Modeling of salt solubilities in mixed solvents

Modeling of salt solubilities in mixed solvents

O. Chiavone Filho^1* and P. Rasmussen²

¹Universidade Federal do Rio Grande do Norte-UFRN, Depto de Engenharia Química (DEQ/CT),

Campus Universitário, Lagoa Nova, 59072-970, Natal-RN, Brazil

E -mail [e-mail]

²Technical University of Denmark (DTU), Department of Chemical Engineering,

IKT, DK-2800 Lyngby, Denmark

E-mail [e-mail]

(Received: April 5, 1999 ; Accepted: October 29, 1999)

Abstract - A method to correlate and predict salt solubilities in mixed solvents using a UNIQUAC+Debye-Hückel model is developed. The UNIQUAC equation is applied in a form with temperature-dependent parameters. The Debye-Hückel model is extended to mixed solvents by properly evaluating the dielectric constants and the liquid densities of the solvent media. To normalize the activity coefficients, the symmetric convention is adopted. Thermochemical properties of the salt are used to estimate the solubility product. It is shown that the proposed procedure can describe with good accuracy a series of salt solubility measurements in aqueous organic solvent mixtures.

Keywords: theory, method of calculation, solid-liquid equilibrium, electrolyte, mixed solvent.

INTRODUCTION

This work describes a method to calculate salt solubilities in mixed solvents. A combination of the UNIQUAC (Abrams and Prausnitz, 1975) and Debye-Hückel (Pitzer, 1980) models is applied. The choice of the pure fused-salt standard state, resulting in the symmetric convention, simplifies the calculation for mixed-solvent salt systems, since it is not necessary neither the evaluation of the standard Gibbs energy of transfer (Lorimer, 1993) nor the use of another formalism, such as MacMillan-Mayer (Zerres and Prausnitz, 1994), using osmotic pressure. Lorimer (1993), for instance, presented a method to calculate salt solubilities in mixed solvents using the infinite dilution standard state for the salt, which requires standard Gibbs energy of transfer data, the binary salt solubilities, and also an activity coefficient model. Loehe and Donohue (1997) presented a review in modeling thermodynamic properties of aqueous strong electrolyte systems.

To test the proposed method of calculation, we use a series of salt solubility measurements in water plus organic solvent (os) mixtures obtained previously by the authors (Chiavone-Filho and Rasmussen, 1993). The salts studied are potassium chloride and potassium bromide, which are unassociated electrolytes (Robinson and Stokes, 1965), i.e., they are considered to be completely dissociated in cations and anions. The organic solvents are ethanol, ethylene glycol and a series of glycol ethers. Three salt solubility isotherms were obtained for the eight ternary systems studied at 298, 323 and 348 K. For all mixtures with a high concentration of glycol ether, salt solubility is observed to decrease with rising temperature; for the other mixtures, salt solubility increases with temperature.

Pinho and Macedo (1996) used the method proposed by Chiavone-Filho (1993, 1994) for calculating salt solubilities for three aqueous-alcohol systems.

IONIZED MOLE FRACTION BASIS

In this work we are only dealing with one-salt/mixed solvent systems. We will however consider dissociation in the model because it can then easily be extended to multiion solutions.

The ionized mole fraction basis is very convenient for modeling electrolyte solutions. Rozen (1979) emphasized that the mole fraction of a pure electrolyte ionized is equal to one, which is desirable for the pure substance standard state. The ionized mole fraction of a salt is defined as the sum of the mole fractions of its constituent species.

(1)

We thus consider the dissociation of a salt molecule into n particles, e.g., n ₁ cations plus n ₂ anions. The mole fraction of the ions is determined as follows as a function of the number of moles (n) of the components for a one-salt solution.

(2)

Where n_OS is the organic solvent number of moles. For water, a solvent species, the mole fraction formula results.

(3)

Equations (2) and (3) may be directly utilized for a thermodynamic model such as UNIQUAC (Abrams and Prausnitz, 1975), considering each ion as a species. It should be noted that associated electrolytes, e.g., hydrated or partially dissociated salts, may also be treated according to the ionized mole fraction basis.

The salt-free mole fraction of water (x¢_water) is defined by equation (4).

(4)

SOLUBILITY EQUATION

The criterion for solid-liquid equilibrium (SLE) is that the chemical potential of the salt in the solid phase be equal to the sum of the product of the stoichiometric coefficients and the chemical potentials of its constituent species, e.g., cation 1 and anion 2.

(5)

If the solid phase consists of pure salt, then its chemical potential is equal to the standard chemical potential in the solid state (s).

(6)

To express the chemical potential of the salt in the liquid phase we applied the Lewis relation for both the cation and the anion, and then combined it with equations (2) and (5).

(7)

where by definition Q and g _± are geometric averages.

(8)

Combining equations (5), (6) and (7), we obtain the solubility product (K_salt).

(9)

The difference between the standard chemical potentials of the salt in the solid and liquid states, equation (9), may be estimated as a function of the temperature and the thermochemical properties of the salt using the following expression:

(10)

where T_m is the melting point, DH_m and DC_p are heat of fusion and heat capacity change at T_m, respectively. T_m, DH_m and DC_p data may be obtained from JANAF thermochemical tables (Chase et al., 1985) (see Table 1).

The resulting salt solubility expression, equation (11), can be used for trial and error calculation of the solubility, since g _± also depends on salt concentration. It should be observed that the addition of an organic solvent to a salt saturated aqueous solution at constant temperature does alter salt solubility because of changes in g _± .

(11)

UNIQUAC+DEBYE-HÜCKEL MODEL

The activity coefficients for both ionic and solvent species are calculated using a sum of short-range and electrostatic contributions.

(12)

The symmetric normalization of the activity coefficients is applied, which means that for a solvent and for a salt the following conditions must be fulfilled:

(13)

The procedure for normalization of the salt requires a better explanation because ionic dissociation is considered. Therefore, an activity coefficient model in its original form, represented by the superscript q , , will be normalized according to the type of species i. For an ionic species i, the normalized activity coefficient, g _i, will be equal to one when x_salt is equal to one, or when x_i is equal to n_i/n . So for a mixture with a mole fraction vector and temperature T, g _i of a cation 1 or an anion 2, taking again the example of a salt with cation 1 and anion 2 (n =n ₁+n ₂), will be calculated according to equation (14), for the symmetric normalization.

(14)

For solvent species, or for an undissociated species, the symmetric normalization states that g _i will be one when x_i is equal to one. If the model is already normalized, e.g., UNIQUAC equation, the second term of equation (15) will be annulled and thus the normalization procedure is only applied to the Debye-Hückel (DH) term for solvent species.

(15)

The UNIQUAC Equation

The UNIQUAC equation (Abrams and Prausnitz, 1975) contains adjustable interaction parameters and is briefly summarized in the ^appendix. It has is applied with temperature-dependent parameters. Ion-ion interactions are neglected. To calculate volume and surface area parameters for the ions, a new and small standard radius, that is an arithmetic average of the alkali, alkali earth and halide ionic radii presented by Pauling (1927) is used. The average ionic radius (0.122 nm) is 67 % smaller than the standard segment chosen by Abram and Prausnitz (1975). Table 2 presents the estimated UNIQUAC structural parameters for the ions. The table also shows the structural parameters of the solvents, which are estimated and used according to Abrams and Prausnitz (1975), retaining the original form of the UNIQUAC model.

The Debye-Hückel Equation

The Debye-Hückel (Pitzer, 1980) equation (see ^appendix) is extended to mixed solvents by using appropriate methods to evaluate the density and dielectric constant of the solvent media as a function of temperature and composition.

Parameter b, equation (A-5), is related to the closest approach of the ions. In order to convert b from the molality basis to the mole fraction basis, factor (1000/MW_ms)^½, where MW_ms is the molecular weight of the mixed solvent, should be multiplied. Parameter b depends on the short-range model used and on the solvent composition, and it may be estimated in the experimental data fitting. However, we have found that the value of 14.9 set by Pitzer (1973, 1980) and also by Chen and Evans (1986) is suitable for aqueous and mixed-solvent salt systems with the UNIQUAC equation. Therefore we have fixed the value of b equal to 14.9.

Similarly to Koh et al. (1985), we have neglected the differentiated terms of the Debye-Hückel parameter in equation (A-5), which is solvent composition dependent via the density and the dielectric constant. The values of A_DH,x are, nevertheless, evaluated as a function of solvent composition, temperature and pure component properties via molar density (r _ms) and dielectric constant (e _ms).

To predict the molar volumes or densities of the pure solvents as a function of temperature, the GCVOL model presented by Elbro et al. (1991) is applied.

(16)

GCVOL is a group contribution method where the component parameters, equation (16), are calculated from the group volume parameters according to molecular structure. However, for water, ethanol, 1,2-ethanediol and 2-methoxyethanol, the required GCVOL component parameters were specially estimated from temperature-density data. Using just A_GCVOL and B_GCVOL a good representation of the data were obtained, except for water, where the three parameters are required for the regression. Table 3 gives the GCVOL parameters used.

The mixed-solvent molar density at a determined temperature can then be calculated assuming zero volume change upon mixing (ideal behavior).

(17)

The solvent mixture dielectric constants are calculated as a function of temperature, density and composition according to Harvey and Prausnitz (1987). This method replaces Oster's mixing rule for polarization with a linear volume-fraction mixing rule in which pure components are mixed isothermally at a constant reduced density.

(18)

Polarization per unit of volume (p) is related to the static dielectric constant (e ) by the following expression of Kirkwood and Onsager (Kirkwood, 1939).

(19)

To estimate pure component polarizations at the desired conditions, the Uematsu and Franck (1980) equation is utilized as a reference model for both water and organic solvents. The Uematsu and Franck equation gives the dielectric constant for water as a function of temperature and density. In order to use this equation for an organic solvent, a reference point is required, where the dielectric constant and the density are known for a given temperature. Thus the pure component polarization at a given temperature and reduced density is calculated as follows:

(20)

where

(21)

The required reference and critical data for the pure components of this algorithm are given in Table 4.

PARAMETER ESTIMATION

The estimation of the interaction parameters is divided into two steps.

(a) Water-ion parameters are estimated from osmotic coefficient and salt solubility data of binary systems only.

(b) Organic solvent-ion and -water parameters are estimated from salt solubility in mixed solvents data, i.e., ternary systems.

In both steps all ion-ion interaction parameters are assigned the value zero.

Water-Ion Parameters

The water-ion interaction parameters are estimated from binary aqueous KCl and KBr solubility and osmotic data. The osmotic coefficient (f ) is calculated by the following expression on the ionized mole fraction basis.

(22)

We found aqueous binary data in the literature for both KCl and KBr in the temperature range of 273-373 K. Salt solubilities were taken from: Chiavone-Filho and Rasmussen (1993), Potter and Clynne (1978), Eddy and Menzies (1940), Shearman and Menzies (1937), Hering (1936), Ricci (1934), Scott and Durham (1930), Flöttmann (1928) and Scott and Frazier (1927). Osmotic coefficient data were obtained from the following authors: Herrington and Jackson (1973), Hamer and Wu (1972), Moore et al. (1972), Humphries et al. (1968) and Johnson and Smith (1941). It should be noted that the isopiestic molalities presented by Moore et al. (1972) and Humphries et al. (1968) for both salts (KCl and KBr) with respect to sodium chloride were converted to osmotic coefficient data with the aid of the reference tabulated values of f _NaCl given by Clarke and Glew (1985), as shown in equation (23).

(23)

The objective function for the simultaneous reduction of salt solubility and osmotic coefficient data is defined by equation (24). It was minimized using a modified Marquardt method for non-linear least square fitting (Fetcher, 1971). Priority was given to the salt solubility measurements so that w_SLE and w_f were set equal to 1 and 0.25, respectively.

(24)

To fit the selected aqueous binary data sets with KCl and KBr, at a total of 290 points, it was necessary to introduce a linear temperature dependence for the water-ion interaction parameters, represented by equations (25) and (26). The UNIQUAC groups are numbered in the way that we consider first solvent and then ionic species.

(25)

(26)

Table 5 gives the estimated water-ion parameters. Figure 1 shows that the proposed UNIQUAC+DH model can represent satisfactorily the aqueous salt solubilities throughout the temperature range. The procedure represented by equation (26) was applied to use a minimal number of adjusted parameters and to show the model fitting capability. But the usual linear temperature dependence (see Hansen et al., 1991) can also be applied.

Organic Solvent -Ion and -Water Parameters

In this section the measured salt solubilities for the eight ternary systems of the water+os+salt type studied previously by the authors, are described by the UNIQUAC+Debye-Hückel model. As we have observed in terms of correlation (Chiavone-Filho, 1993), it is not necessary to introduce temperature dependence to the UNIQUAC interaction parameters, and thus the a_ij,2 coefficients involving organic solvent species were set equal to zero.

The required os-ion and os-water parameters were estimated separately for each ternary system studied, except for the case of water + 2-methoxyethanol mixtures where two data sets, with KCl and with KBr, were included in the reduction. Figure 2 demonstrates that the UNIQUAC+DH model with 6 a_os-ion plus 2 a_os-water optimized parameters (see Table 6) is able to represent both KCl and KBr solubilities. The mean percent deviation (MPD), defined by equation (27), for the two data sets is 1.7 % and of the same order of magnitude as for the one-salt system correlation with six adjustable parameters (a_water,os, a_os,water, a_cation,water, a_anion,water, a_cation,os and a_anion,os, where a_ion,solv = a_solv,ion). It is noteworthy that the water-ion parameters estimated from binary water-salt data (see previous section) were used in this phase of the correlation.

(27)

Table 6

RESULTS AND DISCUSSIONS

Some predictions using the estimated UNIQUAC+DH parameters were performed and compared with data sets from the literature for the aqueous systems of KCl with ethanol (Delesalle and Heubel, 1972; Akhumov, 1940; Ferner and Mellon, 1934; Flatt and Jordan, 1933; Armstrong and Eyre, 1911) and with 1,2-ethanediol (Trimble, 1931). Figure 3 shows that the predictions are in good agreement with experimental values.

We also tried to correlate the KCl and KBr solubilities in aqueous mixtures of 2-methoxyethanol by using and fixing the VLE-based parameters from the two isothermal data sets at 343 and 363 K presented by Chiavone et al. (1993). It turns out that the fit is not satisfactory even with 12 a_solv-ion adjusted parameters. Therefore we need to perform a simultaneous correlation of VLE and salt solubility data in mixed solvents, assigning weights in order to represent better the phase equilibrium type data of greater interest. This procedure has already been applied for water-ion parameter estimation based on the osmotic coefficient and salt solubility data.

The importance of the electrostatic contribution given by the DH model at low salt concentrations was confirmed by the improvement in the correlation of the salt solubilities in aqueous glycol ether mixtures (Chiavone-Filho, 1993). This is due to the fact that A_DH,x is properly evaluated as a function of solvent composition and temperature.

The symmetric convention of normalization of activity coefficients was successfully applied for describing both osmotic coefficient and salt solubility data. However, it is important to verify the behavior of the function ln g _± throughout the salt concentration range. To provide this picture (see Figure 4) we took the water+potassium chloride system and estimated parameters (Table 5), calculating ln g _± at 298.15 K for both the DH and the UNIQUAC+DH models. The shape of the curves is similar to that of the curves in Pitzer (1980), and the value of at salt-zero concentration agrees with theory.

CONCLUSIONS

A method for the calculation of salt solubility in mixed solvents was developed.

The pure fused-salt standard state allows the evaluation of the solubility product to be calculated as a function of the temperature and of the calorimetric properties of the salt, i.e., the melting point and heat of fusion. This is an extrapolation in terms of temperature, since melting points of salts are normally far from ambient temperature, but it can provide an ideal estimation of the solubility product. Deviations are then described by the activity coefficient model, but it is recommended to evaluate the standard chemical potential of the salt in the liquid state through the properties of the ions to avoid the extrapolation procedure (Thomsen et al., 1996).

The aqueous systems studied were well correlated using the UNIQUAC+DH model. All parameters presented were estimated with reasonably small confidence limits, meaning that the model may be used for prediction purposes. Furthermore, the model requires a relatively small number of adjusted interaction parameters.

The original form of the UNIQUAC model was applied. The volume and surface parameters for the ions were calculated from a normalization with an average ion equal to 0.122nm. Recent research showed that a normalization for the ions based on the empirical volume and surface values of the water and its molar mass ratio gives the desired flexibility and compatibility with the existing non-electrolyte interaction parameters (Oliveira et al., 1999).

For the correlation of a salt solubility data set in a water plus organic solvent (os) mixture, it was shown that one adjusted parameter per os-ion interaction is required. The os-water interaction is still characterized by two parameters, but the ion-ion interactions were neglected. To give more flexibility to the modeling of more than one system, two parameters per os-ion interaction were used. This form of the model is also the same as the original UNIQUAC model and may therefore be used for the case of a systematic parameter estimation, also allowing the application of a group contribution method like UNIFAC with a comprehensive data base. The DH parameter solvent composition dependence was introduced by evaluating the density and dielectric constant directly, however it was not considered their differentiated terms in the gamma equation from the excess Gibbs energy expression (Koh et al., 1985). Thereby the DH term does not satisfy the Gibbs-Duhen equation for mixed solvents, but it does give the required electrostatic contribution, specially for salt-diluted solutions.

The systems with 2-isopropoxyethanol and 2-butoxyethanol, which split into two liquid phases, were also well represented by the UNIQUAC+DH model in terms of salt solubilities.

ACKNOWLEDGMENTS

We would like to thank IVC-SEP, Phase Equilibria and Separation Process Research Center, Denmark; CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil; and FAPESP, Fundação de Amparo à Pesquisa do Estado de São Paulo, Brazil for their financial support. We also wish to thank Jens M. Sørensen, Niro A/S, Denmark for his helpful suggestions and discussions.

LIST OF SYMBOLS

A, B, C_GCVOL GCVOL parameters [see equation (16)] A_DH,x Debye-Hückel parameter, defined by equation (A-9) DH Debye-Hückel F objective function, defined by equation (24) I_x ionic strength, defined by equation (A-10) K_salt solubility product of the salt MW molecular weight N_o Avogadro number = 6.0232 10²³ mol^-1 N_SLE number of salt solubility data points Nf number of osmotic coefficient data points Q mean ionic stoichiometric coefficient R universal gas constant = 1.98726 cal mol^-1 K^-1 SLE solid-liquid equilibrium T temperature in Kelvin UNIFAC UNIQUAC Functional Activity Coefficient UNIQUAC Universal Quasi-chemical Activity Coefficient VLE vapor-liquid equilibrium a_i,j UNIQUAC i-j interaction parameter a_i,j,1, a_i,j,2 UNIQUAC i-j interaction coefficients [see equation (26)] b DH parameter related to the closest approach of the ions d density in g cm^-3 e electron charge = 4.8029 10^-10 esu k Boltzmann constant = 1.38045 10^-16 erg K^-1 l_k function of z, r_k and q_k [see equation (A-3)] m molality in moles of solute per kg of solvent n number of moles os organic solvent p polarization per unit of volume p_0,i dielectric polarization of component i (see Table 4) q surface area parameter r volume parameter s solid state solv solvent v molar volume in cm³ mol^-1 w weight factor [see equation (24)] x_i mole fraction of species i z_k valency number of ion k z lattice coordination number = 10

Greek letters

g activity coefficient D H_m heat of fusion in cal mol^-1 D C_p heat capacity change in cal mol^-1 K^-1 Q _k surface area fraction of species k m chemical potential in cal mol^-1 n stoichiometric coefficient r molar density in mol cm^-3 s _i ratio of T_c of component i to water [see equation (21)] t _ij function of a_ij and T [see equation (A-4)] u ratio of v_c of component i to water [see equation (21)] F _k volume fraction of species k f osmotic coefficient e dielectric constant

Superscripts

C combinatorial L liquid phase o reference R residual S solid phase 0 standard ´ salt-free solvent mixture + evaluated at a reduced molar density of the mixture q original activity coefficient model

Subscripts

c refers to critical point m refers to melting point ms mixed solvent os organic solvent ± mean ionic

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APPENDIX

This appendix briefly presents the UNIQUAC and Debye-Hückel models.

UNIQUAC

The UNIQUAC model consists of two parts, a combinatorial (C) part that attempts to describe the dominant entropic contribution, and a residual (R) part that is due mainly to the interactions between the species.

The combinatorial part is determined only by the composition, sizes and shapes of the species.

(A-1)

where

(A-2)

(A-3)

The residual part depends on the intermolecular forces and contains the binary interaction parameters to be regressed.

(A-4)

Debye-Hückel

The electrostatic or long-range forces are described by the Debye-Hückel model converted to the mole fraction basis.

(A-5)

Where A_DH,x and I_x are the Debye-Hückel parameter and the ionic strength, on the mole fraction basis, respectively.

(A-6)

(A-7)

appendix

*

To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  06 July 2000
• Date of issue
  June 2000

History

• Accepted
  29 Oct 1999
• Received
  05 Apr 1999