Thermodynamics of Phase Equilibria in Aqueous Strong Electrolyte

CORRELATION pubs.acs.org/IECR

Thermodynamics of Phase Equilibria in Aqueous Strong Electrolyte Systems Mi-Yi Li,† Li-Sheng Wang,*,† and J€urgen Gmehling‡ † ‡

School of Chemical Engineering and the Environment, Beijing Institute of Technology, 100081 Beijing, China Department of Industrial Chemistry, Institute for Pure and Applied Chemistry, Carl von Ossietzky University Oldenburg, D-26111, Oldenburg, Germany ABSTRACT: The middle range (MR) parameters of the LIQUAC model were improved as a function of temperature to allow the calculation of the activities of all the species in the solution over a wide temperature and concentration range. New parameters were recorrelated by using the experimental data of osmotic coefficients, vapor-liquid equilibria (VLE), mean ion activity coefficients and salt solubilities from the Dortmund Data Bank (DDB). The coordinates of the eutectics, the peritectics, and the ice-water equilibria of the aqueous electrolyte systems were represented with the model. Standard Gibbs energies of formation, standard enthalpies of formation, and heat capacities of various solids were retrieved from the fitting of their solubility products with respect to temperature or estimated using the group contribution method. The schematic phase diagrams for the salts in aqueous systems are presented in this work. A good agreement of calculated results with the published measurements is observed.

1. INTRODUCTION Electrolyte solutions are commonly encountered in many chemical industrial processes including desalination, wastewater treatment, extractive distillation, and biotechnology. Strong electrolyte systems are applied in refrigeration and air conditioning industries, especially the aspects of energy saving and protection of the environment. The widely used lithium salts for those applications are lithium chloride and lithium bromide. They are used in air conditioning for the dehydration of gases, catalyst chemistry, lithium batteries, and refrigeration applications. Based on those applications, the calculation of the thermodynamic properties of the strong electrolytes in aqueous solutions is presented using the LIQUAC model1,2 with adequate experimental data stored in Dortmund Data Bank (DDB).3 In the thermodynamic theories of the electrolyte system, specific discussions had been made by Robinson and Stokes,4 Pitzer,5 and Barthel.6 The knowledge of the electrolyte systems was well presented in those publications. The Debye-H€uckel (DH) limiting theory brought out a starting point of understanding the electrolyte systems, and an extensive amount of research work had been performed to develop equations suitable for the calculation of those systems. Stokes and Robinson4 proposed an ionic hydrate model to apply the DH theory to higher concentrations. Pitzer7 developed a series of semiempirical equations for the calculation of aqueous electrolyte systems by using the pressure equation of statistical thermodynamics and assuming the interaction coefficients of ionic strength dependence, which could represent the experimental results to a high ionic strength. Based on the Ornstein-Zernike equation, Blum8 used the concept of hard sphere solutions and square-well potentials to solve the primitive model of electrolyte solutions and obtained the mean spherical approximation (MSA). Henderson9 applied the statistical associating fluids theory (SAFT) into the non-primitive model of electrolyte systems. It has the ability of describing any molecule of interest with meaningful parameters and appropriate expression. r 2011 American Chemical Society

Furthermore, a set of empirical equations for the electrolyte systems was proposed based on local composition model like UNIQUAC (UNIFAC) or NRTL. Renon et al.10 suggested a system of equations based on the ionic atmosphere theory of DH, Born model contribution, and local compositions of the nonrandom two-liquid (NRTL) model to develop the expressions of isothermal activity coefficients for the completely dissociated electrolyte systems. Chen et al.11 combined the electrostatic function of the Pitzer model with an extension of the NRTL equation, which was based on two assumptions: (1) repulsion assumption of like-charge for the local cell with anion or cation as the central species; (2) local electroneutrality for the cell with noncharged species as the center. Sander et al.12 introduced an extended UNIQUAC term adding a simplified DH term to calculate the vapor-liquid equilibria (VLE), liquid-liquid equilibria (LLE), and solid-liquid equilibria (SLE) behavior for the mixed solvent electrolyte systems. The concentration-dependent UNIQUAC parameters were adjusted for ion-solvent interactions. Kikic et al.13 substituted the UNIQUAC term by the UNIFAC group contribution model to discuss the electrolyte systems. The LIQUAC model was originally proposed by taking into account of all the interactions between species presenting in the solutions from a statistical thermodynamic point of view by Li et al.1,2 This model had been applied to calculate the VLE behavior, osmotic coefficient, and mean ion activity coefficient for a large number of single and mixed solvent electrolyte systems with high accuracy in the past few years. It was also successfully used to predict salt solubilities in aqueous solutions by Li et al.14 The results matched well with experimental results, and the obtained deviations were less than 4%. In 2009, Huang et al.15 deduced a series of theoretical formulas to calculate the salt Received: July 5, 2010 Accepted: January 21, 2011 Revised: December 22, 2010 Published: February 21, 2011 3621

dx.doi.org/10.1021/ie101428j | Ind. Eng. Chem. Res. 2011, 50, 3621–3631

Industrial & Engineering Chemistry Research

CORRELATION

solubilities not only in water but also in organic or mixed organic electrolyte systems. The results in the water-methanol electrolyte systems were in good agreement with measurements. In this work, experimental data for the phase equilibrium behavior of aqueous strong electrolytes (LiCl, LiBr, NaCl, and NaBr) in DDB were used for the further study on the application of the LIQUAC model. The thermodynamic properties of inorganic salts at standard state were calculated. The group contribution model was also used to estimate the thermodynamic properties and compared with the calculated results. The ternary systems with mixed salts were predicted with the correlated parameters obtained from binary systems.

2. THERMODYNAMIC BACKGROUND FOR THE STRONG ELECTROLYTE SYSTEM It is a hard task to extend the thermodynamic theories for the nonelectrolyte system to the electrolyte system. All the empirical models to express the thermodynamic equilibrium behavior of electrolyte systems are based on the construction of an excess Gibbs energy model. The definition of the chemical potential for nonelectrolyte species was also introduced into the electrolyte systems. For solvent s, the chemical potential is expressed as μs ¼ μ°s þ RTlnðxs γs Þ

ð1Þ

Where xs is the mole fraction of solvent in the solution, μs° is the standard chemical potential of the pure liquid at system temperature T and pressure P, and γs is the symmetric activity coefficient, where γs f 1 as xs f 1. For the electrolyte MX, the unsymmetrical convention and molality scale are adopted.6

μMX ¼ μMX þ RT lnðmMX γMX Þ

ð2Þ

By assuming complete dissociation of strong electrolyte MX into cation Mvþ and anion Xv-, the chemical potential of solute MX in the solution is related to those ions by

μMX ¼ vþ ½μMvþ þ RT lnðmMvþ γMvþ Þ

þ v- ½μXv- þ RT lnðmXv- γXv- Þ

ð3Þ

Where vþ and v- are the stoichiometric coefficients. Superscript * indicates the standard-state based on the unsymmetrical convention, m is the concentration in molality scale, and γ* is the molal unsymmetrical activity coefficient. The standard state for ions in the solution is defined as the hypothetical ideal dilute solution in solvent s at unit concentration. Usually the unit concentration for ions means m f 1. In the real solution, the activity coefficients for ions were normalized by γ* f 1 as m f 0. However, only the chemical potential of the electrically neutral salt is experimentally accessible. Thus, the mean ion activity coefficient and standard state of solute MX are used by following convention: v ¼ vþ þ v

ð4Þ

γ( ¼ ðγMvþ vþ 3 γMv- v- Þ1=v

ð5Þ

m( ¼ ðmMvþ vþ 3 mXv- v- Þ1=v

ð6Þ

μMX

¼

vþ μMvþ

þ v- μMvþ

Figure 1. (a) Contributions to the activity coefficient of water in the system of LiCl þ H2O at 298.15 K: —, ln γcalc; 3 3 3 , ln γMR; ---, ln γLR; and - 3 -, ln γSR. (b) Contributions to the mean ion activity MR coefficient of LiCl þ H2O at 298.15 K: —, ln γcalc ( ; 3 3 3 , ln γ( ; ---, LR SR conv ln γ( ; - 3 -, ln γ( ; - 3 3 -, ln γ( ; and 0, 4, experimental data.25

2.1. Vapor-Liquid Equilibrium (VLE). By assuming ideal vapor-phase behavior and neglecting the Poynting correction, VLE can be calculated using the simplified equation:

xs γs Pssat ¼ ys P

ð8Þ

Where Psat and P are the vapor pressure of the pure solvent s and solution in kPa at system temperature, and ys is the vapor-phase mole fraction of solvent s. 2.2. Osmotic Coefficients. In the single solvent system, the molal osmotic coefficient for the solvent s at system T and P is calculated from φ¼ -

ð7Þ 3622

lnðxs γs Þ Ms

∑ mion ion

ð9Þ

dx.doi.org/10.1021/ie101428j |Ind. Eng. Chem. Res. 2011, 50, 3621–3631


CORRELATION

where Ms is the molar mass of the solvent s (kg 3 mol-1), and subscript ion indicates all the ions in the solution. 2.3. Mean Ion Activity Coefficient. Based on the definition of the chemical potential for the solute as mentioned above, the mean ion activity coefficient can be calculated by using eq 5. In the binary system, the mean ion activity coefficient can be correlated from the Gibbs-Duhem equation with molal osmotic coefficient at system temperature T and system pressure P. 2.4. Solid-Liquid Equilibrium (SLE). The equilibrium constant for the solubility of the solid electrolyte MνþXν- 3 nH2O can be derived starting from the following reaction: ΔG

Mvþ Xv- 3 nH2 OðsolidÞ S vþ MðliquidÞ þ v- XðliquidÞ þ nH2 OðliquidÞ

ð10Þ

Where the condition of chemical equilibrium is given ΔG ¼ vþ μMvþ ðliqiudÞ þ v- μXv- ðliquidÞ þ nμH2 OðliquidÞ - μMvþ Xv-

3 nH2 OðsolidÞ

¼0

¼ ¼ ¼

the past, this model was used to describe the activity coefficients of all the species of the electrolyte systems investigated in this work. In the LIQUAC model, the excess Gibbs energy is defined as a sum of three contributions: GE ¼ GELR þ GEMR þ GESR

ðaþ Þvþ ða- Þv- ðaH2 O Þn ðmþ γþ Þvþ ðm- γ- Þv- ðxH2 O γH2 O Þn ðm( γ( Þv ðxH2 O γH2 O Þn

ð12Þ

MR þ ln γSR ln γs ¼ ln γLR s þ ln γs s

ln γLR s ¼

þv

ð14Þ Δh°ðTref Þ ¼ vþ Δf h°Mvþ ðTref Þ þ v- Δf h°Xv- ðTref Þ þ nΔf h°H2 O ðTref Þ - Δf h°MX

3 nH2 O

ðTref Þ

ð15Þ

3 nH2 O

ðTref Þ

0

ð19Þ

ð20Þ

0

0

jsol ¼

I ¼

xsol V sol

∑sol xsol V sol 0

∑

1 mion z2ion 2 ion

ð21Þ

ð22Þ

0:5 A ¼ 1:327757 105 dm =ðDTÞ1:5

ð23Þ

0:5 =ðDTÞ0:5 b ¼ 6:35969dm

ð24Þ

where x0s is the salt free mole fraction of solvent s in the solvent mixture, Vsol (m3 3 mol-1) is the molar volume of pure solvent s, and j0sol is the salt-free volume fraction of solvent, sol, in the liquid phase. The symbol ∑sol indicates the sum of all the solvent components in the solution. T is the absolute temperature, and D represents the dielectric constant for mixed solvents. For binary mixed solvents, Oster’s mixing rule is used

Δc°p ðTref Þ ¼ vþ c°p, Mvþ ðTref Þ þ v- c°p, Xv- ðTref Þ þ nc°p, H2 O ðTref Þ - c°p, MX

∑sol jsol dsol

dm ¼

° Δg°ðTref Þ ¼ vþ Δf gM vþ ðTref Þ ° Δf gX° v- ðTref Þ þ nΔf gH° 2 O ðTref Þ - Δf gMX ðTref Þ 3 nH2 O

pffiffi pffiffi 2AMs dm ½1 þ b I - ð1 þ b I Þ-1 3 b ds pffiffi - 2 lnð1 þ b I Þ

ð18Þ

The LR term is calculated using the Debye-H€uckel theory as modified by Fowler and Guggenheim.17 I is the ionic strength of the solution, Ms (kg 3 mol-1) is the molar mass of the solvent s, ds (kg 3 m-3) is the density of the pure solvent s, and dm is the density of the mixed solvents calculated using the following equations:

Using the Gibbs-Helmholtz equation, the temperature dependence of the solubility product can be expressed by: Δg°ðTref Þ Δh°ðTref Þ 1 1 ln Ksp ðTÞ ¼ RTref R T Tref 2R 3 T Z ° 1 T 4 Tref Δcp ðTÞ5 dT ð13Þ þ R Tref T2

-

ð17Þ

The first term on the right side of the eq 17 represents the long-range (LR) interaction contribution caused by the Coulomb electrostatic forces. The second term represents the middle range (MR) interaction contributions such as charge-dipole interactions and charge-induced dipole interactions. The third term expresses the contribution of noncharge interactions which is identical to the short-range (SR) interaction. The original UNIQUAC model has been chosen to describe the SR term. For solvent s, the activity coefficients based on mole fraction scale are calculated by

ð11Þ

The subscript liquid indicates the ions in the solutions and solid indicates the solid phase of the salt, while n is the stoichiometric number of water in the hydrated crystal. In the liquid phase, the concentration of cations and anions depends on the solubility product, which indicates the solid-liquid equilibrium. Since the activity of solid phase is defined as unity, the solubility product is obtained by substituting eq 3 into eq 11: Ksp

2.5. GE Model. Based on the success of the LIQUAC model in

D ¼ D1 þ ½ðD2 - 1Þð2D2 þ 1Þ=2D2 0

- ðD1 - 1Þj2

ð16Þ

In eqs 14-16, Δf g°(Tref) Δf h°(Tref) and cp°(Tref) are the standard Gibbs energies of formation, standard enthalpies of formation, and standard heat capacities. All these values in the standard state for ions in aqueous solution and for crystalline salts and water are available from references, e.g., from Wagman16 or can be found in data banks such as the DDB.3

ð25Þ

where subscript 1 indicates one solvent in binary mixture, and 2 indicates another one. For a multicomponent mixture, D can be estimated by D¼ 3623

∑sol jsol Dsol 0

ð26Þ



CORRELATION

Table 1. Correlated MR and SR Parameters of the LIQUAC Model in this Work b(0) i,j

b(2) i,j

c(0) i,j

c(1) i,j

521.625

-0.77431

-0.94179

-2272.12

5.04546

0.00532

5218.87

-249.343

1.16286

-0.75684

4423.11

-11.1686

-557.072

-8.73550

0.18154

-3.97308

0.00790

0.00000

1737.02

-4.99788

-133.195

24654.3

0.19300

75.6639

0.00738

-0.36051

-1311.64

1.47555

81502.1

25129.6

H2O

7757.10

-519.656

1028391

-0.0100

-39.6558

0.06355

0.00475

810.5744

Naþ

H2O

-0.01627

-8862.60

-42.5683

-0.00348

85.4429

-0.28251

-195.402

-5068.68

Cl-

H2O

1190346

960.6591

16023.36

0.02900

-3.00596

0.02800

-236.621

-894.110

Br-

H2O

-75.745

1414.93

-0.00183

0.00057

-33.0197

0.09195

-63.1354

689.707

i

j

Liþ

Cl-

0.43416

þ

Br-

0.58325

Naþ

Cl-

þ

Na

Br-

Liþ

Li

b(1) i,j

c(2) i,j

ai,j

aj,i

charged ions and between solvent molecules are ignored. This leads to the following expression for the middle-range term: Ms MR ½Bsol, ion ðIÞ ln γs ¼ Bs, ion ðIÞmion Mm sol ion ion

∑

0

∑∑

0

þ IBsol, ion ðIÞxsol mion - Ms

∑c ∑a ½Bca ðIÞ þ IBca ðIÞmcma 0

ð27Þ Bsol, ion ðIÞ ¼ bsol, ion þ csol, ion expð- 1:2I 1=2 þ 2dsalt IÞ ð28Þ Bc, a ðIÞ ¼ bc, a þ cc, a expð- I 1=2 þ dsalt IÞ Mm ¼

∑xsol Msol 0

ð29Þ ð30Þ

where dsalt is the parameter with adjusted values,20 bsol,ion and c sol,ion are the MR interaction parameters between solvents and ions, and bc,a, cc,a, and dsalt are the parameters between cations and anions. B0 (I) is equal to dB(I)/dI. Mm (kg 3 mol-1) is the mean molar mass of the mixed solvents. The SR term is calculated by the UNIQUAC model: Vs Vs SR ln γs ¼ 1 - Vs þ ln Vs - 5qs 1 - þ ln Fs Fs 8 0 1 2 39 > > qi xi ψi, s = < Bi C 6 qi xi ψs, i 7 þ qs 1 - ln@ A- 4 5 > i qi xi ð qk xk ψk, i Þ > ; : i

∑ ∑

∑

∑k

ð31Þ

Figure 2. Comparison of experimental values of system pressure with calculated results using the LIQUAC model for aqueous binary systems at different temperatures. (a) 0, NaCl; O, NaBr; (b) 0, LiCl; O, LiBr; 1 3 3 3 , Li et al.; and —, this26work. The symbols represent published data obtained from Patil et al., which is stored in DDB.

The MR term was originally proposed by Li et al1,2 with the objective to represent the indirect effects between pair species. On the basis of approximate results for the radial distribution functions, the interactions between equally

V s ¼ rs =

∑i ri xi

ð32Þ

Fs ¼ qs =

∑i qi xi

ð33Þ

ψi, j ¼ expð- ai, j =TÞ

ð34Þ

where ri and qi are the van der Waals volumes and surface areas, ai,j represent the UNIQUAC interaction parameters, whereby ai,j is different from aj,i, and xi is the mole fraction of species i in the solution. In these equations, i and j cover all solvents and ions. 3624



CORRELATION

Table 2. Average Absolute Relative Deviation (AARD) Obtained in this Work AARD (%) molal osmotic coefficient (φ)

mean ion activity coefficients (γ()

system pressure

ice-water equilibrium

salts solubility

LiCl þ H2O

1.5

1.7

1.6

1.3

1.8

LiBr þ H2O

1.6

2.0

1.7

1.1

1.7

NaCl þ H2O

0.3

0.5

0.4

0.2

0.5

NaBr þ H2O

0.4

0.6

0.4

0.4

0.4

Figure 3. Experimental and calculated molal osmotic coefficients for aqueous electrolyte systems at 298.15 K: —, model; 0, NaCl; and 4, NaBr.25

Figure 4. Experimental and calculated mean ion activity coefficients for aqueous electrolyte systems at 298.15 K: —, model; 0, LiCl;and 4, LiBr.25

For ion j, each part of the activity coefficient is given based on the unsymmetrical convention on molality scale. pffiffi z2j A I r, LR ð35Þ pffiffi ln γj ¼ 1þb I

electrolyte solution is the molality scale (mol 3 kg-1). The standard state of ion j is defined as the hypothetical ideal solution at unit molality. In this hypothetical ideal solution, mj = m°j = 1 mol 3 kg-1 and γj* = 1. Using the superscript * as the molality scale, the reference state at infinite dilution for ion j is normalized at x0 sf 1 and I f 0. Where the pure solvent is as γ*f1 j considered as the reference solvent. The variables given in eqs 18-38 were already defined in detail in the paper of Li et al.1,2 Based on molality scale, the complete expression of the activity coefficient of ion j can be obtained from: r, LR r, MR r, SR þ ln γj þ ln γj Þ ln γj ¼ ðln γj

r, MR

" þ

z2j

ln γj #

2Mm þ

¼ ðMm Þ-1

∑sol Bj, sol ðIÞxsol 0

∑sol iion ∑ Bsol, ion ðIÞxsolmion þ ∑ Bj, ion ðIÞmion ion 0

z2j

!

0

∑c ∑a Bca ðIÞmc ma - Bj, s ðIM¼ 0Þ 0

2

ð36Þ

- lnðMs =Mm þ Ms

s

r, SR

ln γj

SR ¼ ln γSR j - ln γj ðBÞ

ð39Þ

ð37Þ

" !# r r r q r q j j j s j s ln γSR þ ln - 5qj 1j ðBÞ ¼ 1 - þ ln rs rs rs qj rs qj þ qj ð1 - ψj, s - ln ψs, j Þ

∑ mion Þ ion

ð38Þ

where, the superscript r indicates the unsymmetrical convention for ions based on the mole fraction scale. In eq 37, the term ln γSR j is the same as in eq 31. The terms lnγSR j (B) and Bj,s(I = 0) /Ms in eq 36 represent the reference state for ion j at infinite dilution based on the mole fraction scale. The subscript s indicates the reference solvent. Moreover, the usual concentration scale of the

3. CALCULATIONS AND RESULTS 3.1. Improvement for the LIQUAC Model and Correlations. Polka et al.2 used the LIQUAC model to calculate 185

mixed and 362 single solvent electrolyte systems. The results showed a considerable improvement at high concentration for binary systems comparing with Pitzer, Chen, and Bromley models. It also showed a better capability in describing the VLE behavior than the models of Sander, Macedo, and Chen. Yan et al.18 found that the LIQUAC model provides superior results comparing with the e-NRTL, e-UNIQUAC, and the electrolyte UNIFAC models when correlating VLE of the system 3625



CORRELATION

Figure 5. Schematic phase diagrams for the aqueous electrolyte systems: (a) O, ice; 0, LiCl*2H2O; 4, LiCl*3H2O; r, LiCl*H2O; and ), LiCl*5H2O; (b) O, ice; 0, LiBr*2H2O; 4, LiBr*3H2O; r, LiBr*H2O; and ), LiBr*5H2O; (c) O, ice; 0, NaCl; and 4, NaCl*2H2O; (d) O, ice; 0, NaBr; 4, NaBr*2H2O; and ), NaBr*5H2O. The symbols represent further published data stored in DDB;3 —, calculated results.

acetone-methanol with KBr and ZnCl2. The LIQUAC model was used to predict the solubilities of salts consisting of various cations (Naþ, Kþ, and NH4þ) and anions (F-, Cl-, Br-, I-, and SO42-) and their mixtures in aqueous solutions by Li et al. The predicted average relative deviations for single salts are less than 4%, and those for mixed salt solutions are less than 10%. Thus the comparison with other models is not included here, and the main object of this work is applying the LIQUAC model to describe the phase behavior of different systems and correlating more reliable parameters and standard thermodynamic properties. For the representation of electrolyte systems, two sets of adjustable parameters are needed in the LIQUAC model. One set of parameters are the Bi,j parameters for the contribution of the MR term, and the others are the ai,j, aj,i parameters for the SR term contribution. The relative magnitudes of contributions to the activity coefficients of solvent and solute at 298.15 K and atmospheric pressure for the LiClþH2O system are shown in Figure 1. The LR term representing the DH theory is important

at low concentrations for solute. However, it only has a little contribution on the activity coefficient of water. The contributions of the MR term to the activity coefficients of solute and solvent are dominant at moderate and high salt concentrations. The SR term also presents a little contribution to the activity in Figure 1b represents the last term in coefficients, and ln γconv ( eq 39. From eqs 27-29 and 36, one can obtain that the MR is temperature independent for all the species in the electrolyte system. To extend the capability of the MR term in describing the activity coefficients to a wider temperature range, an improvement is given 1 Tref ð0Þ ð1Þ 1 ð2Þ þ bi, j ln bi, j ¼ bi, j þ bi, j ð40Þ T Tref T ci, j ¼ 3626

ð0Þ ci, j

ð1Þ þ ci, j

1 1 T Tref

ð2Þ þ ci, j

Tref ln T

ð41Þ



CORRELATION

Table 3. Standard Thermodynamic Properties of Species for the Correlations in this Work Δfg° (kJ mol-1)

Δfh° (kJ mol-1)

LiCl

-384.015a

-408.26a

48.030 a

LiCl 3 H2O

-631.8

-698.499 -712.5a

89.063

LiCl 3 2H2O LiCl 3 3H2O

-996.125

130.095

-1114.98

-1287.94

171.128

-1592.2

-1871.19

253.193

LiCl 3 5H2O

a

-874.244

LiBr

-341.631

LiBr 3 H2O

-595.885

LiBr 3 2H2O

cp° (J mol-1 K-1)

-350.912

48.941a

-594.3a

-641.751

89.974

-842.221 -840.5a

-662.6 -951.953

131.006

a

-962.7a LiBr 3 3H2O LiBr 3 5H2O NaCl

-1240.66

172.039

-1562.09

-1828.45

254.104

-384.024a

-411.12a

50.503a

-858.372

-997.493

NaBr

-349.267a

-361.414a

NaBr 3 2H2O

-828.622 -828.29a

-950.197 -951.94a

NaCl 3 2H2O

NaBr 3 5H2O H2O (aq) H2O

b

þ

Li (aq) þ

a

-1084.9

-1541.96

-1813.75

-237.141a

75.29a 41.033

-239.35 -237.2823

-298.93324

-244.31724

-293.31

-278.49a

Na (aq) Cl- (aq)

-261.905 -131.228a

Br- (aq)

-104.16a

133.958 257.056

-290.53

a

51.893a

-285.83a

-290.4223 a

132.568

68.6a

-240.120 -167.159a

176.7 -136.4a

-119.57a

-72.3a

a

a

3

The values are taken from DDB. The values in DDB for standard-state thermodynamic properties are neglected for the correlation when there are two values in the table. b It represents the water molecules in the crystalline structure. It is used in eq 46, 48, and 49 to estimate the thermodynamic properties for the hydrate salts. (0) (1) (1) (2) (2) where b(0) i,j , ci,j , bi,j , ci,j , bi,j , ci,j are adjustable parameters, and Tref and T indicate the reference temperature 298.15 K and the system temperature, respectively. Substituting eqs 40 and 41 into the expressions of the MR term, the temperature-dependent MR term is improved. In the LIQUAC model, the MR interactions between like-charged ions and solvents are ignored, and dsalt is set to 0.125. For the short rang term (SR), the UNIQUAC equation is used to represent the noncharge interactions of all the species in the solution. Another improvement is carried out for the SR term by following the assumption of local like-charge repulsion and local electroneutrality proposed by Chen11 to describe the real local composition of the electrolyte solutions, which is equivalent to the assumption that UNIQUAC parameter ai,i of the like-charge ions is much greater than other interaction energies. Thus, there are no like-charge neighborhoods around the center of cations or anions. This assumption is applied in the residual part of the UNIQUAC equations, and the combinatorial term is kept without changing. Within the present study, four binary electrolyte systems (LiCl þ H2O, LiBr þ H2O, NaCl þ H2O, and NaBr þ H2O) were investigated to check the capability of the improvements for the

LIQUAC model in describing the phase behavior over a wide temperature range. New parameters were correlated with 300 data sets stored in the DDB covering molal osmotic coefficients (φ), molal mean ion activity coefficients (γ*() and vapor-liquid and solid-liquid equilibria data types. The temperature range is from 200 to 473 K, and the concentration is up to 25 mol 3 kg-1 in some systems. The parameters were determined by minimization of the following objective function: !2 Qexp - Qcalc wQ Fðai, j , aj, i , bi, j , ci, j Þ ¼ 3 100 Qexp np nt

∑∑

¼ min

ð42Þ

where Q represents the respective value of φ, γ*(, T, P, and m, wQ is a weighting factor for Q, and np and nt refer to the number of data points and data types, respectively. The subscripts “exp” and “calc”, respectively, refer to experimental and calculated values. The van der Waals volumes and surface areas for the ions were taken directly from Kiepe.20 The short-range interaction parameters and the volume and surface area parameters for the solvents were taken directly from the parameters of the UNIQAC model. The new parameters, which take into account all the related systems are listed in Table 1. The overall results for the salts (LiCl, LiBr, NaCl, and NaBr) in water calculated by the LIQUAC model are shown in Figures 2-5. The quality of the correlation by using this model can be made by calculating the average absolute relative deviation (AARD): np calc exp 1 Qi - Qi ð43Þ 100 AARD ¼ exp 3 np i ¼ 1 Qi

∑

Table 2summarizes the AARD values obtained in this work by using the LIQUAC model. 3.2. Discussion. Vapor pressures for LiCl þ H2O, LiBr þ H2O, NaCl þ H2O, and NaBr þ H2O at different temperatures were recorrelated with new temperature-dependent MR parameters. Comparing with the original LIQUAC model, smaller deviations were obtained especially at higher temperatures and concentrations, as shown in Figure 2. The molal osmotic coefficients of water in the strong electrolyte solutions were investigated at 298.15 K, as shown in Figure 3. The diagram in the upper left corner of Figure 3 presents the zooming in the low-concentration range. The calculated results matched well with the experimental data from diluted region to the saturated concentration. Usually, experimental data of mean ion activity coefficients at 298.15 K were available for the correlation, as shown in Figure 4. Although the mean ion activity coefficient data can be derived from molal osmotic coefficients data using the Gibbs-Duhem equation for the binary systems, the calculated results using the LIQUAC model can be used to check the coincidence. For the prediction of solid-liquid equilibria, an activity coefficient model and the thermodynamic knowledge of the solubility products for all possible solutes are required in the considered system. For a single salt in aqueous solution, the values of the thermodynamic solubility product calculated with eq 12 using the model should be equal to the solubility products, as described by eq 13. The calculation of the solubility product as a function of temperature using eqs 13-16 requires the standard Gibbs energies of formation, the standard enthalpies of 3627



CORRELATION

Table 4. Salt Solubilities and Temperature of the Eutectic and the Peritectics Points in these Aqueous Electrolyte Systems this work

ice þ LiCl 3 5H2O (eutectic)

literature data

T (K)

molality (mol 3 kg-1)

T (K)

molality (mol 3 kg-1)

193.88

8.134

194.9521

8.00221

22

197.3 LiCl 3 5H2O þ LiCl 3 3H2O

205.02

9.513

21

205.95

22


254.18

14.129

ice þ LiBr 3 5H2O (eutectic) LiBr 3 5H2O þ LiBr 3 3H2O LiBr 3 3H2O þ LiBr 3 2H2O LiBr 3 2H2O þ LiBr 3 1H2O

ice þ NaCl 3 2H2O (eutectic) NaCl 3 2H2O þ NaCl ice þ NaBr 3 5H2O (eutectic)

NaBr 3 5H2O þ NaBr 3 2H2O NaBr 3 2H2O þ NaBr

293.86

19.726

14.20921

22

14.6022

21

19.55221

22

19.5722 7.39021

253.65 293.05

206.22

7.515

222.59

10.509

222.4521

10.60321

15.227

21

15.77721

21

22.33121

275.41 316.77

21.124

251.58

5.181

271.91

6.051

243.52

6.469

248.76 324.44

6.770 11.301

Atomic heat 26J 3 mol-1 3 K -1

ð44Þ

(b) The second rule, due to Kopp, applies to solid compounds and may be expressed by Heat capacity ¼ sum of the atomic heats of the constituent atoms ð45Þ In the absence of measured values, Kopp’s rule can prove extremely useful. With these rules, the missing heat capacities of the hydrate salts are estimated with eq 46, and the values are listed in Table 3. The heat capacity values of H2O as water of crystallization in solid substances are 41.03 J 3 mol-1 3 K-1. 3 nH2 O

9.8222

293.94 202.8521

formation, and the standard heat capacities of all the species. The specific heat capacities of ions and anhydrous crystals can be obtained from DDB or other chemical engineering handbooks. But the specific heat capacities of solid substances for the salt hydrates, such as lithium chloride monohydrate, are not available. A reasonable degree of accuracy by combing two empirical rules was adopted:19 (a) The first, due to Dulong and Petit, expresses a term called the atomic heat which is defined as the product of the relative atomic mass and the specific heat capacity. For all solid elemental substances, the atomic heat is assumed to be roughly constant:

c°p, MX

9.65921

21


7.8722

ðTref Þ ¼ c°p, MX ðTref Þ þ n 3 c°p, H2 O ðTref Þ

ð46Þ

Where the subscript MX indicates the salt, and n is the molecular number of hydrate water. Some values of the standard Gibbs energy of formation and the standard enthalpy of formation for the hydrate crystals are also absents in the DDB, thus they were correlated with the SLE data, and the values were listed in Table 3, comparing with the published values. With the calculated standard thermodynamic properties and the LIQUAC model, the

277.15 322.25

SLE for aqueous electrolyte system are represented, and the phase diagrams are shown in Figure 5. To determine the transition temperatures between hydrates and anhydrates of the corresponding saturated solution composition, the transition points are fixed by intersection of curves representing two different precipitated solids. The peritectic points calculated from the LIQUAC model are listed in Table 4. The values have 1-3 K deviations, comparing with the result of Patek and Klomfar.21 For the large scatter or systematic deviations of experimental data were observed for LiCl þ H2O and LiBr þ H2O systems at higher temperature and concentrations, the correlation for these systems higher than 323.15 K were not investigated in the present work. No data are available for solutions in phase equilibria with anhydrous LiBr.21 Comparing with these systems; a better fitting in a wide temperature range is obtained for the systems of NaCl þ H2O and NaBr þ H2O. For the water-ice equilibria, the normal SLE-equation developed by Schr€oder and van Laar to describe the melting points was used: ! ΔHm, i 1 1 lnðxi γi Þ ¼ ð47Þ T Tm, i R The values of thermodynamic properties for the calculation are the enthalpy of fusion ΔHm,i and the melting temperature Tm,i of the pure component i. These values are stored in DDB3 or other pure component property tables. By assuming the enthalpy of melting as a constant value which is correct for small changes in temperature, the water-ice equilibrium for the aqueous electrolyte system can be described in Figure 5. The eutectic point obtained using the LIQUAC model and the literature values are listed in Table 4 for comparison. It has been observed for LiCl 3 nH2O22 that the contribution of each water molecule to the standard enthalpy of formation and standard Gibbs energy of formation of a hydrated solid is approximately constant. This result may be interpreted in terms of group contributions, which states that the thermodynamic properties of a hydrated solid phase are the sum of the 3628



CORRELATION

Figure 6. (a) The standard enthalpy of formation of the solid salt hydrates versus the number of water molecules in the crystalline structure. (b) The standard Gibbs energy of formation of the solid salt hydrates versus the number of water molecules in the crystalline structure: 0, LiCl; 4, LiBr; (, NaCl; and •, NaBr; the symbols represent the correlated results given in Table 3; the lines indicate the linear fitting.

corresponding quantities for the cation in aqueous solution and those for the anion and for the water molecules in the crystalline structure.23 This leads to a linear trend when the standard Gibbs energy of formation and standard enthalpy of formation are plotted versus the number of hydration waters, which are observed for the LiCl 3 nH2O, LiBr 3 nH2O, NaCl 3 nH2O, and NaBr 3 nH2O, as shown in Figure 6. The tendency of the hydrate salts is approximately the same. The slope of the linear fit represents the group contribution of hydration water to the standard Gibbs energy of formation and standard enthalpy of formation. This method can be used to estimate missing standard thermodynamic properties for complex salts. For example, the values for NaBr 3 5H2O and LiBr 3 5H2O have not been reported in the literature yet. By using following equations: Δf g°ðMX 3 nH2 OÞ ¼ Δf g°ðMþ , aqÞ þ Δf g°ðX- , aqÞ þ nΔf g°ðH2 OÞ

ð48Þ

Figure 7. Predicted results of SLE for one salt against another in aqueous systems: (a) NaCl þ LiCl þ H2O: 0, LiCl, 298.15 K; (b) NaCl þ NaBr þ H2O: 0, NaCl, 298.15 K; O, NaCl, 323.15 K; 9, NaBr 3 2H2O, 298.15 K; b, NaBr 323.15 K; the symbols represent further published data stored in DDB;3 —, calculated results.

Δf h°ðMX 3 nH2 OÞ ¼ Δf h°ðMþ , aqÞ þ Δf h°ðX- , aqÞ ð49Þ þ nΔf h°ðH2 OÞ The predicted values of Δf g°298.15 K and Δf h°298.15 K for LiBr 3 5H2O are -1594.81 kJ 3 mol-1 and -1850.37 kJ 3 mol-1, respectively. Comparing with the correlated results in Table 3, the absolute errors are 2.1 and 1.2%, respectively. It is also emphasized by Mostafa24 that significant errors are found for the single metallic oxides and the halides, such as NaCl, KI, CaI2, etc., to predict heats and free energies of formation for these simple salts using eqs 48 and 49. Fortunately the experimental values are available for these salts. For the mixed salt 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 3629



CORRELATION

As usual, we performed the test of the capability to predict the apparent relative molar enthalpy hE, which is expressed as28 ∂ ð50Þ ðφ - ln γ( Þ ∂T For NaCl aqueous system, data could be found in the literature 27. The results in this work were compared with the original LIQUAC1 and MSA-NRTL model,28 as shown in Figure 8. It is seen that the predicted results are improved greatly than original LIQUAC and better than the MSA-NRTL at high concentration (Figure 9). Because the new parameters are mainly obtained from solubilities data and short of temperature dependence activity coefficients data at low concentrations, the predicted results with large deviations were observed. However, the model certainly has predictability by taking into account all the data type and temperature and concentration-dependence experimental data. hE ¼ vRT 2

Figure 8. Comparison of predicted results with experimental data29 of molal osmotic coefficient in the mixed salts system at 298.15 K.

4. CONCLUSIONS From the overview of the calculated results, the LIQUAC model has a good capability in describing the phase equilibrium behavior of electrolyte systems. From the SLE results for various single or mixed salts in aqueous solutions, it can be concluded that the LIQUAC model can be used to predict the salt solubilities in aqueous. The LIQUAC model can be applied in a wide temperature range by introducing the local composition model and improving the temperature dependent MR parameters as mentioned above. The average relative deviation of salt solubilities is less than 2%, comparing to 4% obtained with original LIQUAC model parameters.14 A group contribution model was also used to estimate the thermodynamic properties of hydrated salts, such as heat capacities, standard Gibbs energies of formation, and standard enthalpies of formation with a good agreement. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel.: þ86-010-68912660. Figure 9. Apparent relative molar enthalpy as a function of molality for aqueous NaCl solution: 9, NaCl, 298.15 K; b, NaCl, 323.15 K; and 2, NaCl, 348.15 K;27 —, this model; ---, original LIQUAC model; 3 3 3 , MSA-NRTL28

the solution. A procedure similar to that for pure salts can be applied to predict the solubilities of salt mixtures in water at a particular temperature. In this work, aqueous NaCl þ LiCl (LiCl 3 nH2O) and NaCl þ NaBr (NaBr 3 nH2O) solutions were studied in detail. It can be seen that the calculated SLE results have a good agreement with the experimental data, as shown in Figure 7, even the point of phase transition is presented well. The lines represent the predicted results by using parameters and thermodynamic properties obtained from binary systems. Considering the scattering of the experimental data, the observed deviations can be accepted with AARD less than 3%. The molal osmotic coefficients of water in the mixed salts system at 298.15 K were obtained with absolute average relative deviations less than 1.8% in Figure 8. It proves that the LIQUAC model has the capability to predict the phase behavior by applying the binary interaction parameters into ternary or multicomponent systems.14,20

’ ACKNOWLEDGMENT The authors thank the Deutsche Forschungsgemeinschaft for financial support of the ongoing research project. We also thank the DDBST GmbH (Oldenburg, Germany) for providing the latest version of the Dortmund Data Bank for the model comparison. ’ REFERENCES (1) Li, J.; Polka, H.-M.; Gmehling A gE Model for Single and Mixed Solvent Electrolyte Systems. 1. Model and Results for Strong Electrolytes. Fluid Phase Equilib. 1994, 94, 89. (2) Polka, H.-M.; Li, J.; Gmehling A gE Model for Single and Mixed Solvent Electrolyte Systems. 2. Results and Comparison with Other Models. Fluid Phase Equilib. 1994, 94, 115. (3) Dortmund Data Bank, version 2008; DDBST Software and Separation Technology GmbH: Oldenburg, Germany, 2008; www.ddbst.de. (4) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; 2nd ed., Butterworth: London, 1970. (5) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions; 2nd ed.; CRC Press: Boca Raton, FL, 1991. 3630



CORRELATION

(6) Barthel, J.; Krienke, H.; Kunz, W. Physical Chemistry of Electrolyte Solutions: Modern Aspects; Steinkopff/Springer: Darmstadt/New York, 1998. (7) Pitzer, K. S. Thermodynamics of Electrolytes. I: Theoretical and General Equations. J. Phys. Chem. 1973, 77, 268. (8) Blum, L. Mean spherical model for asymmetric electrolytes I. Method of solution. Mol. Phys. 1975, 30, 1529. (9) Henderson, D. Perturbation Theory, Ionic Fluids, and the Electric Double Layer. ACS Advances in Chemistry Series 1983, 204, 47. (10) Renon, H.; Cruz, J. L. A New Thermodynamic Representation of Binary Electrolyte Solutions Nonideality in the Whole Range of Concentrations. AIChE J. 1978, 24, 817. (11) Chen, C. C.; Britt, H. I.; Boston, J. F. Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. AIChE J. 1982, 28, 588. (12) Sander, B.; Fredenslund, A. Calculation of Vapor-Liquid Equilibria in Mixed Solvent/Salt Systems using an Extended UNIQUAC Equation. Chem. Eng. Sci. 1986, 41, 1171. (13) Kikic, I. UNIFAC Prediction of Vapor-Liquid Equilibria in Mixed Solvent-Salt Systems. Chem. Eng. Sci. 1991, 46, 2775. (14) Li, J.; Lin, Y.; Gmehling, J. gE Model for Single- and MixedSolvent Electrolyte Systems. 3. Prediction of Salt Solubilities in Aqueous Electrolyte Systems. Ind. Eng. Chem. Res. 2005, 44, 1602. (15) Huang, J.; Li, J.; Gmehling, J. Prediction of Solubilities of Salts, Osmotic Coefficients and Vapor-Liquid Equilibria for Single and Mixed Solvent Electrolyte Systems using the LIQUAC Model. Fluid Phase Equilib. 2009, 275, 8. (16) Wagman, D. D.; Williams, H. E. The NBS Tables of Chemical Thermodynamic Properties: Selected Values for inorganic and C1 and C2 organic Substances in SI Units. J. Phys. Chem. Ref. Data 1982, 11, suppl. no. 2. (17) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, U.K., 1949. (18) Yan, W.; Topphoff, M.; Rose, C.; Gmehling, J. Prediction of Vapor-Liquid Equilibria in Mixed-Solvent Electrolyte Systems using the Group Contribution Concept. Fluid Phase Equilib. 1999, 162, 97. (19) Mullin, J. W.; Crystallization; 3rd ed.; Butterworth: London, 1992. (20) Kiepe, J.; Noll, O.; Gmehling, J. Modified LIQUAC and Modified LIFAC - A Further Development of Electrolyte Models for the Reliable Prediction of Phase Equilibria with Strong Electrolytes. Ind. Eng. Chem. Res. 2006, 45, 2361. (21) Patek, J.; Klomfar, J. Solid-Liquid Phase Equilibrium in the Systems of LiBr-H2O and LiCl-H2O. Fluid Phase Equilib. 2006, 250, 138. (22) Monnin, C.; Simonin, J. P. Thermodynamics of the LiClþH2O System. J. Chem. Eng. Data 2002, 47, 1331. (23) Li, J.; Li, B.; Gao, S. Calculation of Thermodynamic Properties of hydrated Broates by Group Contribution Method. Phys. Chem. Miner. 2000, 27, 342. (24) Mostafa, A. T.; Eakman, J. M.; Yarbro, S. L. Prediction of Standard Heats and Gibbs Free Energies of Formation of Solid Inorganic Salts from Group Contributions. Ind. Eng. Chem. Res. 1995, 34, 4577. (25) Hamer, W. J.; Wu, Y.-C. Osmotic Coefficients and Mean Activity Coefficients of Uni-UniValent Electrolytes in Water at 25 °C. J. Phys. Chem. Ref. Data 1972, 4, 1047. (26) Patil, R.; Tripathi, A. D.; Pathak, G.; Katti, S. S. Thermodynamic Properties of Aqueous-Electrolyte Solutions. 1. Vapor-Pressure of Aqueous-Solutions of LiCl, LiBr, and LiI. J. Chem. Eng. Data 1990, 35, 166. (27) Gibbard, H. F.; Scatchard, G.; Rousseau, R. A. Liquid-Vapor Equilibrium of Aqueous Sodium Chloride, from 298 to 373 K and from 1 to 6 mol kg-1, and related properties. J. Chem. Eng. Data 1974, 19, 281. (28) Simonin, J. P.; Bernard, O.; Papaiconomou, N.; Kunz, W. Description of Dilution Enthalpies and Heat Capacities for Aqueous Solutions within the MSA-NRTL Model with Ion Solvation. Fluid Phase Equlib. 2008, 264, 211. (29) Robinson, R. A.; Wood, R. H.; Reilly, P. J. Calculation of Excess Gibbs Energies and Activity Coefficients from Isopiestic Measurements on Mixtures of Lithium and Sodium Salts. J. Chem. Thermodyn. 1971, 3, 461. 3631


Thermodynamics of Phase Equilibria in Aqueous Strong Electrolyte

Recommend Documents