(563b) Thermodynamic Frameworks for Electrolyte Thermodynamic Models: Single- and Mixed- Solvent Electrolyte Solutions
AIChE Annual Meeting
2010 Annual Meeting
Engineering Sciences and Fundamentals
Thermophysical Properties and Phase Behavior III
Wednesday, November 10, 2010 - 3:36pm to 3:57pm
For almost a century the Debye-Hückel equation  and its many extensions have been used for the prediction of the deviation from ideality of electrolyte solutions due to electrostatic interactions. However, much criticism has been leveled at the validity and accuracy of the fundamental equation of Debye and Hückel (DH) because of its inherent assumptions about the role of solvent and the ensemble in which the theory was developed. This presentation will focus on the details of the assumptions and the ensemble in which the Debye-Hückel theory was developed and will discuss rigorous extensions of the DH theory to the Lewis-Randall ensemble (in which experimental measurements are taken). Relative considerations are accounted for the so-called Born equation , the applicability and accuracy of which will be examined. On the basis of typical engineering assumptions regarding the pressure and composition dependence of the volume of the solution and of the dielectric constant of the solvent, the activity coefficients converted to the Lewis-Randall ensemble are criticized and conclusions on effect of the various treatments are drawn.
Typically, thermodynamic properties described at the McMillan-Mayer  level are explained and understood using a hypothetical osmotic cell [4-8]. The solution of interest is assumed to be in equilibrium with a pure solvent compartment of the cell that is in contact with the solution to be studied through a semi-permeable membrane that allows only solvent(s) to equilibrate between the two compartments of the cell. The concept is that, despite the non-idealities in the solution of interest, the solvent chemical potential remains constant and equal to that of the pure solvent compartment. This description suits very well the description of electrolyte solutions. In this presentation, the approach taken is to consider several different charging processes that describe the contribution of electrostatic interactions to non-ideality not as an excess function but as an additional energy , since the uncharged solution is not ideal. This is consistent with the original theory of Debye and Hückel, where the electrostatic contribution is expressed as an additional energy contribution and of course with more modern theories, where models of short-range interactions are used to describe non-ideality. In the charging processes considered the variables kept constant depend on the thermodynamic framework considered. Lewis-Randall activity coefficients are calculated at constant system temperature, pressure and mole numbers; volume and chemical potentials are dependent variables. McMillan-Mayer activity coefficients are calculated at constant temperature, volume, solute mole numbers and solvent chemical potential; pressure, solute chemical potentials and solvent mole number are dependent variables. The difference (hence, the conversion between the two) has to account for the difference in independent variables.
At the McMillan-Mayer level the solvent is cast as a continuum; hence, a reduced Gibbs-Duhem equation applies . In mixed-solvent electrolyte solutions at the Lewis-Randall level one needs to account for the composition and pressure dependence of the dielectric constant and of the volume of the solution. When comparing model predictions to experimental data an inconsistency arises: the Gibbs-Duhem consistency is different at the two ensembles. One school of thought is considering solvent properties of the solvent mixture in the calculation of free energies, but neglects their composition dependent derivatives. Another school of thought is taking into account the derivatives of dielectric constant and volume with respect to composition. Using Legendre transforms and other fundamental thermodynamic manipulations, rigorous expressions for the conversion between the McMillan-Mayer and Lewis-Randall frameworks are derived for single- and mixed-solvent electrolyte solutions. These expressions are useful for understanding the effect of the dependence of volume and dielectric constant on pressure and composition, but are not of much use for engineering models. A number of simplifying engineering assumptions are applied to test the effect of these dependences on the activity coefficients at the Lewis-Randall level.
This presentation will focus on the effect of the aforementioned assumptions on the prediction of free energies and activity coefficients in single- and mixed-solvent electrolyte solutions. In single-solvent solutions the effect of considering a pressure and composition dependent dielectric constant is examined based on several literature equations for the effect of electrolyte composition and pressure on the relative permittivity of the solvent. The applicability of the Born equation is discussed and rigorous expressions are derived for the conversion between different thermodynamic frameworks. Next, expressions are derived for the conversion in mixed-solvent solutions. These equations are simplified using approximate mixing rules for estimating the mixed-solvent volume and dielectric constant and then the effect of these approximations is discussed. The capability of each approach to predict the so-called salting effect is demonstrated and discussed.
1. Debye P, Hückel E. Zur Theorie der Elektrolyte. I. ?Gefrierpunktserniedrigung und verwandte Erscheinungen?. Physikalische Zeitschrift. 1923;24:185-206.
2. Born M. Volumen und Hydratationswärme der Ionen. Zeitschrift für Physik. 1920;1:45-48.
3. McMillan WG, Mayer JE. The statistical thermodynamics of multicomponent systems. Journal of Chemical Physics. 1945;13:276?305.
4. Curtis RA, Newman J, Blanch HW, Prausnitz JM. McMillan-Mayer solution thermodynamics for a protein in a mixed solvent. Fluid Phase Equilibria. 2001;192:131-153.
5. de M. Cardoso MJE, O'Connell JP. Activity coefficients in mixed solvent electrolyte solutions. Fluid Phase Equilibria. 1987;33:315-326.
6. Friedman HL. Lewis-Randall to McMillan-Mayer conversion for the thermodynamic excess functions of solutions. Part I. Partial free energy coefficients. Journal of Solution Chemistry. 1972;1:387-412.
7. Haynes CA, Newman J. On converting from the McMillan-Mayer framework: I. Single-solvent system. Fluid Phase Equilibria. 1998;145:255-268.
8. Lee LL. Thermodynamic consistency and reference scale conversion in multisolvent electrolyte solutions. Journal of Molecular Liquids. 2000;87:129-147.
9. Breil MP, Mollerup JM. The McMillan-Mayer framework and the theory of electrolyte solutions. Fluid Phase Equilibria. 2006;242:129-135.
10. de M. Cardoso MJE, Demedeiros JL. A thermodynamic framework for solutions based on the osmotic equilibrium concept - 1: General formulation. Pure and Applied Chemistry. 1994;66:383-386.