenkel summarized the main points of our theory nicely. Also, he offered his smaller-ion-shell (SiS) model3, 4 as an alternative to our approach. In this response, we categorize our thoughts along a few main issues. (1) The most important issue for us is the question whether the (concentration-dependent) dielectric constant, (c), of the solution should be used in the model, or the dielectric constant of the pure solvent (that is equal to the dielectric constant of the solution at infinite dilution). We argue that is a macroscopic physical parameter whose value is well-defined and measurable for a given thermodynamic state point (different concentration corresponds to a different state point). In the implicit solvent model of electrolytes the ions are modeled on the molecular level, therefore, they are characterized by microscopic parameters such as the ionsize parameters (ISP). The solvent, on the other hand, is modeled on the basis of its average dielectric response, which is characterized by a macroscopic physical parameter: the dielectric constant. In our view, physical variables that enter the calculations as external parameters, such as the dielectric constant, should have the experimental values for the given state point. The structure of the solvent is influenced by the presence of ions and decreases with increasing concentration (dielectric saturation). Unfortunately, measurements of the static dielectric constant of an electrolyte have several technical difficulties. The extrapolation to zero frequency based on dielectric relaxation data over a frequency range requires special instrumentation to determine the dielectric loss. For example, newer experiments by Buchner et al.5 for NaCl gave (c) values that are larger than those reported by Barthel et al. previously.6 The absence of experimental data for (c) makes the applicability of our approach limited and, at the same time, from a practical point of view, necessarily requires the use of theories applying the dielectric constant of the solvent. The absence or availability of experimental data, however, does not influence the correctness of the physical picture behind our model. Our opinion is that solvation should be taken into account by a concentration-dependent dielectric constant rather than by an increased, solvated ionic radius. (2) Our paper might give the (false) impression that the question whether adjustable parameters are used or not is a central issue for us. It is not. It depends on the intention of the investigator: if one wants to adjust molecular parameters to fit with experiments, it is legitimate to do so. Actually, that is what we do in our ion channel studies, for example.7 There are parameters, however, whose values are well-defined so they should not be adjusted. Even if we adjust a parameter, the resulting value should have a sensible physical meaning. In general, we can shed a light on our standpoint regarding modeling if we cite Occam’s razor that “admonishes us to choose from a set of otherwise equivalent models of a given phenomenon the simplest one. In any given model, Occam’s razor helps us to ‘shave off’ those concepts, variables, or constructs that are not really needed to explain the phenomenon. By doing that, developing the model will become much easier, and there is less chance of introducing inconsistencies, ambiguities, and redundancies.”8 Paraphrasing this idea for the case of adjustable parameters, we can offer a less strong statement: “In a model, we should minimize the number of adjustable parameters.” (3) The third issue is the question of the statistical– mechanical method with which the results for the activity coefficient [and the ion–ion (II) term in our approach] are calculated. Our opinion is that the best method should be used even if it is computationally expensive (especially in the age of fast computers). Technically simpler methods might be justified by computational convenience, but when scientific knowledge is sought, the most accurate method is favorable. According to the complexity of the studied many-body systems, more complex methods usually provide more accurate results. (Alas, Occam’s razor does not help here.) For a well-defined molecular model of a many-particle system, computer simulations, if performed properly, provide exact results apart from statistical uncertainties and system-size effects. Theories necessarily use approximations. Their results, therefore, should be tested against simulation data. The simulation results contain only the error due to the imperfectness of the model, while the theoretical results also contain the error due to the imperfectness of the approximation to derive the theory. Separation of these two errors is important for the activity coefficient of electrolytes as demonstrated in our paper using grand canonical Monte Carlo (MC) simulations9 and the mean spherical

Comment on “The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion–ion correlations” [J. Chem. Phys. 133, 154507 (2010)

Comment on “The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion–ion correlations” [J. Chem. Phys. 133, 154507 (2010)

The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion-ion correlations

We propose a simple model to explain the nonmonotonic concentration dependence of the mean activity coefficient of simple electrolytes without using any adjustable parameters. The primitive model of electrolytes is used to describe the interaction between ions computed by the adaptive grand canonical Monte Carlo method. For the dielectric constant of the electrolyte, we use experimental concentration dependent values. This is included through a solvation term in our treatment to describe the interaction between ions and water that changes as the dielectric constant changes with concentration. This term is computed by a Born-treatment fitted to experimental hydration energies. Our results for LiCl, NaCl, KCl, CsCl, NaBr, NaI, MgCl(2), CaCl(2), SrCl(2), and BaCl(2) demonstrate that the principal reason of the nonmonotonic behavior of the activity coefficient is a balance between the solvation and ion-ion correlation terms. This conclusion differs from previous studies that assumed that it is the balance of hard sphere repulsion and electrostatic attraction that produces the nonmonotonic behavior. Our results indicate that the earlier assumption that solvation can be taken into account by a larger, "solvated" ionic radius should be reconsidered. To explain second order effects (such as dependence on ionic size), we conclude that explicit water models are needed.

The mean activity coefficients of 2:2 electrolyte solutions: An integral equation study of the restricted primitive model

We apply the closure theory ZSEP (self-consistent zero-separation based closures) developed earlier to the restricted primitive model (RPM) of 2:2 electrolytes in order to (i) obtain the activity coefficient information via the direct formula for chemical potentials [L. L. Lee, J. Chem. Phys. 97, 8606 (1992)] and (ii) test the performance of this flexible ZSEP closure at high-coupling strengths (i.e., high valency and low temperatures) for cases of 2:2 electrolytes where other closure schemes have encountered difficulties [e.g., the hypernetted chain (HNC) equation]. In particular, we shall remedy the shortcomings of the HNC theory at low concentrations (from 0.001M to 0.2M). The ZSEP closure is found to perform well at coupling strengths beta(') = absolute value(z(1)z(2))e(2)/(epsilon(m)kTd) approaching approximately 10 where some other closure theories cease to give good results. In addition, by applying the direct chemical potential formula, we demonstrate numerically that, in the RPM cases examined, the logarithm of the mean activity coefficients of electrolytes are closely approximated by the electrostatic internal energy, an easily accessible quantity, a fact that shall afford ready access to the chemical potentials for phase equilibrium and electrochemical calculations on electrolytic systems.

Correlation of Activity Coefficients in Aqueous Solutions of Ammonium Salts Using Local Composition Models and Stochastic Optimization Methods

In this study, several local composition models and stochastic optimization methods have been used and compared in data fitting of activity coefficients in aqueous electrolytes. We have utilized the electrolyte-NRTL model of Chen et al. and the modified Wilson models proposed by Xu and Macedo, and Zhao et al. to fit the activity coefficients of several quaternary ammonium salts in water at 25°C. These electrolytes have interesting properties for their application as ionic liquids. However, the modeling of their thermodynamic behavior using local composition models is a global optimization problem. In this study, several stochastic optimization methods have been used to solve this optimization problem and their numerical performances have been compared. Specifically, we have tested the classical Simulated Annealing and the hybrid methods: Direct Search Simulated Annealing, Simplex Coding Genetic Algorithm, Simulated Annealing Heuristic Pattern Search, and Directed Tabu Search. Our results show that Simulated Annealing is a suitable tool for data fitting of the activity coefficients of aqueous electrolytes. Finally, the tested models can satisfactorily correlate the mean activity coefficients of electrolytes treated in this study, and are suitable for process design.

A three-characteristic-parameter correlation model for strong electrolyte solutions

In this study, a characteristic-parameter correlation model was developed for the aqueous strong electrolyte solutions. This correlation model consisted of the long-range ion–ion interaction described by Pitzer–Debye–Huckel model and short-range ion-solvent molecule interaction attributed to solvation effect. The proposed approach is based on our previous paper of Lin et al. [C.-L. Lin, L.-S. Lee, H.-C.Tseng, Fluid Phase Equilibria, 90 (1993) 57–79.]. However, the significant improvement was made that the differential equation was solved analytically instead of numerically. This proposed model contains three parameters of physical significance and has been used to estimate the mean activity coefficients of electrolytes and osmotic coefficients of 144 aqueous strong electrolyte solutions from literature. Comparisons were made to the models of Chen et al. [C.C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE, 28 (1982) 588–596; C.C. Chen, L.B. Evans, AIChE, 32 (1986) 444–454.], Ananth and Ramachandran [M.S. Ananth, S. Ramachandran, AIChE, 36 (1990) 370–386.], and Lin et al. [C.-L. Lin, L.-S. Lee, H.-C. Tseng, Fluid Phase Equilibria, 90 (1993) 57–79.]. The results also showed that the present model is at least as good as above models for 1-1 and 1-2 electrolyte solutions and improved very much for 2-1, 2-2, 3-1, and 3-2 electrolyte solutions.

Ionic behavior across charged membranes in methanol–water solutions. I: Membrane potential

The membrane potential across cation- and anion-exchange membranes was measured for LiCl methanol–water solutions. The experimental data were analyzed on the basis of the Donnan equilibrium and the Nernst–Planck equation of ion flux, considering the effect of mean activity coefficient of electrolytes in the external solution, and the theoretical predictions agreed well with the experimental data. The effective membrane charge densities and the cation-to-anion mobility ratios in membranes were determined by a non-linear regression method. The results showed that the effective membrane charge density would decrease with the increase in the weight fraction of methanol. It is postulated that the counter-ions prefer to make ion pairs with the fixed charge groups in the mixed solvent system since the dielectric constant of methanol is smaller than water. Thus the ion pairs between the counter-ions and fixed charge groups will increase with the increase in methanol concentration, and the effective membrane charge density will decrease.

Modification of the Pitzer model to calculate the mean activity coefficients of electrolytes in a water-alcohol mixed solvent solution

The approach tested in this work consists in adapting the Pitzer model, initially designed for aqueous solutions of electrolytes, to the case of solutions with a mixed solvent, without systematically readjusting the coefficients. This modified model was applied successfully to the calculation of the mean activity coefficients of NaBr in the mixed solvent H2O+MeOH, H2O+EtOH and compared with the experimental values obtained from electrode potential measurements.

A program for calculation of activity coefficients at selected concentrations and temperatures

A program which calculates the mean activity coefficient of electrolytes in aqueous solutions has been written for the Texas Instruments TI-59 calculator. Using previously published [1] compact tables of three parameters, the activity coefficients at 25°C for 227 electrolytes can be calculated at selected concentrations. The equations and parameters employed give results in good agreement with empirical determinations of mean activity coefficients over an extended range of concentrations, often ≥ 5m. The program uses the Debye-Hückle equation to estimate the mean activity coefficient at other temperatures. Good agreement with empirical results is obtained up to moderate concentrations at temperatures ranging from 15 to 80°C.