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Calculation of Pitzer Parameters at High Ionic Strengths Charles F. Weber† Oak Ridge National Laboratory,‡ P.O. Box 2008, Oak Ridge, Tennessee 37831-6370
A technique is developed for univalent salts that allows calculation of activity coefficients at very high ionic strengths using solubility data. Pitzer parameters for binary solutions are extended to ionic strengths exceeding the salt solubility limits. Pitzer mixing parameters are obtained from solutions with ionic strengths much higher than the usual valid range for the Pitzer method. The method is demonstrated on several salt systems at 25 °C. Introduction
Ksp ) aMaX ) mMmXγ(2
The Pitzer model of electrolyte thermodynamics1,2 has been used frequently for the prediction of salt solubilities, with such diverse applications as precipitation of natural minerals, desalination of seawater, and processing of hazardous waste. It was originally formulated for aqueous solutions under 6 m and is most successfully applied in this range. While it has been applied to more concentrated solutions, certain problems arise in doing so. This work provides enhancement to the estimation of Pitzer modeling parameters in concentrated solutions of univalent electrolytes. Standard measurement of activity coefficients for pure salt systems must be restricted to concentrations below the solubility limit. However, applications in mixed salt solutions often involve higher ionic strengthssbeyond the region where pure salt activity data are available. In such cases, binary solution parameters may not be known reliably, yet they are still needed to calculate solubilities, ion association, etc. In particular, it has been shown that the Pitzer method does not extrapolate well.3 Hence, one of the motivations of this work is to extend the range of ionic strength for which binary Pitzer parameters can be applied. A second motivation for this work is to estimate Pitzer mixing parameters by using solubility data at high ionic strength. Harvie and co-workers4-6 have demonstrated the ability to estimate mixing parameters using solubility data at moderate ionic strengths (usually I < 6 m). At higher ionic strengths, the results may not be generally applicable, because the error in the binary parameters may contribute to significant error in the mixing terms. For some mixed salt systems, solubility data involves very high ionic strengths, possibly exceeding I ) 20 m. Application of the Pitzer model to such solutions requires special care and special methodology.
(1)
where a and m denote activity and molality, respectively, and γ( is the mean activity coefficient. One representation for γ( involves a Debye-Hu¨ckel term and power series in I, the ionic strength:7
ln γ( ) -
AxI 1 + BxI
+ CI + DI2 + EI3 + ...
(2)
The activity of water is most conveniently represented by the osmotic coefficient, for which the following representation is consistent with eq 2:7
φ-1)
A [-(1 + BxI) + 2 ln(1 + BxI) + B3I 2 1 3 (1 + BxI)-1] + CI + DI2 + EI3 + ... (3) 2 3 4
In eqs 2 and 3, the coefficient A depends on the DebyeHu¨ckel constant, known for given temperature and solution components and the ionic charges (A ) 1.176 16 for singly charged ions at 25 °C). The coefficients B, C, D, etc., are obtained for each electrolyte from statistical fitting of data. As many parameters as necessary can be used to represent activity and osmotic coefficients accurately through extremely wide ranges of ionic strength, in some cases exceeding I ) 20. References 7 and 8 list the values of these coefficients for several electrolytes of interest. Within the past 25 years, the procedure used by Pitzer and co-workers1,2 has become quite popular. For a binary system of singly charged ions, the analogues of eqs 2 and 3 are
ln γ( ) -A1
[
]
xI 2 + ln(1 + bxI) + mBγ + m2Cγ b 1 + bxI (4)
Salt Solubility If an aqueous solution containing ions M and X is in equilibrium with the solid precipitate MX, then the distribution is described by the solubility product. For the case of singly charged ions, this quantity is †
To whom correspondence should be addressed. Phone: 865 576-4475. Fax: 865 576-3513. E-mail:
[email protected]. ‡ Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.
and
φ-1)-
A1xI 1 + bxI
+ mBφ + m2Cφ
(5)
where A1 is the modified Debye-Hu¨ckel constant (A1 ) 0.3915 at 25 °C) and b ) 1.2 is a fixed constant. The terms Bγ, Bφ, Cγ, and Cφ depend on the adjustable parameters β(0), β(1), and C (which are fit to data) according to the relations
10.1021/ie000411o CCC: $19.00 © 2000 American Chemical Society Published on Web 10/07/2000
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4423
Bφ ) β(0) + β(1)e-RxI, B ) β(0) + β(1)g(RxI), 2 g(x) ) 2[1 - (1 + x)e-x] x and
Table 1. Optimal Parameters for the NaCl Systema,b parameter
value
parameter
value
B C D
1.250 8 0.120 45 -0.014 113
E F ln Ksp (NaCl)
0.003 175 5 -1.625 3 × 10-4 3.615 5
a Results valid to 12 m. b Calculated using the following mixing parameters from Pitzer:2 θH,Na ) 0.036, θCl,OH ) -0.050, θCl,NO3 ) 0.016, ψH,Na,Cl ) -0.004, ψNa,Cl,OH ) -0.006, and ψNa,Cl,NO3 ) -0.006.
3 Bγ ) B + Bφ, Cφ ) 2C, Cγ ) Cφ, 2 R ) 2 (kg/mol)1/2 The Pitzer formulation is generally applicable in the range 0 < I < 6 m, although it may often be used at higher ionic strengths. It requires only three adjustable parameters and is, therefore, easier to use than eqs 2 and 3. In addition, the Pitzer formulation is easily generalized to multicomponent solutions, which is one reason for its popularity. However, the method does not extrapolate wellsit may have a large error outside the range used to derive the parameters.3 Pitzer’s Method in Multicomponent Solutions The three binary parameters β(0), β(1), and C account for interactions between a single cation-anion pair. For many applications these will provide a credible description of behavior in systems containing additional ions. However, this is not always the case, and a few additional parameters may be necessary. The “mixing rules” of the Pitzer procedure are best illustrated with a ternary system (i.e., two salts that have a common anion or a common cation). For singly charged ions, a solution of salts MX and NX (present with solute mole fractions 1 - y and y, respectively), the activity and osmotic coefficients can be represented as
{
ln γMX ) f γ + m BMX + (1 - y)BφMX + yBφNX + 3 y m - y CφMX + yCφNX + y ΦMN + 1 - mψMNX 2 2
[(
] [
)
(
)
]} (6)
and
φ - 1 ) f φ + (1 - y)m(BφMX + 2mCMX) + ym(BφNX + 2mCNX) + y(1 - y)m(ΦφMN + mψMNX) (7) where f γ and f φ are the Debye-Hu¨ckel terms in eqs 4 and 5:
fφ)-
A1xI
, fγ ) f φ -
1 + bxI
2A1 ln(1 + bxI) b
If both MX and NX are 1-1 salts, then mixed solution properties can be expressed in terms of binary activity and osmotic coefficients, together with “mixing terms” involving Φ and ψ:
ln γMX ) ln γbMX + y(φbNX - φbMX) +
[
(
yI ΦMN + I 1 -
y ψ 2 MNX
)
)] (8)
and
φ - 1 ) y(φbMX - 1) + (1 - y)(φbNX - 1) + y(1 - y)I(ΦMN + IψMNX) (9)
Figure 1. Solubilites in solutions containing NaCl: s, calculation (this study); b, data (Linke12).
where the superscript b is now used to indicate the binary solution representations (4) and (5). Analogous expressions also hold for the binary solution of salt NX. Equations 8 and 9 are useful because they identify those mixing rules which utilize the parameters ΦMN and ψMNX and involve general binary solution activity and osmotic coefficients. Their special nature does not require use of Pitzer binary representations (4) and (5) nor even knowledge of the binary parameters β(0), β(1), and C for salts MX and NX. That is, some other formalism for binary coefficients [e.g., eqs 2 and 3] can be used in eqs 8 and 9 to estimate the Pitzer mixing parameters ΦMN and ψMNX. (An alternative approach is to add more virial terms to the binary Pitzer equations, as was done by Pitzer et al.9 and Holmes et al.10)
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Table 2. Optimal Parameter Values for the KCl Systema,b parameter
value
parameter
value
B C D E
1.439 1 -0.050 932 0.034 925 -0.006 085 3
F ln Ksp (KCl) ψK,H,Cl ψK,OH,Cl
3.327 5 × 10-4 2.014 8 -0.010 99 -0.003 17
a Results valid to 9.3 m. b Calculated using the following mixing parameters from Pitzer:2 θK,H ) 0.005, θCl,OH ) -0.050, θK,Na ) -0.012, and ψK,Na,Cl ) -0.0018.
This is especially helpful if the range of ionic strength exceeds that for which the binary Pitzer parameters are valid (generally I e 6). It is important also to note that eqs 8 and 9 were derived for a 1-1 electrolyte. Such simple relations do not hold generally unless the absolute charges of all ions are the same. However, similar results can also be obtained for ternary 1-1 systems involving a common cation. Parameter Estimation Using eqs 2 and 3 for the binary coefficients in eq 8 yields an activity coefficient in the mixed system. This can be substituted into eq 1 to predict salt solubility. If the coefficients B, C, D, etc., in eqs 2 and 3 are known, together with the mixture parameters ψ and Φ in eq 8, the system is well defined and accurate prediction can
be obtained. Because these parameters are not known a priori, they must be obtained by nonlinear leastsquares fitting to solubility data. This has been done for several systems of univalent salt mixtures involving a common ion. The actual estimation of parameters is straightforward and utilizes both Gauss-Newton and BFGS algorithms, as described by Fletcher.11 All solubility data occur at 25 °C, and most were obtained from the compendium by Linke.12 NaCl to High Ionic Strengths. Using solubility data for the ternary systems Na-H-Cl, Na-Cl-OH, and Na-Cl-NO3, the range of NaCl binary parameters was extended from the salt solubility limit (about 6.1 m) to ionic strengths of about 12 m. In lieu of actual experimental measurements, the published values of Hamer and Wu7 for binary activity and osmotic coefficients were used as data points below the solubility limit. The results of parameter estimation are shown in Table 1. Satisfactory values of ternary mixing parameters were already available, as listed in Table 1. Figure 1 illustrates the usefulness of these results in calculating solubilities. KCl to High Ionic Strengths. The binary parameters had previously been fit to data up to the solubility limit (about 4.8 m). This range is extended to ionic
Figure 2. Solubilities in mixed systems containing NO3- ions: s, calculation (this study); 402b, data from Linke,12 (c) O, data from Cornec and Krombach,13 (d) 9, data from Flatt and Bocherens.14
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4425 Table 3. Optimal Parameters for the Nitrate Systema,b parameter
value
parameter
value
NaNO3 E F ln Ksp
0.003 1755 -1.625 3 × 10-4 3.615 5
KNO3 F G ln Ksp
-1.974 5 × 10-4 3.823 5 × 10-6 -0.195 45
B C D
1.250 8 0.120 45 -0.014 113
B C D E
0.635 79 -0.079 415 -0.019 942 0.003 855 5
θOH,NO3 ψK,OH,NO3 ψNa,OH,NO3 ψK,H,NO3
Pitzer Mixing Parameters -0.054 66 ψNa,H,NO3 -0.002 74 -0.003 21 ψK,Cl,NO3 -0.003 12 0.000 19 ψNa,K,NO3 -0.005 96 -0.010 34
a Validity exceeds I ) 20 m. b Calculated using the following mixing parameters from Pitzer:2 θK,H ) 0.005, θCl,NO3 ) 0.016, θNa,H ) 0.036, and θNa,K ) -0.012.
Table 4. Binary Pitzer Parameters at Extended Ionic Strengths solute
β(0)
β(1)
Cφ
range
KCl KNO3 NaCl NaNO3
0.059 57 -0.098 27 0.067 43 -0.000 55
0.1782 0.2118 0.3301 0.2110
-0.004 33 0.001 99 0.002 63 0.000 82
Ie8 I e 11 I e 8.5 I e 8.5
strengths of 9.3 by using solubility data for ternary systems K-H-Cl, K-Cl-OH, and K-Na-Cl. The published results of Hamer and Wu7 were used in place of experimental data below the solubility limit. The parameter estimation results are given in Table 2. Ternary mixing parameters ψK,H,Cl and ψK,OH,Cl were included in the optimization, and the resulting estimates differ somewhat from the values listed in ref 2. The values in Table 2 allowed better computation of solubilities and, thus, were used in the remainder of this work. NaNO3 and KNO3. These two salts were optimized simultaneously because of the mixing parameter ΦOH,NO3, which is common to both K+ and Na+ systems. Previously, KNO3 binary parameters were valid only up to I ) 3.5. This limit is extended considerably in the present work. Also, the range for NaNO3 binary parameters is extended, and several mixture parameters are estimated. Again, binary data from Hamer and Wu7 (revised values from Wu and Hamer8 for NaNO3) were used below the solubility limits. Optimal parameters are given in Table 3 and model calculations compared with data in Figure 2.
Figure 3. Water activity in supersaturated NaCl solutions: 9, data from Chan et al.;15 2, data from An et al.;16 s, calculation (this study).
centration range of binary Pitzer parameters and for determining mixture parameters from solubility data of high ionic strength. The extension of binary parameters beyond the solubility limit constitutes a measure of supersaturated solution properties. A comparison of this approach with other methods is shown in Figure 3, which depicts water activity in supersaturated NaCl solutions. The solid line was calculated using the values in Table 1 and is slightly below the data of Chan et al.15 These authors attained supersaturation by the evaporation of water from aerosol droplets. Also shown is the data of An et al.,16 which is well above both our own calculations and the data of Chan et al.15 These authors measured vapor pressures of mixed solutions of high ionic strength. Figure 3 indicates a divergence of the three methods as ionic strength increases. This illustrates the inherent difficulties in thermodynamic measurement of systems not in equilibrium and the definition of model parameters which cannot be directly measured. None of these methods can claim unique superiority over the others, although each may claim some special relevance. The model described in this work is consistent with the Pitzer procedure and has proved useful in predicting solubilities at high ionic strength. Acknowledgment This work was funded by the U.S. Department of Energy through the Office of Science and Technology’s Tanks Focus Area. Nomenclature Roman Symbols
Results and Conclusions The optimal parameters in Tables 1-3 are combined with polynomial coefficients for NaOH, KOH, HCl, and HNO3 from refs 7 and 8 to produce a model which can predict solubilities at high ionic strengths. The results in Figures 1 and 2 indicate good agreement with data at 25 °C. Binary Pitzer parameters at high ionic strength can be obtained by fitting to the polynomial representation,2 as if it were actual data. This has been done for salts whose coefficients appear in Tables 1-3. These are shown in Table 4, together with the valid concentration range. The method is valid only for solutions where all ionic charges have the same magnitude. In the present example, this has been demonstrated for univalent salts. However, the method is useful for extending the con-
a ) chemical activity A, A1 ) Debye-Hu¨ckel coefficients (eqs 2 and 5, respectively) B, C, D, ... ) empirical constants (eqs 2 and 4) B, Bγ, Bφ ) Pitzer coefficients (eq 5) C, Cγ, Cφ ) Pitzer coefficients (eq 5) f γ, f φ ) Debye-Hu¨ckel terms I ) Ionic strength Ksp ) solubility product m ) concentration (m) y ) solute mole fraction Greek Symbols β(0), β(1) ) Pitzer parameters γ ) activity coefficient φ ) osmotic coefficient Φ, Ψ ) Pitzer mixing parameters
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Subscripts M, N ) cation indices X ) anion index
Literature Cited (1) Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268. (2) Pitzer, K. S. Ion-Interaction Approach: Theory and Data Correlation. In Activity Coefficients in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, FL, 1991. (3) Zemaites, J. F., Jr.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte ThermodynamicssTheory and Application; AIChE: New York, 1986. (4) Harvie, C. E.; Weare, J. H. The Prediction of Mineral Solubilities in Natural WaterssThe Na-K-Mg-Ca-Cl-SO4H2O System from 0 to High Concentration at 25 °C. Geochim. Cosmochim. Acta 1980, 44, 981. (5) Harvie, C. E.; Eugster, H. P.; Weare, J. H. Mineral Equilibria in the 6-Component Seawater System Na-K-Mg-CaSO4-Cl-H2O at 25 °C. 2. Compositions of the Saturated Solutions. Geochim. Cosmochim. Acta 1982, 46, 1603. (6) Harvie, C. E.; Moller, N.; Weare, J. H. The Prediction of Mineral Solubilities in Natural WaterssThe Na-K-Mg-Ca-HCl-SO4-OH-HCO3-CO3-CO2-H2O System to High Ionic Strengths at 25 °C. Geochim. Cosmochim. Acta 1984, 48, 723. (7) Hamer, W. J.; Wu, Y. C. Osmotic Coefficients and Mean Activity Coefficients of Uni-univalent Electrolytes in Water at 25 °C. J. Phys. Chem. Ref. Data 1972, 1 (4), 1047-99. (8) Wu, Y. C.; Hamer, W. J. Revised Values of the Osmotic Coefficients and Mean Activity Coefficients of Sodium Nitrate in Water at 25 °C. J. Phys. Chem. Ref. Data 1980, 9 (2), 513-18. (9) Pitzer, K. S.; Wang, P. M.; Rard, J. A.; Clegg, S. L. Thermodynamics of Electrolytes. 13. Ionic Strength Dependence
of Higher Order Terms; Equations for CaCl2 and MgCl2. J. Solution Chem. 1999, 28 (4), 265-282. (10) Holmes, H. F.; Busey, R. H.; Simonson, J. M.; Mesmer, R. E.; Archer, D. G.; Wood, R. H. The Enthalpy of Dilution of HCl(aq) to 648 K and 40 MPasThermodynamic Properties. J. Chem. Thermodyn. 1987, 19 (8), 863-890. (11) Fletcher, R. Practical Methods of Optimization, 2nd ed.; John Wiley & Sons: New York, 1987. (12) Linke, W. F. Solubilities, 4th ed.; Van Nostrand Co.: Princeton, NJ, 1958; Vol. I and Linke, W. F. Solubilities, 4th ed.; American Chemical Society: Washington DC, 1965; Vol. II. (13) Cornec, E.; Krombach, H. Contribution to the Study of Equilibria between Water, Nitrates, Chlorides, and Sulfates of Sodium and Potassium. Ann. Chim. (Paris) 1929, 12 (10), 20334. (14) Flatt, R.; Bocherens, P. On the Ternary System K+-H+NO3--H2O. Helv. Chim. Acta 1962, 45, 195-97. (15) Chan, C. K.; Liang, Z.; Zheng, J.; Clegg, S. L.; Brimblecombe, P. Thermodynamic Properties of Aqueous Aerosols to High Supersaturation: IsMeasurements of Water Activity of the System Na+-Cl--NO3--SO42--H2O at ∼298.15 K. Aerosol Sci. Technol. 1997, 27 (3), 324-344. (16) An, D. T.; Teng, T. T.; Sangster, J. M. Vapour pressures of CaCl2-NaCl-H2O and MgCl2-NaCl-H2O at 25 °C. Prediction of water activity of supersaturated NaCl solutions. Can J. Chem. 1978, 56, 1853-1855.
Received for review April 13, 2000 Revised manuscript received August 21, 2000 Accepted August 29, 2000 IE000411O
