Temperature Dependence of a Modified Pitzer ... - ACS Publications

Nov 25, 2003 - Chemical Engineering Department, Texas A&M University, College ... We have established the temperature dependence for a modified Pitzer...
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6962

Ind. Eng. Chem. Res. 2003, 42, 6962-6969

Temperature Dependence of a Modified Pitzer Equation for Strong Electrolytes Systems Fernando Pe´ rez-Villasen ˜ or† and Gustavo A. Iglesias-Silva* Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya, Celaya Guanajuato C.P. 38010, Mexico

Kenneth R. Hall Chemical Engineering Department, Texas A&M University, College Station, Texas 77843

We have established the temperature dependence for a modified Pitzer model. This model adequately correlates osmotic and activity coefficient data together with the dilution enthalpy for single-electrolyte solutions. For multisalt aqueous solutions, the modified model can successfully predict the behavior of the osmotic coefficient and the enthalpy at different temperatures. For the new model, the total average percentage error in the osmotic and activity coefficients is 3.1%. Introduction

GEm

The presence of electrolytes in industrial processes requires knowledge of the nonideality of these solutions in terms of the osmotic and activity coefficients. Over the past 30 years, many researchers1-7 have developed equations to predict these properties for electrolyte solutions applicable at 298.15 K. Desalination, geochemical, and environmental processes require thermodynamic properties at temperatures other than 298.15 K. To address this problem, Pitzer8,9 described a procedure to accommodate both temperature and pressure effects; however, he did not report the parameter values for the resulting temperature- and pressure-dependence expressions. Recently, Pe´rez-Villasen˜or10 reported a set of temperature-dependence parameters for the Pitzer model. They based their parameters on dilution enthalpies and osmotic and activity coefficients of 1:1 and 1:2 aqueous single-electrolyte solutions. Their work covers a wide temperature range that, in some cases, goes up to 623 K. Pe´rez-Villasen˜or et al.11 performed a detailed analysis of the Pitzer model and proposed a modification that improves the model predictions. Also, Pe´rez-Villasen˜or et al.12 used the model successfully to predict the osmotic and activity coefficients of multicomponent solutions. The objective of this work is to account for the temperature effects of the modified Pitzer model, compare the modified model to the original model with the original temperature dependence,8,9 and predict the osmotic and activity coefficient for multicomponent systems at temperatures greater than 298.15 K. This work does not incorporate pressure effects because the relevant experimental data do not exist.

wwRT

) F(I) +

∑c ∑a mcma[Bca + (Σmz)CMX]

where F(I) is the Debye-Hu¨ckel term as considered by Pe´rez-Villasen˜or et al.12 for multicomponent solutions q

F(I) ) -4AφI

[

yk

∑ k)1 b

* To whom correspondence should be addressed. Tel.: 52 461 17802. Fax: 52 461 17744. E-mail: [email protected]. † Present address: Departamento de Ingenierı ´a y Tecnologı´a, Universidad Auto´noma de Tlaxcala, Apizaco, Tlaxcala C.P. 90300, Mexico.

ln(1 + bkI1/2)

k

]

(2)

For single-electrolyte-solvent solutions, the excess Gibbs free energy for the modified Pitzer model of Pe´rezVillasen˜or et al.11 is

4IAφ GE )ln(1 + bMXI1/2) + 2mMmX(BMX + wwRT bMX mMzMCMX) (3) In eq 3, bMX, BMX, and CMX are characteristic parameters for each salt solution; Aφ is the Debye-Hu¨ckel coefficient;8-10,13 and I is the ionic strength

I)

1

mizi2 ∑ 2 i

(4)

Using eq 3, the expression for the osmotic coefficient is

φ - 1 ) |zMzX|f φ +

( )

2νMνX mBMX + ν 2(νMνX)3/2 2 m CMX (5) ν

Modified Pitzer Model The general expression for a modified Pitzer equation describing excess Gibbs free energy is

(1)

[

]

and for the activity coefficient, we use the definition for the mean ionic activity coefficient

1 ln γ( MX ) (νM ln γM + νX ln γX) ν

(6)

Thus, the expression for the mean ionic activity coefficient expression is

10.1021/ie030251r CCC: $25.00 © 2003 American Chemical Society Published on Web 11/25/2003

Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6963 γ ln γ( MX ) |zMzX|f +

( )

2νMνX mBγMX + ν 2(νMνX)3/2 2 γ m CMX (7) ν

(

)

In eqs 6 and 7, the Debye-Hu¨ckel terms for the osmotic and activity coefficients are 1/2

fφ)and

[

AφI

1/2

I 2 + ln(1 + bMXI1/2) f ) -Aφ 1/2 b 1 + bMXI MX γ

(8)

1 + bMXI1/2

constant P and I and

( )

AL ∂Aφ ) 4T RT ∂T

ww

q

) ALI

[

yk

∑ k)1 b

BγMX ) 2BMX

(10)

CγMX ) 3|zMzX|1/2CMX

(11)

]

]

ln(1 + bkI1/2) -

k

2RT

respectively. In the modified Pitzer equation, the second and third virial-type coefficient contributions to the activity coefficient are

)

with  and V being the dielectric constant and the volume of the solvent, respectively. Using the expressions for the osmotic and activity coefficients in multicomponent mixtures given by Pe´rez-Villasen˜or et al.,12 the relative apparent molal enthalpy is

L

(9)

∂ ln  + ∂T P,I T ∂ ln V (16) 3 ∂T P,I

(

φ

]

P,I

[ ( )

) -6Aφ 1 + T

∑c ∑a mcma[BLca + (Σmz)CLca]

2

(17)

where BLca and CLca are the partial derivatives of Bca and Cca, respectively, with respect to temperature at constant P and I; bk is the maximum approach parameter; and yk is the ionic strength fraction of salt k in the mixture.

and Parameter Optimization

respectively. It recently came to our attention that Lietzke and Stoughton14 also considered the apparent virial coefficients (up to the fourth) independent of the ionic strength. However, for the long-range interaction, they considered the Debye-Hu¨ckel osmotic function. To preserve simplicity in the equation, we consider bMX to be independent of temperature and BMX and CMX to be simple polynomial functions

BMX(T) ) q1 + q2(T - Tr) + q3(T 2 - Tr2) + q4(T3 Tr3) + q5(T4 - Tr4) (12) and

CMX(T) ) q6 + q7(T - Tr) + q8(T 2 - Tr2) + q9(T3 Tr3) + q10(T4 - Tr4) (13) In the above equations, Tr is 298.15 K, and the qi are adjustable parameters. In this work, we have calculated the dilution enthalpy of a solution containing 1 mol of solute from concentration m1 to concentration m2. This thermodynamic quantity is related to the apparent molal enthalpy through

∆Hdil(m1 f m2) ) φL2 - φL1

(14)

with the relative apparent molal enthalpy given by

φ

L)-

( )

2

T n

GEm T ∂T

∂

AL ) ν|zMzX| ln(1 + bMXI1/2) P,m 2bMX

2νMνXRT 2[mBLMX + m2νMzMCLMX] (15) where BLMX and CLMX are the partial derivatives of BMX and CMX, respectively, with respect to temperature at

We optimized the qi adjustable parameters using a least-squares method developed by Stewart et al.15 We used experimental dilution enthalpies (when available) and osmotic and activity coefficients of aqueous electrolyte solutions to determine the characteristic parameters. Table 1 contains the experimental data sources, temperature and molality ranges, and number of data points for the 32 electrolytes considered in this work. For simplicity, we fixed q1 ) BMX|298.15K and q6 ) CMX|298.15K. These parameters (bMX, BMX, and CMX) were obtained from Pe´rez-Villasen˜or et al.11,12 and appear in Table 2. Parameters for the Pitzer model at 298 K are reported in Table 3. Also, we calculated the parameters of the Lietzke-Stoughton model14 truncated at the apparent third virial coefficient. These parameters also appear in Table 2. To test the model’s prediction capabilities, we included no dilution enthalpy data in the regression analyses for H2SO4 and Ca(NO3)2. In this work, we used the temperature-dependence expressions from Ferna´ndez et al.16 for the Debye-Hu¨ckel coefficient and for the dielectric constant and its derivatives. Results and Discussion We compare the modified model and the LietzkeStoughton model14 truncated at the third virial at 298.15 K. The average percentage deviations of the modified model and Lietzke-Stoughton model are 1.23 and 3.8%, respectively. These results agree with the findings of Pitzer6 when he replaced the traditional Debye-Hu¨ckel function with the Debye-Hu¨ckel osmotic function in his equation. He also found that the traditional Debye-Hu¨ckel function better correlates the osmotic and mean activity coefficients data. The parameter A of the Lietzke-Stoughton model corresponds to the bMX parameter in the modified Pitzer model. These parameters should show a trend for cations of the same type according to their positions in the periodic table. For the modified Pitzer model, the bMX parameters for CaCl2 and LiBr show an anomaly because of the

6964 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 Table 1. Systems Used in This Work and Corresponding Experimental Sources system

variable(s)

data set size

mmax

temperature range (K)

experimental source(s)

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

γ, φ φ φ γ, φ γ γ, ∆Hdil γ, φ γ, φ φ γ, φ, ∆Hdil γ, φ γ, φ γ, φ γ, φ φ γ, φ, ∆Hdil φ φ γ, φ φ γ, φ φ γ, φ, ∆Hdil φ, ∆Hdil φ φ φ γ γ

229 40 25 114 117 295 317 106 40 278 213 41 251 310 40 255 30 40 52 40 55 25 240 83 40 40 40 48 52

8.277 8.590 2.595 5.921 1.000 16.000 7.434 4.000 5.648 17.000 5.387 7.219 7.981 4.000 8.398 7.298 6.949 3.398 1.500 4.596 7.031 2.915 20.000 4.801 3.340 3.203 4.156 1.000 0.800

273.15-523.15 298.15-343.15 298.15-343.15 298.15-473.15 273.15-333.15 273.15-623.15 273.15-498.15 273.15-313.15 298.15-343.15 273.15-623.15 298.15-473.15 298.15-473.15 298.15-498.15 273.15-373.15 298.15-343.15 273.15-623.15 298.15-343.15 298.15-343.15 298.15-333.15 298.15-343.15 298.15-343.15 298.15-343.15 273.15-393.15 298.15-573.15 298.15-343.15 298.15-343.15 298.15-343.15 283.15-313.15 278.15-313.15

Patil et al.,25 Holmes and Mesmer27 Patil et al.25 Patil et al.25 Holmes and Mesmer22 Harned and Owen23 Harned and Owen,23 Fuangswasdi et al.28 Robinson and Stokes,24 Patil et al.,25 Holmes and Mesmer27 Harned and Owen,23 Patil et al.25 Patil et al.25 Holmes and Mesmer22, Harned and Owen,23 Fuangswasdi et al.28 Holmes and Mesmer27 Holmes and Mesmer22 Harned and Owen,23 Patil et al.,25 Holmes and Mesmer27 Harned and Owen,23 Robinson and Stokes24 Patil et al.25 Holmes and Mesmer22, Harned and Owen,23 Fuangswasdi et al.28 Patil et al.25 Patil et al.25 Patil et al.25 Patil et al.25 Ananthaswamy and Atkinson,13 Patil et al.,25 Baabor et al.19 Patil et al.25 Oakes et al.29 Patil et al.,25 Wang et al., 21 Baabor et al.20 Patil et al.25 Patil et al.25 Patil et al.25 Harned and Owen23 Harned and Owen23

Table 2. Parameters for the Modified Pitzer Model and the Lietzke and Stoughton Model at 298 K system CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

bMX

BMX

CMX

1.47723 0.01030 0.00118 1.34596 0.02923 -0.00019 1.42147 0.01336 -0.00126 2.64030 0.13357 -0.00748 0.38021 0.35774 -0.00446 1.45056 0.20339 -0.00196 2.06975 0.03407 0.00000 1.99609 0.02699 0.00043 2.30629 0.04747 -0.00101 0.67967 0.21040 -0.00211 0.10632 0.37681 -0.00402 0.64805 0.10607 -0.00557 2.06796 0.08821 0.00000 2.22718 0.05383 0.00134 2.14921 0.11339 0.00000 0.43331 0.20075 -0.00212 1.80000 0.02535 0.00008 2.621266 0.19530 0.00000 2.546986 0.14481 -0.00116 2.735398 0.25292 0.00587 1.108089 0.44947 -0.00608 2.994201 0.30217 0.00649 2.28619 0.10395 -0.00077 2.37916 0.28486 0.00331 2.771101 0.20708 0.00629 2.616823 0.18048 0.00330 3.043715 0.26538 0.00753 3.53559 -0.02209 0.00323 4.96649 0.20142 -0.00579

A

B

C

0.97166 -0.01225 0.00447 0.85923 0.01363 0.00035 0.92823 -0.00890 -0.00011 1.73408 0.09309 -0.00189 0.00426 0.44117 -0.01113 0.84771 0.19491 -0.00371 1.29754 0.01321 0.00195 1.26026 0.00544 0.00297 1.37172 0.02930 -0.00026 0.39812 0.20698 -0.00424 0.00134 0.41078 -0.00836 0.46031 0.08681 -0.00984 1.24252 0.07391 0.00076 1.38543 0.03325 0.00453 1.21548 0.10556 0.00010 0.28427 0.19402 -0.00418 1.08156 0.01230 0.00080 0.00236 0.24907 -0.00851 0.43333 0.09537 -0.00301 0.13004 0.18190 0.00140 0.00454 0.25412 -0.00387 0.00220 0.30043 -0.00470 0.66735 0.02007 0.00000 0.15016 0.17992 0.00033 0.00225 0.25236 -0.00428 0.19428 0.13942 -0.00091 0.00209 0.28240 -0.00350 0.53983 -0.00122 0.00152 7.72563 0.08486 0.33293

range of molality in the selected database. When the molality range is reduced up to 6 m, the values follow the expected trend. However, in the Lietzke-Stoughton model, when the activity and osmotic coefficients are fitted simultaneously, the value of in some cases is anomalously low, even if the molality range is reduced to 3 m. This is not the case for the modified Pitzer model, as shown in Table 2. We used experimental osmotic and activity coefficients of aqueous electrolyte solutions to determine the temperature dependences of the modified and original

Table 3. Parameters for the Pitzer Model at 298 K system

β(0)

β(1)

CMX

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

0.01975 0.03389 0.02174 0.12091 0.23370 0.20783 0.05504 0.04646 0.07337 0.18576 0.31088 0.06680 0.10853 0.07598 0.13282 0.17384 0.04558 0.31335 0.26297 0.34611 0.43638 0.43240 0.13827 0.56756 0.33022 0.28663 0.39675 0.09122 0.32891

0.0828550 0.0477280 0.0638000 0.4473350 0.1332960 0.1158180 0.2349190 0.2200390 0.2640060 -0.1057150 -1.1844110 -0.1735890 0.2357630 0.2739420 0.2643940 -0.5768310 0.1463240 1.5788450 1.5024390 1.8209030 0.0228170 1.8655360 1.9827100 -0.3035640 1.7025980 1.6267950 1.9038420 2.3841580 3.1307720

-0.0008230 -0.0003180 -0.0016340 0.0034750 0.0000000 -0.0020200 -0.0007220 -0.0002100 -0.0020130 -0.0017010 -0.0034650 -0.0037520 -0.0006160 -0.0006420 -0.0004880 -0.0018670 -0.0007390 -0.0055090 -0.0068560 0.0033950 -0.0058510 -0.0005190 -0.0010850 -0.0129130 0.0005650 -0.0005360 0.0015160 -0.0000720 -0.0092500

Pitzer models. For the original Pitzer model, we used the original temperature dependence proposed by Pitzer.9,10 Tables 4 and 5 list the values of temperature coefficients and their corresponding asymptotic standard errors for the modified and original Pitzer models, respectively. The absence of a parameter value from the table indicates that the parameter is not statistically significant. We also compared our model to the original Pitzer model, and Table 6 presents the average percentage

Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6965 Table 4. Temperature-Dependence Parameters and Corresponding Asymptotic Standard Errors for the Modified Pitzer Model system

q2

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

3.44798 × 5.51910 × 10-3 -1.58290 × 10-1 8.12357 × 10-3 1.56776 × 101 -1.71368 × 10-1 -1.39717 × 10-1 2.86795 × 100 -1.40394 × 100 -1.41453 × 10-1 3.81940 × 10-3 -1.00559 × 100 1.92418 × 10-2 1.54152 × 10-2 3.33596 × 10-1 -2.62622 × 10-1 -1.54079 × 10-2 1.26846 × 100 3.91315 × 100 9.64662 × 10-1 -3.92082 × 10-1 -3.91664 × 10-1 1.45810 × 10-2 1.84710 × 10-2 9.54242 × 10-1 -1.07279 × 10-1 6.63419 × 10-1 -2.06830 × 10-1 -1.98874 × 100

σ q2 10-3

3.7 × 1.4 × 10-2 1.9 × 10-2 3.7 × 10-4 1.7 × 10-2 8.6 × 10-4 3.5 × 10-3 2.3 × 10-3 1.3 × 10-2 1.5 × 10-3 1.8 × 10-3 5.5 × 10-4 6.4 × 10-4 9.3 × 10-5 1.6 × 10-3 4.5 × 10-3 6.5 × 10-3 1.0 × 10-2 8.0 × 10-3 8.6 × 10-3 8.2 × 10-3 2.5 × 10-2 1.6 × 10-3 1.4 × 10-3 1.9 × 10-2 2.1 × 10-2 1.1 × 10-2 5.2 × 10-2 5.4 × 10-2

q3

σ q3

10-4

system

q7

σ q7

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

-3.63658 × 10-4 -1.17697 × 10-2 6.97457 × 10-2 9.76520 × 10-4 5.67697 × 100 3.62497 × 10-2

7.5 × 10-6 1.0 × 10-3 4.6 × 10-3 5.2 × 10-5 1.2 × 10-2 3.5 × 10-4

-1.18033 × 100 1.22543 × 10-1 8.79494 × 10-4 -1.14129 × 10-3 1.37049 × 10-1

4.9 × 10-4 1.4 × 10-3 1.6 × 10-4 5.9 × 10-5 7.2 × 10-5

-1.51455 × 10-3 5.55635 × 10-2 1.96540 × 10-2 -1.00847 × 10-1 -1.97782 × 100 -7.00288 × 10-2 1.32632 × 10-2 -6.44254 × 10-2 -5.78334 × 10-5 1.04754 × 10-3 -6.66828 × 10-2 3.96774 × 10-2 -7.63711 × 10-2 8.40930 × 10-2 1.08518 × 101

q4

σ q4

q5

-1.76563 × -2.88421 × 10-6 1.71573 × 10-6 -4.94910 × 10-8 1.74340 × 10-4 -1.26902 × 10-6 -9.75020 × 10-7 1.96825 × 10-5 1.33470 × 10-5 -8.45455 × 10-7 -1.59279 × 10-8 -6.98567 × 10-6 1.07644 × 10-7

2.3 × 4.0 × 10-7 2.6 × 10-7 4.2 × 10-9 3.2 × 10-6 5.9 × 10-9 1.5 × 10-8 8.7 × 10-8 3.0-06 7.9 × 10-9 1.2 × 10-8 2.0 × 10-7 4.3 × 10-9

2.54330 × 10-6 -1.54031 × 10-6 1.33543 × 10-7 -1.22984 × 10-5 2.55933 × 10-5 -9.14636 × 10-6 3.72552 × 10-6 3.47861 × 10-6 6.03552 × 10-8 -7.42604 × 10-11 -9.17246 × 10-6 1.03770 × 10-6 -6.31051 × 10-6

6.4 × 10-7 6.1 × 10-8 2.1 × 10-10 2.5 × 10-6 1.5 × 10-8 4.6 × 10-6 7.8 × 10-9 8.1 × 10-7 3.7 × 10-9 7.8 × 10-13 4.3 × 10-9 9.0 × 10-8 2.0 × 10-6

1.97369 × 10-5

8.9 × 10-7

2.05694 × 4.42562 × 10-9 -2.79937 × 10-9 5.60720 × 10-11 -1.44124 × 10-7 8.01161 × 10-10 6.28988 × 10-10 -1.10704 × 10-8 -2.05282 × 10-8 4.97848 × 10-10 1.65555 × 10-11 4.54139 × 10-9 -6.37667 × 10-11 4.00346 × 10-11 -1.72652 × 10-9 8.97269 × 10-10 -1.93693 × 10-10 1.91069 × 10-8 -1.39595 × 10-8 1.40457 × 10-8 -5.73831 × 10-9 -5.18255 × 10-9 -2.51596 × 10-11 4.02481 × 10-11 1.41966 × 10-8 -1.60533 × 10-9 9.71043 × 10-9 -3.58267 × 10-10 -3.63247 × 10-8

10-8

-7.86411 × 10-2 7.12043 × 10-4 5.58156 × 10-4 -1.16453 × 10-2

1.5 × 10-4 2.3 × 10-5 7.1 × 10-6 3.3 × 10-3

5.24524 × 10-4

1.6 × 10-5

3.99330 × 10-3 -6.83198 × 10-5 -3.15788 × 10-5 -1.38841 × 10-3 9.65776 × 10-4

7.9 × 10-6 3.3 × 10-6 2.5 × 10-6 5.0 × 10-4 7.8 × 10-6

-1.55304 × 10-2

7.7 × 10-4

-4.63650 × 10-5 -3.98138 × 10-5

3.6 × 10-6 7.7 × 10-7

4.07242 × 10-4 9.70131 × 10-4

8.3 × 10-5 5.0 × 10-5

q8

10-9

σq5 10-11

2.7 × 10-12 1.1 × 10-10 1.5 × 10-10 1.8 × 10-12 1.4 × 10-10 7.1 × 10-12 7.9 × 10-12 2.3 × 10-11 9.8 × 10-11 8.3 × 10-12 1.5 × 10-11 2.7 × 10-12 5.4 × 10-12 6.5 × 10-13 1.3 × 10-11 5.8 × 10-12 5.1 × 10-11 8.0 × 10-11 7.5 × 10-11 6.7 × 10-11 6.5 × 10-11 1.9 × 10-10 5.6 × 10-12 2.1 × 10-12 1.5 × 10-10 1.7 × 10-10 8.2 × 10-11 4.9 × 10-10 5.3 × 10-10

σ q8

q9

σq9

q10

σq10

-2.84963 × 10-2 -1.51584 × 10-4

8.3 × 10-4 3.6 × 10-5

2.05444 × 10-9 1.09935 × 10-7 -7.02679 × 10-7 -6.28545 × 10-9 6.38510 × 10-5 2.70682 × 10-7

1.1 × 10-10 2.3 × 10-8 5.8 × 10-9 4.9 × 10-11 1.1 × 10-5 3.6 × 10-9

-2.45520 × 10-12 -1.67793 × 10-10 1.11197 × 10-9 8.16841 × 10-12 -5.38724 × 10-8 -1.70632 × 10-10

3.2 × 10-14 7.8 × 10-12 3.6 × 10-11 2.6 × 10-13 1.1 × 10-10 5.9 × 10-12

5.39210 × 10-3

4.0 × 10-6

8.70833 × 10-7

1.0 × 1030

-5.41237 × 10-4

5.5 × 10-5

-1.08022 × 10-5 -1.16256 × 10-6 -9.93412 × 10-9 5.73726 × 10-9 9.41541 × 10-7

1.2 × 10-6 9.2 × 10-9 1.3 × 10-9 4.2 × 10-10 1.0 × 10-8

7.98018 × 10-9 1.78641 × 10-9 1.22786 × 10-11 -6.40421 × 10-12 -6.08920 × 10-10

4.8 × 10-12 1.1 × 10-11 1.9 × 10-12 2.6 × 10-13 3.8 × 10-13

1.9 × 10-5

3.19312 × 10-6

4.0 × 10-9

-4.33948 × 10-12

1.3 × 10-13

3.6 × 10-4 5.6 × 10-4 9.9 × 10-4 2.0 × 10-3 5.8 × 10-4 3.2 × 10-4 2.6 × 10-3 4.6 × 10-6 8.0 × 10-4 1.8 × 10-3 2.1 × 10-3 7.9 × 10-4 2.1 × 10-2 2.6 × 10-2

-2.02706 × 10-4

3.4 × 10-6

8.95964 × 10-3

2.0 × 10-3

-1.85344 × 10-10 2.89586 × 10-10 -1.52033 × 10-9 1.31282 × 10-8 -1.01694 × 10-9 1.95304 × 10-10 -9.74767 × 10-10 -5.59601 × 10-13 -2.15045 × 10-11 -9.92137 × 10-10 5.82939 × 10-10 -1.11025 × 10-9 2.29195 × 10-10 -7.33067 × 10-8

2.1 × 10-12 4.4 × 10-12 7.7 × 10-12 1.9 × 10-11 4.5 × 10-12 2.6 × 10-12 2.0 × 10-11 2.9 × 10-14 1.4 × 10-12 1.4 × 10-11 1.6 × 10-11 6.2 × 10-12 1.9 × 10-10 2.5 × 10-10

-2.16694 × 10-9 -9.59579 × 10-6

1.2 × 10-10 1.6 × 10-6

-1.82754 × 10-4 -4.98912 × 10-2

3.6 × 10-5 8.4 × 10-3

error for each electrolyte system. The average percentage error for the modified model is smaller than that for the original Pitzer model in 60% of the cases. For aqueous solutions of cadmium salts and sulfuric acid, the new model produces high relative errors because the osmotic and mean molal activity coefficients calculated on the basis of complete dissociation exhibit negative deviations from the Debye-Huckel limiting law; therefore, these systems were not included in this work. For the electrolytes considered in this work, the modified model has an average percentage error not greater than 7.5%, whereas the error of the original Pitzer model sometimes exceeds 10%.

3.20691 × 10-7 -1.87809 × 10-7 9.78241 × 10-7 -1.78334 × 10-5 6.62525 × 10-7 -1.26651 × 10-7 6.26732 × 10-7 4.16363 × 10-10 2.58882 × 10-8 6.40896 × 10-7 -3.78671 × 10-7 7.22917 × 10-7

1.9 × 10-9 2.4 × 10-8 1.9 × 10-10 7.0 × 10-8 2.6 × 10-8 3.5 × 10-10 2.9 × 10-9 1.4 × 10-11 6.8 × 10-9 1.2 × 10-7 2.4 × 10-8 5.8 × 10-8

9.98789 × 10-5

7.2 × 10-6

Figure 1 shows the osmotic coefficient of KOH vs molality at different temperatures. In general, both models adequately predict the experimental osmotic coefficient data. For the dilution enthalpy, both models predict the experimental data successfully except at 623.15 K, as shown in Figure 2. This could occur because the pressure dependence is ignored or because the model tends to fail for such systems in the high-temperature, dilute region. The NaOH aqueous system exhibits similar behavior. Figure 3 shows that both models can correlate the dilution enthalpies of the HCl and MgCl2 systems at high temperatures; however, in the Pitzer model for

σ q7

1.27 × 10-3 1.04 × 100 1.85 × 100 1.27 × 10-2 5.83 × 10-2 1.94 × 10-2 2.35 × 10-3 2.43 × 10-2 7.93 × 10-1 7.19 × 10-3 1.62 × 10-2 5.11 × 10-3 3.57 × 10-4 3.72 × 10-4 2.21 × 10-1 2.86 × 10-3 6.52 × 10-2 2.60 × 100 3.15 × 10-1 2.50 × 10-1 5.70 × 10-1 3.82 × 100 3.86 × 10-2 1.82 × 100 4.68 × 100 3.15 × 10-1 1.87 × 100 1.08 × 100 1.06 × 100

q7

7.03060 × 10-3 3.40927 × 100 -2.12962 × 100 2.45236 × 10-2 -1.90335 × 10-1 -2.45145 × 10-2 -1.00020 × 10-2 -2.42319 × 10-1 -1.00232 × 100 2.81317 × 10-2 9.74868 × 10-2 1.77974 × 10-2 2.36421 × 10-3 -1.61181 × 10-3 -6.21613 × 10-1 1.20131 × 10-2 4.55257 × 10-1 2.88525 × 100 8.70493 × 10-1 1.34433 × 100 8.39615 × 10-1 4.77954 × 100 7.91771 × 10-2 2.56524 × 100 7.24339 × 100 -1.91068 × 100 4.22116 × 100 3.15542 × 100 1.78233 × 100

system

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

q8 1.64 × 10-6 1.64 × 10-3 7.68 × 10-5 1.73 × 10-5 9.23 × 10-5 3.19 × 10-5 2.44 × 10-6 4.06 × 10-4 1.25 × 10-3 9.91 × 10-6 2.15 × 10-5 7.01 × 10-6 4.78 × 10-8 5.67 × 10-7 3.49 × 10-4 3.90 × 10-6 1.03 × 10-5 4.10 × 10-3 5.21 × 10-4 3.95 × 10-4 9.01 × 10-4 6.03 × 10-3 5.74 × 10-5 2.94 × 10-3 7.39 × 10-3 4.97 × 10-4 2.96 × 10-3 1.80 × 10-3 1.79 × 10-3

σ q8

3.39 × 100

7.04428 × 100 3.26513 × 102 2.73901 × 102 -3.54944 × 104 9.38579 × 101 -2.67895 × 101 -7.38821 × 100 -5.17270 × 101 1.63200 × 101 1.41707 × 102 -1.33160 × 103 -5.15967 × 103 2.76009 × 104 -5.18739 × 102 5.74203 × 103 -5.88913 × 103 3.04374 × 104 -7.07056 × 100 -6.45820 × 103 3.70520 × 104 -9.93699 × 103 1.00888 × 104 -1.10426 × 104 -9.36112 × 104

2.83 × 101 3.30 × 102 2.55 × 100 2.03 × 101 5.79 × 100 2.71 × 100 5.86 × 100 4.51 × 101 3.08 × 101 1.46 × 102 2.86 × 102 5.20 × 101 1.69 × 102 4.93 × 101 2.97 × 102 8.13 × 10-1 1.52 × 102 6.70 × 102 5.39 × 102 1.51 × 102 3.74 × 103 5.33 × 103

1.02 × 100 1.57 × 102 6.64 × 102 1.72 × 100

σq10

q12

1.33 × 10-1 2.70 × 100 4.51 × 10-2 3.14 × 10-2 1.45 × 10-1 6.62 × 100 4.36 × 101 1.84 × 100 6.62 × 100 5.70 × 101 5.14 × 10-3 5.14 × 10-2 6.08 × 101 1.49 × 100 1.88 × 100 1.14 × 100 6.93 × 100

-2.21115 × 102 5.55519 × 10-1 -2.71639 × 10-1 -7.30417 × 100 -3.12842 × 101 1.74228 × 102 3.39531 × 101 -3.65578 × 101 1.94252 × 102 -4.39714 × 10-2 -3.62627 × 101 2.36229 × 102 -6.30728 × 101 6.31836 × 101 -4.63998 × 100 -6.42133 × 102

4.27 × 10-2 3.59 × 10-1 6.05 × 100

σq12

q5

7.65123 × 10-5 -2.03898 × 10-3 3.29004 × 10-3 3.43764 × 10-1 -8.01055 × 10-4 -1.79258 × 10-4 -1.63355 × 10-5 3.37558 × 10-4 1.15776 × 10-4 1.34335 × 10-3 9.90442 × 10-3 4.72979 × 10-2 -2.74613 × 10-1 -6.16716 × 10-3 -5.00482 × 10-2 5.66630 × 10-2 -3.09350 × 10-1 6.51750 × 10-5 5.04277 × 10-2 -3.75897 × 10-1 9.98828 × 10-2 -9.88592 × 10-2 -1.09718 × 10-1 1.09929 × 100

-3.20575 × 10-4 -4.25985 × 10-2 7.01539 × 10-2 4.35250 × 10-5

q13

8.71744 × 10-6 -2.23180 × 10-3 6.37895 × 10-3 2.28229 × 10-4 -1.24377 × 10-2 -2.17337 × 10-6 1.83121 × 10-4 3.06653 × 10-3 1.00066 × 10-3 3.94969 × 10-5 -8.23561 × 10-5 -2.88772 × 10-5 -1.39926 × 10-5 4.78830 × 10-5 -1.08200 × 10-3 -1.04861 × 10-4 -2.61881 × 10-3 -1.65104 × 10-3 7.45029 × 10-4 4.97888 × 10-3 1.77277 × 10-3 6.07983 × 10-6 -1.53369 × 10-5 -5.89060 × 10-3 -3.23072 × 10-3 -8.28054 × 10-3 4.96431 × 10-3 -2.89231 × 10-2 -3.38057 × 10-2

1.64885 × 100

2.62699 × 10-1 2.50668 × 101 -2.88212 × 101

3.10 × 10-4 1.59 × 100 4.57 × 100 1.81 × 100 2.77 × 10-3 3.39 × 10-1 1.63 × 100

3.01484 × 10-2 1.10072 × 101 1.12246 × 101 1.56246 × 101 -8.87217 × 100 5.29928 × 101 5.96906 × 101

σ q4 3.27 × 10-4 2.21 × 10-1 2.10 × 100 9.26 × 10-2 1.26 × 10-3 6.46 × 10-5 1.19 × 10-1 1.91 × 10-1 3.76 × 100 2.09 × 10-2 2.76 × 10-2 2.55 × 10-2 1.08 × 10-3 4.18 × 10-2 1.31 × 10-2 3.28 × 10-2 5.18 × 10-1 4.58 × 10-1 9.06 × 10-1 5.96 × 100 4.59 × 10-1

q4 -1.82612 × 10-2 5.32883 × 100 -1.24750 × 101 -5.10881 × 10-1 2.22970 × 101 -8.99019 × 10-5 -4.09260 × 10-1 -5.57574 × 100 -5.90068 × 100 -6.72468 × 10-2 1.99753 × 10-1 7.40305 × 10-2 3.29065 × 10-2 -9.00255 × 10-2 2.13106 × 100 1.76389 × 10-1 5.10753 × 100 6.82591 × 100 -1.70913 × 100 -9.19031 × 100 -4.94573 × 100

5.45907 × 101 3.65477 × 103 -2.03133 × 103 8.98966 × 100

q10

7.21 × 101 1.13 × 102 2.09 × 101 2.41 × 102 1.90 × 101 7.25 × 10-1 3.53 × 102 1.13 × 102 8.54 × 100 1.49 × 101 4.84 × 102 3.37 × 101

-3.33486 × 103 6.49569 × 102 2.79545 × 103 2.08327 × 103 7.36315 × 101 -9.87435 × 100 -3.35986 × 103 -5.14332 × 103 -4.92756 × 103 2.63945 × 103 -1.58750 × 104 -1.74847 × 104

σq3 5.57 × 10-1 2.15 × 101 1.29 × 102 3.98 × 100 4.82 × 100 5.39 × 10-2 2.06 × 100 7.69 × 100 3.25 × 101 1.01 × 100 5.90 × 100 1.97 × 100 2.60 × 100 2.07 × 101 1.49 × 10-2 2.02 × 100

q3 6.17581 × 100 -2.00187 × 103 3.94709 × 102 1.88165 × 101 -6.66082 × 103 -1.71549 × 10-1 1.49447 × 102 1.66638 × 102 3.17103 × 102 1.74088 × 101 -7.74562 × 101 -3.10276 × 101 -1.28633 × 101 2.70808 × 101 -7.13134 × 102 -1.64763 × 103

-6.74996 × 10-6 -5.30095 × 10-3 3.32188 × 10-3 -3.46121 × 10-5 3.18668 × 10-4 4.38637 × 10-5 1.59538 × 10-5 4.23280 × 10-4 1.53703 × 10-3 -3.44193 × 10-5 -1.15523 × 10-4 -2.26142 × 10-5 -9.08989 × 10-7 3.47818 × 10-6 9.59032 × 10-4 -5.87910 × 10-6 -7.11129 × 10-4 -4.55027 × 10-3 -1.46694 × 10-3 -2.10004 × 10-3 -1.25844 × 10-3 -7.49035 × 10-3 -9.40100 × 10-5 -3.96116 × 10-3 -1.12702 × 10-2 3.02384 × 10-3 -6.56433 × 10-3 -5.29124 × 10-3 -3.01003 × 10-3

σ q2 1.78 × 101 1.67 × 103 1.85 × 103 1.04 × 102 5.29 × 102 5.35 × 101 5.11 × 102 1.41 × 103 2.44 × 103 9.71 × 101 1.19 × 102 3.77 × 101 6.30 × 101 2.96 × 101 4.57 × 102 4.74 × 102 1.32 × 103 2.42 × 103 1.93 × 103 1.83 × 103 8.26 × 102 1.21 × 103 1.62 × 102 1.63 × 103 5.41 × 103 4.20 × 103 1.50 × 103 9.40 × 103 1.11 × 104

q2

6.31756 × 102 1.78324 × 101 1.66891 × 103 1.84722 × 103 1.04288 × 102 5.28679 × 102 5.34822 × 101 5.11232 × 102 1.40788 × 103 2.44375 × 103 9.71231 × 101 1.19255 × 102 3.76539 × 101 6.29676 × 101 2.95857 × 101 4.57217 × 102 4.73976 × 102 1.32494 × 103 2.42398 × 103 1.93380 × 103 1.82603 × 103 8.25892 × 102 1.20777 × 103 1.62465 × 102 1.62760 × 103 5.41101 × 103 4.20359 × 103 1.50201 × 103 9.39953 × 103

system

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

Table 5. Temperature-Dependence Parameters and Corresponding Asymptotic Standard Errors for the Pitzer Model σ q5

3.81 × 10-5 3.81 × 10-5 3.00 × 10-4 3.31 × 10-3 2.43 × 10-5 1.49 × 10-5 4.74 × 10-6 2.05 × 10-5 5.53 × 10-5 4.51 × 10-4 2.40 × 10-4 1.46 × 10-3 2.87 × 10-3 5.69 × 10-4 1.69 × 10-3 4.96 × 10-4 3.60 × 10-4 7.35 × 10-6 1.54 × 10-3 6.71 × 10-3 5.40 × 10-3 1.51 × 10-3 4.22 × 10-2 6.17 × 10-2

7.46 × 10-6 1.58 × 10-3 6.81 × 10-3 1.35 × 10-5

σq13

2.86 × 10-7 2.64 × 10-5 3.13 × 10-5 1.12 × 10-6 7.55 × 10-6 9.95 × 10-7 2.74 × 10-6 2.47 × 10-5 3.86 × 10-5 1.84 × 10-6 1.15 × 10-6 4.25 × 10-7 8.51 × 10-7 4.26 × 10-7 7.23 × 10-6 3.73 × 10-6 2.09 × 10-5 3.83 × 10-5 3.48 × 10-5 2.89 × 10-5 1.32 × 10-5 1.95 × 10-7 1.18 × 10-6 2.64 × 10-5 8.56 × 10-5 6.65 × 10-5 2.37 × 10-5 1.78 × 10-4 2.17 × 10-4

6966 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003

Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6967

Figure 1. Osmotic coefficient plot of KOH(aq) comparing Pitzer and modified Pitzer models at different temperatures.

Figure 2. Dilution enthalpy plot of KOH(aq) comparing Pitzer and modified Pitzer models at different temperatures.

Table 6. Average Percentage Errors and Standard Deviations of the Osmotic and Mean Activity Coefficients from Pitzer and Modified Pitzer Models error (%)

σstd system

modified model

Pitzer model

modified model

Pitzer model

CsBr CsCl CsI CsOH HBr HCl KBr KCl KI KOH LiBr LiOH NaBr NaCl NaI NaOH RbCl BaBr2 BaCl2 CaBr2 CaCl2 CaI2 Ca(NO3)2 MgCl2 SrBr2 SrCl2 SrI2 ZnCl2 ZnI2

1.7 × 10-2 8.7 × 10-2 3.0 × 10-2 4.2 × 10-2 3.2 × 10-2 7.3 × 10-1 7.1 × 10-2 7.8 × 10-3 5.3 × 10-2 2.5 × 10-1 5.4 × 10-2 3.5 × 10-2 2.6 × 10-2 2.5 × 10-3 3.2 × 10-2 7.2 × 10-2 3.0 × 10-2 3.7 × 10-2 2.2 × 10-2 3.9 × 10-2 7.3 × 10-2 6.2 × 10-2 4.7 × 10-2 5.0 × 10-2 6.5 × 10-2 6.9 × 10-2 4.5 × 10-2 2.1 × 10-2 3.0 × 10-2

2.3 × 10-2 5.2 × 10-2 3.0 × 10-2 4.1 × 10-2 1.5 × 10-2 2.5 × 10-1 7.3 × 10-2 1.7 × 10-2 3.8 × 10-2 2.7 × 10-1 2.4 × 10-2 2.8 × 10-3 2.9 × 10-2 2.0 × 10-2 1.1 × 10-2 7.8 × 10-2 3.0 × 10-2 3.1 × 10-2 7.2 × 10-2 6.3 × 10-2 7.3 × 10-2 7.2 × 10-2 1.4 × 10-1 1.7 × 10-1 9.1 × 10-2 7.0 × 10-2 3.1 × 10-2 5.6 × 10-2 8.1 × 10-2

1.9 5.9 2.2 2.7 2.6 2.2 4.4 1.0 4.5 4.4 4.0 5.0 2.0 0.2 2.2 6.8 0.3 2.1 2.3 1.6 3.6 3.0 4.7 2.5 3.3 4.4 1.8 3.5 3.7

2.4 3.0 2.3 2.4 1.3 2.1 5.2 2.2 3.1 4.7 1.7 0.4 2.3 2.4 0.7 6.9 0.3 1.7 12.3 2.4 3.1 3.3 11.8 11.3 3.7 4.4 1.4 8.1 7.5

MgCl2, inclusion of the dilution enthalpy data in the regression analysis produces a lack of fit in the osmotic and activity coefficient at low temperatures. For the Ca(NO3)2 system, the results are significantly better for the modified model than for the Pitzer model, as shown in Figure 4. Table 7 presents the average percentage error and standard deviation for the dilution enthalpy for each electrolyte system. For multicomponent systems, we tested our results on four systems: NaCl + KCl + H2O at 318.15 K, CaCl2 + MgCl2 + H2O and LiCl + MgCl2 + H2O at 313.15 K, and NaCl + MgCl2 + H2O at 373.15 K. Mixing terms in the Pitzer equation17 (θij and ψijk) at 298.15 K were not considered because we are not certain that these terms remain valid at different temperatures. In the NaCl + KCl + H2O system, we used osmotic coefficient data reported by Flesia et al.,18 and our model worked better than the original Pitzer model. Figure 5

Figure 3. Dilution enthalpy plot of HCl(aq) and MgCl2(aq) comparing Pitzer and modified Pitzer models at 573.15 K.

Figure 4. Dilution enthalpy plot of Ca(NO3)2(aq) comparing Pitzer and modified Pitzer models at 372.15 K.

shows these results. The maximum relative percentage errors are 5 and 2.1% for the original and modified models, respectively. Figure 6 presents the results for the CaCl2 + MgCl2 + H2O system,19 for which the average percentage errors are 14 and 5.4% for the original and modified Pitzer models, respectively. Too few data exist to develop temperature functions for LiCl, so calculations for this sytem were performed at 313.15 K with β(0) ) 0.14349, β(1) ) -0.33093, and CMX ) 0.00167 for the Pitzer model and bMX ) 0.41585, BMX ) 0.27034, and CMX ) -0.00340 for the modified model. Figure 7 shows that the modified model predicted the osmotic coefficient20 behavior better than the origi-

6968 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003

Figure 5. Osmotic coefficient plot of NaCl + KCl + H2O comparing the predictive behaviors of the Pitzer and modified Pitzer models at 318.15 K.

Figure 6. Osmotic coefficient plot of CaCl2 + MgCl2 + H2O comparing the predictive behaviors of the Pitzer and modified Pitzer models at 313.15 K. Table 7. Standard Deviations and Average Percentage Errors of the Pitzer and Modified Pitzer Models for the Dilution Enthalpy error (%)

σstd system

modified model

Pitzer model

modified model

Pitzer model

Ca(NO3)2 HCl KOH MgCl2 NaOH

0.7 2.8 4.7 1.0 4.5

2.3 4.3 5.1 0.9 4.3

27.7 7.4 17.6 4.4 17.4

81.0 8.1 18.8 4.1 16.8

nal Pitzer model. The average percentage errors are 11 and 1.9% for the original and modified models, respectively. Finally, we predicted the mixing enthalpy data for the NaCl + MgCl2 + H2O system at yB ) 0.5. The experimental data were obtained from Wang et al.,21 and Figure 8 shows the predictions obtained using the modified model. The results obtained using the original Pitzer model are not presented because they are very poor and fall outside the scale. The average percentage error for our model is 7.23% with a standard deviation of 26.8.

Figure 7. Osmotic coefficient plot of MgCl2 + LiCl + H2O comparing the predictive behaviors of the Pitzer and modified Pitzer models at 313.15 K.

Figure 8. Prediction of the mixing enthalpy of NaCl + MgCl2 + H2O at 373.15 K using the modified Pitzer model.

systems, the correlation of the mean activity and osmotic coefficients at different temperatures using the modified model is an improvement compared to that of the original Pitzer model with temperature dependence. We could not find sufficient experimental data to account for pressure effects in electrolyte solutions, so the effect of pressure has not been considered in the model predictions for the electrolytes tested here. As mentioned by Pe´rez-Villasen˜or et al.,12 for multicomponent systems, the modified model does not require mixing terms, whereas the Pitzer model does require such terms, which must be calculated from experimental data. The modified model provides better predictions than the Pitzer model when mixing terms are not considered. Consequently, the modified model is a good alternative for accurately predicting the dilution enthalpy and the osmotic and activity coefficients for multicomponent aqueous systems using information from single-electrolyte systems. Acknowledgment Texas A&M University, Instituto Tecnolo´gico de Celaya, and Conacyt provided financial support for this work.

Conclusions

Notation

We have extended the modified Pitzer model to account for temperature dependence. In binary aqueous

Aφ ) Debye-Hu¨ckel coefficient AL ) first derivative of Aφ with respect to temperature

Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6969 bk ) maximum approach parameter in the Debye-Hu¨ckel term CMX ) third virial-type coefficient CφMX ) third virial-type coefficient form for the osmotic coefficient CLMX ) first derivative of CMX with respect to temperature BMX ) second virial-type coefficient BφMX ) second virial-type coefficient form for the osmotic coefficient BLMX ) first derivative of BMX with respect to temperature GE ) excess Gibbs free energy on a molality basis f φ ) ionic strength function form of the Debye-Hu¨ckel term used for the osmotic coefficient f γ ) ionic strength function form of the Debye-Hu¨ckel term used for the activity coefficient I ) ionic strength (mol/kg of solvent) L ) relative enthalpy (J/kg) φL ) relative molal enthalpy (J/mol) ∆Hdil ) dilution enthalpy (J/mol) ∆Hmix ) mixing enthalpy (J/kg) ni ) number of moles in solution of ion i mi ) molality of species i (mol/solvent kg) R ) universal gas constant T ) absolute temperature (K) ww ) solvent weight (kg) yk ) ionic strength fraction of salt k in a mixture zi ) ion valence Greek letters ν ) stoichiometric coefficient φ ) osmotic coefficient γ( MX ) mean ionic activity coefficient for MX neutral electrolyte Subscripts a, a′ ) any anion c, c′ ) any cation i, j ) any ionic species k, p ) any salt in mixture M ) cation MX ) neutral electrolyte X ) anion

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Received for review March 20, 2003 Revised manuscript received September 23, 2003 Accepted October 1, 2003 IE030251R