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Ind. Eng. Chem. Res. 2005, 44, 1602-1609
gE Model for Single- and Mixed-Solvent Electrolyte Systems. 3. Prediction of Salt Solubilities in Aqueous Electrolyte Systems Jiding Li,† Yangzheng Lin,† and Ju 1 rgen Gmehling*,‡ Department of Chemical Engineering, Tsinghua University, Beijing 100084, China, and Lehrstuhl fu¨ r Technische Chemie, Universita¨ t Oldenburg, D-26111 Oldenburg, Germany
The LIQUAC and LIFAC models have been widely used to predict osmotic coefficients, mean ion activity coefficients, the vapor-liquid equilibrium behavior, and the solubility of gases in single- and mixed-solvent electrolyte systems. In this paper, starting from the solubility product calculated using standard thermodynamic properties, the LIQUAC model is used to predict the solubilities of salts consisting of various cations (Na+, K+, and NH4+) and anions (F-, Cl-, Br-, I-, and SO42-) and their mixtures in aqueous solutions as a function of temperature. 1. Introduction Various industrial and natural processes require the knowledge of the real behavior of aqueous electrolyte solutions, such as precipitation and crystallization processes; desalination of water and water pollution control; salting-in or salting-out effects in extraction, absorption, and distillation; food processing; production of fertilizers; and so on. The solubilities of nearly insoluble salts in water can be calculated directly using their solubility product constants because at low ion concentrations the required activity coefficients are very close to unity. At higher concentrations, the mean activity coefficients of the ions have to be taken into account (γ( * 1.0). In 1994, Li et al.1,2 developed the LIQUAC model for systems with strong electrolytes based on the results from statistical thermodynamics taking into account the interactions between all species present in electrolyte solutions. Later, Yan et al.3 introduced the groupcontribution method UNIFAC into the LIQUAC model, called the LIFAC model. A large number of the required interaction parameters for the two models have been fitted with the help of a comprehensive data bank (Dortmund Data Bank4,5). Both models have been used to calculate the vapor-liquid equilibrium (VLE) behavior, osmotic coefficients, and mean ion activity coefficients for a large number of single- and mixedsolvent electrolyte systems with high accuracy.1-3 Li et al.6 also used these models in the gE-mixing rules of the Soave-Redlich-Kwong equation of state to predict the solubility of gases in aqueous electrolyte solutions. Yan et al.7,8 used several electrolyte solution models to correlate the VLE of the system acetone-methanol with KBr and ZnCl2, respectively. They found that the LIQUAC model provides superior results when compared with the electrolyte nonrandom two-liquid (NRTL) model,9 the extended UNIQUAC model,10 and
the electrolyte UNIFAC group-contribution model.11 Topphoff et al.12 and Yan et al.13 also successfully applied the LIQUAC model to correlate their experimental VLE data for mixed-solvent electrolyte solutions. Huh and Bae14 extended the LIQUAC model to describe the water activities in aqueous polymer electrolyte solutions at the operating conditions of polymer electrolyte fuel cells. Ming and Russell15 mentioned the LIQUAC model in their study on the hygroscopic growth of sea salt aerosols. Other researchers16-18 stated that until now the LIQUAC model was not used for the prediction of liquid-liquid and solid-liquid equilibria. Therefore, in this paper the results for salt solubilities in water predicted by the LIQUAC model using the already published interaction parameters1 are presented. 2. LIQUAC Model In the LIQUAC model,1 the excess Gibbs energy for single- or mixed-solvent electrolyte systems is calculated as the sum of three contributions:
GE ) GELR + GEMR + GESR
(1)
The first term represents the long-range (LR) interaction contribution caused by charge-charge interactions. The second term describes the middle-range (MR) interaction contribution caused by charge dipole and charge-induced dipole interactions. The third term expresses the contribution of the noncharge interactions. According to the thermodynamics relation
ln γi )
1 ∂GE RT ∂ni
(2)
the activity coefficients can be expressed as * To whom correspondence should be addressed. Tel.: ++49 (0)441 798 3831. Fax: ++49 (0)441 798 3330. E-mail:
[email protected]. † Tsinghua University. ‡ Universita¨t Oldenburg.
MR + ln γSR ln γi ) ln γLR i + ln γi i
where i indicates all of the solvents and ions.
10.1021/ie049283k CCC: $30.25 © 2005 American Chemical Society Published on Web 01/29/2005
(3)
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1603
On the basis of mole fraction, for solvents, each part of the activity coefficient can be calculated as follows:
ln γLR s )
) ln γMR s
( ) 2AMsd 3
b ds
∑ ion
MR ln γ′j ) (ln γLR + ln γSR j + ln γj j ) - ln(Ms/Mm +
[1 + bxI - (1 + bxI)-1 -
IB′s,ion(I)]x′smion - Ms γSR s
( )∑∑ Ms
Mm
∑c ∑a [Bca(I) + IB′ca(I)]mcma
) ln
γCs
{ ( )
ln γRs ) qs 1 - ln
∑i qixiΨis
+ ln
γRs
[
Vs Vs + ln Fs Fs
-
∑i qixi
(5)
(6)
(7)
[ ]} qixiΨsi
(
(8)
∑k qkxkΨki)
[ ]∑ ∑ zj2
2Mm
zj2AxI
(9)
1 + bxI
Bj,sol(I) x′sol + ∑ sol
B′′sol,ion(I) x′solmion +
()
sol ion
zj2 2
Bj,s(I) Ms
C C R R ln γSR j ) ln γj - ln γj (B) + ln γj - ln γj (B)
[
ln γCj ) 1 - Vj + ln Vj - 5qj 1 -
{ ( )
ln γRj ) qj 1 - ln
ln[γCj (B)] ) 1 -
∑i qixiΨij ∑i qixi
-
∑i
( )]
Vj Vj + ln Fj Fj
[ ]} qixiΨji
(
∑k qkxkΨki)
() [
(10)
(11)
(12)
∆G
Mν+Xν-‚nH2O(s) T ν+M(aq) + ν-X(aq) + nH2O(l) (19) ∆G ) ν+µM(aq) + ν-µX(aq) + nµw(aq) µMν+Xν-‚nH2O(s) ) 0 (20) Substituting eqs 17 and 18 into eq 20, one obtains
-
∆G°(T) ) ν+ ln(mMγ′M) + ν- ln(mXγ′M) + n ln(xwγw) RT (21)
where
(13)
( )]
rj rjqs rj rjqs + ln - 5qj 1 + ln rs rs rsqj rsqj
(14) ln[γRj (B)] ) qj[1 - Ψjs - lnΨsj]
(18)
where γ′i is the asymmetrical rational activity coefficient of ion i. µ/i is the standard state of the chemical potential for ion i based on the asymmetrical convention and molality concentration scale at system temperature. This is the hypothetical ideal state at unit molality in solvent s. For aqueous electrolyte systems, the equilibrium between the aqueous phase and the solid salt Mν+Xν-‚ nH2O(s) consisting of ν+ cations M, ν- anions X, and n water molecules, can be expressed as
Bj,ion(I) mion + ∑ ion
∑c ∑a B′ca(I) mcma -
(17)
where µ°s is the standard state of the chemical potential for the pure liquid solvent s at system temperature and pressure and γs is the activity coefficient of solvent s. According to the symmetrical convention, γs is unity for the pure liquid (xs ) 1) at all temperatures. The asymmetrical convention for ions based on molality is used in this work. Thus, the chemical potential for ion i can be written as
µi ) µ/i + RT ln(miγ′i)
For ions, each part of the activity coefficient is calculated as follows:
ln γLR j ) -
The symmetrical convention for solvents based on mole fraction is used in this work. Thus, the chemical potential for a solvent s can be written as
µs ) µ°s + RT ln(xsγs)
( )]
∑i
(16)
3. Solid-Liquid Equilibria
[Bs,ion(I) +
s ion
ln γCs ) 1 - Vs + ln Vs - 5qs 1 -
) (Mm)-1 ln γMR j
mion) ∑ ion
Ms
2 ln(1 + bxI)] (4)
Bs,ion(I) mion -
ln
molalities, the activity coefficient of ion j can be obtained from
(15)
The variables given in eqs 4-15 were already defined in detail in the paper of Li et al.1 On the basis of
∆G°(T) ) ν+µ/M + ν-µ/X + nµ°w - µMν+Xν-‚nH2O(s) (22) Using the Gibbs-Helmholtz equation, the temperature dependence of ∆G° can be described by the following equation:
( ) ∫ ∫[ ]
∆G°(T) ∆G°(Tr) ∆H°(Tr) 1 1 + ) RT RTr R T Tr T
1 R
T
Tr
Tr
∆c°p(T) dT T2
dT (23)
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Table 1. Heat Capacity and Gibbs Energy and Enthalpy of Formation for Water H2O(l)19 cp (J mol-1 K-1) 298.15 K 300 K 373.15 K 75.288
75.280
75.946
∆fG298.15K (kJ mol-1)
∆fH298.15K (kJ mol-1)
-237.141
-285.830
where Tr is the reference temperature (298.15 K). / / ∆G°(Tr) ) ν+∆fGM(aq) (Tr) + ν-∆fGX(aq) (Tr) + n∆fG°w(l)(Tr) - ∆fG°Mν+Xν-‚nH2O(s)(Tr) (24) / / (Tr) + ν-∆fHX(aq) (Tr) + ∆H°(Tr) ) ν+∆fHM(aq) n∆fH°w(l)(Tr) - ∆fH°Mν+Xν-‚nH2O(s)(Tr) (25) ,
,
°* °* (Tr) + ν-cp,X(aq) (Tr) + nc°p,w(l)(Tr) ∆c°p(Tr) ) ν+cp,M(aq) c°p,Mν+Xν-‚nH2O(s)(Tr) (26)
In eqs 24-26, ∆fG°ion(aq), ∆fH°ion(aq), and c°p,ion(aq) are the Gibbs energy of formation, the enthalpy of formation, and the heat capacity in the standard state. It should be noted that (i) the standard state mentioned above is the hypothetical ideal solution at unit molality in water and (ii) the solid salt is the crystalline solid. Substituting eq 23 into eq 21, considering that the concentration scale of the ion is molality, one obtains
( ) ∫ ∫[ ]
∆G°(Tr) ∆H°(Tr) 1 1 + RTr R T Tr T
1 R
T
Tr
∆c°p(T) dT
Tr
T2
dT ) -[ν+ ln(mMγ′M) +
ν- ln(mXγ′X) + n ln(xwγw)] (27)
4. Results and Discussion The salt solubilities in aqueous solutions can be predicted using eq 27 and the LIQUAC model when the Gibbs energies (∆fG) of formation, enthalpies (∆fH) of formation, and heat capacities ∆c°p for the salt and ions are available. In this paper, the solubilities of various single and mixed salts in aqueous solutions are predicted. The heat capacities and Gibbs energies and enthalpies of formation for water, crystal salts, and ions in aqueous solutions used for the calculation are listed in Tables 1-3. It should be pointed out that the c°p values that
are not given in the tables were set to zero in this work. Often ∆c°p (298.15 K) had to be used instead of ∆c°p (T) in eq 27 because only c°p (298.15 K) values were available for the ions and some salts. a. Aqueous Single-Solvent Electrolyte Solutions. The procedure to predict the solubility of a single salt in aqueous solutions at a particular temperature is the following: (1) Calculate the left-hand side of eq 27 with the values given in Tables 1-3. (2) Estimate a concentration of the salt for the given temperature. (3) Calculate the right-hand side of eq 27 with the LIQUAC model. (4) Use the Newton method to determine the concentration of the salt, so that the difference between the right- and left-hand sides obtained from steps 1 and 3 is equal to 0. This concentration is the solubility of the salt in the aqueous solution. The predicted solubilities might be influenced by the interaction parameters in the LIQUAC model and the values of the Gibbs energy and enthalpy of formation of the salts. The interaction parameters are taken as the original ones from ref 1. It was found that the predicted results are very sensitive to the ∆fG and ∆fH values, which are given in Tables 1-3. In a few cases, the ∆fG and ∆fH values were slightly modified to obtain more reliable results when compared with the experimental solubility data. If there are two values of the thermodynamic properties cp, ∆fG and ∆fH in Table 2, the values with asterisks were used for the calculations. The predicted solubilities using the LIQUAC model for seven salts (NaF, NaCl, KCl, KBr, KI, K2SO4, and NH4Cl) in aqueous solutions in equilibrium with the anhydrous crystals are shown in Figure 1. From the results, it can be seen that not only the absolute values of the solubilities but also the temperature dependence are described quite accurately. The predicted average deviations for the solubilities of NaF, NaCl, KCl, KBr, KI, K2SO4, and NH4Cl in aqueous solutions are 0.54%, 1.27%, 2.25%, 2.05%, 2.28%, 2.66%, and 2.14%, respectively, as listed in Table 4. In Figure 2, the saturation solubilities of sodium bromide and sodium sulfate in equilibrium with the hydrated and anhydrous crystals are shown. The predicted solubilities consist of two smooth parts. In Figure 2a, the left part at low temperature is predicted by assuming NaBr‚2H2O as the precipitate. The right-hand part at higher temperature is predicted by assuming anhydrous NaBr as the precipitate. At the singular point, the precipitates are
Table 2. Heat Capacity and Gibbs Energy and Enthalpy of Formation for Various Crystals20 cp (J mol-1 K-1) salt NaF NaCl NaBr NaI Na2SO4 KCl KBr KI K2SO4 NH4Cl NaBr‚2H2O NaI‚2H2O Na2SO4‚10H2O a
298.15 K 46.853 50.503 51.893 52.225 128.151 51.713 52.306 52.780 131.193 84.100
300 K 46.923 50.544 51.918 52.260 128.487 51.710 52.361 52.803 131.584
400 K 49.598 52.374 53.249 53.793 145.101 52.483 54.103 53.934 147.904
-246.60
Values used for the calculations (modified in this work).
500 K 51.260 53.907 54.580 55.047 54.209 55.055 55.357 160.164
∆fH298.15K (kJ mol-1) -573.647a
-575.384, -411.120 -361.414 -287.859, -292.000a -1387.816 -436.684, -431.300a -393.798 -327.900 -1437.790, -1431.800a -314.430, -314.000a -951.940 -883.096 -4327.26
∆fG298.15K (kJ mol-1) -545.080, -543.497a -384.024 -349.267 -284.519, -281.500a -1269.848 -408.754, -409.340a -380.470 -323.024, -324.892a -1319.684, -1321.300a -202.97, -203.60a -828.290 -771.100 -3646.850
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Figure 1. Solubilities of various salts in water. The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (a) (0);21 (b) (0, 4, O);22-28 (c) (0)22 and (4);29 (d) (0);22 (e) (0);22 (f) (0);30,31 (g) (0)32 and (4).33 Table 3. Heat Capacity and Gibbs Energy and Enthalpy of Formation for Ions in Aqueous Solutions19 ion (1 mol kg-1, aq)
c298.15K p (J K-1 mol-1)
∆fG298.15K (kJ mol-1)
∆fH298.15K (kJ mol-1)
Na+(aq) K+(aq) NH4+(aq) F-(aq) Cl-(aq) Br-(aq) I-(aq) SO42-(aq)
46.4 21.8 79.9 -106.7 -136.4 -141.8 -142.3 -293.0
-261.905 -283.27 -79.31 -278.79 -131.228 -103.96 -51.57 -744.53
-240.12 -252.38 -132.51 -332.63 -167.159 -121.55 -55.19 -909.27
Table 4. Mean Average Deviations for the Solubility of Single-Salt Systems system
no. of data sets
∆m (mol kg-1)
∆m (%)
temp range (°C)
NaF-H2O NaCl-H2O KCl-H2O KBr-H2O KI-H2O K2SO4-H2O NH4Cl-H2O NaBr-H2O Na2SO4-H2O
7 42 13 28 25 17 16 28 27
0.005 0.084 0.110 0.117 0.241 0.023 0.168 0.121 0.068
0.54 1.27 2.25 2.05 2.28 2.67 2.14 1.06 3.64
10-35 0-200 0-100 -6.2 to +100 25.6-113.7 15-101.1 0-75 10-171.2 0.7-170
NaBr‚2H2O and NaBr. In Figure 2b, the left part at low temperature is predicted by assuming Na2SO4‚10H2O as the precipitate. The right part is predicted by
Figure 2. Solubilities of hydrated and anhydrous salts in water. The curves are predicted with the LIQUAC model. The solid symbols are the experimental data for the hydrated crystals. The open symbols are the experimental data for the anhydrous salt. The symbols denote the data from the following references: (a) (9, 0, 4);22,34-36 (b) (9, 0, 4).21,26,30,35,37-39
assuming anhydrous Na2SO4 as the precipitate. Again a very good agreement with the experimental findings is achieved. Also, the singular points are described quite well. It can be found from Table 4 that the predicted average errors for the solubilities of Na2SO4 and NaBr in aqueous solutions are 3.64% and 1.06%, respectively. The solubilities vary remarkably for the anhydrous and hydrated crystals. However, as can be seen from Figure 2, the LIQUAC model describes the solubilities
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Figure 5. Water solubilities of various salts with the same anion (SO42-). The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (9, 0),21,26,30,35,37-39 (4).30,31 Figure 3. Water solubilities of various salts with the same anion (Cl-). The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (O),32,33 (0),22-28 (4).22,29
Figure 4. Water solubilities of various salts with the same anion (Br-). The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (9, 0),22,34-36 (4).22
reliably using the standard thermodynamic properties of formation listed in Tables 1-3. In Figures 3-7, the predicted and experimental solubilities for different electrolytes with the same cation or anion in aqueous solutions are presented. Of course, for the common ion, not only the same thermodynamic values of formation but also the same group interaction parameters of the LIQUAC model are used. In Figure 3, the solubilities of KCl, NH4Cl, and NaCl as a function of temperature are shown. It can be seen that the solubilities and the temperature dependence predicted is in very good agreement with the experimental data; e.g., it is predicted that KCl shows the lowest solubility in the temperature range 0-70 °C, while NaCl shows the lowest solubility at higher tem-
Figure 6. Water solubilities of various salts with the same cation (Na+). The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (b, O),22,34-36 (9, 0),21,26,30,35,37-39 (4).22-28
peratures (>80 °C) from the three salts investigated. In Figures 4-7, the predicted solubilities for various salts (NaBr, Na2SO4, NaCl, KI, KBr, KCl, and K2SO4) in aqueous solutions are compared. In all cases from the investigated salts with the same anion, the sodium salts show higher solubility than the potassium salts. Depending on the anions, the following solubilities are predicted: NaBr > NaCl > Na2SO4 and KI > KBr > KCl > K2SO4, which is in agreement with the experimental findings. b. Aqueous Mixed-Salt Solutions. For the mixedsalt systems, the solubility of a salt depends on the concentrations of the other salts in the solution. To obtain the solubilities, one has to know which salt will precipitate from the solution.
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1607
Figure 7. Water solubilities of various salts with the same cation (K+). The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (4),22 (O),22,29 (+),22 (0).30,31
A procedure similar to that for pure salts can be applied to predict the solubilities of salt mixtures in water at a particular temperature: (1) Calculate the left-hand side of eq 27 with the values given in Tables 1-3. (2) Estimate an appropriate concentration of the salts at the given temperature. (3) Calculate the right-hand side of eq 27 with the LIQUAC model. (4) Use a nonlinear regression method to obtain the concentrations of the two salts in such a way that the difference between the right- and left-hand sides obtained from steps 1 and 3 is 0. From the results, it can be judged which deposit (salt) is in equilibrium with the solution. In this work, aqueous NaCl-Na2SO4 (Na2SO4‚10H2O), NaCl-KCl, KCl-K2SO4, and NH4Cl-KCl solutions were studied in detail. The results are shown in Figures 8-11. In Figure 8a, the predicted curve consists of two smooth parts; the left part, AB1, indicates that the solution is in equilibrium with an anhydrous NaCl deposit. The right part, B1C, indicates that the solution is in equilibrium with a Na2SO4‚10H2O crystal. At the cross point (singular point), B1, the precipitates are the anhydrous NaCl and Na2SO4‚10H2O. In Figure 8b, the predicted curve consists of three smooth parts; the left and first part, AB2, denotes that the solution is in equilibrium with an anhydrous NaCl deposit. The middle and second part, B2B3, denotes that the solution is in equilibrium with an anhydrous Na2SO4 precipitate. The right and third part, B3C, indicates that the solution is in equilibrium with Na2SO4‚10H2O crystals. At the cross point, B2, of the first and second parts, the deposits are the anhydrous NaCl and Na2SO4. At the cross point, B3, of the second and third parts, the precipitates are the anhydrous Na2SO4 and Na2SO4‚10H2O. In Figure 8c, the predicted curve consists of two smooth parts. The
Figure 8. Solubilities of NaCl and Na2SO4 in aqueous solutions. The solid symbols are the experimental data with NaCl as the precipitate. The open symbols are the experimental data with Na2SO4 as the precipitate. The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (a) (9, 0);40 (b) (b, O, 4);21,40 (c) (9, 0);40 (d) (b, O).40
left part, AB4, indicates that the solution is in equilibrium with an anhydrous NaCl deposit. The right part, B4C, denotes that the solution is in equilibrium with an anhydrous Na2SO4 precipitate. At the cross point, B4, the precipitates are anhydrous NaCl and Na2SO4. The curves in Figures 8d and 9-11 have the same meanings as those in parts a and c of Figure 8. The deviations in the figures are partly caused by the scattering data of the different authors. The predicted relative deviations for the mixed-salt systems can be found in Table 5. The relative average errors for the aqueous NaCl-Na2SO4, NaCl-KCl, KClK2SO4, and NH4Cl-KCl solutions are 8.32%. 7.60%, 6.88%, and 3.63%, respectively. The differences between the predicted solubilities and the experimental data might be caused by the interaction parameters fitted to a limited database for the LIQUAC model, the quality of the values for the Gibbs energy and enthalpy of formation, or the questionable experimental data. Considering the scattering of the experimental data, the observed deviations can be accepted. 5. Conclusion In this paper, a method for the prediction of single or mixed salts in aqueous solutions as a function of temperature is introduced. To apply this method, the Gibbs energies and enthalpies of formation and heat capacities for the considered salts and ions have to be known to be able to calculate the solubility product for a given temperature. To obtain reliable results, the activity coefficients of the ions and water have to be known reliably. In this paper, the LIQUAC model with the already published interaction parameters was applied to calculate the required activity coefficients.
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Figure 11. Solubilities of NH4Cl and KCl in aqueous solutions. The solid symbols are the experimental data for NH4Cl as the precipitate. The open symbols are the experimental data for KCl as the precipitate. The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (9, 0, b, O).32 Table 5. Mean Average Deviations for the Solubility of Mixed-Salt Systems two salts and water system salt 1 salt 2 Figure 9. Solubilities of NaCl and KCl in aqueous solutions. The solid symbols are the experimental data with NaCl as the precipitate. The open symbols are the experimental data with KCl as the precipitate. The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (a) (9, 0, b, O);29 (b) (b, O);40 (c) (2, 4);40 (d) (9, 0).40
NaCl NaCl KCl NH4Cl
no. of data sets
∆m (mol kg-1)
∆m (%)
concn scale (mol kg-1)
93 90 22 20
0.086 0.369 0.038 0.112
8.32 7.60 6.88 3.63
0-7.0 0-7.5 0-6.0 0-10.0
Na2SO4 KCl K2SO4 KCl
The calculation of the salt solubility in other solvents or mixed solvents is in progress. Acknowledgment The authors thank K.-C. Wong Foundation for financial support. The fees of the international transportation between China and Germany were covered from the Special Funds for Major State Basic Research Program of China (973 Program; Grant 2003CB615701), the National High Technology Research and Development Program of China (863 Program; Grant 2003AA328020), and the National Natural Science Foundations of China (Grant 20276034). List of Symbols
Figure 10. Solubilities of mixed KCl and K2SO4 in aqueous solutions. The solid symbols are the experimental data with KCl as the precipitate. The open symbols are the experimental data with K2SO4 as the precipitate. The curves are predicted with the LIQUAC model. The symbols denote the data from the following references: (9, 0, b, O).40-42
From the results for various single or mixed salts in aqueous solutions, it can be concluded that the LIQUAC model is used to reliably predict the solubility of strong electrolytes in aqueous solutions. The predicted average relative deviations for single salts are less than 4%, and those for mixed-salt solutions are less than 10%.
a ) activity cp ) heat capacity (J mol-1 K-1) G ) Gibbs energy (J mol-1) H ) enthalpy (J mol-1) M ) molar mass (kg mol-1) m ) molality (mol kg-1) n ) mole number R ) general gas constant (J mol-1 K-1) T ) absolute temperature (K) x ) liquid mole fraction y ) vapor mole fraction Greek Letters γ ) activity coefficient (mole fraction scale) γ′ ) activity coefficient (molality scale) µ ) chemical potential φ ) osmotic coefficient
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1609 Superscripts E ) excess property LR ) long range MR ) middle range SR ) short range Subscripts a ) anion c ) cation i ) component i s ) component s j ) component j sol ) solvent ion ) ion M ) cation X ) anion w ) water
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Received for review August 9, 2004 Revised manuscript received December 7, 2004 Accepted December 13, 2004 IE049283K
