5702
Ind. Eng. Chem. Res. 2003, 42, 5702-5707
New Modified Wilson Model for Electrolyte Solutions Xin Xu and Euge´ nia A. Macedo* Laboratory of Separation and Reaction Engineering, Departamento de Engenharia Quı´mica, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
A new model is proposed to model the activity coefficients of electrolytes in aqueous solutions. The model consists of two contributions: one due to long-range forces represented by the unsymmetric Pitzer-Debye-Hu¨ckel equation and the other due to short-range forces described by a new local composition expression. This last term is obtained from work presented in a previous paper for polymer aqueous solutions. With only two adjustable parameters per electrolyte, the new expression for the short-range excess Gibbs energy has the same form as the Wilson model for nonelectrolytes. The new equation is tested with experimental data available in the literature. The mean ionic activity coefficients of electrolytes are represented very satisfactorily. The performance of the new equation is compared with the performances of the other three models. 1. Introduction
2. Thermodynamic Model
Electrolytes play an important role in many applications that typically occur in the areas of corrosion, water pollution control, salting-in and salting-out effects in extraction and distillation, food processing, oilfield, paper production, and bioseparation, among others.1-4 Numerous attempts have been made to develop models to represent thermodynamic properties of electrolyte solutions. Reviews of the present understanding of the thermodynamics of electrolytes have appeared in the literature,5,6 including the local composition models7-11 and equations for mixed solvents and mixed electrolytes.8,12-14 The most well-known local composition model for electrolytes is the nonrandom two-liquid model (NRTL) proposed by Chen,7,8 using Pitzer’s modification of the Debye-Hu¨ckel theory15,16 for activity coefficients for long-range forces and a NRTL equation for short-range forces. Using only two binary adjustable parameters, the model can correlate and predict the activity coefficients for electrolyte solutions successfully. Zhao et al.17 have extended the Wilson model for nonelectrolytes18 to electrolyte solutions, with the shortrange interactions represented with a local composition expression. The main difference between this model and the extensions of the NRTL equation to electrolytes available is the assumption that the short-range energy parameter between species in a local cell has an enthalpic rather than commonly assumed Gibbs energy nature.17 This model represents the mean activity coefficients of electrolytes very well. In this study, an expression extended from the residual contribution of the modified Wilson model for polymer aqueous solutions19 is proposed to represent the short-range interactions in electrolyte solutions.
As usual, the excess Gibbs energy is written as a sum of two contributions:
* To whom correspondence should be addressed. Tel.: 35122-5081653. Fax: 351-22-5081674. E-mail:
[email protected].
Gex* Gex*,LR Gex*,SR ) + RT RT RT
(1)
Hence, two terms of the unsymmetric mean ionic (mole fraction based) activity coefficient are obtained:
ln γi* ) ln γi*,LR + ln γi*,SR
(2)
The first term represents the long-range Coulombic interactions, while the second term, based on the local composition concept, refers to the short-range interactions in electrolyte solutions. 2.1. Long-Range Contribution. In this work, the Pitzer-Debye-Hu¨ckel equation proposed by Pitzer16 is used to represent the contribution of the long-range ion-ion interactions (see the Appendix). 2.2. Short-Range Contribution. In this study, the term for the residual contribution of the modified Wilson model for polymer solutions published earlier19 is extended to represent the short-range excess energy in electrolyte solutions:
G E HE 1 ) RT RT R
∫∞TT1
( ) ∂HE ∂T
P,x
dT
(3)
in which the expression of HE can be easily obtained following previous derivation treatments.8,20,21 The effective mole fractions of ions are used, Xj ) xjCj (Cj ) Zj for ions and Cj ) 1 for molecules), and the excess Gibbs energy function is described as
GE 1 ) - {Xm ln[Xm + (Xa + Xc)Hem] + RT R Xc ln(Xa + XmHme) + Xa ln(Xc + XmHme)} (4)
10.1021/ie030514h CCC: $25.00 © 2003 American Chemical Society Published on Web 10/04/2003
Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5703
which is very similar to the Wilson equation18 in formula. It must be normalized to the infinitely dilute reference state for the ions, according to the so-called unsymmetric convention. The normalization equation is
GE* GE ) - xc ln γ∞c - xa ln γ∞a RT RT
(5)
where γ∞c and γ∞a are the infinite-dilution activity coefficients of the cation and anion of the electrolyte, respectively. Proper partial differentiation of the excess Gibbs energy expression is used to obtain the expressions for the activity coefficients of the different species in the solution. The activity coefficient expression for the anion is
ln γ/a ) -
[
Figure 1. Comparison between measured and calculated mean ionic activity coefficients for several aqueous electrolytes at 298.15 K.
Za Xc XmHem + + R Xm + (Xa + Xc)Hem Xa + XmHme
]
ln(Xc + XmHme) - Hem - ln Hme (6) and for the cation
ln γ/c ) -
[
Zc Xa XmHem + + R Xm + (Xa + Xc)Hem Xc + XmHme
]
ln(Xa + XmHme) - Hem - ln Hme (7)
3. Results 3.1. Minimization. The mean ionic activity coefficients available in the literature were used to evaluate the capabilities of the new model proposed in this work for electrolyte solutions. The least-squares correlation was carried out on deviations between calculated values and experimental data: NP
SSQ )
exp 2 (ln γcal ∑ ( - ln γ( ) j)1
(8)
where NP is the number of data points. The conversion of the values of the activity coefficients between the true mole fraction and the molality scale is given as follows:
The standard deviations obtained with the model proposed in this work are also included in Table 1. They are compared with the other standard deviations of the NRTL model,7 the NRTL-NRF model,23 and the extension of the Wilson model17 for electrolytes. The average standard deviations are summarized in Table 2, by electrolyte type. As can be seen, with the same number of parameters, i.e., two parameters for each electrolyte system, the new model under study can represent, in most cases, the experimental data superiorly than the NRTL. For 2-2 electrolyte systems, the NRTL-NRF model seems to give better results. From Tables 1 and 2, it can be seen that the standard deviations for the model proposed in this work are almost comparable to those calculated with the extension of the Wilson model for electrolytes.17 However, the expression of the model presented here is much simpler, with the same form as the Wilson equation.18 The nonrandom factor could be considered as another adjustable parameter. For example, the average standard deviation of the fit for 1-1 electrolytes can be reduced from 0.012 to 0.0084 by adjusting the value of the nonrandom parameter within the range 0.008-7. Thus, the modified Wilson model not only can successfully describe the experimental data for electrolytes but also has flexibility comparable to that of the extension of the Wilson model.17
ln γ(,m ) ln γ(,x - ln[1 + 0.001Mw(vc + va)m] (9) 4. Conclusions where Mw is the molecular weight of water and γ(,m is the molal mean activity coefficient. 3.2. Results. Some of the results are shown in Figure 1. It can be seen that the results are in good agreement with the experimental data. The binary parameters are presented in Table 1, with the standard deviations σ.
σ)
[
(ln ∑ j)1
]
1/2
NP
γcal (
- ln
NP
2 γexp ( )
(10)
A new activity coefficient model for electrolyte solutions is proposed in this work. The short-range interaction term is extended from the residual contribution of the modified Wilson model published in a previous paper19 calculating the solvent activities in polymer solutions. The new model contains two adjustable energy parameters, which were correlated from experimental data using the nonrandom factor fixed at 0.1. The modified Wilson model can describe the activity coefficients for electrolyte solutions successfully.
5704 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 Table 1. Energy Parameters Correlated and the Standard Deviations of the Modified Wilson Model, Extension of the Wilson Model,17 NRTL-Electrolyte Model,7 and NRTL-NRF Model23,24 to Mean Ionic Activity Coefficient Data at 298.15 K with r ) 0.1 σthis work
σWilson17
σNRTL7
σNRTL-NRF23,24
-8.0927 -2.7921 -10.0565 -10.168 -9.7659 -9.9828 0.9692 2.2949 13.260 -2.1443 -1.7147 -3.627 1.9083 59.134 -8.9008 -10.7002 -8.3650 -8.5421 -9.5219 -9.995 -1.6977 -8.3946 -12.2839 -0.3260 -13.0519 -12.5356 -0.8826 -2.5004 -0.2187 -10.6692 0.3107 6.1992 -8.5053 -8.3426 -8.1995 20.4919 2.1973 2.9993 -3.1107 -9.4085 -3.1982 -9.4451 -10.0354 -9.9011 1.3158 -9.1786 -11.6417 -11.6351 -4.8486 4.9800 -9.1298 -2.9794 -9.9291 -10.1031 -10.0457 -8.9966 -9.6212
1-1 Electrolyte 18.5675 0.0080 1.8215 0.0071 21.0443 0.0037 21.2234 0.0034 20.5505 0.0058 21.9123 0.0029 -55.040 0.0120 -483.250 0.0289 26.6146 0.0720 -8.4104 0.0141 1.2319 0.0072 4.2772 0.0067 -1.2240 0.0031 -12.8330 0.0029 18.3093 0.0027 21.6223 0.0066 17.1370 0.0023 18.2525 0.0053 19.9029 0.0030 21.9447 0.0030 3.8443 0.0041 18.9307 0.0063 24.6581 0.0290 -0.6090 0.0037 26.1878 0.0590 25.1045 0.0460 -10.5848 0.0164 -3.5240 0.0209 -3.3204 0.0050 22.3636 0.0219 1.7101 0.0122 -16.1988 0.0052 18.2056 0.0017 14.8123 0.0062 15.0918 0.0606 -29.4375 0.0102 -0.2520 0.0045 -3.4646 0.0072 4.9417 0.0102 19.4873 0.0002 6.27125 0.0042 19.6410 0.0030 20.6846 0.0058 21.3508 0.0090 -7.4068 0.0074 19.2780 0.0050 23.5621 0.0370 26.7249 0.0438 6.8807 0.0057 -4.6000 0.0016 19.5695 0.0091 2.3673 0.0073 20.4887 0.0035 20.8005 0.0041 20.7588 0.0032 20.0450 0.0086 20.3684 0.0064
0.0050 0.0088 0.0030 0.0078 0.0044 0.0028 0.0103 0.0176 0.0665 0.0143 0.0074 0.0070 0.0020 0.0010 0.0017 0.0023 0.0029 0.0076 0.0090 0.0027 0.0045 0.0049 0.0210 0.0038 0.0520 0.0314 0.0167 0.0214 0.0056 0.0198 0.0127 0.0054 0.0021 0.0052 0.0242 0.0082 0.0045 0.0072 0.0101 0.0006 0.0045 0.0032 0.0032 0.0134 0.0077 0.0021 0.0176 0.0448 0.0061 0.0019 0.0075 0.0073 0.0048 0.0053 0.0075 0.0070 0.0039
0.0100 0.0100 0.0060 0.0060 0.0070 0.0030 0.0150 0.0350 0.0630 0.0180 0.0080 0.0080 0.0040 0.0010 0.0020 0.0050 0.0020 0.0050 0.0030 0.0040 0.0050 0.0080 0.0230 0.0050 0.0500 0.0400 0.0230 0.0240 0.0130 0.0260 0.0130 0.0090 0.0020 0.0070 0.0240 0.0180 0.0050 0.0100 0.0100 0.0000 0.0040 0.0020 0.0020 0.0030 0.0100 0.0020 0.0290 0.0420 0.0060 0.0010 0.0120 0.0090 0.0020 0.0020 0.0030 0.0120 0.0120
0.014a 0.008 0.005 0.013 0.005 0.000 0.040a 0.024a 0.026 0.062a 0.018a 0.008 0.004 0.021a 0.003 0.006a 0.003 0.004 0.003 0.002 0.005 0.004 0.039a 0.005 0.045 0.052a 0.017a 0.020 0.016a 0.021a 0.014 0.061a 0.002a 0.026 0.015a 0.011a 0.005a 0.009 0.039a 0.002 0.008 0.001 0.002 0.003a 0.028a 0.072a 0.057a 0.058 0.008 0.002a 0.010a 0.008 0.001 0.003a 0.002 0.007 0.010
2.8191 24.7138 -10.7542 5.8484 32.2224 7.1798 -11.2965 -10.5745 24.2154 -10.6275 15.0158
1-2 Electrolyte -3.0750 0.0077 -17.1881 0.0160 22.2201 0.0071 -12.0044 0.0156 -26.8840 0.0480 -22.4237 0.0057 23.2221 0.0148 21.8676 0.0390 -18.0824 0.0310 22.1226 0.0163 -11.9715 0.0068
0.0077 0.0140 0.0147 0.0157 0.0358 0.0047 0.0148 0.0263 0.0168 0.0561 0.0069
0.0090 0.0220 0.0080 0.0230 0.0570 0.0030 0.0200 0.0240 0.0300 0.0170 0.0090
electrolyte
mmax
τem
AgNO3 CsAc CsBr CsCl CsI CsNO3 HBr HCl HClO4 HI HNO3 KAc KBr KCl KCNS KF KH adipate KH malonate KH succinate KH2PO4 KI KNO3 KOH LiAc LiBr LiCl LiClO4 LiI LiNO3 LiOH LiTol NaBr NaBrO3 Na butyrate Na caproate NaCl NaClO3 NaClO4 NaCNS NaF Na formate NaH malonate NaH succinate NaH2PO4 NaI NaNO3 NaOH Na pelargonate Na propionate NH4Cl NH4NO3 RbAc RbBr RbCl RbI RbNO3 TiAc
6.0 3.5 5.0 6.0 3.0 1.4 3.0 6.0 6.0 3.0 3.0 3.5 5.5 4.5 5.0 4.0 1.0 5.0 4.5 1.8 4.5 3.5 6.0 4.0 6.0 6.0 4.0 3.0 6.0 4.0 4.5 4.0 2.5 3.5 1.8 6.0 3.5 6.0 4.0 1.0 3.5 5.0 5.0 5.0 3.5 6.0 6.0 2.5 3.0 6.0 6.0 3.5 5.0 5.0 5.0 4.5 6.0
Cs2SO4 K2CrO4 K2SO4 Li2SO4 Na2CrO4 Na2 fumarate Na2 maleate Na2SO4 Na2S2O3 (NH4)2SO4 Rb2SO4
1.8 3.5 0.7 3.0 4.0 2.0 3.0 4.0 3.5 4.0 1.8
τme
Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5705 Table 1 (Continued) σthis work
σWilson17
σNRTL7
σNRTL-NRF23,24
-3.7417 -1.3083 -3.0139 -0.9389 1.0582 -1.5066 -2.4962 1.4799 -12.6993 -12.0385 -14.0238 -0.6515 0.0963 -2.3316 2.2814 -3.4956 3.0269 -2.0952 8.3224 -1.4923 -0.1212 -2.4772 -2.7126 1.5060 -2.0145 1.3270 1.6836 -10.5024 -1.7301 2.5982 -0.0960 -2.6172 -1.4140 -4.0724 -4.5745 -12.3229 -0.7073 -2.5837 1.7425
2-1 Electrolyte 3.3198 0.0232 -5.2853 0.0408 -2.4877 0.0285 -669.35 0.3400 -864.49 0.1602 -1886.49 0.2097 -7.7339 0.0368 -6.0264 0.0321 28.5924 0.2165 26.8359 0.1704 32.3979 0.3317 -405.033 0.0963 -14.7591 0.0274 -2941.2 0.2138 -1704.9 0.0461 3.9911 0.031 -32.5806 0.0381 -2.7007 0.0242 -24.9946 0.0101 -333.185 0.2361 -598.622 0.194 -1877.92 0.2041 -974.127 0.3284 -1386.22 0.0591 -2.0499 0.0238 -414.882 0.0416 -3967.55 0.0561 22.3383 0.0172 -5.0970 0.0302 -2486.62 0.0536 -250.00 0.0831 -5.4059 0.0381 2.7245 0.0259 2.1253 0.0336 -1032.0 0.4855 23.8274 0.0452 -0.2180 0.1042 -2432.5 0.2340 -491.13 0.0535
0.0232 0.0419 0.0285 0.2790 0.1815 0.1815 0.0362 0.0336 0.1967 0.1512 0.3013 0.0784 0.027 0.1721 0.0418 0.0357 0.0391 0.0239 0.0096 0.2111 0.1793 0.1804 0.2761 0.0483 0.0235 0.0314 0.0436 0.0165 0.0299 0.0438 0.0800 0.0375 0.0270 0.0343 0.4431 0.0415 0.1069 0.2099 0.0507
0.0260 0.0720 0.034 0.3510 0.2050 0.2720 0.0460 0.060 0.2580 0.2140 0.3740 0.1410 0.0550 0.2420 0.1080 0.0380 0.1130 0.0290 0.0130 0.2410 0.2020 0.2080 0.3160 0.1250 0.0470 0.0920 0.1470 0.0220 0.0360 0.0880 0.1680 0.0460 0.0290 0.0400 0.4470 0.0410 0.1190 0.2110 0.1480
0.020 0.021b 0.015 0.072 0.021 0.005 0.007 0.046 0.365 0.333 0.466 0.039 0.045 0.100 0.026 0.048 0.035 0.019 0.072 0.025 0.018 0.026 0.046 0.022 0.067 0.052 0.028 0.064 0.013 0.020 0.042 0.006 0.041 0.024 0.029 0.094 0.029 0.019 0.021
4.0 3.0 4.0 2.5 1.4 3.5 3.5 6.0
-12.0393 -11.5578 -11.7022 -11.4888 -11.6883 -11.7459 -11.5035 -12.0012
2-2 Electrolyte 24.7568 0.0831 23.7217 0.0431 24.1765 0.0768 23.7164 0.0463 24.1490 0.0507 24.2745 0.0684 23.7835 0.0538 24.8986 0.1099
0.0822 0.0514 0.0710 0.0414 0.0449 0.0659 0.0488 0.0889
0.0390 0.0360 0.0370 0.0310 0.0370 0.0380 0.0370 0.0500
0.059 0.052 0.051 0.046 0.046 0.049 0.055 0.096
AlCl3 CeCl3 CrCl3 Cr(NO3)3 EuCl3 LaCl3 NdCl3 PrCl3 ScCl3 SmCl3 YCl3
1.8 2.0 1.2 1.4 2.0 2.0 2.0 2.0 1.8 2.0 2.0
-1.3068 -1.7278 -6.8055 -5.7060 -1.8702 -2.2079 -1.5299 -1.8307 -2.3526 -1.8227 -0.7492
3-1 Electrolyte -273.55 0.0846 -8.6120 0.0642 8.8598 0.0511 6.1042 0.0499 -9.2051 0.0703 -6.1138 0.0651 -9.9591 0.0629 -7.9128 0.0638 -8.8951 0.0603 -9.0152 0.0663 -20.8865 0.0658
0.0802 0.0634 0.0458 0.0488 0.0690 0.0640 0.0830 0.0624 0.0586 0.0624 0.0646
0.1150 0.0840 0.0690 0.0540 0.0910 0.0820 0.0830 0.0820 0.0780 0.0870 0.0930
0.080 0.063 0.073 0.070 0.068 0.063 0.061 0.062 0.059 0.064 0.064
Al2(SO4)3 Cr2(SO4)3
1.0 1.2
-0.2946 -3.6644
3-2 Electrolyte -5.1396 0.0672 3.1868 0.1471
0.0648 0.1374
0.0750 0.1290
0.051 0.132
electrolyte
mmax
τem
BaBr2 Ba(ClO4)2 BaI2 CaBr2 CaCl2 Ca(ClO4)2 CaI2 Ca(NO3)2 CdBr2 CdCl2 CdI2 CoBr2 CoCl2 CoI2 Co(NO3)2 CuCl2 Cu(NO3)2 FeCl2 MgAc2 MgBr2 MgCl2 Mg(ClO4)2 MgI2 Mg(NO3)2 MnCl2 NiCl2 Pb(ClO4)2 Pb(NO3)2 SrBr2 SrCl2 Sr(ClO4)2 SrI2 Sr(NO3)2 UO2Cl2 UO2(ClO4)2 UO2(NO3)2 ZnCl2 Zn(ClO4)2 Zn(NO3)2
2.0 5.0 2.0 6.0 6.0 6.0 2.0 6.0 4.0 6.0 2.5 5.0 4.0 6.0 5.0 6.0 6.0 2.0 4.0 5.0 5.0 4.0 5.0 5.0 6.0 5.0 6.0 2.0 2.0 4.0 6.0 2.0 4.0 3.0 5.5 5.5 6.0 4.0 6.0
BeSO4 MgSO4 MnSO4 NiSO4 CuSO4 ZnSO4 CdSO4 UO2SO4
a
τme
Maximum molality is larger than that of the other two models. b Maximum molality is smaller than that of the other two models.
5706 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 Table 2. Summary of the Regression Results (the Mean-Average Standard Deviation) for Different Types of Aqueous Electrolyte Systems electrolyte type no. of systems this work Wilson17 NRTL7 NRTL-NRF23,24
1-1
1-2
2-1
2-2
3-1
3-2
57 0.012 0.010 0.012 0.011
11 0.019 0.019 0.020 0.017
39 0.113 0.103 0.139 0.064
8 0.066 0.062 0.038 0.057
11 0.064 0.064 0.083 0.066
2 0.107 0.101 0.102 0.092
m ) molality M ) molecular weight n ) number of moles NP ) number of experimental data points P ) pressure SSQ ) sum of squares R ) universal gas constant T ) absolute temperature x ) mole fraction X ) effective mole fraction Greek Letters
Acknowledgment X.X. acknowledges the support provided by FCT (Fundac¸ a˜o Para a Cieˆncia e Tecnologia), Ministe´rio da Cieˆncia e Tecnologia, Lisboa, Portugal. Appendix The Pitzer-Debye-Hu¨ckel equation is normalized to mole fractions of unity for solvent and zero for electrolyte:
Gex*,pdh
) -(
RT
4AφIx
∑k xk)
1/2
FMs
ln(1 + FIx1/2)
(A1)
∑zi2xi
1 2
(A2)
Hence, the expression of the activity coefficient is derived as
) ln γ/,pdh i
( ) [
1000 Ms
1/2
]
zi2Ix1/2 - 2Ix3/2 2zi2 1/2 Aφ ln(1 + FIx ) + F 1 + FI 1/2 x
(A3) The Debye-Hu¨ckel parameter Aφ for aqueous electrolyte systems is calculated using the following expression:7
Aφ ) -61.4453 exp[(T - 273.15)/273.15] + 2.864468{exp[(T - 273.15)/273.15]}2 + 183.5379 ln(T/273.15) - 0.6820223(T - 273.15) + 0.0007875695[T
2
- (273.15)2] + 58.95788(273.15/T) (A4)
The closest-approach parameter F is set constant and equal to 14.9 for all electrolytes, as suggested by Pitzer.16 List of Symbols Aφ ) Debye-Hu¨ckel constant in mole scale G ) Gibbs energy H ) enthalpy Ix ) ionic strength in mole fraction scale
Subscripts a ) anion c ) cation m ) solvent e ) electrolyte Superscripts
where Aφ is the usual Debye-Hu¨ckel parameter, Ms is the molecular weight of the solvent, F is the closest approach parameter, and Ix is the ionic strength on a mole fraction basis:
Ix )
R ) nonrandom factor τ ) binary interaction parameter ) energy parameter γ ) activity coefficient ν ) stoichiometric number F ) closet-approach parameter σ ) standard deviation ∂ ) partial derivative ∞ ) infinity
cal ) calculated exp ) experimental E ) notation of excess quality LR ) long range pdh ) Pitzer-Debye-Hu¨ckel SR ) short range * ) Henry’s conversion
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Received for review June 19, 2003 Revised manuscript received September 16, 2003 Accepted September 19, 2003 IE030514H
