Osmotic and Activity Coefficients of 1:1, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2 and 4

6366

Ind. Eng. Chem. Res. 2007, 46, 6366-6374

CORRELATIONS Osmotic and Activity Coefficients of 1:1, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2 and 4:1 Strong Electrolytes at 298.15 K Using a Modified Pitzer Equation Fernando Pe´ rez-Villasen˜ or* and M. L. Bedolla-Herna´ ndez Departamento de Ciencias Ba´ sicas, Ingenierıá y Tecnologıá, UniVersidad Auto´ noma de Tlaxcala, Apizaco, Tlaxcala, C.P. 90300, Mexico

Gustavo A. Iglesias-Silva Departamento de Ingenierıá Quı´mica, Instituto Tecnolo´ gico de Celaya, Celaya Gto. C.P. 38010, Mexico

We have correlated simultaneously experimental data for osmotic and activity coefficients of 122 aqueous solutions of pure electrolytes at 298.15 K using a modified Pitzer model. We have compared our results with well-known models such as the Pitzer and Lietzke models. For electrolytes with monovalent ions, the modified model compares favorably to the original Pitzer and Lietzke models. For electrolytes without monovalent ions such as 2:2 and 3:2, the Pitzer equation correlates better at the expense of using four adjustable and three fixed parameters whereas the modified model requires only three adjustable parameters. Introduction Design and simulation of chemical processes that involve electrolytes such as geothermal systems,1 seawater desalination,2 residual water treatment,3 distillation4 (salting-out, salting-in), food processing,5 salt recovery,6 and crystallization7,8 require rigorous models to represent nonideal mixtures.9 Several theories and empirical correlations have been developed to represent excess properties of electrolyte solutions, for example those proposed by Meissner and Tester,10 Pitzer,11 Chen et al.,12 Haghtalab and Vera,13 Jaretum and Aly,14 Zhao et al.,15 and Masoudi et al.16 One of the most used models to represent the activity and osmotic coefficient of electrolyte solutions is the Pitzer equation. This model is relatively simple, and it performs electrolyte calculations with accuracy, especially for concentrations under 6 molal. For electrolytes with monovalent ions uses two nonlinear parameters that must be fixed and three linear parameters adjustable from experimental data.11 For application of the Pitzer equation to electrolytes with higher valence, an extended second virial coefficient is needed, and the model requires four adjustable and three fixed parameters.17-19 In a previous work, Pe´rez-Villasenõr et al.20 performed a parametric analysis for the Pitzer equation. They found that it is possible to reduce the number of parameters of the model to three if one considers the closest-approach parameter from the electrostatic term as a fitting parameter. They also showed that their modified model could represent the activity and osmotic coefficients at molalities as high as 25 with better accuracy than the original Pitzer model. However, their results are only valid for 1:1 and 1:2 electrolytes, although they correlated the osmotic coefficient of the system [KCl + MgSO4] within 2% of the experimental data using single electrolyte-solvent parameters only.21 In this work, we have calculated the parameters for the modified Pitzer equation simultaneously using experimental data * To whom correspondence should be addressed. Phone: (241) 41 72544. Fax: (461) 41 75844. E-mail: [email protected].

Figure 1. Osmotic (a) and activity (b) coefficients plots of 2:2 electrolytes comparing the Pitzer (dashed lines) and modified Pitzer (solid lines) models. b, MnSO4; 2, NiSO4; [, UO2SO4.

for osmotic and activity coefficients of 1:1, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2, and 4:1 aqueous electrolyte solutions. Performance of the equation is compared with that of the Pitzer and Lietzke22 equations.

10.1021/ie070228w CCC: $37.00 © 2007 American Chemical Society Published on Web 08/21/2007

Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6367 Table 1. Systems Used in This Work and Corresponding Experimental Sources electrolyte

variable(s)

Nobs

mmin

mmax

variable(s)

Nobs

mmin

mmax

(CH3)4(guanadinium)Cl (CH3)4NCl (CH3)4NF (CH3NH3)ClO4 (CH3NH3)NO3 (n-C3H7)4NF (n-C4H9)4NF (guanadinium)Cl LiBrO3 LiClO3 NaCF3SO3

φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ

24 36 24 19 30 21 13 29 21 20 26

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.01

6.5 19 7 4 9.5 5 1.6 12 5 4.2 5.3735

43 39 40 41 44 40 40 46 42 42 45

K3Co(CN)6

φ, γ

15

0.01

1.311

32

[N(CH3)4]4Mo(CN)8

φ, γ

12

0.01

1.44

33

MgBr2 MgCl2 MgI2 Ni(C6H4O6S2)2 NiCl2 Ni(ClO4)2 Pb(ClO4)2 Pb(NO3)2 Sr(ClO4)2 Sr(NO3)2 UO2(ClO4)2 UO2(NO3)2 UO2Cl2 Zn(C6H4O6S2)2 Zn(C7H7O3S)2 Zn(ClO4)2 Zn(NO3)2 ZnBr2 ZnCl2 ZnI2

φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ

21 21 21 16 21 18 23 15 23 19 21 22 17 16 21 19 23 23 23 23

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.001 0.1 0.1 0.1 0.1 0.1

5 5 5 2.5 5 3.5 6 2 6 4 5 5.5 3 2.5 3 4 6 6 6 6

28 28 28 27 28 47 28 28 28 28 28 28 28 27 29 28 28 28 28 28

NiSO4 UO2SO4 ZnSO4

φ, γ φ, γ φ, γ

16 23 18

0.1 0.1 0.1

2.5 6 3.5

28 28 28

Lu(NO3)3 LuCl3 Nd(ClO4)3 Nd(NO3)3 NdCl3 Pr(ClO4)3 Pr(NO3)3 PrCl3 ScCl3 Sm(ClO4)3 Sm(NO3)3 SmCl3 Tb(ClO4)3 Tb(NO3)3 TbCl3 Tm(ClO4)3 Tm(NO3)3 TmCl3 Y(NO3)3 Yb(ClO4)3 Yb(NO3)3 YbCl3 YCl3

φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ

30 26 29 37 15 29 37 15 14 29 27 15 29 28 23 29 35 25 42 29 44 26 88

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

5 4.1239 4.6822 6.2598 2 4.7182 6.2861 2 1.8 4.6621 4.2774 2 4.6196 4.5320 3.5733 4.6215 5.9526 3.8814 7.2103 4.6400 7.9115 4.0018 4.0843

34 31 30 38 28 30 38 28 28 30 37 28 30 37 31 30 37 31 36 30 37 31 28, 36

[Co(ethylenediamine)3]2(SO4)3

φ, γ

17

0.1

1.884

32

refs

electrolyte

refs

1:1 (C2H5)4NCl (C2H5)4NF (C3H7)4NCl (C3H7)4NI (C4H9)4NBr (C4H9)4NCl (CH3)2NH2CIO4 (CH3)2NH2NO3 (CH3)3NHClO4 (CH3)3NHNO3 (CH3)4(guanadinium)Br

φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ

26 22 35 9 38 32 26 23 14 28 28

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

9 5.5 18 0.5 27 15 7.5 6 1.8 8.5 8.5

39 40 39 39 39 39 41 44 41 44 43

K3Fe(CN)6

φ, γ

12

0.1

1.4

28

K4Fe(CN)6

φ, γ

9

0.1

0.9

28

1:3 1:4 2:1 Ba[COOCH3]2 BaI2 (C8H22N2)(C6H4O6S2)2 C8H22N2Cl2 C8H22N2I2 Ca(ClO4)2 Cd(C7H7O3S)2 Cd(ClO4)2 Cd(NO2)2 Cd(NO3)2 CoBr2 CoCl2 Co(ClO4)2 CoI2 Co(NO3)2 Cu(C6H4O6S2)2 Cu(NO3)2 FeCl2 Mg(ClO4)2 Mg[COOCH3]2

φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ

18 15 11 23 22 23 44 28 25 16 21 19 18 20 21 16 23 15 18 19

0.1 0.1 0.1 0.1 0.1 0.1 0.001 0.001 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

3.5 2 1.2 4.25 4 6 6 1 4.75 2.5 5 4 3.5 4.5 5 2.5 6 2 4 4

28 28 35 29, 35 29, 35 28 29 29 29 28 28 28 47 28 28 27 28 28 28 28

BeSO4 CuSO4 MnSO4

φ, γ φ, γ φ, γ

19 12 19

0.1 0.1 0.1

4 1.4 4

28 28 28

AlCl3 CeCl3 Co(ethylenediamine)3(NO3)3 Co(propylenediamine)3(ClO4)3 Cr(NO3)3 CrCl3 Dy(ClO4)3 Dy(NO3)3 DyCl3 Er(ClO4)3 Er(NO3)3 ErCl3 Eu(NO3)3 EuCl3 Gd(ClO4)3 Gd(NO3)3 GdCl3 Ho(ClO4)3 Ho(NO3)3 HoCl3 La(NO3)3 LaCl3 Lu(ClO4)3

φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ φ, γ

14 15 8 9 12 11 29 30 24 29 41 24 37 15 29 27 23 29 30 24 38 15 29

0.1 0.1 0.01 0.01 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

1.8 2 0.25 0.2612 1.4 1.2 4.6042 5 3.6302 4.6221 7.2 3.784 6.3858 2 4.6215 4.3701 3.5898 4.6314 5 3.6987 6.4739 2 4.6313

28 28 32 32 28 28 30 34 31 30 37 31 36 28 30 37 31 30 34 31 38 28 30

Al2(SO4)3 Cr2(SO4)3

φ, γ φ, γ

10 11

0.1 0.1

1 1.2

28 28

Th(NO3)4

φ, γ

21

0.1

5

28

2:2

3:1

3:2

4:1

6368

Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007

Table 2. Modified Pitzer Model Parameters electrolyte

bMX

BMX

CMX

(C2H5)4NCl (C2H5)4NF (C3H7)4NCl (C3H7)4NI (C4H9)4NBr (C4H9)4NCl (CH3)2NH2CIO4 (CH3)2NH2NO3 (CH3)3NHClO4 (CH3)3NHNO3 (CH3)4(guanadinium)Br

0.59179 2.37561 1.05518 1.52822 0.26518 2.64648 0.87321 1.76533 0.78944 1.22322 1.03331

0.13071 0.36119 0.16376 -0.78888 0.01505 0.01474 -0.02811 -0.01620 -0.09427 -0.00834 -0.01149

0.00059 0.00574 -0.00317 0.26111 0.00012 0.00000 0.00073 0.00088 0.00659 0.00091 0.00202

K3Fe(CN)6

2.39550

0.09329

-0.00174

electrolyte

bMX

BMX

CMX


1.70402 0.88396 1.5696 1.25691 1.75516 0.23404 4.27177 1.54758 2.14586 2.04807 2.85129

0.02258 0.08128 0.27642 -0.03802 -0.02302 0.76142 0.51805 -0.04169 0.06651 0.15124 0.10712

0.00219 -0.00081 -0.00127 0.00207 0.00063 -0.01668 0.00480 0.00138 0.00084 -0.00212 -0.00509

K3Co(CN)6

2.1806

0.18212

-0.01093

[N(CH3)4]4Mo(CN)8

2.50941

0.05787

0.00216


2.44928 2.37916 2.51096 0.8446 2.07708 3.18984 2.63493 1.36463 2.43391 2.4238 3.22672 2.47625 2.78837 0.75713 2.40771 3.02493 2.71345 4.35701 3.5986 4.96659

0.35950 0.28486 0.42172 0.51189 0.33233 0.36878 0.23530 -0.06145 0.33872 0.01061 0.52513 0.38645 0.29079 0.54428 0.15721 0.37884 0.23362 0.09782 -0.02505 0.20141

0.00293 0.00331 0.00434 -0.00288 -0.00442 0.01041 -0.00025 0.00320 -0.00210 0.00054 0.00612 -0.01123 -0.00550 -0.00659 -0.07554 0.00887 -0.00048 -0.00164 0.00332 -0.00579

NiSO4 UO2SO4 ZnSO4

1.60418 1.5417 1.50271

0.19536 0.25344 0.26151

0.00000 -0.00301 -0.00128


2.82943 2.28938 2.20423 2.76956 2.74300 2.02194 2.83241 2.74381 2.76291 2.32674 2.66490 2.82451 2.34684 2.86440 2.53632 2.45838 3.00925 2.34706 3.10800 2.44146 3.15541 2.27213 2.85393

0.31182 0.50957 0.72718 0.17198 0.32842 0.75999 0.16031 0.32632 0.41190 0.70965 0.19844 0.32875 0.75559 0.21974 0.46544 0.74678 0.27345 0.50185 0.24952 0.74972 0.26464 0.51347 0.33435

-0.00364 -0.00190 -0.00070 -0.00102 0.00533 -0.00204 -0.00076 0.00463 0.00474 -0.00003 -0.00126 0.00592 -0.00122 -0.00145 -0.00200 -0.00016 -0.00283 -0.00199 -0.00201 -0.00022 -0.00244 -0.00218 0.00849


1.98613

0.15929

-0.00358

1:1

1:3 1:4 K4Fe(CN)6

2.52205

0.09039

-0.01113 2:1

Ba[COOCH3]2 BaI2 (C8H22N2)(C6H4O6S2)2 C8H22N2Cl2 C8H22N2I2 Ca(ClO4)2 Cd(C7H7O3S)2 Cd(ClO4)2 Cd(NO2)2 Cd(NO3)2 CoBr2 CoCl2 Co(ClO4)2 CoI2 Co(NO3)2 Cu(C6H4O6S2)2 Cu(NO3)2 FeCl2 Mg(ClO4)2 Mg[COOCH3]2

1.95491 2.72555 1.15438 1.20529 0.89491 1.84149 2.53041 3.05118 0.47019 2.71545 1.95668 2.39854 3.18691 2.82947 2.59838 0.9035 2.51228 2.59176 3.27465 1.86009

0.17516 0.29867 0.11002 0.09592 -0.02793 0.45351 0.06713 0.30534 0.16092 0.16516 0.43485 0.28062 0.38383 0.40001 0.21359 0.50493 0.19363 0.21772 0.36626 0.14628

-0.01085 0.00000 0.00477 0.00117 0.00299 -0.00303 -0.03634 0.00122 -0.00278 -0.00353 -0.00500 -0.00337 0.00879 0.00338 0.00045 -0.00426 -0.00056 0.00247 0.00817 -0.00250

BeSO4 CuSO4 MnSO4

1.62901 1.21646 1.54931

0.28356 0.71142 0.23511

0.00000 -0.07680 -0.00116


3.14422 2.80815 1.90128 1.55581 2.80594 2.83861 2.3258 2.95873 2.51151 2.2354 3.12057 2.42591 2.65717 2.80548 2.64357 2.77216 2.52170 2.35995 3.00279 2.53590 3.03398 2.81010 2.44515

0.35961 0.31123 0.05376 0.42535 0.39310 0.43572 0.76659 0.22719 0.47859 0.78107 0.24100 0.49252 0.21148 0.34895 0.69588 0.21081 0.45285 0.76030 0.23861 0.48046 0.13688 0.31927 0.75092

0.01850 0.00604 -0.06554 -0.21562 0.00000 0.00234 -0.00122 -0.00172 -0.00222 -0.00139 -0.00198 -0.00199 -0.00150 0.00436 0.00042 -0.00116 -0.00183 -0.00111 -0.00206 -0.00191 -0.00044 0.00472 -0.00009

Al2(SO4)3 Cr2(SO4)3

2.22332 2.82842

0.34042 0.10035

0.00013 0.01213

Th(NO3)4

4.32411

0.25513

-0.00304

2:2

3:1

3:2

4:1

Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6369

Figure 2. Residuals for the osmotic and activity coefficients of 2:2 and 3:2 electrolytes using the Pitzer and modified models.

Equations

where

According to the Pitzer and modified models, the total excess Gibbs energy for a single electrolyte in an aqueous solution is

GE ) f(I) + 2mMmX[BMX + zMmMCMX] wsRT

f )φ

AφI 1/2

(5)

1 + bMXI 1/2

(1) 1/2 1/2 (1) (2) B φMX ) β (0) MX + β MX exp(-R1I ) + β MX exp(-R2I )

(6)

where the Debye-Hu¨ckel function is given as and

4AφI ln(1 + bMXI1/2) f(I) ) bMX

(2)

In this work, an expression for the coefficient BMX proposed by Pitzer and Silvester18 has been used

BMX(I) ) β (0) MX +

[1 - e-R2I (1 + R2I1/2)] (3)

RT

[ ] ( ) ( ∂GE

∑i mi

∂wS

2νMνX νMX

ln γmi )

[ ]

∂GEm 1 RTwS ∂ni

(8)

P,T,wS,nj

1/2

and it can be used for any electrolyte. Then, the osmotic coefficient can be calculated with

1

(7)

Similarly, the activity coefficient for an individual ion can be obtained from

2β (1) 1/2 MX [1 - e-R1I (1 + R1I1/2)] + R12I 2β (2) MX R22I

φ-1)-

C φMX ) 2|zMzX|1/2CMX

Consequently the mean ionic activity coefficient is νM νX 1/νMX ) |zMzX| f γ + ln γ ( MX ) (γM γX )

( )

6(νMνX)3/2 4νMνX mMXBMX + |zMzX|mMX2CMX (9) νMX νMX

) |zMzX|f φ +

P,T,ni

mBφMX +

2(νMνX) νMX

)

with

3/2

m2C φMX (4)

f γ ) -Aφ

(

I 1/2 2 + ln(1 + bMXI 1/2) 1/2 b 1 + bMXI MX

)

(10)

6370


Figure 3. Residuals for the osmotic and activity coefficients of electrolytes with monovalent ions using the Pitzer and modified models.

Figure 4. Closest approach parameter for all electrolytes. b, Inorganic electrolytes; 9, organic electrolytes. The shading areas are where most electrolytes lie down.

BγMX ) 2β (0) MX + β (1) MX

and

1 2 2 1 - (2 - R12I + 2R1I 1/2) exp(-R1I 1/2) + 2 R1 I 2

[

β (2) MX R22I

]

[1 - 21(2 - R

]

I + 2R2I 1/2) exp(-R2I 1/2) (11)

2 2

3 C γMX ) C φMX 2

(12)

In the modified Pitzer model,20 we remove the terms depending upon the ionic strength in eqs 3, 6, and 11 and allow

Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6371 Table 3. Average Percentage Error for the Pitzer Model and the Modified Model % error electrolyte

Pitzer model

% error

modified model

electrolyte

Pitzer model

modified model


0.61 0.79 0.68 0.18 0.93 2.48 0.42 1.29 0.12 0.13 0.42

0.60 0.76 0.52 0.16 0.96 2.25 0.32 1.06 0.08 0.10 0.31

K3Co(CN)6

1.81

1.64

[N(CH3)4]4Mo(CN)8

15.29

2.73


0.27 0.38 0.48 27.41 1.69 0.40 0.29 0.28 0.50 0.86 2.98 1.51 0.44 28.94 0.06 0.57 0.76 5.95 1.67 8.34

1.18 1.42 1.72 28.33 2.50 0.89 1.08 0.26 1.19 0.46 2.77 1.38 0.35 30.25 0.05 0.88 0.66 4.86 1.36 7.26

NiSO4 UO2SO4 ZnSO4

0.35 0.51 0.31

4.26 2.04 6.05


3.48 0.91 2.03 4.54 1.23 2.12 4.74 1.01 0.82 1.62 2.82 1.22 2.19 3.55 0.78 2.51 4.86 1.07 5.83 2.50 6.85 1.07 1.34

0.49 3.67 4.69 0.30 0.67 4.32 0.33 0.61 0.47 3.75 0.42 0.52 4.68 0.52 3.16 4.60 1.04 2.53 0.85 4.52 1.74 3.69 0.62


5.82

8.42

1:1 (C2H5)4NCl (C2H5)4NF (C3H7)4NCl (C3H7)4NI (C4H9)4NBr (C4H9)4NCl (CH3)2NH2CIO4 (CH3)2NH2NO3 (CH3)3NHClO4 (CH3)3NHNO3 (CH3)4(guanadinium)Br

1.71 1.21 5.57 0.51 6.38 4.06 0.67 0.66 0.16 0.75 1.12

1.50 1.17 5.53 0.51 2.70 3.83 0.58 0.66 0.15 0.74 1.04

K3Fe(CN)6

1.35

0.95

1:3 1:4 K4Fe(CN)6

2.07

1.65

2.08 0.36 1.20 2.40 0.62 1.87 0.19 0.30 2.17 0.32 1.63 0.75 0.44 1.40 0.38 35.65 1.67 0.27 0.86 0.73

1.80 0.47 0.99 1.54 0.52 3.01 0.12 0.20 0.69 0.24 2.01 1.16 0.79 1.32 0.57 37.25 0.78 0.24 0.46 0.42

BeSO4 CuSO4 MnSO4

0.33 0.17 0.50

4.17 4.52 5.57


1.17 1.65 0.50 0.23 0.66 0.71 2.71 4.05 0.91 2.89 6.05 1.05 4.78 1.03 1.79 2.86 0.82 3.24 4.27 0.96 5.04 1.18 2.83

0.64 0.84 0.42 0.29 0.52 0.68 4.48 0.38 3.09 4.52 1.17 2.69 0.69 0.74 4.01 0.95 2.83 5.87 0.58 3.03 0.42 0.50 4.97

Al2(SO4)3 Cr2(SO4)3

0.51 1.16

4.83 1.49

Th(NO3)4

8.28

4.58

2:1 Ba[COOCH3]2 BaI2 (C8H22N2)(C6H4O6S2)2 C8H22N2Cl2 C8H22N2I2 Ca(ClO4)2 Cd(C7H7O3S)2 Cd(ClO4)2 Cd(NO2)2 Cd(NO3)2 CoBr2 CoCl2 Co(ClO4)2 CoI2 Co(NO3)2 Cu(C6H4O6S2)2 Cu(NO3)2 FeCl2 Mg(ClO4)2 Mg[COOCH3]2

2:2

3:1

3:2

4:1

6372


the parameter bMX to be adjustable, therefore

φ - 1 ) |zMzX|f φ +

( )

(

)

2νMνX 4(νMνX)3/2 mβ (0) + |zMzX|m2CMX (13) MX νMX νMX

and γ ln γ( MX ) |zMzX|f +

( )

4νMνX mMXβ (0) MX + νMX 6(νMνX)3/2 |zMzX|mMX2CMX (14) VMX

Also in this work, we have compared the Pitzer models with the equations proposed by Lietzke and Stoughton22 for the osmotic and activity coefficients:

φ-1)-

Aφ bMX3I

(

1 + bMXI 1/2 - 2 ln(1 + bMXI 1/2) -

)

1 (1) 2 (2) 3 + β (0) MXI + β MXI + β MXI (15) 1 + bMXI1/2 ln γ( MX ) -

AφI 1/2 1 + bMXI1/2

+ 3 (1) 2 4 (2) 3 2β (0) MXI + β MXI + β MXI (16) 2 3

In summary, the Lietzke model has four adjustable parameters; the Pitzer model has four adjustable parameters and three fixed parameters; and the modified Pitzer model has three adjustable parameters. Results Experimental sources and molality ranges for each aqueous electrolyte solution are shown in Table 1. Characteristic parameters for all models are estimated by minimizing

better than the modified Pitzer model. Average percentage deviations from the experimental values are 1.08 and 4.62%, respectively. In Figure 1, we compare the osmotic and activity coefficients using the original Pitzer model and the modified Pitzer model for 2:2 electrolytes. The lack of fit of the modified Pitzer model for the osmotic coefficient is more evident. This fact is clearly observed in a residual log-linear plot as proposed for Holste et al.25 Figure 2 shows that residuals fluctuate about (10% and that the predictions are better for the osmotic coefficient than for the activity coefficient with both models. However, for the Pitzer model most of the residuals are around (1% while for the modified Pitzer model the residuals are over the entire range. For electrolytes with monovalent ions, the modified Pitzer model compares well with the Pitzer model. Average percentage deviations are 2.42 and 2.74%, respectively. The modified Pitzer model performs better in 65% of the systems. These results agree with the finding of Pe´rez-Villasenõr et al.20 for 1:1 and 1:2 electrolytes. As before, residuals are about (10%, but error distributions are quite similar for both models, as shown in Figure 3. At molalities between 2.5 and 5, deviations of the Pitzer model for most of the systems are within 10% while deviations in the modified model for many systems are within (1%. Again,20 both models correlate the data in a similar way, but when the Pitzer model cannot correctly correlate the data, the modified model does a better job. An example of this behavior is the aqueous solution of Th(NO3)4, as shown in Table 3. In the modified Pitzer model, the most important parameter is the closest approach parameter, bMX; however, we have not found a definitive correlation between the parameter bMX and the valence of either ionic species forming the electrolyte. For the 2:2 electrolytes, the value of bMX is around 1.5. Also, we can find two well-defined regions for the value of the closest approach as shown in Figure 4 if we do not consider the values for the 2:2 electrolytes. One region is between 0.9 and 2.0 for most organic electrolytes, and the other region is between 2 and 3.1 for inorganic electrolytes. Conclusions

n

S(y) )

pred 2 (yobs ∑ i - yi ) i)1

(17)

using a least-squares method developed by Stewart et al.23 In eq 17, yi denotes either the osmotic or the activity coefficient. Table 2 shows the parameter values for the modified Pitzer model. In this model, when the minimization procedure shows that the parameter CMX is insignificant, its value has been fixed to zero as shown in Table 2. Asymptotic standard errors of the parameters are 10-30% of the parameter value. Parameter values for the Pitzer and Lietzke models are calculated by Bedolla-Hernańdez24 using the same data sets. The Lietzke model performs poorer than the Pitzer and modified Pitzer models. This model has deviations under 4% with respect to the activity coefficient values for 1:1 and 1:2 electrolytes,26 but if absolute charge is greater than 2 for any ion, relative deviations from the experimental values can be as high as 10%. Their worst prediction is for electrolytes without monovalent ions. As mentioned before for aqueous electrolyte solutions without monovalent ions, the Pitzer model requires an extended form for the apparent virial coefficient, BMX, with four adjustable and three fixed parameters. As expected, for all electrolytes without monovalent ions, the Pitzer model fits the experimental data

We have used the modified Pitzer equation to the correlate experimental data of activity and osmotic coefficients of strong electrolyte solutions. For the electrolytes analyzed in this work, the modified Pitzer model correlates better the data of electrolyte solutions with monovalent ions than the original Pitzer model. For electrolytes without monovalent ions, the original Pitzer model is better but it requires additional adjustable parameters. It is remarkable that the modified Pitzer model correlates the data well without additional terms. Acknowledgment Universidad Autońoma de Tlaxcala, Instituto Tecnolo´gico de Celaya, PROMEP, and FOMIX-Tlaxcala-CONACyT have provided financial support for this work. Nomenclature Aφ ) Debye-Hu¨ckel coefficient bMX ) closest approach parameter in the Debye-Hu¨ckel term CMX ) third virial-type coefficient CφMX ) third virial-type coefficient form for the osmotic coefficient BMX ) second virial-type coefficient

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BφMX ) second virial-type coefficient form for the osmotic coefficient Gex ) excess Gibbs free energy on a molality basis f(I) ) ionic strength function form of the Debye-Hu¨ckel term fφ ) ionic strength function form of the Debye-Hu¨ckel term used for the osmotic coefficient fγ ) ionic strength function form of the Debye-Hu¨ckel term used for the activity coefficient I ) ionic strength (mol/kg of solvent) 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) wS ) solvent weight (kg) zM ) cation valence zX ) anion valence Greek Letters R1,R2 ) nonlinear parameters for second virial coefficient in the Pitzer equation ν ) stoichiometric coefficient (1) (2) β (0) MX,β MX, β MX ) adjustable parameters for the Pitzer, Lietzke, or modified Pitzer models φ ) osmotic coefficient γ( MX ) mean ionic activity coefficient for MX neutral electrolyte Subscripts M ) cation X ) anion MX ) neutral electrolyte Literature Cited (1) Bartels, J.; Ku¨hn, M.; Pape, H.; Clauser, C. A new aquifer tool for coupled flow, heat transfer, multi-species transport and Chemical waterrock interaction. Proceedings World Geothermal Congress, Japan 2000, 3997-4001. (2) Byrne, R.; Laurie, H. Influence of pressure on chemical equilibria in aqueous systems with particular reference to sea water. Pure Appl. Chem. 1999, 71, 871-980. (3) Pilinis, C. Modeling atmospheric aerosols using thermodynamic arguments. A reView. Global Nest: The International Journal 1999, 1, 5-13. (4) Edmonds, B.; Moorwood, R. A. S.; Szczepanski, R. A practical model for the effect of salinity on gas hydrate formation. Presented at European Production Operations Conference and Exhibition, Stavanger, Norway, April 16-17, 1996. Published by Petroleum Engineers Inc.; paper number 35569-MS. (5) Roa, V.; Tapia, M. Estimating water activity in systems containing multiple solutes based on solute properties. J. Food Sci. 1998, 63, 559564. (6) Chong, T.; Sheikleslami, R. Thermodynamics and Kinetics for mixed calcium carbonate and calcium sulfate precipitation. Chem. Eng. Sci. 2001, 56, 5391-5400. (7) Louhi-Kultanen, M.; Kallas, J.; Partanen, J.; Sha, Z.; Oinas, P.; Palosaari, S. The influence of multicomponent diffusion on crystal growth in electrolyte solutions. Chem. Eng. Sci. 2001, 56, 3505-3515. (8) Ji, X.; Chen, D.; Wei, T.; Lu, X.; Wang, Y.; Shi, J. Determination of dissolution kinetics of K2SO4 crystal with ion selective electrode. Chem. Eng. Sci. 2001, 56, 7017-7024. (9) Takano, K.; Gani, R.; Ishikawa, T.; Kolar, K. Computer aided design and analysis of separation processes with electrolyte systems. Comput. Chem. Eng. 2000, 24, 645-651. (10) Meissner, H. P.; Tester, J. W. Activity Coefficients of Strong Electrolytes in Aqueous Solutions. Ind. Eng. Chem. Proc. Des. DeV. 1972, 11, 128-133. (11) Pitzer, K. Thermodynamics of Electrolytes. I. J. Phys. Chem. 1973, 77, 268-277.

(12) Chen, C.; Britt, H.; Boston, J.; Evans, L. Local Composition Model for the Excess Gibbs Energy of Aqueous Electrolyte Systems. AIChE J. 1982, 28, 588-596. (13) Haghtalab, A.; Vera, J. Nonrandom Factor Model for Electrolyte Solutions. AIChE J. 1991, 37, 147-149. (14) Jaretum, A.; Aly, G. New Local Composition Model for Electrolyte Solutions. Fluid Phase Equilib. 1999, 163, 175-193. (15) Zhao, E.; Yu, M.; Sauve´, R.; Khoshkbarchi, M. Extension of the Wilson Model to Electrolyte Solutions. Fluid Phase Equilib. 2000, 173, 161-175. (16) Masoudi, R.; Arjmandi, M.; Tohidi, B. Extension of ValderramaPatel-Teja equation of state to modelling single and mixed electrolyte solutions. Chem. Eng. Sci. 2003, 58, 1743-1749. (17) Pitzer, K.; Mayorga, G. Thermodynamics of Electrolytes. III. Activity and Osmotic Coefficients for 2:2 Electrolytes. J. Sol. Chem. 1974, 3, 539-546. (18) Pitzer, K.; Silvester, L. Thermodynamics of Electrolytes. 11 Properties of 3:2, 4:2 and Other High-Valence Types. J. Phys. Chem. 1978, 82, 1239-1242. (19) Kim, H.; Frederick, W. Evaluation of Pitzer Ion Interaction Parameters of Aqueous Electrolytes at 25 °C. 1. Single Salt Parameters. J. Chem. Eng. Data 1988, 33, 177-184. (20) Pe´rez-Villasenõr, F.; Iglesias-Silva, G.; Hall, K. Osmotic and Activity Coefficients Using a Modified Pitzer Equation for Strong Electrolytes 1:1 and 1:2 at 298.15 K. Ind. Eng. Chem. Res. 2002, 41, 10311037. (21) Pe´rez-Villasenõr, F.; Iglesias-Silva, G.; Hall, K. Prediction of Osmotic and Activity Coefficients Using a Modified Pitzer Equation for Multicomponent Strong Electrolyte Systems at 298 K. Ind. Eng. Chem. Res. 2003, 42, 1087-1092. (22) Lietzke, M. H.; Stoughton, R. W. The Calculation of Activity Coefficients from Osmotic Coefficient Data. J. Phys. Chem. 1962, 66, 508. (23) Stewart, W.; Caracotsios, M.; Sorensen, J. GREG; Department of Chemical Engineering, University of WisconsinsMadison, U.S.A., 1990. (24) Bedolla-Hernańdez, M. L. Osmotic and Activity Coefficients of Strong Electrolytes 1:1, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2 and 4:1 at 298.15 K using a Modified Pitzer Equation. Master Thesis, Universidad Autońoma de Tlaxcala, Mexico, 2007. (25) Holste, J.; Hall, K.; Iglesias-Silva, G. Log-Linear Plots for Data Representation. AIChE J. 1996, 42, 296-297. (26) Pe´rez-Villasenõr, F.; Iglesias-Silva, G.; Hall, K. Temperature Dependence of a Modified Pitzer Equation for Strong Electrolytes Systems. Ind. Eng. Chem. Res. 2003, 42, 6962-6969. (27) Bonner, D.; Carey, R.; Torres, A. The Osmotic and Activity Coefficients of Some Bolaform Electrolytes. J. Phys. Chem. 1968, 72, 4290-4295. (28) Robinson R.; Stokes R. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1965. (29) Goldberg, R. N. Evaluated Activity and Osmotic Coefficient for Aqueous Solutions: Bi-Univalent Compounds of Zinc, Cadmium, and Ethylene Bis(Trimethylammonium) Chloride and Iodide. J. Phys. Chem. Ref. Data 1981, 10, 1-52. (30) Rard, J.; Weber, H.; Spedding, F. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25 °C. 2. The Rare Earth Perchlorates. J. Chem. Eng. Data 1977, 22, 187-201. (31) Spedding, F.; Weber, H.; Saeger, V.; Petheram, H.; Rard, J.; Habenschuss, A. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25 °C. 1. The Rare Earth Chlorides. J. Chem. Eng. Data 1976, 2, 341-360. (32) Wynveen, R.; Dye, J.; Brubaker, C. Activity Coefficient and Conductivity Measurements of High-charge (3-1, 1-3, 3-2) Electrolytes. I. J. Am. Chem. Soc. 1960, 82, 4441-4445. (33) Goves, K.; Dye, J.; Brubaker, C. Activity Coefficients and Conductances of High-charge (4-1, 1-4, 1-2) Electrolytes. II. J. Am. Chem. Soc. 1960, 82, 4445-4448. (34) Rard, J.; Spedding, F. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare-Earth Electrolyte Solutions at 25 °C. 5. Dy(NO3)3, Ho(NO3)3, and Lu(NO3)3. J. Chem. Eng. Data 1981, 26, 391395. (35) Bonner, D.; Kim, S. The Osmotic and Activity Coefficients of Some Dibolaform Electrolytes. J. Phys. Chem. 1969, 73, 1367-1370. (36) Rard, J.; Spedding, F. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare-Earth Electrolyte Solutions at 25 °C. 6. Eu(NO3)3, Y(NO3)3, and YCl3. J. Chem. Eng. Data 1982, 27, 454-461. (37) Rard, J.; Shiers, L.; Heiser, D.; Spedding, F. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25 °C. 3. The Rare Earth Nitrates. J. Chem. Eng. Data 1977, 22, 337-347.

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(38) Rard, J.; Miller, D. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25 °C. 4. La(NO3)3, Pr(NO3)3, and Nd((NO3)3. J. Chem. Eng. Data 1979, 24, 348354. (39) Lindenbaum, S.; Boyd, G. Osmotic and Activity Coefficients for the Symmetrical Tetraalkyl Ammonium Halides in Aqueous Solutions at 25 °C. J. Phys. Chem. 1964, 68, 911-917. (40) Wen, W.; Saito, S.; Lee, C. Activity and Osmotic Coefficients of four Symmetrical Tetraalkylammonium Fluorides in Aqueous Solutions at 25 °C. J. Phys. Chem. 1966, 70, 1244-1248. (41) Bonner, D. Osmotic and Activity Coefficients of Methyl-Substituted Ammonium Perchlorates at 298.15 K. J. Chem. Eng. Data 1982, 27, 6264. (42) Bonner, D. Osmotic and Activity Coefficients of Lithium Chlorate and Lithium Bromate J. Chem. Eng. Data 1979, 24, 210-211. (43) Bonner, D. Osmotic and Activity Coefficients of Some Tetramethylguanidinium Salts at 298.15 K. J. Chem. Eng. Data 1979, 24, 211212. (44) Bonner, D. Osmotic and Activity Coefficients of Methyl-Substituted Ammonium Nitrates at 298.15 K. J. Chem. Eng. Data 1981, 26, 148-149.

(45) Rard, J.; Palmer, D.; Albright, J. Isopiestic Determination of the Osmotic and Activity Coefficients of Aqueous Sodium Trifluoromethanesulfonate at 298.15 K and 323.15 K, and Representation with an Extended Ion-Interaction (Pitzer) Model. J. Chem. Eng. Data 2003, 48, 158-166. (46) Macaskill, J.; Robinson, R.; Bates, R. Osmotic Coefficients and Activity Coefficients of Guanidinium Chloride in Concentrated Aqueous Solutions at 25°C. J. Chem. Eng. Data 1977, 22, 411-412. (47) Libus, Z.; Sadowska, T. Coordination and Association Equilibria in Aqueous Electrolyte Solutions. I. Osmotic and Activity Coefficients of Divalent Metal Perchlorates. J. Phys. Chem. 1969, 73, 3229-3236.

ReceiVed for reView February 8, 2007 ReVised manuscript receiVed May 29, 2007 Accepted June 8, 2007 IE070228W

Osmotic and Activity Coefficients of 1:1, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2 and 4

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