+
+
Ind. Eng. Chem. Res. 1996, 35, 1777-1784
1777
Prediction of Activity Coefficients of Electrolytes in Aqueous Solutions at High Temperatures Xiaohua Lu, Luzheng Zhang, Yanru Wang, and Jun Shi Nanjing University of Chemical Technology, Nanjing 210009, People’s Republic of China
G. Maurer* Lehrstuhl fu¨ r Technische Thermodynamik, Universita¨ t Kaiserslautern, D-67653 Kaiserslautern, Germany
A recently published model for the excess Gibbs energy of aqueous solutions containing mixed electrolytes (Lu and Maurer, 1993) is extended from 298 K to temperatures up to 573 K. The extension is achieved by introducing a universal relation for the influence of temperature on some model parameters. Although parameters are still ion specific, the influence of temperature on those parameters is universal. No additional parameters are required for describing aqueous solutions of mixed electrolytes. The model accurately predicts activity coefficients at high temperatures and at concentrations up to the solubility limit in electrolyte aqueous solutions containing, for example, K+, Na+, NH4+, SO42-, Cl- and NO3-. Introduction Phase equilibria in electrolyte solutions are important in industrial chemistry and related fields. Methods describing phase equilibria usually depart from an expression for the excess Gibbs energy which has been designed and tested for aqueous solutions of single electrolytes at around room temperature. An extension to mixed electrolytes and higher temperatures is often unreliable. Such extensions are required for the design of equipment for, for example, waste water treatment, sea water desalination, manufacturing of inorganic chemicals, and hydrometallurgical processes. Recently, Lu and Maurer (1993) presented a new model for electrolyte solutions. It combines physical interactions with solvation equilibria, i.e. chemical equilibria. The model parameters were determined from experimental results for single electrolyte aqueous solutions at 298.15 K, but it has been applied successfully without any information on multicomponent systems to predict activities in aqueous solutions of two strong electrolytes. Here, that model is extended to higher temperatures. The influence of temperatures on the model parameters is expressed by empirical equations requiring no ion specific parameters, i.e. although the model parameters are specific for each ion, the temperature dependence of the model parameters are universal. The model is tested by comparing predicted to measured activities and solubilities of salts in concentrated aqueous solutions of mixed electrolytes up to 573 K. Model Description The model is to describe the excess Gibbs energy and related properties like, for example, the osmotic coefficient φ and mean activity coefficients γ of dissolved electrolytes in aqueous solution. It is assumed that dissolving strong electrolytes in water results in a mixture of water molecules, unsolvated and solvated ions. Solvation equilibria are used to calculate the true concentrations of solvated and unsolvated ions from the overall concentrations of the dissolved electrolytes. Physical interactions between all species are taken into account by combining the Debye-Hu¨ckel law with the UNIQUAC model (Abrams and Prausnitz, 1975). * Author to whom correspondence should be addressed.
Solvation Equilibria. When an electrolyte MνcXνa is dissolved in water, it completely dissociates to νc cations M (charge number Zc) and νa anions X (charge number Za).
MνcXνa f νcMZc + νaMAa
(1)
Ions are solvated according to
MZc + hcH2O ) MZc(H2O)hc
XZa + haH2O ) XZa(H2O)ha (2)
Superscripts Zc and Za designate the charge number of cations and anions, respectively. Solvated cations are designated by hc (hc ≡ MZc(H2O)hc) and solvated anions by ha (ha ≡ XZa(H2O)ha) where hc and ha are the number of water molecules in a solvated cation or anion respectively. Thus dissolving a strong electrolyte in water gives a mixture of water molecules as solvated and unsolvated ions. The “true” composition of the aqueous mixture is determined by chemical equilibria for these solvation reactions expressed by chemical equilibrium constants Kc and Ka:
Kc )
ahc acahwc
)
zhc
γ*hc
zczhwc
γ*cγhwc
Ka )
aha aaahwa
)
zhz
γ* ha
zazhwa
ha γ* aγw
(3) where ai, zi, and γi are the activity, true mole fraction, and activity coefficient of species i, respectively. The activity of water, aw, is normalized according to Raoult’s law while the activity ak of any dissolved species k (unsolvated and solvated ion) is normalized according to Henry’s law on mole fraction scale. Activity is expressed through “true” mole fraction z and activity coefficient (γw for water and γi* for solute species i, respectively) resulting in
aw ) zwγw
(4)
ai ) ziγ*i
(5)
and
Superscript * is used to distinguish between the different normalizations. Activity coefficients are cal-
+
+
1778 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
culated from an expression for the excess Gibbs energy GE of the aqueous mixture. Excess Gibbs Energy and Derived Properties. Two contributions are being considered to account for nonideal mixing. The Debye-Hu¨ckel expression is adopted to account for long range electrostatic interactions between charged species (solvated and unsolvated ions), whereas short range interactions are accounted for by the UNIQUAC equation, which is usually only used in nonelectrolyte solutions: E GE ) GEDH + GUNIQUAC
(6)
For a charged specie k (i.e. unsolvated or solvated ion), the activity coefficient γk* therefore is
ln γ* k ) ln γ* k,DH + ln γ* k,UNIQUAC
(7)
The Debye-Hu¨ckel contribution to the activity coefficient of a species k, γ*k,DH, is
ln γ* k,DH )
-Z2kAxIm
1 2
∑i
(9)
where mi is the molality of species i. dk is a size parameter of species k, and A and B are so-called Debye-Hu¨ckel constants (Robinson and Stokes, 1959). The short range interaction contribution γ*k,UNIQUAC is the ratio of two activity coefficient of species k both calculated from UNIQUAC: ∞ γ* k,UNIQUAC ) γk,UNIQUAC/γk,UNIQUAC
(10)
γk,UNIQUAC is the activity coefficient of species k normalized according to Raoult’s law and calculated from UNIQUAC:
[
]
E ∂GUNIQUAC ∂nk
T,pni*k
is the corresponding, but limiting, activity coefficient of species k in pure water.
lim zkf0;zwf1
γk,UNIQUAC
(12)
The activity coefficient of water is similarly a sum of contributions resulting from both parts contributing to the excess Gibbs energy
[ (
ln γw ) ln γw,DH + ln γw,UNIQUAC Mw
2(I+J)
A
) 1000 B3I
m
∑k
mkZ2k d3k
2 ln(1 + BdkxIm) -
1 + BdkxIm 1
hk
ln Kk
1.840 1.840 0.810 0.530 0.025 0.110 5.050 4.120 5.600 1.030 1.547 1.835 2.215 2.067 0.019 0.074
4.367 4.643 5.281 10.00 16.00 7.630 4.680 5.720 4.260 16.50 5.708 8.123 4.483 7.111 7.960 14.80
(T/K ) 298.15), cf. Table 1) were extended to higher temperatures by the empirical relation:
298.15 ln Kk (T/K ) 298.15) T/K
)]
+ 1 + BdkxIm ln γw,UNIQUAC (13)
where Mw is the molecular weight of water and mk is the molality of species k. Calculation of activity coefficients requires the following information: (A) Chemical equilibrium constants Kk(k ) a,c), cf. eq 3. Chemical equilibrium constants reported by Lu and Maurer (1993) for 298.15 K (i.e. Kk
(14)
(B) Debye-Hu¨ckel constants A and B for water. The correlation of Chen et al. (1982) was used to calculate Debye-Hu¨ckel constant A at temperatures from 273 to 573 K. Property B is related theoretically to property A by B(T)/B (T/K ) 298.15) ) [A(T)/A (T/K ) 298.15)]1/3 or B ) 0.31163A1/3. (C) Ion size parameters dk were calculated using the empirical equations proposed by Lu and Maurer (1993). No distinction is made between solvated and unsolvated ions:
dhci ) dci ) rhci +
λc ,a rha ∑ j)1 i
j
( ) ( ) m j aj
J
j
dhaj ) daj ) rhaj +
∑ i)1
λci,ajrhci
baj
J
∑ mj a
k)1
I
(11)
∞ γk,UNIQUAC
∞ γk,UNIQUAC )
rk (Å) 0.208 0.348 0.080 2.520 2.950 2.750 0.150 2.530 3.470 0.020 1.721 2.675 4.113 0.009 1.610 0.010
ln Kk(T) )
miZ2i
RT ln γk,UNIQUAC )
ion k H+ Li+ Na+ K+ Cs+ NH4+ Ca2+ Mg2+ Sr2+ Ba2+ ClBrIOHNO3SO42-
(8)
1 + BdkxIm
Im is the ionic strength on molality scale:
Im )
Table 1. Pure Component Parameters of the Model Proposed by Lu and Maurer (1993) at 298.15 K
(15)
k
m j ci
bci
I
∑ mj c
k)1
k
where rhci and rhaj are size parameters for solvated cation hci and solvated anion haj which are approximated by
4 4 π(r )3 ) υj Whk + π(rk)3 3 hk 3
(16)
where rk is the radius of ionic species k, hk is the number of water molecules in the solvated ion k, and υj W is the volume occupied by such a water molecule (υ j W ) 2.9910 × 10-29 m3). λci,aj ) λaj,ci is a binary size correction parameter and m j ci and m j aj are the overall stoichiometric molalities of cations and anions. Ion size parameters rk and binary size correction parameter λci,aj ) λaj,ci were taken from Lu and Maurer (1993). Parameter b is to correct for effects observed with some special ions. Generally bci ) baj ) 1.0 except for ci ) H+ and aj ) OH-: bH+ ) 0.5 and bOH- ) 1.5. In the absence of theoretical guidelines, we assumed that the influence of temperature on solvation number hk can be described by
hk(T)/hk (T/K ) 298.15) ) 1 + C1(T/K - 298.15) + C2((T/K)2 - 298.152) (17) where C1 and C2 are universal parameters (i.e. they do
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1779
expressed through the properties mentioned before (see for example Lu and Maurer (1993)):
(νc + νa) ln γ((m) ) νc ln ahc + νa ln aha j νc cm j νaa) (νchc + νaha) ln aw - ln(m (νc + νa) ln
Figure 1. The relationship h(T)/h (T/K ) 298.15) for some strong electrolytes in aqueous solutions.
not depend on species k). They were determined by a least square fit to experimental results for the mean ionic activity coefficient and the osmotic coefficient of 13 single electrolytes in aqueous solutions at temperatures between 323.15 and 573.15 K resulting in C1 ) 0.003 and C2 ) -7.0 × 10-6. Numbers for solvation parameter hk at 298.15 K were again taken from Lu and Maurer (1993). The temperature dependence of solvation parameter hk is shown in Figure 1. Points shown in that figure represent results from an isothermal fit of hk(T)/hk (T/K ) 298.15) to data sets for single aqueous electrolyte systems. (D) All UNIQUAC parameters (size, surface, and interaction parameters) were taken from Lu and Maurer (1993). Extending the model from 298 K to higher temperatures is therefore achieved by taking into account the influence of temperature on the following. (i) The equilibrium constants Kk for the solvation of ionic species k. As it was to be expected, that equilibrium constant decreases with increasing temperature. For example, raising the temperature from 298 to 400 or 500 K reduces ln Kk by about 25 or 40%, respectively. (ii) Both Debye-Hu¨ckel constants A and B and (iii) the number of water molecules in the solvation shell of a solvated ion. As it was to be expected, increasing the temperature reduces the number of water molecules in a solvated species. For example, increasing the temperature from 298 to 400 or 500 K is accompanied by a decrease of the number of water molecules by about 20 or 48%, respectively. Comparison between Experimental and Calculated Data In electrolyte solutions it is common practice to express the deviation from ideal mixing behavior through the osmotic coefficient φ:
-1000 φ)
I+J
Mw
ln(γwzw)
(18)
∑ mj k
k)1
or the mean ionic activity coefficient of a strong electrolyte MνcXνa, on molality scale γ((m), which can be
Mw (19) 1000
Aqueous Solutions of Single Electrolytes at High Temperatures. Calculated results for the mean ionic activity coefficient and the osmotic coefficient of 13 single electrolytes in aqueous solutions at temperatures between 323.15 and 573.15 K are compared with experimental results in Table 2. Calculated results are from the present work as well as from the compilation of Zemaitis et al. (1986) and from Pabalan and Pitzer (1991). In an extensive investigation on the thermodynamics of aqueous solutions of strong electrolytes, Zemaitis et al. applied the methods of Bromley, Meissner, Pitzer, and Chen to describe mean ionic activity coefficients of aqueous solutions of single strong electrolytes: HCl at 323.15 K, KCl at 353.15 K, KOH at 353.15 K, NaCl at 373.15 and 573.15 K, NaOH at 308.15 K, CaCl2 at 382 and 475 K, Na2SO4 at 353.15 K, and MgSO4 also at 353.15 K. In order to enable a direct comparison with the tabulated results by Zemaitis et al., the new model was used to calculate mean ionic activity coefficients for the same systems. A summarized comparison is shown in Table 3. As it was to be expected, the best agreement between calculated activity coefficients and the compilation of Zemaitis et al. is achieved by Pitzer’s model with parameters reported by Pitzer and Pabalan (1991). The average standard deviation between calculated and compiled data for the electrolytes shown in Table 3 is 0.012 for that model, while it is 0.064 for the present study. All other methods, including the often applied short form of Pitzer’s method recommended by Zemaitis et al., result in average standard deviations of more than 0.168. Zemaitis et al. used a special simplification of Pitzer’s method as they combined interaction parameters determined for 298 K with temperature dependent Debye-Hu¨ckel parameters to calculate the thermodynamic properties of aqueous electrolyte solutions. Due to its simplicity that model has gained much interest in industrial engineering calculations. The most current state of Pitzer’s model (see for example Pabalan and Pitzer, 1991) requires temperature dependent interaction parameters. However, those temperature dependent parameters have to be determined for each single aqueous electrolyte system, and currently that information is only available for a limited number of electrolytes. Therefore in applied science, especially in chemical engineering, there is still an interest in simpler models. The unavoidable reduction in accuracy is often accepted for the sake of getting reliable estimates also when no high-temperature experimental data are available. The simplification of Pitzer’s method by Zemaitis et al. (1986) is included in the comparisons of the present work as it does not require experimental information on aqueous electrolyte solutions at higher temperatures. Additionally, some selected comparisons are also shown in Figures 2-4. Figure 2 presents calculated results for the mean ionic activity coefficient of CaCl2 in aqueous solutions at 382 and 475 K as reported by Zemaitis et al. (1986) for their modification of Pitzer’s model as well as from the present study in comparison
+
+
1780 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 Table 2. Root Mean Square Deviations of Mean Ionic Activity Coefficients and Standard Deviations of Osmotic Coefficient of Single-Electrolyte Solutions Predicted from the Model Compared with Experimental Data at Elevated Temperatures electrolyte
T/K
σγ(a
σlnγ(b
HCl LiCl LiCl LiCl LiNO3 Li2SO4 Li2SO4 Li2SO4 Li2SO4 NaCl NaCl NaOH Na2SO4 NH4NO3 KCl KOH K2SO4 K2SO4 K2SO4 K2SO4 CaCl2 CaCl2 MgSO4
323.15 373.15 413.15 473.15 373.15 348.15 398.15 448.15 498.15 373.15 573.15 308.15 353.15 363.15 353.15 353.15 348.15 398.15 448.15 498.15 382.00 475.00 353.15
0.0148 1.013 0.0058 0.0134
0.017 0.096 0.008 0.023
0.0281 0.0380 0.0303 0.0160 0.0373 0.0343 0.0138 0.0259
0.125 0.228 0.274 0.249 0.052 0.080 0.019 0.112
0.0066 1.0432 0.5149 0.0103 0.0112 0.0103 0.0212 0.0224 0.0033
0.011 0.178 0.021 0.044 0.081 0.114 0.037 0.077 0.063
SDφc 0.085 0.027 0.043 0.033 0.077 0.148 0.214 0.281
0.078 0.020 0.080 0.148 0.223
Imax
property
reference
2.0 18.0 3.5 3.5 25.0 7.5 9.0 7.5 7.5 6.0 6.0 4.0 4.8 23.5 4.0 17.0 6.0 7.5 7.5 7.5 10.5 10.5 8.0
ln γ( ln γ( and φ ln γ( and φ ln γ( and φ φ ln γ( and φ ln γ( and φ ln γ( and φ ln γ( and φ ln γ( ln γ( ln γ( ln γ( φ ln γ( ln γ( ln γ( and φ ln γ( and φ ln γ( and φ ln γ( and φ ln γ( ln γ( ln γ(
Harned and Owen, 1958 Gibbard et al., 1973 Holmes et al., 1981 Holmes et al., 1981 Sacchetto et al., 1981 Holmes et al., 1986 Holmes et al., 1986 Holmes et al., 1986 Holmes et al., 1986 Silvester and Pitzer, 1976 Silvester and Pitzer, 1976 Harned and Owen, 1958 Snipes et al., 1975 Sacchetto et al., 1981 Snipes et al., 1975 Harned and Owen, 1958 Holmes et al., 1986 Holmes et al., 1986 Holmes et al., 1986 Holmes et al., 1986 Holmes et al., 1978 Holmes et al., 1978 Snipes et al., 1975
a
σγ( ) b
x
SDφ )
N
∑(γ N
σln γ( ) c
1
(,cal
- γ(,exp)2i
i)1
x
N
1
∑(ln γ N
x
(,cal
- ln γ(,exp)2i
i)1
N
1
∑(φ N-1
cal
- φexp)2i
i)1
Table 3. Comparison of Standard Deviationa of ln γ((m) for Aqueous Electrolyte Solutions at Elevated Temperatures with Data Compiled by Zemaitis et al., 1986b electrolyte HCl KCl KOH NaCl NaCl NaOH MgSO4 Na2SO4 CaCl2 CaCl2 avg SD
T/K
N
323.15 353.15 353.15 373.15 537.15 308.15 353.15 353.15 382.00 475.00
15 19 13 38 38 11 13 13 42 42
Bromley (1973)
Meissner (1972)
Zemaitis et al. (1986)
Pabalan+Pitzer (1991)
Chen et al. (1982)
this work
0.021 0.053 0.117 0.051 0.842 0.034 0.795 0.267 0.317 1.050 0.355
0.009 0.017 0.101 0.032 0.671 0.035 0.324 0.052 0.391 1.018 0.265
0.014 0.079 0.898 0.056 0.066 0.014 0.061 0.156 0.109 0.232 0.168
0.003 0.003
0.021 0.058 0.321 0.031 0.231 0.034 0.070 0.124 0.365 0.627 0.188
0.018 0.011 0.185 0.053 0.081 0.020 0.068 0.118 0.037 0.037 0.064
0.002 0.032 0.014 0.023 0.005 0.018 0.005 0.012
a
SDln γ( ) b
x
1
N
∑(ln γ N-1 i)1
(,cal
- ln γ(,exp)2i )
x
N
σln γ(
N-1
All comparisons with literature models, besides that with Pabalan and Pitzer, are from Zemaitis et al., 1986.
to data compiled by Zemaitis et al. from experimental work. The results of such compilations of experimental data are called “compiled” data throughout the present work. In Figure 3 mean ionic activity coefficients of LiCl and K2SO4 calculated with the new model at temperatures between 398 and 523 K are compared to compiled data. Figure 4 shows a similar comparison for the
osmotic coefficients of concentrated aqueous solutions of LiCl, LiNO3, and NH4NO3 at around 373 K. Solubilities of Salts in Mixed Electrolyte Solutions at High Temperatures. For many mixed electrolyte aqueous solutions experimental results neither for the osmotic coefficients nor for activity coefficients are available at elevated temperatures. The solubility
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1781
Figure 2. Mean ionic activity coefficient of CaCl2 in water at 382 and 475 K. Figure 4. Osmotic coefficients of some single aqueous electrolyte solutions at elevated temperatures. Table 4. Temperature Coefficients of Solubility Product Kspa of Some Salts for the Precipitation from Aqueous Solutions Based on the Model Proposed by Lu and Maurer (1993) and Determined from Single-Electrolyte Solubility in Pure Water As Reported by Linke and Seidell (1965) and Silcock (1979) salt
T/K
U1
U2
U3
U4
NaCl KCl NH4Cl NaK3(SO4)2 K2SO4 Na2SO4 Na2SO4‚ 10H2O KNO3 NaNO3
273-462 273-462 273-363 288-373 273-373 298-473 273-303
3.61572 2.08949 3.00445 -8.47979 -4.18629 0.661999 -2.70124
2187.41 7.12599 1.93139 5.98201 79646.1 17669.8 230805.
18.8623 18.6402 16.9631 54.9217 443.046 106.384 1699.95
0.0396581 0.0414204 0.0282449 -0.148065 0.622767 -0.175698 -3.00855
a
Figure 3. Mean ionic activity coefficients of LiCl and K2SO4 in water at elevated temperatures.
of salts in aqueous solutions are therefore often used for testing new models. Lu and Maurer (1993) have shown that the model is able to predict salt solubilities in mixed electrolyte aqueous solutions at 298 K also at high salt solubilities. When a solid electrolyte MνcXνa(H2O)h precipitates from an aqueous solution the concentrations of anionic and cationic species of that electrolyte in the liquid phase are determined by its solubility product: c+νa) h Ksp ) mνMc mνXaγ(ν ((m) aw
(20)
Solubility products Ksp are calculated from the solubility of a single electrolyte in pure water applying an adequate model for describing activity coefficients. Ksp depends on temperature. The following equation is used to express that influence:
ln Ksp(T) ) U1 + U2[1/(T/K) - 1/298.15] + U3 ln[(T/K)/298.15] + U4[(T/K) - 298.15] (21) Parameters U1, U2, U3, and U4 as determined from
273-373 0.263300 273-373 2.89788
40880.8 230.932 0.351406 5515.11 42.5647 0.0663410
Compare eq 21.
single-electrolyte solubility in pure water as reported by Linke and Seidell (1965) and Silcock (1979) are listed in Table 4. To test the model also at high concentrations and elevated temperatures, the solubility of some salts in seven binary electrolyte solutions (where both electrolytes have a common ion) was predicted and compared with experimental data taken either from Linke and Seidell (1965) or from Silcock (1979), cf. Table 5. The comparisons are shown in Figures 5-11. Figure 5 gives results for the system NaCl-KCl-H2O at 273, 373, and 463 K. Figure 6 gives results for the system KClK2SO4-H2O. Those two figures show also a comparison with the current version of Pitzer’s method taken from Pabalan and Pitzer (1991). As it was to be expected that method also gives a good agreement with the experimental data. However, it must be mentioned that the calculation by Pabalan and Pitzer is based upon specific binary ion-interaction parameters fitted not only to high-temperature data but also to experimental results for binary electrolyte aqueous systems. Whereas in the present work individual interaction parameters were fitted neither to experimental results at elevated temperature nor to results for binary electrolyte aqueous solutions. Figures 7 and 8 show comparisons for systems containing either KCl and NH4Cl or Na2SO4
+
+
1782 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
Figure 5. Measured and calculated solubilities in the system KCl-NaCl-H2O at 273, 373, and 463 K.
Figure 6. Measured and calculated solubilities in the system KCl-K2SO4-H2O at 288, 323, and 373 K. Table 5. Sources for Salt Solubility Data for Ternary Aqueous Solutions system
T/K
Imax
source
Na+-K+-Cl--H2O NH4+-K+-Cl--H2O K+-Cl--SO42--H2O Na+-K+-SO42--H2O K+-Cl--NO3--H2O Na+-Cl--NO3--H2O Na+-K+-NO3--H2O
273-363 273-363 273-373 298-373 298-373 273-373 273-333
14 14.5 9.5 22.5 25 21 27
Linke and Seidell, 1965 Linke and Seidell, 1965 Linke and Seidell, 1965 Silcock,1979 Silcock,1979 Silcock,1979 Silcock,1979
and K2SO4. In the later systems under certain conditions double-salt NaK3(SO4)2 precipitates. Figures 9 to 11 show comparisons for some very soluble nitrates. For those salts Clegg and Pitzer (1992a,b) recently proposed a model based on the mole fraction equations of Pitzer and Simonson (1986) instead of the virial (molalitybased) model of Pitzer (1973), but they applied that method only to 298 K. As shown in Figures 9-11, the
Figure 7. Measured and calculated solubilities in the system K2SO4-Na2SO4-H2O at 298, 323, and 373 K.
Figure 8. Measured and calculated solubilities in the system KCl-NH4Cl-H2O at 273, 318, and 363 K.
new model gives a reasonable prediction for the solubilities of the nitrates at elevated temperatures with individual parameters determined only from information on the properties of single-electrolyte aqueous solutions at 298 K. Conclusion The Lu-Maurer model (1993) for mixed electrolyte aqueous solutions is extended to temperatures up to 573 K. It is shown that by using parameters correlated from single-electrolyte aqueous systems at 298 K together with a generalized expression for the influence of temperature on some parameters, the activity coefficients in electrolyte aqueous solutions at high temperatures can be predicted with good accuracy up to the solubility limit, e.g. at very high ionic strength.
+
+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1783
Figure 9. Measured and calculated solubilities in the system KCl-KNO3-H2O at 298, 323, and 373 K.
Figure 10. Measured and calculated solubilities in the system NaCl-NaNO3-H2O at 273, 298, and 373 K.
Acknowledgment X.L. thanks the Alexander-von-Humboldt Stiftung of Germany for providing a research fellowship at the Universita¨t Kaiserslautern and The National Natural Science Foundation of P. R. China and Fok Ying-Tong Education Foundation for financial support. He also thanks Dongyun Lu and Jian Zhou for their assistance in the calculations. Notation a ) true activity A ) Debye-Hu¨ckel constant B ) Debye-Hu¨ckel constant baj ) parameter in eq 15 bci ) parameter in eq 15 C1, C2 ) constants in eq 17 d ) size parameter
Figure 11. Measured and calculated solubilities in the system KNO3-NaNO3-H2O at 273, 298, and 333 K.
GE ) excess Gibbs energy ha ) solvated anion hc ) solvated cation h ) number of water molecules in a complex I ) number of cationic species Im ) ionic strength on molality scale J ) number of anionic species i,j,k ) species i,j,k K ) solvation equilibrium constant Ksp ) solubility product m ) molality mk ) true molality of species k m j k ) overall (stoichiometric, apparent) molality of component k Mw ) molecular weight of water M ) cation n ) mole number N ) number of components p ) pressure rk ) ionic radius of species k R ) universal gas constant SD ) standard deviation T ) thermodynamic temperature U1-U4 ) constants in eq 21 υ j W ) volume occupied by a water molecule X ) anion z ) true mole fraction Z ) charge number of ionic species Greek Letters γk ) true activity coefficient of specie k on mole fraction scale normalized according to Raoult’s law γ*k ) true activity coefficient of specie k on mole fraction scale normalized according to Henry’s law γ((m) ) mean ionic activity coefficient on molality scale λci,aj ) binary size correction parameter between cation ci and anion aj σ ) root mean square deviation ν ) stoichiometric number φ ) osmotic coefficient Subscripts a ) anion c ) cation cal ) calculated DH ) contribution from Debye-Hu¨ckel law
+
+
1784 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 exp ) experimental ha ) solvated anion hc ) solvated cation hk ) solvated species k i ) solute k ) species k m ) on molality scale max ) maximum sp ) solubility product UNIQUAC ) contribution from UNIQUAC equation w ) water Superscripts * ) normalized according to Henry’s law ∞ ) limiting (infinite dilution)
Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116. Chen, C. C.; Britt, H. I.; Boston, J. F.; Evans, L. B. Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. AIChE J. 1982, 28 (4), 588. Clegg, S. L.; Pitzer, K. S. Thermodynamics of Multicomponent, Miscible, Ionic Solutions: Generalized Equations for Symmetrical Electrolytes. J. Phys. Chem. 1992, 96, 3513. Clegg, S. L.; Pitzer, K. S.; Brimblecombe, P. Thermodynamics of Multicomponent, Miscible, Ionic Solutions 2: Mixtures Including Unsymmetrical Electrolytes. J. Phys. Chem. 1992, 96, 9470. Gibbard, H. F., Jr.; Scatchard, G. Liquid-Vapor Equilibrium of Aqueous Lithium Chloride, from 25 to 100 °C and from 1.0 to 18.5 molality, and Related Properties. J. Chem. Eng. Data 1973, 18 (3), 293. Harned, H. S.; Owen, B. B. Physical Chemistry of Electrolyte Solutions; Reinhold: New York, 1958. Holmes, H. F.; Mesmer, R. E. Isopiestic Studies of Aqueous Solutions at Elevated Temperatures: VI. LiCl and CsCl. J. Chem. Thermodyn. 1981, 13, 1035. Holmes, H. F.; Mesmer, R. E. Thermodynamics of Aqueous Solutions of the Alkali Metal Sulfates. J. Solution Chem. 1986, 15 (6), 495. Holmes, H. F.; Baes, C. F.; Mesmer, R. E. Isopiestic Studies of Aqueous Solutions at Elevated Temperatures I. KCl, CaCl2 and MgCl2. J. Chem. Thermadyn. 1978, 10, 983.
Linke, W. F.; Seidell, A. Solubilities of Inorganic and Metal-organic Compounds; American Chemical Society: Washington, DC, 1965. Lu, X.; Maurer, G. Model for Describing Activity Coefficients in Mixed Electrolyte Aqueous Solutions. AIChE J. 1993, 39 (9), 1527. Mayrath, J. E.; Wood, R. H. Enthalpy of Dilution of Aqueous Solutions of LiCl, NaBr, KCl, KBr, and CsCl at 373, 423, and 473K. J. Chem. Thermodyn. 1982, 14, 563. Pabalan, R. T.; Pitzer, K. S. Mineral Solubilities in Electrolyte Solutions. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boston, MA, 1991; Chapter 7. Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77 (2), 268. Pitzer, K. S.; Simonson, J. M. Thermodynamics of Multicomponent, Miscible, Ionic Systems: Theory and Equation. J. Phys. Chem. 1986, 90, 3005. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: Boston, MA, 1959. Sacchetto, G. A.; Bombi, G. G.; Macca, C. Vapor Pressure of Very Concentrated Electrolyte Solutions. J. Chem. Thermodyn. 1981, 13, 31. Silcock, H. L. Solubilities of Inorganic and Organic Compounds; Pergamon Press: Oxford, 1979; Vol. 3. Slivester, L. F.; Pitzer, K. S. Thermodynamics of Geothermal Brine I. Thermodynamic Properties of Vapor-Saturated NaCl(aq) Solutions from 0-300 °C. Report LBL-4456; Lawrence Berkeley Laboratory: Jan. 1976. Snipes, H. P.; Manly, C.; Ensor, D. D. Heats of Dilution of Aqueous Electrolytes: Temperature Dependence. J. Chem. Eng. Data. 1975, 20 (3), 287. Zemaitis, F. D., Jr.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics: DIPPR, AIChE: New York, 1986.
Received for review July 28, 1995 Revised manuscript received February 15, 1996 Accepted February 15, 1996X IE950474K
X Abstract published in Advance ACS Abstracts, April 1, 1996.
