J. Phys. Chem. 1992,96,9470-9479
9470
Thermodynamics of Multicomponent, Mlsclble, Ionlc Solutions. 2. Mlxtures Includlng Unsymmetrical Electrolytes Simon L. Clegg,*It Kenneth S. Pitzer,**tand Peter Brimblecombet School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, U.K..and Department of Chemistry, University of California. Berkeley, California 94720 (Received:March 16, 1992: In Final Form: June 23, 1992)
Model equations for the excess Gibbs energy and solvent and solute activity coefficients (given previously for symmetrical salt systems) are here developed for mixtures containing an indefinite number of ions of arbitrary charge, over the entire concentration range. The equations are expressed on a mole fraction basis and comprise a Debye-Hiickel term extended to include the effects of unsymmetrical mixing, and a Margules expansion carried out to the four suffix level. The model is tested using activity coefficient and salt solubility data for the systems H-(Al,Mg,Ca,Sr)41-H20, H-(La,Ca)-N03-Hz0, Na-Mfll-SOd-H20, Mg-Ca-K-Cl-H20, and Na-K-MeO4-H20 at 298.15 K. Thermodynamicproperties of quaternary aqueous solutions are predicted using only parameters determined from binary and ternary mixtures. Salt solubilities in both systems involving HNOBare satisfactorily represented from aqueous salt solution to pure liquid HN03.
I. Introductioa Pitzer and Simonson' recently developed a molefraction-based thermodynamic model, for mixtuna containing ions of symmetrical charge type, that is applicable Over the entire concentration range. The excess G i b b energy of the mixture is assumed to consist of two components: short-range forces accounted for by a Margules expansion in mole fraction (to the three suffm level) and a long-range Debye-Hiickel term. The equations have been used to represent water activities in mixtures with molten salts,2 salt solubilities in Na-K-CI-H20 solutions,3and osmotic and activity coefficients in HN03-H20 mixture^.^ Clegg and Pitzer' extended the model by introducing composition-dependent terms into the DebytHiickel expression, and an additional short-range parameter for the interaction between the solvent and a single anion and cation in highly concentrated solutions. Tests showed that the model was able to represent, within experimental error, osmotic and activity coefficientsin single salt and ternary ion solutions containing the species H+, Na+, K+, C1-, and NO3-. By comparison, the local composition model of Chen6 and Chen and Evans' represented the data less well. A model restricted to ions of a single charge type is severely limited in application. Here we generalize the model to include ions of arbitrary charge. The resulting equations, which for the symmetricalcase reduce to those given previouslyIs are tested using activity coefficient and salt solubility data for the systems H(Al,Mg,Ca,Sr)-Cl-H20, H-(La,Ca)-NOB-H20, Na-Mg-ClS04-H20, Mg-Ca-K-Cl-H20, and Na-K-Mg-S04-H20.
II. Tbeoreticpl Development The basis of the present model has been described by Clegg and Pitzer,' and the same symbols and definitions are generally adopted here. Mole fractions are calculated on the basis of complete dissociation of all salts; thus the mole fraction xj of species j is given by xi = nj/Cini,where ni is the number of moles of species i present, with both cations and anions included. We E,&. The define the total mole fraction of ions, xI,as excess Gibbs energy for any amount of material is GE and per mole of particles is 8. Activity coefficientsf; on the mole fraction scale are related to the excess Gibbs energy by
(zjC +
The focus of the present work is upon aqueous systems containing salts of finite, though high, solubility. We therefore adopt
a reference state infiinitely dilute with respect to the solvent, water, as the convention for expressing the activity coefficients of solute species. The symbolf;* is used for the activity coefficient on this basis, thusfi* 1 as xi 0 in the pure solvent. For the solvent iklf,fi 1 as XI --c 1; and for all solution componentsf; = a,/x,. Activity and osmotic coefficients in aqueous solution are commonly represented in molal units, and for convenience a number of conversions to the corresponding mole fraction quantities are now given. The activity of the solvent (al) is related to the rational osmotic coefficient g, and molal osmotic coefficient t$ by
-- -
In (all = g In (x1)
(3)
In (al) = -(M1/1000)t$Cmj J
(4)
where M Iis the molar mass (grams) of the solvent and m4 the molal concentration of solute speciesj . The mole fraction aavity coefficientf,* (infinite dilution reference state) is related to the corresponding molal value (rj)by8
fi*
yj(1 + ( M 1 / 1 m ) C m j ) I
(5)
In the model the excess Gibbs energy per mole is assumed to consist of short-range force (8)and long-range Debye-Hackel (gDH) components:
sE=8+$H
(6)
From the application of eq 1 to eq 6 it is clear that expressions for the solvent and solute activity coefficient are also the sum of terms for short-range forces and the Debye-Hackel effect. In subsections 1 and 2, equations for 8 and gDH are developed for solutions containing indefinite numbers of neutral species and ions of arbitrary charge. These are then differentiated to yield equations for solute and solvent activity coefficients. (1) Short-Range Contributioos to the Excess Cibbs Energy. Pitzer and Simonson' presented an expression for 8 / R T based upon a threesuffix Margules expansion. This involves two parameters for a binary system consisting of a solution of a single salt or mixture of two neutral species. However, application of the model to osmotic and activity coefficient data for aqueous H N 0 3 showed that a further term was useful for representing thermodynamic properties to very high concentration.' Clegg and PitzerS therefore extended the Margules expansion to include quaternary coefficients, though ultimately retaining only one of the additional terms-for the interaction of two molecules of a neutral species (generally the solvent) and a single anion and cation. The basic expression for 8 / R T is ~ S / R T = ~ C X ~ +Xuij(xi ~ [ -W xi)]~+~ i< J
University of East Anglia. *Universityof California.
zxixjxk[cijk
i
