4772
Ind. Eng. Chem. Res. 1996, 35, 4772-4780
A Unified and Predictive Model for Mixed-Electrolyte, Aqueous Mixed-Solvent Systems Using Parameters for Ions and Solvents Chul Soo Lee,* Sung Bin Park, and Yon Sik Shim Department of Chemical Engineering, Korea University, Seoul 136-701, Korea
A unified excess Gibbs function model is proposed for mixed-electrolyte mixed-solvent systems by superposing a short-range interaction derived from a new lattice-hole-based Helmholtz free energy on the electrostatic interaction. The short-range interaction energy is represented by the sum of electrostatic and nonelectrostatic interactions, the former being expressed by a universal function of distance parameter of ions. With two parameters for solvents and three parameters for ions, the model was found to describe various properties with average root mean square errors from 2-4%. For a system with some complex oxy anions, a counterion parameter is needed. The model was found useful for various predictions including vapor-liquid equilibria. Introduction The Debye-Hu¨ckel theory for dilute electrolyte solutions has been extended to concentrated solutions by various investigators. Frequently cited models include those of Bromley, Meissner, Pitzer, and Chen. The Pitzer model (Pitzer, 1973; Pitzer and Mayorga, 1973) gives generally excellent results for single-salt aqueous systems with three electrolyte parameters. However, for mixed-electrolyte aqueous systems additional ternary ionic parameters are required to achieve the comparable accuracy (Pitzer and Kim, 1974). This model is considered probably the best-fitting method (Rafal et al., 1993). Methods of Bromley (1973) and Meissner (Meissner and Tester, 1972; Meissner and Kusik, 1972) use one salt parameter and are extended to mixed-electrolyte aqueous systems without any additional parameters. Although often coarse, a correlation is available for calculation of Bromley coefficients from two ionic constants for each ion. The Meissner method is known to give relatively good results when extrapolated to very highly concentrated solutions. By adding an NRTL term to the Debye-Hu¨ckel term, Chen and Evans (1986) fit nonidealities of aqueous electrolyte solutions with two parameters. All these methods are not extended to mixed-solvent systems. Phase equilibria involving mixed solvents or volatile electrolytes have attracted considerable interests. In these efforts the excess Gibbs function is usually represented by the sum of a Debye-Hu¨ckel term and a short-range interaction term. For the short-range interaction, the NRTL term (Cruz and Renon, 1978; Chen et al., 1982; Mock et al., 1986), the UNIFAC term (Kikic et al., 1991; Achard et al., 1994), or the UNIQUAC term (Polka et al., 1994; Li et al., 1994; Macedo et al., 1990; Sander et al., 1986) is employed. The works of Li et al. and Polka et al. give impressive results, with four parameters between ions and two between solvents. They applied the model up to very high electrolyte concentrations and also presented extensive tables of parameters. These models are not applied to mixedelectrolyte systems. Thus, different models are applied for solute activity coefficients and for solvent activity coefficients in mixedsolvent systems. Also there appear to be few practical * Author to whom correspondence is addressed. Phone: 822-920-1515. Fax: 82-2-926-6102. E-mail: cslee@ prosys.korea.ac.kr.
S0888-5885(96)00294-1 CCC: $12.00
means to handle mixed-electrolyte, mixed-solvent systems. A further review on various models reveals that a short-range interaction term is needed in addition to an electrostatic term for vapor-liquid equilibria and that electrostatic effects are not included separately in describing short-range interactions. Since the shortrange interaction between ions is both nonelectrostatic and electrostatic, a scheme of superposing the latter on the former may be of help in developing a unified model. In the present work we seek a unified model which is applicable to mixed-solute and mixed-solvent systems based on a plausible scheme of superposing electrostatic interaction on the short-range nonelectrostatic interactions. With the minimum number of interaction parameters, we also seek the model that serves the prediction purpose. Excess Gibbs Function Model As is usually done, the excess Gibbs function is expressed as a sum of an electrostatic interaction term and a short-range interaction term. We use the classical expression of the Debye-Hu¨ckel term in the asymmetric convention for electrostatic interactions,
GE* DH ) -
[
]
2
(aK) RTV ln(1 + aK) - aK + 3 2 4πa Na
(1)
where a is the distance of the closest approach of ions, V is the volume of solvent, and K is the reciprocal of the Debye length given by
(
)
8πe2NaI K) DskT
1/2
(2)
where e is the charge of an electron, Na is Avogadro’s number, Ds is the dielectric constant of solvent, and I is the ionic strength defined by
I)
1
salt
∑ni(ν+iz+i2 + ν-iz-i2)
2Vi)1
(3)
In a multielectrolyte multisolvent system the electrostatic contributions to the activity coefficient are © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4773
obtained by De M. Cardoso and O’Connell (1987). The results are for solvents
V hj
{
by short-range interactions
}
ln γA,j ) ln
aK aK ln γDH,j ) ln(1 + aK) 3 2 2(1 + aK) 4πa Na
(4) where the partial molar volume is on the solute-free basis. The expression for solutes is
ln γ* DH,(i ) -
|z+iz-i|
(aK)3 16πa3NaI 1 + aK
θi
)
zqj 1 - ln( 2
ln γR,j )
θkτjk
zNq 2
qMri
∑
∑
θi ln(
θjτji)
}
∑θkτkj) - ∑ θ τ ∑ l lk
(15)
The solvent activity coefficient is expressed by a sum of contributions due to electrostatic and short-range interactions
(6)
(7)
Here Ni is the number of molecules or ions, θi is the surface area fraction, xi is the mole fraction of species i, and β equals 1/kT. Other quantities are defined below.
(16)
The osmotic coefficient is related to the solvent activity coefficient in a pure solvent system by the relation salt
φ ) -nsol ln xiγi/
∑k ν(knk
(17)
The different contributions due to short-range interactions to the mean ionic activity coefficient of solute i are
ln γ* A,(i )
ln γ* R,(i ) -
( )(∑ )
νi θix∞i z ln + 1ν(i θ∞x 2
∑ z 2
i
{ ∑∑
νiqi ln ν(i
∑
i
θ∞j τji
riνi qMr∞M ln ∞ ν(i qMrM (18)
θjτji
θjτij
+
∑ θτ ∑ k kj
θ∞j τij
∑ θ∞τ ∑ k kj
The residual contribution becomes identical to that of the UNIQUAC model.
βGER ) -
(14)
ln γj ) ln γDH,j + ln γA,j + ln γR,j
∑Ni ln xi + (1 - 2)∑Niqi ln rMqi z
(
(5)
We consider a mixture of solvents and ions, each of which has segment number ri, surface area parameter qi, and the nonelectrostatic interaction energy parameter ii(n). The short-range interaction term is derived from the Helmholtz function of a new lattice-hole theory (Yoo et al., 1995; Shin et al., 1995). The excess Helmholtz function may be considered equivalent to the excess Gibbs function in the high-pressure limit (Wong and Sandler, 1992). In this limit holes vanish and the excess volume becomes zero. The expression for the excess Gibbs function may be represented by a sum of the athermal contribution and the residual contribution. The athermal contribution in the symmetric convention is written as
βGEA )
{
θj qMrj z + rj 1 - ln xj 2 rMqi
}
(19)
where ν(i is the sum of ν+i and ν-i. Short-range interactions in the asymmetric conventions were obtained using the relation
ln γ* j ) ln γj - lim ln γj
(20)
xif0
The solute activity coefficient is expressed by a sum of all contributions
∑Niqi
(8)
ln γ* (j ) ln γ* DH,(j + ln γ* A,(j + ln γ* R,(j
θi ) Niqi/Nq
(9)
∑xiqi
The solute activity coefficient in the molality scale γm* is related to that of the mole fraction scale by the i relation
(10)
Nq )
qM ) rM )
∑
xiri
τij ) exp[β(ij - jj)]
(11) (12)
where ij is the total interaction energy. In lattice models ri and qi are not independent but are related by
zqi ) (z - 2)ri + 2
salt
ln γm* (i ) ln γ* (i - ln(1 +
sol
ν(ink/
nj) ∑ j)1
(22)
In the lattice model only nearest-neighbor interactions are considered. The interaction potential between ions is the sum of electrostatic and nonelectrostatic interaction energies (n) ij ) (e) ij + ij
(13)
where z is the coordination number and is usually set to 10. Taking partial molar properties of eqs 6 and 7, we have the contribution to the solvent activity coefficients
∑
k)1
(21)
for interactions between ions (23)
while the interaction involving uncharged species is nonelectrostatic
ij ) (n) ij
for interactions with uncharged species (24)
4774 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 1. Parameters for Ions and Solvents at 298.15 K rs,j (10-10 m)
rj (10-10 m)
ij (kJ‚K/mol)
3.463 4.045 2.327 1.595 1.813 1.031 6.569 6.790 7.397 6.188
3.201 2.625 2.554 2.800 3.200 3.717 10.16 7.585 5.208 3.761
5.635 6.797 5.521 4.419 2.718 3.146 2.667 4.368 6.930 9.918
H+ Li+ Na+ K+ Rb+ Cs+ Mg2+ Ca2+ Sr2+ Ba2+
rs,j (10-10 m)
rj (10-10 m)
ij (kJ‚K/mol)
1.901 3.017 2.339 1.847 0.235 728.0 2.217 6.882
2.604 0.484 0.308 0.336 0.042 0.500 0.115 0.304 2.500 2.968
4.200 1.362 1.048 0.016 0.983 16.46 6.842 10.16 2.062 0.875
NH4 ClBrIClO4OHNO3SO4water methanol +
Table 2. Parameters for Counterion Interactions OHNO3SO42-
H+
Li+
Na+
K+
Cs+
NH4+
Mg2+
Ca2+
Sr2+
Ba2+
0.0
-0.004 0.060 0.0
0.027 0.0 0.0
0.34 -0.039 0.0
-0.110 0.0 0.0
0.0 0.0
0.0 0.01
0.0 0.1
0.0
0.0
The nonelectrostatic part is the sum of all interactions other than charge-charge interactions. It includes interactions of dispersion, induction, charge-dipole, etc. The nonelectrostatic interaction between unlike species is assumed to be given by (n) (n) 1/2 (n) ij ) (ii jj ) (1 - kij)
(25)
where kij is a binary interaction parameter. The electrostatic potential between ions i and j at contact is readily found from the Debye-Hu¨ckel theory (Pitzer, 1973),
φij ) -
zizje2 aDs(1 + aK)
(26)
Since ij(e) is the electrostatic potential between ions, we need to convert this into the potential between segments of ions. There appears to be no rigorous way of doing this conversion. Also it is not clear how the solvent dielectric constant affects the short-range interaction. Therefore, we adopt an empirical approach and assume the following form for the potential between counterions:
|z+iz-j| (e) ij ) C (rs,+i + rs,-j)
for counterion interactions (27)
where C is an empirical universal constant and rs,+i or rs,-j is the radius of cations or anions. No electrostatic interaction is considered between solvent molecules or between solvents and ions. To use the present model for prediction, we need to reduce the number of binary parameters to a minimum. In principle eq 27 is also applicable to co-ion interactions. Then we need kij for co-ion interactions in eq 25. To avoid this, we separately assume
(e) ij ) -∞
for co-ion interactions
(28)
We still have binary parameters between counterions, between ion and solvent, and between solvents in mixedsolvent systems. The last one is determined independently by fitting mixed-solvent data without electrolytes. It turns out that we need to determine kij between some cation and complex oxy anions to improve accuracies. Other than these, we set all binary parameters to zero. Parameters We have ri and ii for each ionic and molecular species. We also have rs,+i or rs,-j for ionic species. Thus, each
ion has three parameters and each molecular species has two parameters. a in the Debye-Hu¨ckel term is usually assumed constant and we use the value of 4.1 × 10-10 m. In the first phase of parameter determination, data for solute activity coefficients and osmotic coefficients of 56 systems (Robinson and Stokes, 1955) were fitted to determine ri, ii, rs,+i, rs,-j, and C in eq 25. The fitted C value was 1.615 × 10-9 kJ‚m‚K/mol, and the maximum ionic strength was 6.5. In the second phase, kij for cation-anion interactions were fitted using the same data set, with ionic and solvent parameters determined in the first phase. Some systems with oxy anion showed marked improvements when the binary parameter is used. For other systems the binary parameter is set to zero. Parameters for pure species and binary parameters are listed in Tables 1 and 2. Pure parameters for methanol and for methanolwater interaction were determined from vapor-liquid equilibrium data. For nonelectrolyte systems, activity coefficients are represented by the sum of eqs 14 and 15 in the symmetric convention. The pure water and methanol parameters are also listed in Table 1, and kij for water-methanol interaction is -0.0295. Also required are dielectric constant and partial molar volume of solvent or solvent mixtures. For a water-methanol mixture volumetric data and dielectric constant are found in the literature (Sandler, 1989; Conway, 1952). Molar volume data of mixtures were fitted by a cubic function of mole fraction, which was then used for partial molar volume calculation by the usual method. The dielectric constant of a mixed solvent was represent by a linear function of weight fraction. Single-Electrolyte Aqueous Systems As noted above the parameter determination is based on data for single-electrolyte aqueous systems. Unfortunately there are indications that some data sets are not quite reliable. Pitzer and Mayorga (1973) did not report errors for acids or for hydroxides and noted that some or all of the data sets are other than isopiestic measurements. The Na2SO4 data in the Robinson and Stokes book differ from the isopiestic measurements of Filippov et al. (1987) by more than 2% at some points. Rard and Miller (1982) noted that various osmotic coefficient data for CsCl and SrCl2 differ systematically by more than 1%. Errors tend to be higher at high salt concentrations. Smoothed correlations by NBS (Hamer et al., 1972; Goldberg et al., 1981) also deviate significantly at some points from data in the Robinson and Stokes book. These probably reflect that less than 1%
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4775 Table 3. Summary of Root Mean Square Errors for Osmotic Coefficients in Single-Electrolyte Aqueous Systems HCl HBr HI HClO4 HNO3 LiCl LiBr LiI LiClO4 LiOH LiNO3 Li2SO4 NaCl NaBr NaI NaClO4 NaOH NaNO3 Na2SO4 KCl KBr KI KOH KNO3 K2SO4 CsCl CsBr CsI
max. I
σ%
6.0 1.0 3.0 6.0 3.0 6.0 6.0 3.0 4.0 4.0 6.0 6.0 6.0 4.0 3.5 6.0 6.0 6.0 6.0 4.8 5.0 4.5 6.0 3.5 2.1 6.0 5.0 3.0
2.26 0.54 0.73 3.90 0.49 3.18 4.51 1.74 1.08 1.03 0.38 0.72 1.46 0.64 0.44 3.95 4.34 2.08 2.08 0.46 0.84 2.59 4.30 0.85 1.66 1.06 0.93 1.74
max. I
σ%
1.0 1.4 5.4 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.4 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 5.4 6.0 6.0 6.0 1.2
0.28 1.73 1.61 0.73 1.84 4.05 2.94 2.62 2.19 3.66 1.61 5.20 1.44 1.97 3.26 2.85 5.01 1.47 2.17 2.85 3.37 5.73 1.96 1.64 1.73 5.46 1.86 2.20
CsOH CsNO3 Cs2SO4 NH4Cl NH4NO3 (NH4)2SO4 MgCl2 MgBr2 MgI2 Mg(ClO4)2 Mg(NO3)2 MgSO4 CaCl2 CaBr2 CaI2 Ca(ClO4)2 Ca(NO3)2 SrCl2 SrBr2 SrI2 Sr(ClO4)2 Sr(NO3)2 BaCl2 BaBr2 BaI2 Ba(ClO4)2 Ba(NO3)2 average
Table 4. Comparison of Root Mean Square Errors in Activity Coefficients by Various Models for Single-Electrolyte Aqueous Systemsa
a
system
max. molality
present
NBS
Bromley
Meissner
Pitzer
Chen
HCl KCl KOH NaOH CaCl2 Na2SO4 NaCl average
6.0 4.5 6.0 6.0 3.0 3.0 6.0
4.4 0.7 7.2 7.5 5.5 3.5 2.1 4.4
0.2 0.4 2.5 0.9 1.0 1.3 0.1 0.9
0.7 0.4 1.9 2.6 3.5 5.5 0.6 3.4
3.3 9.9 3.6 4.0 10.7 4.0 2.4 5.4
0.6 0.4 1.7 0.8 0.7 3.1 0.2 2.9
3.6 0.3 3.7 3.2 19.6 4.9 1.8 6.2
σγ %.
error can be obtained only by fitting a specific data set. Since the present scheme is a generalized one, such an accuracy is hardly expected. The average root mean square (rms) errors for osmotic coefficients and activity coefficients of the 56 systems we used in the parameter determination were 2.2 and 3.1%, respectively. We summarized rms errors in osmotic coefficients in Table 3. In Table 4 we compared errors in the activity coefficients of the present model with results of the Bromley, Meissner, Pitzer, and Chen models for the systems as compared in a book (Zemaitis et al., 1986). The comparison is based on identical data sets and shows that the Pitzer and Bromley models are better than others and that the present model is better than the Meissner and Chen models. Pitzer uses three binary parameters, and other models use one binary parameter, while the present model uses one binary parameter only for some systems with some oxy anions. For magnesium sulfate the Bromley model uses an extended form with three parameters. The table also shows about 1% error between the NBS smoothed data and the data in the Robinson and Stokes book. Mixed-Electrolyte Aqueous Systems For mixed-electrolyte aqueous systems the calculation is predictive. With parameters given in Tables 1 and 2, osmotic coefficients of aqueous mixed-electrolyte
systems are predicted. The results are summarized in Table 5 with an average rms error of 2.3%. The accuracy for single-salt systems is preserved in the extension to mixed electrolytes. In Table 6 the present model is compared with models of Pitzer and Chen for some systems. Chen and Evans reported only on 1:1 electrolyte mixtures with the exception of the Na2SO4 system. Without salt-salt parameters, these models give comparable results with the present results. With two ternary ionic parameters Pitzer and Kim reported better results than the present ones. It is seen from Table 6 that without ternary ionic parameters or binary salt parameters models of Pitzer (Pitzer and Kim, 1974) and of Chen and Evans (1986) give results similar to those of the present model. Predicted activity coefficients of solutes in mixedelectrolyte aqueous systems are summarized in Table 7. The average rms error is 3.1%. The data are from Harned and Owen (1968). They reviewed early studies on the mixed-electrolyte systems. Data of Filippov et al. (1987) for mixtures of alkali-metal sulfate were not included in the comparison since their osmotic coefficients for pure solute were consistently higher than the set we used for parameter determination. Errors for these systems are slightly higher. Also shown in the table are results of the Pitzer method without ternary ionic parameters. They are similar to the present results.
4776 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 5. Summary of Root Mean Square Errors for Osmotic Coefficients in Mixed-Electrolyte Aqueous Systems max. I
σ%
ref
6.0 5.8 4.3 6.3 6.0 4.8 4.1 5.0 4.7 6.4 6.5 4.3 5.0 4.3 4.3 5.7 3.7 3.7 6.3 5.9 5.0 5.4 6.4
2.66 2.53 2.73 1.46 1.83 1.41 1.51 1.38 2.08 1.06 1.66 1.19 1.72 1.07 0.70 1.49 0.91 1.13 2.92 3.95 8.03 2.02 3.11
a b c d e f c g f g d i a i i j j j b k a l l
NaCl-LiCl NaCl-LiCl LiCl-BaCl2 NaCl-CaCl2 CsCl-NaCl NaCl-BaCl2 CsCl-BaCl2 KCl-CaCl2 KCl-BaCl2 NaCl-KCl CaCl2-MgCl2 NaCl-NaBr CsCl-KCl NaBr-KBr KCl-KBr NaCl-NaNO3 KCl-KNO3 NaNO3-KNO3 NaNO3-LiNO3 CaCl2-Ca(NO3)2 CsCl-LiCl HClO4-NaClO4 HClO4-NaClO4
max. I
σ%
ref
3.9 4.4 6.0 4.2 3.6 4.5 5.2 5.0 3.7 5.8 4.5 6.2 6.0 4.3 2.1 2.5 3.6 3.5 6.4 6.5 6.2 6.4
1.74 0.69 2.62 0.91 1.12 2.45 1.57 4.18 0.68 2.40 1.49 3.13 4.89 2.63 2.73 0.86 0.81 2.93 2.96 9.58 2.24 4.44 2.34
k m a h m l l a j j i k k k m m m n n n o o
MgCl2-Mg(NO3)2 KCl-Na2SO4 LiCl-LiNO3 KCl-NaBr Na2SO4-K2SO4 HClO4-LiClO4 LiClO4-NaClO4 KCl-LiCl NaCl-KNO3 KCl-NaNO3 NaCl-KBr MgCl2-Ca(NO3)2 Mg(NO3)2-Ca(NO3)2 Mg(NO3)2-CaCl2 KCl-K2SO4 NaCl-Na2SO4 NaCl-K2SO4 K2SO4-Cs2SO4 Na2SO4-Cs2SO4 Li2SO4-Cs2SO4 Li2SO4-Na2SO4 Li2SO4-K2SO4 average
a Robinson and Lim, 1953. b Robinson et al., 1971. c Lindenbaum et al., 1972. d Robinson and Bower, 1966b. e Robinson, 1952. f Robinson and Bower, 1965. g Robinson, 1961. h Robinson and Bower, 1966a. i Covington et al., 1968. j Bezboruah et al., 1970. k Platford, 1971. l Rush and Johnson, 1968. m Robinson et al., 1972. n Filippov et al., 1987. o Filippov et al., 1989.
Table 6. Comparison of Osmotic Coefficients by Various Models for Mixed-Electrolyte Aqueous Systems σ% system
max. I
present
Pitzer
Chen
NaCl-LiCl NaCl-CaCl2 CsCl-NaCl NaCl-BaCl2 CsCl-BaCl2 KCl-CaCl2 KCl-BaCl2 NaCl-KCl NaCl-NaBr CsCl-KCl NaBr-KBr KCl-KBr NaCl-NaNO3 KCl-KNO3 NaNO3-KNO3 NaNO3-LiNO3 CaCl2-Ca(NO3)2 CsCl-LiCl HClO4-NaClO4 MgCl2-Mg(NO3)2 Na2SO4-K2SO4 HClO4-LiClO4 LiClO4-NaClO4 KCl-LiCl NaCl-Na2SO4 average
6.0 6.3 6.0 4.8 4.1 5.0 4.7 6.4 4.3 5.0 4.3 4.3 5.7 3.7 3.7 6.3 5.9 5.0 5.4 3.9 3.6 4.5 5.2 5.0 2.5
2.66 1.46 1.83 1.41 1.51 1.38 2.08 1.06 1.19 1.72 1.07 0.70 1.49 0.91 1.13 2.92 3.95 8.03 2.02 1.74 1.12 2.45 1.57 4.18 0.86 2.02
0.2 0.4 2.7 0.1 2.4 2.5 1.8 1.4 0.1 0.3 0.9 0.2 0.7 0.3 0.8 1.4 1.4 10.0 2.5 0.8 1.1 0.6 0.3 4.5 0.4 1.5
1.7
1.9 0.9 0.5 0.3 0.6 1.1 1.4 1.8
2.1
1.2
As a further test of activity calculations, predicted solubilities in mixed-salt systems in water are compared in Table 8. The comparisons for the NaCl-Na2SO4 system and for the MgSO4-CaSO4 system are shown in Figures 1 and 2. These aqueous systems are extensively discussed by Zemaitis et al. (1986). They compared various models for extended ranges of solute concentration. Solubility is sensitive to activity coefficients of electrolyte. The solubility is also very sensitive to solvent activity coefficients when solids form hydrates as in Na2SO4‚10H2O since the activities of solvent are raised to large powers in the equilibrium relation. The discontinuities in Figure 1 indicate the invariant points for three-phase equilibrium of solute(1)-solute(2)-solution. Thus, the upper point is for
Table 7. Summary of Root Mean Square Errors for Activity Coefficients of Hydrogen Chloride and Hydrogen Bromide in Mixed-Electrolyte Aqueous Systems σ% HCl-KCl HCl-NaCl HCl-LiCl HCl-CsCl HCl-BaCl2 HCl-SrCl2 HBr-KBr HBr-NaBr HBr-LiBr average
max. I
present
Pitzer
3.5 3.0 4.0 3.0 3.6 6.0 3.0 3.0 3.0
1.76 1.92 3.14 8.16 2.64 2.21 1.14 4.08 2.71 3.08
1.4 4.0 2.3 8.2 1.1 3.4 3.0 2.8 2.7 3.2
NaCl-Na2SO4-solution and the lower point for Na2SO4-Na2SO4‚10H2O-solution. These points correspond to states of the minimum Gibbs free energy when excess amounts of both solutes are present. Solubility product values of Na2SO4, Na2SO4‚10H2O, and CaSO4‚2H2O were fitted. Although solubility product values for calcium sulfate are available in the literature, they are widely different. The present model is seen to be similar to the Pitzer model. The results for calcium sulfate systems shown in Figure 2 are not very good, but other models including the Pitzer model also give poor results for these systems. It is noted that the calculated solubilities for CaSO4‚2H2O show a weak maximum, as do experimental data. In a electrolyte(1)-electrolyte(2)-solvent system with a common ion, the solubility of electrolyte(1) tends to decease by the presence of electrolyte(2) due to the common ion effect. The solubility is also affected by electrolyte(2) due to its effect on the activity coefficient of electrolyte(1). The combined effect generally results in an initial decrease of the solubility, sometimes followed by a gradual increase as the concentration of electrolyte(2) increases. The increasing tendency may further be followed by the solubility decrease, and a solubility maximum can be observed as in the MgSO4CaSO4‚2H2O system.
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4777 Table 8. Summary of Root Mean Square Errors in Solubilities of Comp(2) in the Presence of Comp(1) for Water-Methanol Mixed-Solvent Systemsc σ% Comp(1)
Comp(2)
methanol wt. fract.
max. I
present
Pitzer
ref
NaCl HCl HCl NaOH NaCl CaCl2 Na2SO4 NaCl NaCl NaCl NaCl NaCl average
KCl NaCl KCl Na2SO4‚10H2O CaSO4‚2H2O CaSO4‚2H2O CaSO4‚2H2O KCl KCl KCl KCl KCl
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5
6.2 6.4 6.1 6.2 3.1 3.2 5.4 5.6 4.1 3.5 2.7 2.5
3.9 14.7 4.0 1.0 19.2 28.1 13.4 5.5 5.1 8.5 3.5 15.9 5.9
3.4 4.0 7.3 7.1
a a a a a a a b b b b b
5.5
a
Link and Seidell, 1965. b Shim and Lee, 1991. c Solubility Product Values: NaCl (aq, 37.66), KCl (aq, 7.81), KCl (10% MeOH, 4.05), KCl (20%, MeOH, 1.89), KCl (30%, MeOH, 0.884), KCl (40% MeOH, 0.438), KCl (50% MeOH, 0.084), Na2SO4‚10H2O (aq, 0.052), Na2SO4 (aq, 0.28), CaSO4‚2H2O (aq, 3.9 × 10-5).
Figure 1. Comparison of solubilities for NaCl, Na2SO4, and Na2SO4‚10H2O in the H2O-Na2SO4-NaCl system by various methods at 25 °C.
Mixed-Solvent Systems For mixed-solvent systems two pure parameters for each solvent plus one interaction parameter between solvents are needed. These are determined from solutefree solvent mixture data (Ciparis, 1966). The interaction energy between an ion and a nonaqueous solvent is determined by eq 25 with kij ) 0. Thus, we do not need to determine this parameter separately. Fitted methanol parameters are listed in Table 1, and kij ) -0.0295 for water-methanol interaction. rms fitting errors for a water-methanol mixture were 0.05 kPa and 0.003 in the total pressure and vapor mole fractions, respectively. The activity coefficient of a solvent component in a solute-free system is given by the sum of eqs 14 and 15. With these parameters activity coefficients of electrolytes in a mixed-solvent system are predicted. The average rms error is 3.0%, and the error for each system is given in Table 9. The present result is better than Rastogi and Tassios’ (1987) value of 6.1% for NaCl-water-methanol systems. No other calculation results are known to the authors. The activity calculation appears acceptable even in pure methanol.
Figure 2. Comparison of solubilities for CaSO4‚2H2O in the H2OMgSO4-CaSO4 system by various methods at 25 °C. Table 9. Summary of Root Mean Square Errors in Solute Activity Coefficients for Water-Methanol Mixed-Solvent Systems NaCl NaCl KCl KCl CsCl NaCl-KCl average d
max. methanol wt. fract.
max. I
σ%
ref
0.9 0.9 0.8 1.0 0.8 0.4
0.2 2.4 0.2 3.6 0.2 1.0
2.58 2.59 1.18 4.97 2.91 4.01 4.10
a b a c a d
a Feakins and Voice, 1972. b Lanier, 1965. c Popovich, 1982. Shim et al., 1991.
Although not shown here, a preliminary calculation for aqueous ethanol solvent systems indicates that the prediction is indeed possible with comparable accuracy. Predicted solubilities of mixed salts in watermethanol mixed-solvent systems are also summarized in Table 8. Solubility product values for NaCl and KCl are from Feakins and Voice (1972). They are also given in Table 8. No other methods are known to the authors for predictions in such systems. When the solubility is low compared to concentrations of the other component present, the error is approximately proportional to the square of errors in activity coefficients.
4778 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 10. Summary of Root Mean Square Errors for Vapor-Liquid Equilibria of Solvents in Electrolyte Mixed-Solvent Systems methanol wt. fract.
max. I
σp (kPa)
σy
ref
0.242 0.430 0.612 0.806 0.976 1.000 0.236 0.423 0.640 0.806 0.941 1.000
1.0 1.0 1.0 1.0 1.0 5.0 4.0 5.7 4.0 2.8 1.9 1.0
0.006 0.08 0.45 0.68 0.29 0.006 0.60 0.65 0.07 0.36 0.20 0.05 0.29
0.009 0.006 0.006 0.002 0.003 1.e-5 0.028 0.049 0.012 0.007 0.002 1.e-5 0.010
a a a a a a a a a a a a
LiCl LiCl LiCl LiCl LiCl LiCl NaBr NaBr NaBr NaBr NaBr NaBr average
Table 11. Summary of Results for Predictions NH4ClO4 RbCl RbBr RbI RbNO3 Rb2(SO4) average a
max. I
σφ %
σγ %
kij
remark
ref
2.1 5.0 5.0 5.0 4.5 5.4
6.27 1.56 2.56 4.06 1.99 (21.6) 2.46 (8.98) 1.8
9.23 1.56 5.84 8.04 5.23 (32.1) 3.81 (17.3) 5.64
0 0 0 0 -0.16 -0.06
predicted fitted predicted predicted fitted (predicted) fitted (predicted)
a b b b b b
Esval and Tyree, 1962. b Robinson and Stokes, 1955.
When vapor-liquid equilibrium is predicted for electrolyte mixed-solvent systems, the average rms errors in the total pressure and mole fraction are 0.3 kPa and 0.01, respectively, as given in Table 10. The calculation is predictive in the sense that all properties are calculated from parameters determined for single-electrolyte aqueous systems and for solute-free mixed-solvent systems. As in aqueous systems we have set all kij’s between ions and methanol to zero. The total pressure in vapor-liquid equilibria appears more sensitive to data sets used for the parameter determination of solute-free mixtures. Parameters determined from other sets of data in Gmehling et al. (1977) for a methanol-water mixture would give a somewhat larger error in the total pressure. Mock et al. (1986) give average errors of 0.08 kPa in pressure and 0.008 in vapor mole fraction for LiCl and NaBr systems. Their results are better than the present results, but they fitted each system with six parameters for ion-solvent interactions. Although error is not presented for each system, the overall results of Polka et al. (1994) are reported to be somewhat better than that for isothermal systems studied by Mock et al. (1986). Mock et al. were concerned only with vapor-liquid equilibria. Other models are proposed solely for vapor-liquid equilibria of solvents in mixed-solvent electrolyte systems. Mock et al. (1986) analyzed existing experimental VLE data sets with salts and concluded that the standard errors for total pressure and vapor mole fraction measurements are 0.5 kPa and 0.005, respectively. The present errors are comparable to these values. Predictions The present model allows two levels of prediction. In the first level, properties for systems of pure or mixed solutes with any combination of cation and anion whose parameters are given in Tables 1 and 2 are calculated. Thus, mixture properties calculation does not require any further information. As discussed in two preceding sections, properties of mixed-electrolyte and/or mixedsolvent systems are predicted. It is noted that binary parameters in Table 2 are fitted values for single
electrolyte-single solvent systems. The second level of prediction involves properties of systems for which not all parameters are known. Three examples are considered. In the first example osmotic coefficients for an aqueous NH4ClO4 system (Esval and Tyree, 1962) are predicted to give an rms error of 6.2% using parameters in Table 1. In the next example, we predict properties of RbBr, RbI, RbNO3, Rb2SO4, and RbCl. Data for these systems are given in the Robinson and Stokes book, and we deliberately did not include these sets in parameter fitting to test the predictability. Since we do not have Rb parameters, we need to determine ii, ri, and rs,i for Rb from the chloride data, for example. Looking up Table 2, we expect that binary ionic parameters may be needed for the sulfate and the nitrate. The results summarized in Table 11 show that the predictions are reasonably good for halides but that nitrates and sulfates need binary parameters. Further Discussions The Debye-Hu¨ckel term is widely used to represent excess Gibbs function models for electrolyte systems. The present short-range interaction term was derived from the author’s previous works (Shin et al., 1995) in which the equation of state form is applied successfully. Although the UNIQUAC model is similar in nature, no equation of state has been studied based on this model. We use less parameters compared with those in UNIQUAC. We hope to develop an equation of state model for electrolyte solutions based on this approach. The present model is unique in that the effect of electrostatic interactions on the short-range order is conceptually incorporated in a generalized method. It results in a universal electrostatic term in the shortrange interactions so that properties of electrolyte mixtures are, in principle, calculated from ionic properties. No binary parameters are needed except for the interactions between some cations and some oxy anions. We have shown that all mixture properties including those of mixed-solvent systems follow from parameters of single-electrolyte aqueous solutions.
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4779
The present results are surprisingly good if we consider the underlying assumptions of the present model. One assumption is that all electrolytes are fully dissociated. Sulfates of both 2:1 and 2:2 types are only partially dissociated, the latter being much less dissociated (Robinson and Stokes, 1955). Sulfates are also known to form complexes (Zemaitis et al., 1986). The present model gives unreliable results for cadmium salts, whereas cobalt salts, for example, are adequately represented. Cadmium is known to form a variety of complex halide ions. The other assumption is that properties of ions are independent of solvents. Since cations are solvated, cation parameters are dependent on solvents, in principle. In aqueous mixed-solvent systems, however, hydrated cations usually dominate (Burgess, 1978). Then a single set of ionic parameters can be used regardless of solvent compositions. Although an increased error is expected as water molecules become insufficient for hydration, the model does not appear to give excessive errors even for pure methanol electrolyte systems. The present work is applicable at isothermal conditions. It gives good results when the total ionic strength is less than 6. The present method may be applied to isobaric systems when the parameters are made temperature dependent. Conclusion A unified excess Gibbs function model for mixedelectrolyte mixed-solvent systems was proposed. In the model a short-range interaction term derived from a new lattice-hole Helmholtz free energy was superposed on the classical Debye-Hu¨ckel term. A scheme for the superposition of electrostatic and nonelectrostatic interactions on the short-range interaction was proposed. The model can be used with two parameters for each solvent and with three parameters for each ionic species to predict properties of single or mixed electrolytes. For interactions between cations and some complex oxy anions, binary ionic parameters are required. The average rms errors of various properties range from 2 to 4% for the maximum ionic strength 6. The average rms errors in vapor-liquid equilibrium of watermethanol electrolyte mixtures were about 0.3 kPa in the total pressure and 0.01 in vapor mole fraction. Acknowledgment C.S.L. is grateful to Korea University for granting a sabbatical year. He is also grateful to Korea Atomic Energy Research Institute and to Korea Science and Engineering Foundation (94-0562-06-01-3) for funding related researches. Nomenclature a ) distance of the closest approach between ions k ) Boltzmann constant q ) surface area parameter r ) segment number or size parameter rs ) saturated cavity radius of ions z ) coordination number Ds ) dielectric constant of solvent G ) Gibbs function I ) ionic strength K ) reciprocal of the Debye length Na ) Avogadro’s number Nq ) surface area of the system
R ) gas law constant T ) absolute temperature V ) solvent volume V h i ) partial molar volume of component i in the solvent mixture Greek Letters β ) 1/kT γ ) activity coefficient ) absolute value of the nearest-neighbor interaction potential parameter θ ) surface area fraction φ ) osmotic coefficient ν ) stoichiometric number σ ) rms error τ ) nonrandomness factor Superscripts (e) ) electrostatic contribution m ) molality scale (n) ) nonelectrostatic contribution E ) excess properties * ) infinite dilution reference ∞ ) infinite dilution state Subscripts sol ) solvent i, j, k ) molecular or ionic species ij ) interaction between species i and j (i ) electrolyte i +i ) cation of electrolyte i or cation i -i ) anion of electrolyte i or anion i A ) athermal contribution DH ) Debye-Hu¨ckel contribution M ) mixture R ) residual contribution
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Received for review May 24, 1996 Revised manuscript received September 13, 1996 Accepted October 7, 1996X IE960294Q
X Abstract published in Advance ACS Abstracts, November 15, 1996.
