Comparison among Pitzer-type Models for the Osmotic and Activity

CORRELATION pubs.acs.org/IECR

Comparison among Pitzer-type Models for the Osmotic and Activity Coefficients of Strong Electrolyte Solutions at 298.15 K F. P
erez-Villase~nor* and S. Carro-S
anchez Facultad de Ciencias B
asicas Ingeniería y Tecnología, Universidad Aut
onoma de Tlaxcala, Apizaco, Tlax. C.P. 90300, M
exico

G. A. Iglesias-Silva Departamento de Ingeniería Química, Instituto Tecnol
ogico de Celaya, Celaya, Gto. C.P. 38010, M
exico

bS Supporting Information ABSTRACT: In this work three Pitzer-type models were compared: the original one, the Archer model (an extended Pitzer model) with ionic strength dependence in the third virial coefficient, and the PIH model that uses as an adjusting parameter the closest approach constant. The performance of these models in correlating the osmotic and activity coefficient of 1:1, 1:2, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2, and 4:1 aqueous electrolyte solutions was tested using experimental data sets for 245 single electrolyte aqueous solutions. The absolute average percentage deviations were 0.78, 1.18, and 3.56% for the Archer, PIH, and Pitzer models, respectively.

’ INTRODUCTION Modeling and optimization of industrial processes with electrolyte solutions1
7 require rigorous activity models in order to take into account nonidealities in this kind of systems. Several theories and equations have been developed with this objective.8
15 Among them, the most employed models are those developed by Pitzer9
11 and Chen et al.12 P
erez-Villase~ nor et al.16,17 proposed a modified Pitzer model (PIH model) considering the closest approach parameter in the Debye
H€uckel term as an adjusting parameter. This modification improves the correlative capacity of the equation even when the second parameter of the virial coefficient BMX is removed. Archer18 proposed an extended Pitzer model (Archer model) adding the ionic strength dependence into the third virial coefficient. With this extension, the Archer model correlates accurately the activity and osmotic coefficients of aqueous solutions of NaBr,18 NaCl,19,20,24,25 MgSO4,21 KCl,22 NaNO3,23 and K2SO4.25 Rard and Wijesinghe26 developed a useful method to convert the parameters of the Archer modification to those of the Pitzer model. However, these parameters are not the optimum values for the representation of the nonideality. In this work, the osmotic and activity coefficients for 245 single electrolytes aqueous solutions were correlated to the Archer model. The performance of this equation was compared with that of the Pitzer and PIH models.

Table 1. Parameters for Three Different Variants of Pitzertype Models parameter

G ¼ f ðIÞ þ 2mM mX ½BMX þ zM mM CMX  wS RT

ð1Þ

This equation is also the basis of all modifications of the Pitzer model. From eq 1, the osmotic coefficient can be obtained from r 2011 American Chemical Society

Archer

PIH

bMX

1.2

1.2

fit to experimental

R1

2.0/1.4b

2.0c/1.4

not required

R2 R3

12a not required

12
32Aϕ 0.4
2.5c

not required not required

βMX(0)

fit to experimental

fit to experimental

fit to experimental

data

data

data

βMX(1)

fit to experimental

βMX(2)

fit to experimental

fit to experimental

data

data not required

data fit to experimental

dataa

not required

data

CMX(0)

fit to experimental data

fit to experimental data

fit to experimental data

CMX(1)

not required

fit to experimental

not required

data a Not required for electrolytes with univalent ions. b For electrolytes without univalent ions. c Most common value reported.

its first derivative with respect to the solvent weight: ϕ
1 ¼


’ EQUATIONS Pitzer expressed the excess Gibbs energy of a single electrolyte in an aqueous solution as E

Pitzer

þ

1

RT

∑i




∂Gex mi ∂wS



¼ jzM zX jf ϕ þ P, T, ni

2ðvM vX Þ3=2 ϕ mMX 2 CMX vMX



 2vM vX ϕ mMX BMX vMX

ð2Þ

Received: December 11, 2010 Accepted: August 2, 2011 Revised: July 28, 2011 Published: August 02, 2011 10894

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CORRELATION

Table 2. Reported Values for Parameters Used in BMX and CMX Exponential Expansions R1

electrolyte

a

R2

R3

R4a

ref

1.34

53

νM νX 1=νMX ln γ( ¼ jzM zX jf γ MX ¼ ðγM γX Þ



 2νM νX 2ðνM νX Þ3=2 þ mMX 2 CγMX mMX BγMX þ νMX vMX

Ca(NO3)2

2.0

CaCl2 Cs2SO4

1.4 2.0

2.5/1.0 2.5

54 64

Cu(NO3)2

2.0

1.5

58

Fe2(SO4)3

2.0

2.5

71

H2SO4

2.0

2.5

60,71

K2SO4

2.0

2.5

25,63

Mg(NO3)2

2.0

MgCl2

2.0

MgSO4 NaBr

2.0/1.4 2.0

2.5/1.0 2.0/1.7

NaCF3SO3

2.0

2.5

38

NaCl

2.0

2.5

20,25,62,65,69

NaNO3

2.0

2.5

23

NdCl3

0.5

1.5

67

(NH4)2SO4

2.0

2.5

57

Rb2SO4

2.0

2.5

64

SrCl2 ZnCl2

2.0 1.1

2.5 2.5

69 70

ZnSO4

1.4

32Aϕ

2.5

2.0 30.65Aϕ

ð3Þ Functions f(I), fϕ, and fγ arising in eqs 1
3 correspond to the electrostatic interactions expressed by

68

0.4

12

Also, the mean activity coefficient can be calculated from the definition of the activity coefficient:

0.28

f ðIÞ ¼


59 60,21 18,56

1.1551/1.1

fϕ ¼


ϕ

ð0Þ

ð1Þ

1=2

!

ð6Þ

In general, the second virial coefficient, BMX, and its derivatives are functions of the ionic strength:10,11,21 ð1Þ

ð0Þ

BMX ðIÞ ¼ βMX þ

þ CMX e
R4 l 

2βMX 1=2 ½1
e
R1 I ð1 þ R1 I 1=2 Þ R1 2 I

1=2

ð2Þ

þ Table 3. Archer Model Performance at Different Values for R3 ϕ

2βMX 1=2 ½1
e
R2 I ð1 þ R2 I 1=2 Þ R2 2 I

ð0Þ

ð1Þ

BMX ðIÞ ¼ βMX þ βMX e
R1 I

average error (%) R3 = 2.0

R3 = 2.5

R3 fitted to experimental data

1:1

0.8156

0.7708

0.6053

[1:2 + 2:1]

0.5132

0.6334

0.2862

[1:3 + 3:1]

1.1109

1.4806

1.1972

[1:4 + 4:1]

3.9876

3.5783

2.8897

2:2

0.0884

0.0836

0.0860

3:2

0.5904

1.5911

1.0908

Global

0.7815

0.8949

0.6294

electrolyte data set

ð5Þ

I 1=2 2 þ lnð1 þ bMX I 1=2 Þ 1=2 bMX 1 þ bMX I

f ¼
Aϕ

61,66,70 ð2Þ

ð4Þ

Aϕ I 1=2 1 þ bMX I 1=2

γ

Parameter required when an expanded form of CϕMX is used: CMX ¼ 2jzM zX j1=2 ½CMX þ CMX e
R3 l

4Aϕ I lnð1 þ bMX I 1=2 Þ bMX

BγMX ðIÞ

1=2

ð2Þ

þ βMX e
R2 I

ð7Þ

1=2

ð8Þ

" # ð1Þ 2βMX e
R1 I 1=2 2 1=2 1
¼ þ ð1 þ R1 I
R1 IÞ R1 2 I 2 2 3  
R2 I1=2 ð2Þ 2βMX 4 e ð1 þ R2 I 1=2
R2 2 IÞ5 þ 2 1
2 R2 I ð0Þ 2βMX

ð9Þ

Table 4. Parameters of the Archer Model for Electrolytes without Monovalent Ions electrolyte

β(0) MX

σ(β(0) MX)

β(1) MX

σ(β(1) MX)

β(2) MX

σ(β(2) MX)

C(0) MX

σ(C(0) MX)

C(1) MX

σ(C(1) MX)

2:2 electrolytes BeSO4

0.31939 8.28  10
3 2.84052 1.49  10
1
58.2017 3.34

0.00871 1.83  10
3

MnSO4 NiSO4

0.22299 6.85  10 3.11328 1.28  10
59.4446 3.34 0.19103 7.42  10
3 3.20830 4.24  10
2
59.1036 9.40  10
1


4

UO2SO4

0.31412 3.00  10
3 1.64522 9.08  10
2
23.8120 2.15

0.16963 1.53  10


2

0.00136 5.40  10
4
0.00371 2.10  10
3
1

CuSO4


2

2.63361 2.53  10


3


3


45.1019 9.21  10


1


1

ZnSO4

0.17429 8.19  10

2.84574 1.23  10


54.7203 2.93

Al2(SO4)3

0.61964 9.19  10
3 10.45783 3.35  10
1
106.5028 1.10  101

0.00871 1.21  10
3

0.00306 4.41  10
0.38506 1.69  10
1 0.00757 6.40  10
4
0.50026 1.03  10
1
0.00401 1.42  10
4

0.22768 1.16  10
1


4

0.21342 1.88  10
1

0.00879 5.77  10

3:2 electrolytes Cr2(SO4)3


2

0.58651 1.77  10

[Co(Ethylene-diamine)3]2(SO4)3 0.367358 0.02184 fixed parameters

b = 1.2


1

8.90254 6.36  10

53.5839 2.10  10

4.076823 0.530765

31.81012 5.656867 R1 = 1.4 10895

1


0.00217 7.80  10
4
0.74607 3.98  10
1
0.00275 1.36  10
3

1.88548 7.02  10
1


0.00672 0.00111

5.643153 0.9625531

R2 = 12

R3 = 2.0

dx.doi.org/10.1021/ie102466b |Ind. Eng. Chem. Res. 2011, 50, 10894–10901

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CORRELATION

(0) Figure 1. Relationship of β(1) MX and βMX for electrolytes using the Pitzer model.

(0) Figure 2. Relationship of β(1) MX and βMX for electrolytes using the Archer model.

(

Similarly, the third virial coefficient11,21 is calculated as ð0Þ

CMX ðIÞ ¼ CMX þ

ð1Þ

4CMX 1=2 ½6
e
R3 I ð6 þ 6R3 I 1=2 R3 4 I 2

þ 3R3 2 I þ R3 3 I 3=2 Þ ϕ

CγMX ðIÞ

ð0Þ

ð1Þ

1=2

ð11Þ

ð0Þ

 4CMX h 1=2 6
e
R3 I 6 þ 6R3 I 1=2 4 2 R3 I ð1Þ

CMX þ

þ 3R3 2 I þ R3 3 I 3=2


ð10Þ

CMX ðIÞ ¼ 2jzM zX j1=2 ½CMX þ CMX e
R3 I 

¼ 3jzM zX j

1=2

R3 4 I 2 2

 ð12Þ

Table 1 shows the fitting and fixed parameters for the models used in this work. 10896

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CORRELATION

(0) Figure 3. Relationship of C(1) MX and CMX for electrolytes using the Archer model.

Figure 4. Osmotic (a) and activity (b) coefficient plots of 1:1 electrolytes comparing the Archer model



. b, HNO3 ; 2, NH 4NO 3 ; 9, KOH.

Figure 5. Residuals for the osmotic and activity coefficients of electrolytes with monovalent ions using the Archer, PIH, and Pitzer models. 10897

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CORRELATION

Table 5. Average Percentage Error for Pitzer Model and its Variants model

ϕ percent error

γ percent error

Archer

0.6951

0.8650

0.7800

Pitzer PIH

3.0564 0.8686

4.1483 1.3113

3.6024 1.0899

average percent error

electrolytes with monovalent ions

electrolytes without monovalent ions Archer

1.0853

0.5993

0.8423

Pitzer

1.5828

0.5891

1.0859

PIH

5.4707

8.0478

6.7592

Archer

0.7042

0.8588

0.7815

Pitzer

3.0318

4.0889

3.5604

PIH

0.9454

1.4237

1.1846

global

the Pitzer model (see Table 1). For the parameter R 3 , three different cases were considered: to fix the parameter to a value of 2.5, to fix it to a value of 2.0 and, finally, to consider it as a fitting parameter. To quantify the model performance, the absolute average percent deviation was used erroravg

Figure 6. Residuals for the osmotic and activity coefficients of 2:2 and 3:2 electrolytes using the Archer, PIH, and Pitzer models.

’ RESULTS Experimental sources and molality ranges for each aqueous electrolyte solution are shown in Table 2s of the Supporting Information. Adjusting parameters of the Archer model have been estimated using a nonlinear least-squares method in GREG27 subroutine by minimizing the objective function: SðyÞ ¼

n

cal 2 ∑ ðyobs i
yi Þ i¼1

ð13Þ

In eq 13, yi is either the osmotic coefficient, the mean ionic activity coefficient, or both. The collection of parameters for Pitzer and PIH models has been reported early16,17,28,29 using the same data sets. When Archer model is used the values of the set of parameters R 1 , R 2 , and R 3 are not the same for each electrolyte solution reported in Table 2, in contrast with Pitzer model, which has well-defined values for R 1 and R 2 (see Table 1). However, the parameter R 1 has the same value than the Pitzer model in 75% of the systems. Although there are few values for the parameter R 2 , it is around 12.0. The parameter R 3 has a value of 2.5 for half of the reported electrolyte solutions. Taking into account these facts, the parameters R 1 and R 2 have been fixed to the values given in

   100 n yobs
ycal i i  ¼    n i ¼ 1  yobs i

∑

ð14Þ

As shown in Table 3, the best results are obtained if R 3 is fitted; however, in order to maintain the number of adjusting parameters for Archer model as small as possible, it is desirable to fix R 3 . In this case, better results were obtained for the systems used in this work when R 3 = 2.0. Therefore, final calculations were performed with R 3 = R 1 = 2.0. Table 5s of the Supporting Information shows the results for electrolyte solutions with monovalent ions, while Table 4 shows the results for 2:2 and 3:2 electrolyte solutions. Figures 1 and 2 show the relationship between β (0) MX and (1) β MX for the Archer and Pitzer models in which the majority of the parameters lie down between
1 and 1. This agrees with the results of Pitzer and Mayorga. 30 As seen in Figure 3, (1) it seems to be a linear relation between C(0) MX and C MX . However, a detailed analysis shows that parameters have values around (0.05, and the distribution is similar for β(0) MX and β(1) MX as shown in Figures 1 and 2. To evaluate the performance of the three models, results previously reported by P
erez-Villase~nor 28 and BedollaHern
andez 29 for the Pitzer model were considered. The results for the PIH model obtained by P
erez-Villase~ nor et al., 16,17 using the same data sets given in Table 2s in the Supporting Information, were used. Table 5 shows that the best results are achieved when the Archer model is used. This model correlates the experimental data within an absolute average percentage deviation of 0.7815 while for the PIH and Pitzer models it is within 1.1846 and 3.5604%, respectively. However, for electrolytes without monovalent ions, results for Pitzer and Archer models are very close and both of them are clearly better than PIH model. 10898

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CORRELATION

In Figure 4a and b, it is shown that the Archer model follows the correct behavior for the osmotic and activity coefficients of HNO3 , KOH, and NH 4 NO 3 aqueous solutions, even at molalities as high as 25. For electrolyte with monovalent ions, residuals for the osmotic and activity coefficients were compared using the three Pitzer-type models. As shown in Figure 5, the residuals fluctuate around (1% for the Archer model, while for the PIH and Pitzer models, they are around (5 and (10%, respectively. For these kinds of systems, the only difference among the Archer and Pitzer models is the exponential term in the third virial coefficient, and it seems to be the reason of the best performance when the Archer model is used. For electrolyte solutions without monovalent ions, the correlative behavior of the Archer model is slightly better than the Pitzer model. Figure 6 shows the percentage deviations for the three models with respect to the experimental measurements. The residuals for the Archer and Pitzer models are mostly within (1% but at low molalities, they are within (5%. In this case, their performance is clearly better than that of the PIH model.

BMX BMXϕ

’ CONCLUSIONS It was shown that the Archer model correlates the experimental osmotic and activity coefficients of strong electrolyte solutions at 298.15 K better than the Pitzer and PIH models. Although the Archer model contains two more characteristic parameters than the Pitzer model, it has been found that by setting up R3 = R1 = 2.0, the set of fixed parameters can be reduced from four (b, R1, R2, R3) to three (b, R1, R2) without changing the performance of the equation. The PIH model contains only three adjusting parameters and compares well with the Archer model for electrolytes with monovalent ions. The absolute average percentage deviations for the Archer, PIH, and Pitzer models are 0.78, 1.18, and 3.56, respectively. However, for electrolytes without monovalent ions, the average error increases up to 6.75% for PIH model compared with 0.84 and 1.08% for the Archer and Pitzer models, respectively.

Greek letters

’ ASSOCIATED CONTENT

b S

Supporting Information. Experimental data sources for 245 single electrolyte solutions (Table 2s). Parameters for the Archer model for electrolytes with monovalent ions (Table 5s). This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: (52) 241-417-25-44. Fax: (52) 241-417-58-44. E-mail: [email protected].

’ ACKNOWLEDGMENT The authors are grateful to the Universidad Aut
onoma de Tlaxcala, Instituto Tecnol
ogico de Celaya, and CONACyT for the financial support in the realization of this work. ’ NOTATION Aϕ bMX

Debye
H€uckel coefficient Closest approach parameter in the Debye
H€uckel term

Second virial-type coefficient Second virial-type coefficient form for the osmotic coefficient Third virial-type coefficient Third virial-type coefficient form for the osmotic coefficient Excess Gibbs free energy on a molality basis Ionic strength function form of the Debye
H€uckel term Ionic strength function form of the Debye
H€uckel term used for the osmotic coefficient Ionic strength function form of the Debye
H€uckel term used for the activity coefficient Ionic strength (mol/kg of solvent) Molality of species i (mol/solvent kg) Number of moles in solution of ion i Universal gas constant Absolute temperature (K) Solvent weight (kg) Cation valence Anion valence

CMX CMXϕ Gex f(I) fϕ fγ I mi ni R T wS zM zX R1, R2, R3, R4 ν (0) (1) (2) , βMX , βMX βMX (0) (1) , CMX CMX

ϕ ( γMX

Non linear parameters for second and third virial coefficients in Archer and Pitzer Equation Stoichiometric coefficient Adjustable parameters for Pitzer, Archer or PIH models Adjustable parameters for Pitzer, Archer or PIH models Osmotic Coefficient Mean ionic activity coefficient for MX neutral electrolyte

Subscripts

M X MX

Cation Anion Neutral electrolyte

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CORRELATION

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