Ind. Eng. Chem. Res. 2006, 45, 7719-7728
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Ion-Based SAFT2 to Represent Aqueous Single- and Multiple-Salt Solutions at 298.15 K Xiaoyan Ji and Hertanto Adidharma* Soft Materials Laboratory, Department of Chemical and Petroleum Engineering, UniVersity of Wyoming, Laramie, Wyoming 82071-3295
SAFT2 with individual-ion parameters, referred to as ion-based SAFT2, is used to represent the properties of aqueous electrolyte solutions. A new set of parameters for 5 cations (Li+, Na+, K+, Ca2+, and Mg2+) and 6 anions (Cl-, Br-, I-, NO3-, SO42-,and HCO3-) is obtained from the fitting of the experimental mean ionic activity coefficients and liquid densities of 24 aqueous single-salt solutions at 298.15 K. The ion parameters are universal and transferable to different salts containing the same ion. Because of the peculiar segment energy of K+, a mixing rule with a binary interaction parameter for the segment energy describing the shortrange interactions between K+ and other cations is needed. The binary interaction parameter is derived from the osmotic coefficients of chloride solutions. The predictions of the osmotic coefficients, vapor pressures, and liquid densities of single- and multiple-salt solutions including seawater (brine) are found to agree with experimental data. 1. Introduction Most models for describing the nonideality of electrolyte solutions are based on the excess Gibbs free energy. Because new precise equations of state are now available for nonelectrolyte solutions, for example, the statistical associating fluid theory (SAFT),1-3 it became desirable to extend those equations of state to electrolytes. A few equations of state for electrolyte on the basis of SAFT have been proposed recently.4-6 SAFT1-RPM was used to represent the properties of aqueous single- and multiple-salt solutions, but the application was limited to monovalent ions because of the limited range of the square-well width parameter λ.7,8 This limitation was relaxed in SAFT2, and the properties of multivalent electrolyte solutions and seawater (brine) could be well-described.9,10 In both SAFT1-RPM7,8 and SAFT2,9,10 a salt is modeled as a molecule composed of two charged, but nonassociating, spherical segments: the cation and the anion. Each salt has an effective hydrated diameter used in the ionic contribution. Since the effective hydrated diameter is a salt parameter, a nonunique set of salts and their compositions have to be constructed for a given ion composition. Although by using this approach the calculated properties of different sets of salts with the same ion composition were found to be essentially the same,10 it is still worthy to attempt to remove this salt parameter and use all ion parameters because the ion parameters are universal and, hence, transferable to different salts containing the same ion. Such an approach can greatly reduce the number of parameters needed to represent electrolytes. This is reminiscent of group-contribution approaches. The purpose of this work is to explore the use of SAFT2 with individual-ion parameters, referred to as ion-based SAFT2, to represent the properties of aqueous single- and multiple-salt solutions. 2. Modeling 2.1. Equation of State. The ion-based SAFT2 is defined in terms of the dimensionless residual Helmholtz energy as follows, * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (307) 766-2500. Fax: (307) 766-6777.
Table 1. Experimental Data Used for Parameter Fitting and the Corresponding ARDa Cm max, mol/kgH2O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
salt
γ
F
LiCl NaCl KCl MgCl2 CaCl2 LiBr NaBr KBr MgBr2 CaBr2 LiI NaI KI MgI2 CaI2 NaNO3 KNO3 Mg(NO3)2 NaHCO3 KHCO3 Li2SO4 Na2SO4 K2SO4 MgSO4
6.0 6.0 4.5 2.0 2.0 6.0 6.0 5.5 2.0 2.0 3.0 6.0 4.5 2.0 1.9 3.0 3.5 2.0 1.0 1.0 2.0 1.75 0.69 3.6
6.0 6.0 4.5 2.0 2.0 6.0 6.0 5.5 2.0 3.0 6.0 4.5 2.0 1.9 5.5 3.5 2.0 1.2 3.5 2.0 1.75 0.65 2.6
ARD, % ref b b b c c b b b c c b b b c c b b d e f g g g h
γ
F
0.71 0.53 0.13 1.04 0.36 1.32 0.77 0.28 0.41 0.35 1.35 0.56 0.67 0.63 0.48 0.71 1.41 1.24 0.79 0.67 1.16 0.31 0.49 2.37
0.79 0.16 0.64 0.40 0.20 1.02 0.51 0.23 0.59 0.32 1.35 0.96 0.96 0.40 0.99 1.33 2.33 0.41 0.18 0.38 0.40 0.22 1.12
a The average relative deviation is defined as ARD ) 1/ N∑N |Ψcalc n)1 n exp Ψexp × 100%, where N is the number of data fitted, superscript n |/Ψn “calc” is the calculated value, superscript “exp” is the experimental value, and Ψ is the measured and calculated entity. b Hamer and Wu.14 c Goldberg and Nuttall.15 d Zaytsev and Aseyev.13 e Peiper and Pitzer.16 f Roy et al.17 g Goldberg.18 h Rard and Miller.19
a˜ res ) a˜ hs + a˜ disp + a˜ chain + a˜ assoc + a˜ ion
(1)
where the superscripts on the right side refer to terms accounting for the hard-sphere, dispersion, chain, association, and ionic interactions, respectively. The calculation of each part in the right side of eq 1 has been described in detail in our previous work.1,7,9 In general, we assume ions to be hard spheres, and thus, the dispersion term in eq 1 is calculated assuming that the dispersion interactions among electrical charges are zero. The dispersion term accounts only for the water-water and water-ion dispersion interactions. For aqueous salt solutions,
10.1021/ie060649y CCC: $33.50 © 2006 American Chemical Society Published on Web 09/21/2006
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Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006
Table 2. Parameters Derived from Experimental Data of Single-Salt Solutions at 298.15 K v25, cc Li+
Na+ K+ Ca2+ Mg2+
3.1514 1.5441 5.7814 3.8620 5.2108
u25/k, K 2524.16 1647.33 190.158 3833.20 2923.28
λ25,w+ 1.7888 1.8214 1.6820 1.1901 1.9536
d25, Å 5.5945 4.5599 3.2265 5.0730 4.6459
the segment numbers (m) for water and ions are set equal to unity, which makes the chain term equal to zero. The association term in eq 1 originates from the self-association of water molecules; water is modeled as a single-segmented molecule with four association sites (type 4C11). As in SAFT1-RPM and SAFT2, the ionic term is given by the restrictive primitive model (RPM). The ionic term by RPM is, in fact, the McMillan-Mayer framework, while the equation of state is the Lewis-Randall framework. However, the conversion between these two frameworks can be compensated by empirical parameters.12 Each ion is modeled as a nonassociating charged spherical segment with four parameters: segment volume (V), segment energy (u/k), reduced well range (λ) of square-well potential, and effective hydrated diameter (d). Note that, in ion-based SAFT2, the effective hydrated diameter is an ion property. Mixing rules for parameters V, u, and λ between two different segments are the same as those for SAFT1,1 and the mixing rule for the effective diameter d is
d)
∑i x′idi
(2)
where xi′ is the mole fraction of ion i on a solvent-free basis and the summation is over all ions. 2.2 Mean Ionic Activity Coefficient and Osmotic Coefficient. Activity coefficients are usually based on molality and the unsymmetric rule, i.e., the reference state of water is pure water, while that of ion is at infinite dilution. The molality* based unsymmetrically normalized activity coefficients (γi,C m) are given by * γi,C m )
γ/i 1 + 0.001Mw
∑j Cmj
(3)
where Mw is the molecular weight of water, Cm j is the molality of ion j, the summation is over all ions, and γ/i is the mole-
Figure 1. Comparison of the fitted effective ion radii (O) with Pauling radii20 ([), hydrated-ion radii20 (2), and hydrated-ion radii (3) fitted by Jin and Donohue.20
Cl-
BrINO3SO42HCO3-
v25, cc
u25/k, K
λ25,w-
d25, Å
4.6106 6.2397 7.4574 6.5090 2.1266 4.7939
1416.10 1636.54 1491.36 1680.51 5557.46 2292.63
1.1080 1.1032 1.1138 1.0171 1.0754 1.1008
5.4290 6.1093 6.2320 6.3413 2.9999 1.2529
fraction-based activity coefficient of ion i given by
γ/i )
φˆ i
(4)
(φˆ i)xjf0,for all j
where φˆ i is the fugacity coefficient of ion i directly calculated from the equation of state, xj is the mole fraction of ion j, and (φˆ i)xjf0,for all j is the fugacity coefficient of ion i at infinite dilution, for which the ionic contribution is zero. The mean ionic activity coefficient of a salt can be calculated from the activity coefficients of the constituting ions, 1
γ(,Cm ) [(γ*+,Cm)ν+ (γ*-,Cm)ν-]ν++ν-
(5)
where ν+ and ν- are the number of moles of cation and anion per mole of salt, respectively. The molality-based osmotic coefficient (φ) is calculated from
φ)-
ln(aw) 0.001Mw
∑j
ln(xwγw)
)Cm j
0.001Mw
∑j
(6) Cm j
where aw is the water activity, xw is the mole fraction of water, and γw is the mole-fraction-based activity coefficient of water calculated by
γw )
φˆ w (φˆ w)xwf1
(7)
where φˆ w is the fugacity coefficient of water. 3. Results and Discussion 3.1. Single-Salt Solutions. The parameters for pure H2O are taken from our previous work.9 They were fitted to the saturated vapor pressure (P) and liquid density (F) data in the temperature (T) range of 283-473 K. The parameters for ions are obtained from the experimental mean ionic activity coefficients and liquid densities of single-salt solutions, as described below. 3.1.1. Parameter Fitting. Our goal is to describe the properties of seawater (brine), which primarily contains Na+, K+, Ca2+, Mg2+, Cl-, and SO42- and has a total ionic strength 2 I ()0.5ΣCm i zi , where zi is the valence of the charged ion i and the summation is over all ions) of up to 6 mol/(kgH2O). Therefore, the solutions of interest are electrolytes containing Na+, K+, Ca2+, Mg2+, Cl-, and SO42- with an ionic strength of up to 6 mol/(kgH2O). Electrolytes containing Li+, Br-, I-, NO3-, and HCO3- are also added because of their common use in other industries. Parameters are fitted to the experimental mean ionic activity coefficients and liquid densities of single-salt solutions at 298.15 K. For the ions of interest, there are 30 single-salt solutions in total. Unfortunately, to the best of our knowledge, the experimental data are available only for some solutions. We also exclude some data, the reliability of which is uncertain. For
Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7721
Figure 2. Predicted and experimental osmotic coefficients for single-salt solutions and water activities for sulfates at 298.15 K: symbols, experimental data taken from references listed in Table 1; curves, predicted.
sulfates, experimental data for the CaSO4 solution are not available. For bicarbonates, reliable data are available only for NaHCO3 and KHCO3 solutions. For nitrates, reliable data are available only for NaNO3, KNO3, and Mg(NO3)2 solutions. Therefore, 24 single-salt solutions are used in the parameter fitting, and they are listed in Table 1 with the corresponding maximum concentration Cm max. The activity coefficient data are taken from the reference indicated in Table 1, and the density data are all taken from the book by Zaytsev and Aseyev.13 The fitted parameters at 298.15 K (25 °C), i.e., V25, u25/k, and d25 for each ion and the arithmetic mean of the reduced well range for water-ion interactions (λ25,w+ ) 0.5(λ25,w + λ25,+) or λ25,w- ) 0.5(λ25,w + λ25,-), where subscript w refers to water), are listed in Table 2. The overall ARDs of the activity coefficients and the liquid densities are 0.74% and 0.67%, respectively. The ARD for each solution is summarized in Table 1. The model represents not only 1-1 type salts (e.g., LiCl),
but also 2-1 type (e.g., MgCl2), 1-2 type (e.g., Na2SO4), and 2-2 type (e.g., MgSO4) salts. The effective radii of all ions calculated from their effective diameters are compared with the Pauling and hydrated-ion radii taken from Jin and Donohue20 in Figure 1. Since the hydration number of an ion is inversely proportional to the ion size, the Pauling and hydrated-ion radii do not have the same trend for cations. Generally, a cation is more easily hydrated than an anion. For cations, because of strong hydration, the trend of their effective radii fitted in this work is the same as that of the hydrated-ion radii. However, for anions, the trend of their effective radii fitted in this work is the same as that of Pauling radii because the hydration is weak. As shown in Figure 1, the effective radii of simple ions fitted in this work are always between the Pauling and hydrated-ion radii. The hydrated-ion radii fitted by Jin and Donohue20 are also
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Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 Table 3. ARDs for Osmotic Coefficient of Aqueous Two-Salt Solutions without Any Potassium Salt or Containing Only Potassium Salts at 298.15 K ARD, %
Figure 3. Predicted and experimental saturated vapor pressure21 for NaCl solution at 298.15 K: symbols, experimental data; curves, predicted.
depicted in Figure 1 for comparison. For some ions, such as K+, Cl-, Br-, and I-, the fitted hydrated-ion radii are less than the Pauling radii. For complex ions, such as NO3-, SO42-, and HCO3-, the trend of the effective radii is not obvious. The fitted effective radius of NO3- is a bit less than its hydrated-ion radius but much larger than its Pauling radius. As also obtained by Jin and Donohue,20 the fitted effective radius of SO42- is less than its Pauling radius. This may be due to the simplification made for these complex ions; a complex ion is modeled as a spherical segment. 3.1.2. Property Prediction. The model with the parameters obtained from activity coefficients and liquid densities is used to predict osmotic coefficients, water activities, and vapor pressures of single-salt solutions at 298.15 K. Since the reference state of water is by convention the pure water, the activity of the pure solvent is unity. In dilute aqueous electrolyte solutions, the activity and the activity coefficient of water are almost unity, so that the reported values for these properties require a larger number of significant figures. To overcome this problem and to simplify many calculations, the osmotic coefficient, which is a function of water activity coefficient and salt molality, is defined. Therefore, to show the model performance, we also apply our model to predict the osmotic coefficients. The osmotic coefficient data for most of the solutions studied in this work have been reported as reference data. As shown in Figure 2, the comparison of the calculated and experimental osmotic coefficients reveals good agreement for all single-salt solutions studied except for MgSO4. However, the predicted water activity in MgSO4 solution agrees with the experimental data, as shown in Figure 2f. This also demonstrates that the prediction of osmotic coefficient is more model-sensitive than that of water activity. Among the single-salt solutions studied in this work, only one group of saturated vapor pressure data of NaCl solution was reported.21 Figure 3 shows that the predicted vapor pressures agree with the experimental data. This is not surprising, because saturated vapor pressure is related to water activity and osmotic coefficient. 3.2 Two-Salt Solutions. 3.2.1 Solutions without Any Potassium Salt or Containing Only Potassium Salts. Without any additional parameters, ion-based SAFT2 with the parameters obtained from single-salt solutions is used to predict the osmotic coefficients for two-salt solutions without any potassium salt or containing only potassium salts at 298.15 K. The prediction is compared with the experimental data and summarized in Table 3, and the prediction agrees with experimental data except for that of the KCl + KBr solution.
system
SAFT2
this work
ref
LiCl + NaCl LiCl + CaCl2 LiCl + MgCl2 NaCl + NaBr NaCl + NaNO3 NaCl + CaCl2 CaCl2 + MgCl2 MgCl2 + Mg(NO3)2 CaCl2 + Mg(NO3)2 KCl + KBr KCl + K2SO4 KCl + KNO3 NaCl + MgCl2 NaCl + Na2SO4 Na2SO4 + MgSO4 MgSO4 + MgCl2 NaCl + MgSO4 Na2SO4 + MgCl2
0.47 0.99 0.87 0.29 0.98 0.49 0.78 1.28 2.84 4.93 1.18 1.86 1.68 1.02 2.90 2.44 1.82 1.65
0.66 0.93 0.91 0.34 1.44 0.54 0.74 1.29 3.42 4.97 1.37 1.53 0.44 1.10 2.66 2.51 1.66 1.36
a b c d e f g h h d i e j j j j j j
a Robinson and Lim.22 b Long et al.23 c Yao et al.24 d Covington et al.25 e Bezboruah et al.26 f Robinson and Bower.27 g Robinson and Bower.28 h Platford.29 i Robinson et al.30 j Platford.31
Figure 4. Predicted and experimental density for NaCl (1) + CaCl2 (2) solution at 298.15 K: symbols, experimental data;32 solid curves, predicted Table 4. krβ for K+-Na+, K+-Li+, K+-Ca2+, and K+-Mg2+ Interactions ARD, % R-β
solution
ref
kRβ
K+-Na+ K+-Li+ K+-Ca2+ K+-Mg2+
K+-Na+-ClK+-Li+-ClK+-Ca2+-ClK+-Mg2+-Cl-
a, b c d e
0.1921 0.1945 0.1259 0.1343
this work
SAFT1-RPM
with kRβ kRβ ) 1
with lij
0.25 0.63 0.46 0.48
4.77 10.8 7.57 6.30
a Covington et al.25 b Robinson.34 c Robinson and Lim.22 and Covington.35 e Padova and Saad.36
0.997 0.965
d
Robinson
The densities and vapor pressures for NaCl + CaCl2 solution at 298.15 K are also predicted. Compared to the experimental data,32,33 the ARD for the density is 0.25% while that for the vapor pressure is 1.92%. Figure 4 shows the model prediction on the density where xi* is the mole fraction of salt i on a solvent-free basis. 3.2.2. Solutions with Only One Potassium Salt. For twosalt solutions with only one potassium salt, the prediction using the same approach shows discrepancies from experimental data because of the peculiar segment energy of K+; the segment energy of K+ is much lower than that of the other ions. For such solutions, unlike for other aqueous salt solutions, we then allow short-range interactions between K+ and other cations.
Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7723 Table 5. ARDs for Osmotic Coefficients of Several Two-Salt Solutions Containing K+-Na+ solution
ARD,%
KCl + NaBr KBr + NaBr KBr + NaCl
0.44 0.48 0.60
a
Covington et al.25
b
ref a a a
Bezboruah et al.26
c
solution
ARD,%
KCl + NaNO3 KNO3 + NaCl KNO3 + NaNO3
0.63 1.82 0.63
ref b b b
solution
ARD,%
KCl + Na2SO4 K2SO4 + NaCl K2SO4 + Na2SO4
1.78 1.20 0.83
ref c c c
Robinson et al.30
Figure 5. Predicted and experimental osmotic coefficients for KCl (1) + NaBr (2) and NaNO3 (1) + KNO3 (2) solutions at 298.15 K: symbols, experimental data listed in Table 5; solid curves, predicted with kRβ; dashed curves, predicted with kRβ ) 1.
Figure 6. Predicted and experimental densities for NaCl (1) + KCl (2) and KCl (1) + NaBr (2) solutions at 298.15 K: symbols, experimental data;37,38 solid curves, predicted.
Figure 7. Predicted and experimental densities for KCl (1) + CaCl2 (2) and KCl (1) + MgCl2 (2) solutions at 298.15 K: symbols, experimental data;39,40 solid curves, predicted.
Thus, the dispersion term in eq 1 now also includes K+-other cation dispersion interactions. Note that, in this case, we assume that, although the cations have short-range interactions with K+, their long-range Coulombic interactions can still be represented by the restrictive primitive model (RPM). The mixing rule for the segment energy for K+-other cation interaction is uRβ ) uβR ) (uRuβ)1/2(1 - kRβ), where kRβ is the adjustable binary interaction parameter.
3.2.2.1. Parameter Fitting. The parameter kRβ for K+-other cation interaction is obtained from experimental osmotic coefficients. Four combinations of ion-ion interactions, i.e., K+Na+, K+-Li+, K+-Ca2+, and K+-Mg2+, are studied in this work. For K+-Na+ interactions, although the experimental data for salt solutions containing K+-Na+ with various anions are available, kRβ is fitted only to the experimental data of the KCl + NaCl solution25,34 because of the wide range of the ionic
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Table 6. Comparison of the Predicted and Experimental33 Vapor Pressure (P, KPa) for NaCl + KCl and NaBr + KBr Solutions at 298.15 K m CNaCl
Cm KCl
Pexp
Pcal
|Pexp - Pcal|
m CNaBr
m CKBr
Pexp
Pcal
|Pexp - Pcal|
0.645 0.777 0.982 1.985
0.322 0.845 1.125 1.331
3.05 2.97 2.93 2.75
3.08 3.01 2.96 2.82
0.03 0.04 0.03 0.07
0.727 0.866 1.493 1.069
0.635 0.677 0.822 2.964
3.00 2.96 2.91 2.65
3.03 3.01 2.92 2.74
0.03 0.05 0.01 0.09
Table 7. Experimental and Predicted Osmotic Coefficients (O) for NaCl + KCl + LiCl + H2O at 298.15 K Cm, mol‚kg-1
|∆φ|
φ
LiCl
NaCl
KCl
exp.41
0.9650 1.0000 1.0114 1.0321
0.9650 1.0000 1.0114 1.0321
0.9650 1.0000 1.0114 1.0321
1.0553 1.0677 1.0709 1.0802
SAFT1-RPM
this work
SAFT1-RPM
this work
1.0803 1.0885 1.0912 1.0961
1.0502 1.0578 1.0602 1.0648
0.0250 0.0208 0.0203 0.0159
0.0051 0.0099 0.0107 0.0154
strength covered and the large number of experimental data points. For the other three combinations (K+-Li+, K+-Ca2+, and K+-Mg2+), to the best of our knowledge, the only experimental data available for fitting are for chloride solutions. The results are summarized in Table 4. The average relative deviations (ARDs) for these four chloride solutions are
