Experimental Studies and Thermodynamic Modeling of the Solubilities

ARTICLE pubs.acs.org/IECR

Experimental Studies and Thermodynamic Modeling of the Solubilities of Potassium Nitrate, Potassium Chloride, Potassium Bromide, and Sodium Chloride in Dimethyl Sulfoxide Bingwen Long* College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada ABSTRACT: Using appropriate nonaqueous solvents to replace water as reaction medium in chemical industries has gained more and more attention recently. Many of them have the special ability to dissolve some compounds and thus may make the reaction more stable in comparison with a water-containing environment. Dimethyl sulfoxide (DMSO) is probably the most frequently mentioned aprotic nonaqueous solvent with extensive applications because of its high stability and powerful solubility. In this study, the solubilities of four inorganic salts, namely potassium nitrate (KNO3), potassium chloride (KCl), potassium bromide (KBr) and sodium chloride (NaCl), in DMSO are measured in the temperature range of 302 through 354 K using a dynamic method. The solubility order of the salts in DMSO is experimentally determined as KNO3 > KBr > NaCl > KCl. The molality solubilities show linear dependencies on temperature and the temperature effect on the solubility for the salts follows the same order as the solubility result. The solubility products of the salts in DMSO at different temperatures are obtained by estimating the solubility products in water and the Gibbs energy of transfer from water to DMSO. Then electrolyte models of the Wilson, NRTL, and UNIQUAC equations are used to model the solubility of the inorganic salts in DMSO. It is found that the three-parameter E-Wilson equation gives the best correlation results followed by the Pitzer, E-NRTL, and E-UNIQUAC equations, while the two-parameter E-Wilson equation presents the worst results in terms of the overall standard deviation.

1. INTRODUCTION The applications of nonaqueous solvents, especially dipolar aprotic solvents to replace water in the field of chemical reactions, electrochemistry, and separation chemistry are recently growing rapidly. In comparison with water, some nonaqueous solvents will show special solubility which makes some water insoluble compounds become soluble and thus the chemical reactions become feasible and controllable. This special character of nonaqueous solvents remarkably widens their applications in pure and applied chemistry.1 Today, efforts are still being made to study the properties of nonaqueous solvents, such as solubility, thermal and chemical stability, and toxicity, with the purpose to help us find more optimal and environmentally benign media for chemical processes. The quantitative measurements and evaluations of these properties are also of great importance to increase our understandings of the behaviors of these nonaqueous solutions from a thermodynamic perspective. Dimethyl sulfoxide (DMSO, CAS Registry Number: 67-68-5) is an odorless, colorless, aprotic nonaqueous solvent. It is widely used as reagent, ligand, solvent, biologically and medically useful compound in various fields for its high stability, low toxicity, ease of recovery and powerful solubility.2,3 Regarding the solubility, DMSO is a versatile and powerful solvent which can dissolve most aromatic and unsaturated hydrocarbons, organic nitrogen compounds, and organo-sulfur compounds.2
4 DMSO has also exhibited an ability to dissolve many inorganic salts, particularly those of the transition metals or those with nitrates, cyanides, or dichromates as their anions.3 A few quantitative studies of solubility of inorganic salts in DMSO have been reported. r 2011 American Chemical Society

Reynolds and Silesky4 measured the solubility of potassium chloride and sodium iodide in DMSO and (DMSO þ water) mixtures at 298 K. Jones and Musulin5 experimentally determined the solubility of iodine in DMSO at temperatures from 300 to 311 K. Dodd and Gasser6 investigated the solubility of cyanic salts in DMSO, and the results showed that cyanides of sodium, potassium, Cu(I), Ni(II), Zn(II) and Co(II) are insoluble in DMSO but those of Hg(II) and Cd(II) are soluble. Warren and Henein7 also measured the solubility of PbCl2, FeC13, and FeCl2 in DMSO as a function of temperature. In addition, Gaylord Chemical Co. provides solubility data of a lot of compounds in DMSO at 298 K including organic materials, polymers, active pharmaceutical ingredients, and inorganic salts on its Web site.8 However, because of being highly polar and aprotic, DMSO has a very complex salt solubility which is still much more difficult to be predicted in comparison with water. A lot of experimental work is still needed to explore the quantitative solubility of DMSO to specific inorganic salt. Therefore, in this work, the solubilities of four specific inorganic salts, potassium nitrate (KNO3), potassium chloride (KCl), potassium bromide (KBr) and sodium chloride (NaCl), in DMSO are measured in the temperature range of (302 to 354) K using a dynamic method. Received: October 21, 2010 Accepted: April 26, 2011 Revised: March 31, 2011 Published: April 26, 2011 7019

dx.doi.org/10.1021/ie102134g | Ind. Eng. Chem. Res. 2011, 50, 7019–7026

Industrial & Engineering Chemistry Research On the other hand, thermodynamic modeling of the solid
liquid equilibrium of the solutions with electrolytes is of great importance to many industrial processes and such modeling work can help us increase our understandings of the interactions among the ions, salts, and solvent molecules. A number of models have been developed for calculations of the phase equilibrium and other properties of solutions containing electrolyte. Several reviews on the thermodynamic modeling progress for electrolyte solutions are available.9
11 However, in comparison with aqueous electrolyte systems, phase equilibrium of electrolyte in pure organic solvent is less studied and its thermodynamic modeling is still very challenging. Notwithstanding this, such knowledge from both experimental and theoretical perspectives is still much desired for many industrial and natural processes such as nonaqueous solvent extraction processes in the oil recovery and upgrading process, precipitation and crystallization in drilling mud, desalination of water, and salting-in and salting-out effects in extraction and distillation. The thermodynamic functions for modeling the phase equilibrium of electrolyte in pure organic solvent or mixed solvents should be based on those for aqueous systems which have already been quantitatively well determined, and the key is to accurately estimate the Gibbs energy of transfer from pure water to the desire solvent. In this way, we propose a method to calculate the Gibbs energy of transfer from pure water to DMSO and then to calculate the solubility product of the electrolyte in DMSO. The activity coefficient of the salts in DMSO is modeled with the electrolyte solution models of the Pitzer’s, E-NRTL, E-Wilson and E-UNIQUAC equations. The binary interaction parameters of each model are then obtained by correlating the measured data.

2. EXPERIMENTAL SECTION

ARTICLE

Beijing DeTianYou Co.), which is capable of maintaining the temperature of the solution within (0.05 K. Temperatures of the crystal disappearance were measured with a 4-wire platinum resistance probe (Pt-100) and detected visually. The thermometer probe was connected to a Yuguang AI-708 data acquisition unit with the resolution of 0.01 K, which is also connected to a computer to collect the temperature data. This temperature measuring system was previously calibrated on the basis of ITS-90. The accuracy of the temperature measurements was judged to be within (0.05 K. Repeat runs were conducted to ensure the accuracy of the measured saturation temperature. After each measurement, the clear solution was first cooled until the nucleation was observed. Then slow heating was employed with a controllable heating rate of e0.2 K min
1 until the dissolution of the last crystal was observed. At least five repetitive temperature measurements were performed and an average value was taken as the equilibrium temperature. The reproducibility of the experimental point was within (0.2 K.

3. THERMODYNAMIC MODELING DMSO is a typical aprotic solvent with good solubility for many inorganic salts. The hydrogen atoms of DMSO are quite inert, which makes it not a hydrogen donor in hydrogen bonding and thus poorly solvate the anions.1
3 However, DMSO has a rather high dielectric constant of 46.4 at 298.15 K and the salts investigated in this work are all 1:1 type strong electrolytes with low concentration in the solutions. Therefore it is reasonable to assume that the salts are completely ionized in DMSO in this work. 3.1. Solid
Liquid Equilibrium. For 1:1 electrolyte MX dissolved in DMSO, it is completely ionized into cations M þ and anions X
and the dissolution equilibrium can be expressed as MXðSÞ T Mþ þ X


2.1. Materials. Analytical reagent potassium nitrate (KNO3,

>99.0%), potassium chloride (KCl, > 99.8%), sodium chloride (NaCl, >99.5%), potassium bromide (KBr, >99.5%) and dimethyl sulfoxide (DMSO) were purchased from Beijing Chemical Reagent Co. The chemicals were used as received and no further purification was performed. 2.2. Procedure and Equipment. A dynamic (synthetic) approach to the solubility measurements is employed in this work.12,13 This method is based on slowly heating a solution containing suspended crystals with stirring until all the crystals are dissolved. The exact concentrations of the solutions are carefully predetermined. The heating process should be controlled slowly when approaching equilibrium, normally less than 0.2 K min
1, until the last crystal dissolved. At this point, the temperature was determined as the equilibrium saturation temperature corresponding to the solution concentration. The apparatus for the solubility measurements are almost the same as that described in our previous work,14
18 so only some apparatus changes are described here. All of the mixtures were prepared by mass using an analytical balance (type Adventurer AR2140, OHAUS Co.) with an uncertainty of (0.0001 g, and the errors do not exceed 2  10
4 in mass fraction. Samples of known composition were carefully transferred into the inner chamber of a jacket equilibria glass cell. Continuous stirring was adopted, and the stirrer speed was kept around 600 rmp. The temperature of the solution was controlled by circulating the thermostatic water through the cell jacket. The circulating water to the jacket was controlled by a thermostat (type DTY-8A,

ð1Þ

The thermodynamic equilibrium constant, also known as solubility product, for this dissociation reaction is R þ RX
ð2Þ KSP, DMSO ðTÞ ¼ M RMX The relation between Gibbs energy change and equilibrium constant is ΔGDMSO ðTÞ ¼
ln KSP, DMSO ðTÞ RT

ð3Þ

By assuming the solid salt is pure and of activity 1, eq 2 yields KSP, DMSO ðTÞ ¼ RMþ RX
¼ ðγmÞMþ ðγmÞX


ð4Þ

Where m is the molality of the ion while γ is the corresponding activity coefficient of the ion on the molality scale. Hereafter all the thermodynamic functions are on the molality scale in this work unless specially stated. Since the activity coefficient of single ion cannot be actually measured, the mean ionic activity coefficient of neutral electrolyte is usually used to express the relation in eq 4. The mean ionic activity coefficient γ( for 1:1 electrolyte MX is defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ γ( ¼ γMþ γX
and the mean molality for the 1:1 electrolyte is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m( ¼ mMþ mX
7020

ð6Þ

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According to the mass balance, m( should be equivalent to the molality of neutral electrolyte in the solution and therefore to the molality solubility when the solution is saturated. Substituting eqs 5 and 6 into eq 4 gives KSP, DMSO ðTÞ ¼ m2 γ2(

ð7Þ

Therefore with the knowledge of the solubility products of the salts in DMSO and the mean electrolyte activity coefficient, the solubility can be well modeled. However, the solubility product KSP,DMSO cannot be directly estimated from eq 3 because the standard thermodynamics functions used for estimating the Gibbs energy change ΔGDMSO(T) are not available for DMSO and therefore the calculation of solubility directly from eq 7 becomes a bit problematic. On the other hand, such thermodynamics functions for aqueous systems are well determined and extensively documented which provides a foundation to estimate ΔGDMSO(T). 3.2. Gibbs Energy of Transfer. Similar to eq 3, the thermodynamic equilibrium relation for the salts in pure water is ΔGW ðTÞ ¼
ln KSP, W ðTÞ RT

ð8Þ

So the Gibbs energy of transfer of salt from pure water to DMSO is Δtr GW f DMSO ðTÞ ΔGDMSO ðTÞ ΔGW ðTÞ ¼
RT RT RT By a combination of eq 3, 8, and 9, we get " # KSP, DMSO ðTÞ Δtr GW f DMSO ðTÞ ¼
ln RT KSP, W ðTÞ

ð9Þ

ð10Þ

the temperature dependence of ΔtrGWfDMSO(T) at constant pressure is calculated from the value at reference temperature T0 by assuming the entropy of transfer is independent of temperature:19 Δtr GW f DMSO ðTÞ ¼ Δtr GW f DMSO ðT 0 Þ
Δtr SW f DMSO ðT 0 ÞðT
T 0 Þ

ð11Þ

Hefter et al.20 has given a critical review and evaluation on the enthalpies and entropies of transfer of electrolytes and ions from water to organic and aqueous organic solvents. They have tabulated the recommended enthalpy and entropy values of transfer from water to DMSO at 298.15 K for many electrolytes and ions including those related to this work. These data has been extensively assembled and carefully evaluated by the authors and therefore are supposed to be fairly reliable. Therefore the recommended data are adopted in this work to calculate the Gibbs energy of transfer at reference temperature T 0 = 298.15 K: Δtr GW f DMSO ðT 0 Þ ¼ Δtr HW f DMSO ðT 0 Þ
T 0 Δtr SW f DMSO ðT 0 Þ

ð12Þ

It should be noted that the ΔtrSWfDMSO(298.15 K) data reported by Hefter et al.20 was on the molarity scale (c, mol 3 L
1). Therefore the values should be transformed into molality scale first by Δtr Sm, W f DMSO ð298:15 KÞ ¼ Δtr Sc, W f DMSO ð298:15 KÞ
R ln

FDMSO FW

ð13Þ

in which FDMSO and FW are the density of DMSO and water at 298.15 K, which are 1.1010 and 0.9970 (g/cm3),21 respectively. 3.3. Solubility Product of the Salts in Water. Unlike in DMSO, the solubility product of the salts in water can be calculated from eq 8 by directly estimating the Gibbs energy changeΔGW(T) via the Gibbs
Helmholtz equation and some standard thermodynamic functions. For a chemical reaction, the Gibbs
Helmholtz equation is written as 0  1 ΔG B ΔHðTÞ C B RT C C ¼
BD ð14Þ @ DT A RT 2 p

and

Z ΔHðTÞ ¼ ΔHðT 0 Þ þ

T

ΔCP ðTÞ dT

ð15Þ

T0

Substituting eq 15 into eq 14 and integrating eq 14 between the reference temperature T 0 and T with the assumption of constant heat capacity ΔCP(T) = ΔCP(T0), we get the expression for ΔGW(T):   ΔGW ðTÞ ΔGW ðT 0 Þ ΔHW ðT 0 Þ 1 1 ¼

RT RT 0 R T0 T !   ΔCP, W ðT 0 Þ T T0 ln 0 þ
1 ð16Þ
T R T where ΔGW(T0), ΔHW(T0), and ΔCP,W(T0) are the Gibbs energy, enthalpy, and mole heat capacity changes, respectively, of the reaction represented by eq 1 but in water at the reference temperature T0. They are calculated as ΔGW ðT 0 Þ ¼ ΔGf , W, Mþ ðT 0 Þ þ ΔGf , W, X
ðT 0 Þ
ΔGf , MX, S ðT 0 Þ

ð17Þ ΔHW ðT 0 Þ ¼ ΔHf , W, Mþ ðT 0 Þ þ ΔHf , W, X
ðT 0 Þ
ΔHf , MX , S ðT 0 Þ

ð18Þ ΔCP, W ðT 0 Þ ¼ CP, W, Mþ ðT 0 Þ þ CP, W , X
ðT 0 Þ
CP, MX , S ðT 0 Þ ð19Þ where ΔGf, ΔHf, and CP are the Gibbs energy of formation, enthalpy of formation, and heat capacity, respectively. There are extensive databases readily available for such thermodynamic properties of aqueous species at 298.15 K.21 So the reference temperature T0 for calculation of ΔGW(T) is again taken as 298.15 K. The ΔGf, ΔHf, and CP values for the ions and salts are taken directly from the handbook.21 In summary, solubility products of the salts in water at different temperature are calculated from eq 8 and eqs 16
19. Then the Gibbs energy transfer from water to DMSO is calculated from eqs 11 and 12 and solubility products of the salts in DMSO at different temperature can be obtained from eq 10. The calculated solubility products for the salts in water and DMSO at different temperature are also listed in Table 1. Finally, the molality solubility of the salts can be calculated from eq 7 if the mean ionic activity coefficient γ( can be estimated from appropriate electrolyte activity coefficient model with optimized interaction parameters. 7021

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Table 1. Experimental Solubility of Different Salts in DMSO solute

m (mol 3 kg
1)

KNO3

x

T (K)

KSP,W

1.033

8.077  10
2

304.15

1.17

0.107

1.065

8.309  10
2

313.15

1.71

0.103

1.095

8.525  10
2

319.95

2.23

9.92  10
2

325.45

2.73

9.57  10
2

329.05

3.09

9.33  10
2

332.45

3.45

9.09  10
2

337.85

4.09

8.70  10
2


2

8.753  10

1.127


2

8.837  10

1.139


2

9.004  10

1.163


2

9.223  10

1.194

KBr


2

1.226 1.260

9.448  10 9.684  10
2

343.45 348.15

4.83 5.50

8.28  10
2 7.93  10
2

0.4345

2.083  10
2

302.85

16.18

2.27  10
2

307.45

18.07

2.12  10
2


2

2.131  10

0.4448

NaCl


2

0.4600

2.202  10

312.65

20.29

1.94  10
2

0.4743

2.269  10
2

317.55

22.43

1.79  10
2

0.4861

2.324  10
2

323.75

25.17

1.60  10
2

0.4974 0.5111

2.377  10
2 2.441  10
2

328.75 332.85

27.39 29.20

1.47  10
2 1.36  10
2

0.5226

2.495  10
2

339.85

32.22

1.19  10
2

0.5379

2.566  10
2

346.75

35.07

1.04  10
2

4.416  10
2

3.439  10
3

303.25

38.66

8.57  10
4

4.802  10
2

3.739  10
3

308.05

39.35

8.21  10
4

5.406  10
2

4.206  10
3

316.85

40.16

7.53  10
4


2

5.868  10 6.264  10
2


3

4.564  10 4.871  10
3

324.55 331.35

40.43 40.34

6.94  10
4 6.43  10
4

6.761  10
2

5.255  10
3

339.75

39.86

5.81  10
4


2


3

349.15

38.88

5.16  10
4

7.245  10 KCl

KSP,DMSO


3

5.630  10


4


4

2.735  10

2.137  10

302.15

12.29

3.67  10

5.615  10
3

4.386  10
4

307.05

13.59

3.65  10
4


3


4

313.75

15.39

3.59  10
4

8.707  10

6.799  10


2

1.225  10 1.588  10
2

9.563  10 1.239  10
3

319.55 328.65

16.95 19.33

3.51  10
4 3.36  10
4

1.978  10
2

1.543  10
3

335.75

21.09

3.21  10
4


2


3

342.65

22.68

3.06  10
4

354.35

25.05

2.78  10
4

2.211  10


2

2.662  10


4

1.724  10


3

2.075  10

3.4. Electrolyte Activity Coefficient Models. Unlike nonelectrolyte solution, the limit of ionization and electric neutrality should be described with much more detailed and complex criteria for the excess Gibbs free energy of electrolyte solution.22 At present, the activity coefficient models that are commonly employed for electrolyte can be divided into two groups. The first one contains models of empirical extensions of the Debye
H€uckel law,22 such as Bromley’s model23 and Pitzer’s model,24 which correct the deviation from the Debye
H€uckel law by additional terms to account for the ionic strength dependence of the long-range forces among ions. The other group of models assume contributions of the mole excess Gibbs free energy is a combination of long-rang electrostatic force described by the medication form of the Debye
H€uckel law, and short
rang intermolecular force described by the local composition activity coefficient models such as the Wilson model,25,26 NRTL model,27 and UNIQUAC model.28 Some models will also add an additional middle
range term to account for the specific

Figure 1. Molality solubility of salts in DMSO: (black circle) KNO3; (open circle) Treivus;34 (blue triangle) KBr; (open triangle) Gopal;32 (red square) NaCl; (open square) Unni;33 (purple diamond) KCl; (open diamond) Unni.33

ion
ion and ion
molecule interactions.29,30 In this work, the Pitzer’s model, E-Wilson, E-NRTL, and E-UNIQUAC equations are employed to model the activity coefficient of the salt in DMSO at different equilibrium temperature. The reader can refer to the literature24,26
28 for the exact mathematical forms and the meaning of various variables for these models.

4. RESULTS AND DISCUSSION 4.1. Solubility Data. The measured solubility data of potassium nitrate (KNO3), potassium chloride (KCl), potassium bromide (KBr), and sodium chloride (NaCl) at different temperatures are summarized in Table 1. For better understanding and comparison, the experimental solubilities were first reported as molality, m (mol 3 kg
1), which is defined as the moles of the solutes dissolved in a fixed amount solvent of 1000 g at saturation. Figure 1 shows the molality solubility of the four salts in DMSO at different temperature. As expected, the solubilities of the four salts all increase with temperature but differ considerably at constant temperature. KNO3 shows the highest solubility in DMSO while KCl presents the lowest solubility which is almost 3 orders of magnitude lower than that of KNO3 at 302 K. At constant temperature, the solubility order is concluded as KNO3 > KBr > NaCl > KCl. Miller and Parker31 had made general discussions on the solubility of inorganic salts in nonaqueous solvents and the factors affecting the solubility. The salt solubility order determined in this work confirms their conclusion that small anions are very poorly solvated in dipolar aprotic solvents because of the difficulty in forming hydrogen bonds. The measured solubility in this work has been compared with the published literature values whenever possible. Gopal,32 Unni,33 and Treivus34 have measured the solubility of KBr, NaCl, and KCl, and KNO3 in DMSO at different temperature, respectively. Those reported solubility data are also plotted in Figure 1 for comparison. It can be seen from the figure that our measured solubility data of KNO3, KBr, and NaCl agrees with the literature values quite well. The solubility of KCl reported by Unni33 is much higher than ours at low temperature but well consistent with our results at high temperature. Unni measured 7022

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the solubility by a gravimetric method but the uncertainty is not known. As pointed out by Labban,35 trace water in the organic solvent can considerably increase the solubility. It is suspected that the higher solubility may be due to such contamination. In addition, the molality solubility measured in this work shows strong linear dependency on temperature. Thus, the measured molality solubility data was empirically correlated as the linear function on temperature: m ¼ A þ BT

ð20Þ

Table 2. Regressed Parameters and the Squared Correction Coefficients R2 for Equation 20 solute

A

104B (K
1)

R2 0.9892

KNO3


0.5631

52.04

KBr


0.2825

23.73

NaCl


0.1409

6.133

0.9964

KCl


0.1356

4.601

0.9945

0.9960

Linear regressions for eq 20 are conducted and the regressed parameters A and B for each salt are listed in Table 2 together with the squared correction coefficients R2 values of the correlations. Good correlation results are obtained with R2 > 0.99 except for KNO3. However, a greatly improved R2 of 0.9992 is obtained for KNO3 with a second order polynomial regression, and the equation is m = 3.532
(1.992  10
2)T þ (3.847  10
5)T2. The parameter B in eq 1 reflects the influence of temperature on the solubility. Thus it can be concluded that temperature possesses the greatest effect on the solubility of KNO3, followed by that of KBr, while the solubility of KCl shows the weakest dependence on temperature. To model the equilibrium solubility, the solubility products of the salts in DMSO are required. In this work, it is estimated from the solubility products in water KSP,W and the solvent Gibbs energy of transfer from water to DMSO, ΔtrGWfDMSO(T) as shown in eq 10. The calculated KSP,DMSO and KSP,W values for each salt at different temperatures are listed in Table 1 and shown in Figure 2, respectively. Because of the lower solubility of the salts in DMSO, KSP,DMSO for each salt is much smaller than KSP,W.

Figure 2. Temperature dependency of solubility product of the four salts in water and DMSO: (a) KNO3; (b) KBr; (c) NaCl; (d) KCl; (red circle) KSP,W; (blue triangle) KSP,DMSO. 7023

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As can be seen from Figure 2, the solubility products in DMSO decrease with the increase of temperature while those in water increase with the increase of temperature, although the experiments show the solubilities in both water and DMSO increase with temperature. There is a maximum for the solubility product of NaCl in water at around 325 K, but this is not observed for DMSO. Now the mean activity coefficient of the salts can be calculated from eq 7 with the knowledge of molality solubility and solubility product. Since the molality concentration of the salts in DMSO is fairly small, we first tried to model the mean activity coefficient with the simple Debye
H€uckel equation,22 which only accounts for the long-range interaction caused by electrostatic forces. The calculation results at 320 K are presented in Table 3. Big errors for the mean activity coefficients predicted by the Debye
H€uckel equation are observed. This implies that the solubility in our case is controlled not only by the standard-state properties of the ions, salts, and solvent but also by the complex intermolecular interactions, and the Debye
H€uckel equation is inadequate to describe such interactions in the solution. Models such as the Pitzer,24 E-Wilson,25,26 E-NRTL,27 and E-UNIQUAC28 equations that also take the short-range interactions caused by intermolecular interactions into account in addition to the long-range electrostatic forces are considered then. In these models, there are adjustable energy parameters that should be regressed from the experimental data. The unsymmetrical mean activity coefficients of the salts on molality scale at equilibrium are calculated with the models mentioned above, and the binary energy parameters are optimized by fitting the experimental data with the following objective function: Min f2 ¼

NP

calc 2 ∑ ðln γexpt i, (
ln γi, ( Þ i¼1

DMSO. The Levenberg
Marquardt method is used as the optimization algorithm for minimizing eq 21. The temperature-dependent F and D for estimating the Debye
H€uckel coefficient of DMSO are calculated using the expression recommended by Yaws36 and Gabrielian,37 respectively. The volume and surface area parameters of the ions and solvent in the E-UNIQUAC equation are taken from the handbook.38 The optimized parameters for the Pitzer, E-NRTL, and E-Wilson are summarized in Table 4 and those for the E-UNIQUAC equation are listed in Table 5. The goodness of the fit of the models to the experimental data can be expressed by the standard deviations σ, which is calculated according to the following definition: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u NP u expt 2 u ðln γi, (
ln γcalc i, ( Þ ð22Þ ti ¼ 1 σ ¼ NP ðNP
nÞ

∑

The calculated standard deviations σ of the electrolyte activity coefficient models are also listed in Table 4 and shown in Figure 3. As can been seen from the figure, the Pitzer equation gives the best results in terms of standard deviation followed by E-NRTL equation. Although there are three parameters involved for each salt
solvent systems, the E-UNIQUAC presents worse results than the two-parameter models of Pitzer and E-NRTL equation. The E-Wilson equations proposed by Xu and Macedo26 showed the worst results. However, if the nonrandom factor R in the E-Wilson model could be considered as an additional adjustable parameter, as discussed by Xu and Macedo,26 the model accuracy will be greatly improved especially for a 1:1 type electrolyte. Therefore the attempt of adjusting R to increase the accuracy of E-Wilson equation is made then. The optimized parameters of the three parameter E-Wilson equation (E-Wilson2), together with the calculated standard deviations σ for each salt are also listed in Table 4. As expected, E-Wilson2 provides better results than the original one with respect to the standard deviations σ, which is even much smaller than the Pitzer equation. In addition, we also tried to set the nonrandom factor R in the E-NRTL equation as the third adjustable parameter, but no improvement was observed.

ð21Þ

where the superscripts expt and calc stand for the experimental and the calculated activity coefficient, respectively. Np is the number of data points for each system. The natural logarithm of experimental activity coefficient is determined from eq 7 with the experimental solubility and the solubility product of the salt in Table 3. The Mean Activity Coefficients of the Salts in DMSO Calculated by the Debye
H€ uckel Equation at 320 K m

Ksp

γ(a

γ

RD, %

KNO3

1.097

0.099

0.287

0.449


56.50

KBr

0.486

0.017

0.270

0.505


87.19

NaCl

0.051

0.001

0.528

0.694


31.35

KCl

0.024

0.000

0.790

0.753

4.72

DHb

Table 5. Optimized Parameter Values Δuki (=Δuik) of E-UNIQUAC Equation

c

a

Calculated from eq 7. b Calculated by the Debye
H€uckel equation. c RD = (γ(
γDH)/γ(  100.

k

i

þ




Δuki

k þ

i

Δuki

K

Cl

1465.18

Na

DMSO


12956.3

Kþ

Br



23111.6

Cl


DMSO


8011.01

Kþ

NO
3


21972

Br


DMSO


5816.82

Naþ

Cl


1443.23

NO
3

DMSO


4113.75

Kþ

DMSO


5815.93

Table 4. Correlation Results of the Pizter’s, E-NRTL, E-Wilson, and E-UNIQUAC Equations: Parameter Values and The Standard Deviation σ Pizter’s β

β

NaCl


54.83

KCl KNO3 KBr

E-NRTL

E-Wilson

E-Wilson2

10 σ

Δgem

Δgme

10 σ

Δλem

Δλme

10 σ

78.11

2.42


5241.0

10463.9

3.14


1622.13


92399.3

19.61


332.5
1.140

430.3 6.267

12.31 1.15


8221.4
54.93

16409.9 100.75

18.92 0.87


8331.97 1378.58

17862.5 3386.61


5.437

15.78

0.77


342.59

660.43

0.73


592.58

1390.06

0

1

2

2

7024

Δλem

Δλme

102σ

5.300  10
6

11356.9


1.660

0.85

11.93 2.08


7.027  10
3
1.919  10
2


6168.77 3098.72

12575.3 120.54

2.05 0.13

17.54


2.374  10
2


749.764

2491.49

0.17

2

R

dx.doi.org/10.1021/ie102134g |Ind. Eng. Chem. Res. 2011, 50, 7019–7026

Industrial & Engineering Chemistry Research

ARTICLE

Institute of Technology) are also greatly acknowledged for the helpful discussions.

Figure 3. Standard deviations of the thermodynamic models for mean activity coefficient calculations.

5. CONCLUSIONS The solubility of potassium nitrate (KNO3), potassium chloride (KCl), potassium bromide (KBr), and sodium chloride (NaCl) in dimethyl sulfoxide (DMSO) is measured in the temperature range of 302
354 K. At constant temperature, KNO3 shows the highest solubility in DMSO followed by KBr, while the KCl shows the lowest solubility which is about 3 orders of magnitude lower than KNO3 at 302 K in molality. The solubility of all the salts in DMSO increases with temperature and linear dependencies of molality solubility on temperature are observed except for KNO3, which is more suitable to be considered as a second order polynomial dependency. The effect of temperature on solubility for the salts follows the same order as the solubility, that is KNO3 > KBr > NaCl > KCl. The solubility products of the salts in DMSO are estimated from the solubility products in water and the Gibbs energy of transfer of solvent from water to DMSO. The results show that solubility products of the salts in DMSO all decrease with the increase of temperature, which is quite different from those in water. The mean activity coefficients of the salt in DMSO are obtained from the experimental solubility data and the calculated solubility products. The mean activity coefficient is modeled by three-parameter electrolyte models of modified E-Wilson (E-Wilson2) and E-UNIQUAC equations and two-parameter models of the Pitzer, E-NRTL, and E-Wilson equations. The interaction energy parameters for each model are optimized. The results show that E-Wilson2 equation gives the best descriptions of the activity coefficient of the electrolytes in solution in terms of the calculated overall standard deviation, followed by the Pitzer, E-NRTL, E-UNIQUAC, and the two-parameter E-Wilson equation. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: 780-492-5963. Fax: 780-492-2881. E-mail address: [email protected].

’ ACKNOWLEDGMENT B. Long would like to thank Ms. Lyenna Wood for our excellent collaboration in writing this Paper. Prof. John M. Shaw (University of Alberta) and Dr. Miyi Li (Beijing

’ NOMENCLATURE A,B empirical parameters of eq 20 molar heat capacity (J 3 mol
1 3 K
1) CP D dielectric constant Δg adjustable cross interaction energy parameters in NRTL equation (J 3 mol
1) G gibss energy (J 3 mol
1) H enthalpy (J 3 mol
1) Ksp solubility product m molality solubility (mol 3 kg
1) n number of adjustable parameters in the model NP number of data points for each system the squared correction coefficients R2 x mole fraction solubility S entropy (J 3 mol
1 3 K
1) T absolute temperature (K) Δu adjustable cross interaction energy parameters in UNIQUAC equation (J 3 mol
1) Greek Symbols

R

activity, nonrandomness parameter in the E-NRTL and E-Wilson equations β0, β1 Pitzer’s model parameters γ( mean activity coefficient on molality scale Δλ adjustable cross interaction energy parameters (J 3 mol
1) F density (g 3 cm
3) σ standard deviations Superscripts

0 DH calc expt

reference Debye
H€uckel equation calculated experimental

Subscripts

em, me M MX S tr W X i, j, k

interaction between electrolyte and solvent molecule cation salt crystal solvent transfer water anion compound in the model

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