Investigation of the (Solidâ Liquid and Vaporâ Liquid) Equilibrium of

454

Ind. Eng. Chem. Res. 2006, 45, 454-466

Investigation of the (Solid-Liquid and Vapor-Liquid) Equilibrium of the System H2O + CH3OH + Na2SO4 A Ä lvaro Pe´ rez-Salado Kamps, Mathias Vogt, Michael Jo1 decke,‡ and Gerd Maurer* Lehrstuhl fu¨r Technische Thermodynamik, UniVersity of Kaiserslautern, D-67653 Kaiserslautern, Germany

New experimental results for the vapor-liquid equilibrium of the system (water + methanol + sodium sulfate) are presented at (salt-free) solvent mixture methanol mole fractions from about 0.036 to 0.23, at sodium sulfate molalities up to about 1.25 mol‚kg-1 of solvent mixture (but always below the solubility limit of the salt), at temperatures from about 314 to 394 K, and total pressures from about 10 to 390 kPa. An extension of Pitzer’s model for the Gibbs excess energy of aqueous electrolyte solutions to mixed-solvent electrolyte systems is applied to describe the (solid-liquid and vapor-liquid) equilibrium of the ternary system (water + methanol + sodium sulfate). That extended model requires information on the solubility of the salt in pure water, on the vapor-liquid equilibrium of the salt-free binary system and on the (molar) Gibbs energy of transfer of sodium sulfate from pure water to mixtures of water and methanol. Literature data for the solubility of sodium sulfate in mixtures of water and methanol are used to determine that Gibbs energy of transfer. The new model is then able to predict the new experimental results for the vapor-liquid equilibrium of the ternary system (at salt concentrations up to the solubility limit) within experimental uncertainty. Furthermore, the model explains Furter’s empirical equation for the salt-effects (“salting-out” and “salting-in”) on the vaporliquid equilibrium of binary (or multicomponent) solvent mixtures, giving Furter’s “salt-effect parameter” a better physical meaning. Introduction Many experimental results for salt-effects on the phase equilibrium of mixed-solvent systems have been reported in the literature (cf., e.g., refs 1-6). From those results, the influence of a salt on the vapor-liquid and liquid-liquid equilibrium is well-known. For example, adding a salt to a binary liquid mixture of volatile solvents usually results in an increase (decrease) of the concentration of one (the other) volatile component in the vapor phase (“salting-out effect” and “saltingin effect”, respectively). If the binary salt-free solvent system exhibits an azeotrope, by adding a salt, that azeotrope may be broken. The opposite effect is also possible; adding a salt to a binary salt-free nonazeotropic solvent mixture may cause an azeotrope. Furthermore, the mutual solubilities in a binary liquid mixture may dramatically be affected by the addition of a salt. For example, adding a salt to a completely miscible binary solvent mixture can result in a liquid-liquid phase split, where either one or both liquid phases may contain appreciable amounts of the salt. To develop separation equipment, the influence of salts on the phase equilibrium of such mixed-solvent systems has to be properly described by means of well-grounded thermodynamic models. And because most systems of industrial interest are multicomponent (i.e., they may contain many solvent components as well as many electrolyte components), the thermodynamic models shouldsat least to a certain extentsexhibit predictive character. The thermodynamic properties of binary (or multicomponent) liquid mixed-solvent mixtures containing strong (and weak) electrolytes (and gases) are most successfully described by means of an equation for the Gibbs energy. In many applications, the concentrations of the solvent components are high in * To whom correspondence should be addressed. Phone: +49 631 205 2410. Fax: +49 631 205 3835. E-mail: [email protected]. ‡ Current address: BASF AG, 67056 Ludwigshafen, Germany.

comparison to the concentrations of the electrolytes and gases. Those electrolyte and gas components are regarded as solutes, and the reference states for the chemical potentials of the solvent and the solute species are normalized according to the unsymmetric convention. The (small) concentrations of the solute species are often described by means of the molality scale. Following that pattern, Pitzer’s (molality scale based) model for the Gibbs excess energy7,8 of aqueous electrolyte solutions has been recently extended to mixed-solvent electrolyte systems (Pe´rez-Salado Kamps9). The so-called (molar) Gibbs energy of transfer of a solute (from one pure solvent to the solvent mixture) was shown to play a very important role in explaining the salteffects. Many such models either (deliberately or not) do not explicitly take into account that Gibbs energy of transfer (e.g., the extended UNIQUAC model by Iliuta et al.10) orsat least for the ionic speciessthey approximate it by means of the Born equation (e.g., the electrolyte NRTL model; cf. Austgen et al.11). However, that Gibbs energy of transfer can be determined experimentally. For strong electrolytes, it is usually determined either from electromotive force (emf) data or from salt solubility data (cf., e.g., Kalidas et al.12 and Hefter et al.13), whereas for gases it is determined from gas solubility data (cf., e.g., ref 9). By way of example, using such experimentally determined numerical values for the Gibbs energy of transfer of the solute species, Pe´rez-Salado Kamps9 was able to simultaneously describe not only the vapor-liquid equilibrium (VLE) and the solid-liquid equilibrium (SLE) of the system (water + methanol + sodium chloride) but also the mean ionic activity coefficient of sodium chloride in that solvent mixture. In particular, the model was able to predict the influence of NaCl on the VLE of that binary solvent mixture. In the present work, that thermodynamic model is further tested on a second ternary system of the kind (water + organic component + salt), namely on the system (water + methanol + sodium sulfate). New experimental data for the influence of sodium sulfate on the vapor-liquid equilibrium of that aqueous organic mixture are presented for temperatures of about 314,

10.1021/ie0508177 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/08/2005

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 455 Table 1. Solubility Products of Sodium Sulfate and Sodium Sulfate Decahydrate in Pure Water m ln Ksp,i,W (T) ) Ai + Bi/(T/K) + Ci(T/K) + Di ln(T/K)

salt i

Ai

Bi

Ci

Di

T (K)

source

Na2SO4 Na2SO4‚10H2O

-326.526 -381.642

6561.58 1686.72

-0.134415 -0.111416

60.3601 71.3206

308-473 200-450

Rumpf and Maurer20 Pabalan and Pitzer14

354, and 394 K. The salt-free mole fraction of methanol in the solvent mixture extends to about 0.23, the sodium sulfate molality, to about 1.25 mol‚kg-1 of the solvent mixture (but always below the solubility limit of the salt), and the total pressure, to about 0.39 MPa. Correlation of the Solid-Liquid Equilibrium In aqueous (methanol-free) solutions of sodium sulfate and at temperatures below about 305.4 K, sodium sulfate precipitates as sodium sulfate decahydrate {Na2SO4‚10H2O(s)}, whereas, above that temperature, it precipitates as pure sodium sulfate {Na2SO4(s)} (cf., e.g., Pabalan and Pitzer14). The same qualitative behavior has also been observed in the presence of small amounts of methanol. But, beyond a certain concentration of methanol in an aqueous solution, due to the lack of sufficient amounts of water, only the precipitation of pure sodium sulfate is observed (cf., e.g., refs 15-19). To describe the solubility of any of those two salts in a mixture of water + methanol, one has to prove for which of those salts the thermodynamic solid-liquid equilibrium condition is fulfilled (cf. Appendix I): m 2 Ksp,Na (T,p,x˜ M) ) aNa +aSO22SO4 4

(1)

m 10 2 Ksp,Na (T,p,x˜ M) ) aNa +aSO2-aW 2SO4‚10H2O 4

(2)

m m Ksp,Na and Ksp,Na are the (molality scale based) 2SO4 2SO4‚10H2O solubility products of sodium sulfate and sodium sulfate decahydrate in the (water + methanol) mixture, respectively. They depend on temperature, T, pressure, p, and the composition of the salt-free solvent mixture. That composition is here expressed by the mole fraction of methanol x˜ M in the salt-free solvent mixture:

x˜ M )

nM n W + nM

(3)

The influence of the pressure on the solubility products is usually neglected. The solubility products of sodium sulfate and sodium sulfate decahydrate in the (water + methanol) mixture are expressed as m (T,x˜ M) Ksp,Na 2SO4

)

m Ksp,Na (T) 2SO4,W

[

exp -

]

m (T,x˜ M) ∆tGNa 2SO4

RT

(4)

and m Ksp,Na (T,x˜ M) ) 2SO4‚10H2O

[

m (T) exp Ksp,Na 2SO4‚10H2O,W

]

m (T,x˜ M) ∆tGNa 2SO4

RT

(5)

m m respectively, where Ksp,Na and Ksp,Na are the 2SO4,W 2SO4‚10H2O,W (molality scale based) solubility products of these salts in pure

m is the (molality scale based) molar Gibbs water and ∆tGNa 2SO4 energy of transfer of sodium sulfate from pure water to the solvent mixture of water and methanol (cf. Appendix I). m m In the present work, Ksp,Na and Ksp,Na were 2SO4,W 2SO4‚10H2O,W 20 adopted from Rumpf and Maurer and from Pabalan and Pitzer,14 respectively. The influence of temperature on both solubility products is correlated using a four parameter equation (cf. Table 1). The term ai represents the activity of species i in the saturated liquid mixture. It is calculated by applying the Gibbs excess energy model of Pitzer,7,8 in its extension to solvent mixtures as recently proposed by Pe´rez-Salado Kamps.9 For the ionic species,

2 aNa +aSO24

)4

(

m j Na2SO4 m°

)

3

m γ(,Na 2SO4

(6)

m represents the (molality scale based) mean where γ(,Na 2SO4 ionic activity coefficient of sodium sulfate in the solvent mixture m of water + methanol. The resulting equation for γ(,Na 2SO4 follows from Pitzer’s GE model (cf. Appendix II), but the (0) (1) (1) (2) parameters βNa βNa RNa βNa +,SO2-, +,SO2-, +,SO2-, +,SO2-, 4 4 4 4 (2) φ RNa+,SO2, and CNa2SO4 do not depend only on temperature but 4 also on the composition of the salt-free solvent mixture. In the (2) present work, βNa +,SO2- was not required and was set to zero. 4 (2) Therefore, the numerical value for RNa +,SO2- is meaningless. In 4 addition, the influence of the solvent mixture composition on (1) (1) RNa+,SO2was neglected (RNa+,SO2) 2). 4 4

(

)

j Na2SO4 4m (0) [2βNa +,SO2- + 4 3 m° m j Na2SO4 2 (1) φ x x βNa h(2 I)] + 2 2 CNa (7) +,SO22SO4 4 m°

m ) f′ + ln γ(,Na 2SO4

(

)

The Debye-Hu¨ckel termsas modified by Pitzersis

f(I,x˜ M) ) -Aφ and

f′ )

() ∂f ∂I

x˜ M

) -2Aφ

4I ln(1 + bxI) b

(8)

[

]

xI 2 + ln(1 + bxI) b 1 + bxI

(9)

For calculating the Debye-Hu¨ckel parameter Aφ,

(

e2 1 Aφ ) (2πNAdm°)1/2 3 4π0kBT

)

3/2

(10)

information is required for the specific density d and the relative dielectric constant of the salt-free solvent mixture. The terms d and depend on temperature and the composition of the saltfree solvent mixture. Correlation equations for those properties were adopted from ref 9. For aqueous solutions, Pitzer selected b ) 1.2. This numerical value was adopted here for the (water + methanol) mixtures as well. The ionic strength is

456

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006

m j Na2SO4

I)3

(11)

m°

where m j Na2SO4 is the stoichiometric molality of sodium sulfate in the liquid solvent mixture of water + methanol, i.e., the amount of substance of the salt per kilogram of the solvent mixture:

m j Na2SO4 )

njNa2SO4

[ (

)

2

2 x 1- 1+xexp(-x) 2 2 x

]

(13)

( )( )(

)

x m j Na2SO4 M* ∂∆tGNa 2SO4 + m° RT ∂x˜ M T ln γW,UNIQUAC + ln γW,Pitzer + ln γW,conv (14)

1-

] [

φ˜ W + qW θ˜ W

1-

] [

φ˜ M θ˜ M

( )( )(

)

M* is the (mean) relative molar mass of the solute-free solvent mixture divided by 1000. For the binary solvent mixture under consideration in the present work, it is

M* ) M/W + x˜ M(M/M - M/W)

(16)

where M/W ) 0.01801528 (for water) and M/M ) 0.03204216 (for methanol). x The term ∆tGNa is the mole fraction scale based molar 2SO4 Gibbs energy of transfer of Na2SO4 from pure water to the solvent mixture of methanol and water, which can easily be converted to the molality scale:

)

m ∆tGNa 2SO4

( )

M* + 3RT ln / MW

(17)

) (

x ∂∆tGNa 2SO4

∂x˜ M

)

) (

m ∂∆tGNa 2SO4

∂x˜ M

T

M/M - M/W - 3RT M* T

)

(18)

The true mole fractions xi of the solvent components in the liquid mixture of water + methanol + sodium sulfate are calculated from the mole fraction of i in the salt-free solvent mixture x˜ i and the stoichiometric molality of Na2SO4 m j Na2SO4 in the solvent mixture:

xi ) 1+3

x˜ i m j Na2SO4

(

m°

)

(i ) W, M)

(19)

M*

The contributions of the UNIQUAC equation21 to the activity

[( )

]

The coordination number z was set equal to 10. The segment fraction, φ˜ i, and the area fraction, θ˜ i, are given by

φ˜ i )

x˜ iri (i ) W, M) x˜ WrW + x˜ MrM

(22)

θ˜ i )

x˜ iqi (i ) W, M) x˜ WqW + x˜ MqM

(23)

Pure-component UNIQUAC constants were estimated from Bondi’s data22 as proposed by Abrams and Prausnitz21 (rW ) 0.92, qW ) 1.4, rM ) 1.43111, qM ) 1.432). Binary UNIQUAC interaction parameters were fitted in ref 9 to vapor-liquid equilibrium data (for the system water + methanol) in the temperature range from 298 to 423 K {ΨMW ) exp[1.8134 728.69/(T/K)], ΨWM ) exp[-2.1338 + 790.08/(T/K)]}. The contribution of Pitzer’s equation to the activity coefficients of the solvent components water and methanol result in9

ln γW,Pitzer )

-M/W

[

(If′ - f) + 4

(

m j Na2SO4 m°

)

2 (0) {βNa +,SO2- + 4

( ) [( ) ( ) { } ( ) [ ( ) ( ) [( ) ( ) { } ( )

(1) βNa +,SO2- exp(-2xI)} + 4x2 4

M* x˜ M

giving

(

x ∆tGNa 2SO4

( )

]

φ˜ M z φ˜ M φ˜ M +1- qM ln + x˜ M x˜ M 2 θ˜ M θ˜ WΨMW + qM 1 - ln(θ˜ WΨWM + θ˜ M) θ˜ W + θ˜ MΨMW θ˜ M (21) θ˜ WΨWM + θ˜ M

The equation for the activity of methanol, which will be required in a forthcoming section, is given here as well: x m j Na2SO4 M* ∂∆tGNa 2SO4 + ln aM ) ln xM + x˜ W m° RT ∂x˜ M T ln γM,UNIQUAC + ln γM,Pitzer + ln γM,conv (15)

[( )

φ˜ W φ˜ W φ˜ W z +1- q ln + x˜ W x˜ W 2 W θ˜ W θ˜ W 1 - ln(θ˜ W + θ˜ MΨMW) θ˜ W + θ˜ MΨMW θ˜ MΨWM (20) θ˜ WΨWM + θ˜ M

ln γM,UNIQUAC ) ln

For water, the GE model proposed by Pe´rez-Salado Kamps9 results in

ln aW ) ln xW - x˜ M

( )

ln γW,UNIQUAC ) ln

(12)

m ˜W+m ˜M

and

h(x) )

coefficients of water and methanol in the salt-free solvent mixture are

∂f +4 ∂x˜ M I

m j Na2SO4

φ CNa 2SO4

m°

(0) m j Na2SO4 2 ∂βNa +,SO24

m°

]

3

∂x˜ M

+

]

(1) φ ∂βNa m j Na2SO4 3∂CNa +,SO24 2SO4 g(2xI) + 2x2 (24) ∂x˜ M m° ∂x˜ M

ln γM,Pitzer ) -M/M (If′ - f) + 4 (1) βNa +,SO24

m j Na2SO4 m°

exp(-2xI)} + 4x2

M*(1 - x˜ M)

∂f +4 ∂x˜ M I

2

(0) {βNa +,SO2- + 4

m j Na2SO4 m°

3

(0) m j Na2SO4 2 ∂βNa +,SO24

m°

]

φ CNa + 2SO4

∂x˜ M

+

]

(1) φ ∂βNa m j Na2SO4 3∂CNa +,SO24 2SO4 g(2xI) + 2x2 (25) ∂x˜ M m° ∂x˜ M

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 457

where

I3/2 If′ - f ) -2Aφ (1 + bxI)

( ) [

∂f f 1 ∂d 3 ∂ ) ∂x˜ M I 2 d ∂x˜ M ∂x˜ M

(26)

]

(27)

and

g(x) ) h(x) - exp(-x)

(28)

Because Pitzer’s GE equation is based on the molality scale, whereas the UNIQUAC GE equation is based on the mole fraction scale, a so-called conversion term had to be included in eqs 14 and 15:

m j Na2SO4

ln γi,conv ) -3

m°

[

M* + ln 1 + 3

m j Na2SO4 m°

M*

]

(i ) W, M) (29)

For “small” salt concentrations, this term can be neglected. The Gibbs energy of transfer of sodium sulfate (from pure m , and the water to a mixture of water + methanol), ∆tGNa 2SO4 (0) interaction parameters in Pitzer’s equation, βNa +,SO2-, 4 (1) φ βNa+,SO2, and C , depend on temperature and on the saltNa2SO4 4 free solvent composition. That Gibbs energy of transfer may, as usual, be determined from electromotive force (emf) measurements, namely, from the standard emf of an specially arranged galvanic cell, which shall contain a liquid mixture of water + methanol + sodium sulfate. The standard emf is calculated from an extrapolation procedure to a zero salt concentration (cf. ref 9). The interaction parameters are then adjusted to the mean ionic activity coefficients, which may result not only from those emf measurements (which are mostly only reliable at moderate salt concentrations) but also from experimental data for the solubility of Na2SO4 in the binary (water + methanol) mixture. Unfortunately, no emf data for the ternary system (water + methanol + sodium sulfate) could be found in the open literature, probably because no suitable electrodes exist. The interaction parameters were therefore estimated as follows (cf. ref 9): (0) (0) ˜ M) ) (1 - x˜ M)βNa βNa +,SO2-(T,x +,SO2-,W(T) + 4 4 (0) x˜ MβNa +,SO2-,M(T) (30) 4 (1) (1) βNa ˜ M) ) (1 - x˜M)βNa +,SO2-(T,x +,SO2-,W(T) + 4 4 (1) x˜MβNa +,SO2-,M(T) (31) 4

and φ φ φ (T,x˜ M) ) (1 - x˜ M)CNa (T) + x˜ MCNa (T) CNa 2SO4 2SO4,W 2SO4,M (32) (0) (1) φ where βNa +,SO2-,W, βNa+,SO2-,W, and CNa SO ,W are the interaction 2 4 4 4 parameters between sodium and sulfate in pure water. These parameters depend on temperature. They were adopted from the work of Rogers and Pitzer23 for temperatures between 298 and 473 K. As the solubility of sodium sulfate in pure methanol is very small (cf. refs 3 and 15-19), the parameters describing the interactions between sodium and sulfate in pure methanol (0) (1) φ (βNa +,SO2-,M, βNa+,SO2-,M, and CNa SO ,M) were all neglected. 2 4 4 4

The Gibbs energy of transfer of sodium sulfate (from pure water to a mixture of water + methanol) was then determined from salt solubility data alone.3,15-19 As was already noticed by Iliuta et al.,10 the experimental results reported by Okorafor24 do not agree with the literature data and were, therefore, not considered here. The following three parameter correlation equation (cf. ref 9) turned out to be sufficient to describe those data within experimental accuracy (cf. Table 2): m m ∆tGNa (T,x˜ M) ) x˜ M∆tGNa (T) + x˜ M(1 - x˜ M)AKsp,0 2SO4 2SO4,WfM (33)

where m m,o (T) ) ∆tGNa ∆tGNa 2SO4,WfM 2SO4,WfM m ∆tSNa (T - T°) (34) 2SO4,WfM m m,o ∆tGNa (∆tGNa ) is the Gibbs energy of transfer 2SO4,WfM 2SO4,WfM of sodium sulfate from pure water to pure methanol (on the molality scale) at temperature T (T° ) 298.15 K). The influence m of temperature on ∆tGNa was approximated by as2SO4,WfM m suming that the entropy of transfer, ∆tSNa , does not 2SO4,WfM depend on temperature. The following numerical values resulted from that fit:

m,o ) 49.06 kJ‚mol-1 ∆tGNa 2SO4,WfM

(35)

m ) -238.2 J‚mol-1‚K-1 ∆tSNa 2SO4,WfM

(36)

AKsp,0 ) 33.6 kJ‚mol-1

(37)

and

At methanol mole fractions above x˜ M ≈ 0.55, the experimentally determined concentration of sodium sulfate in the saturated liquid mixtures is very small (lower than about 0.008 mol‚kg-1). In that region, the relative experimental uncertainties are large. Therefore, these data were not taken into account in the correlation. The model correctly describes not only the saturation concentration of Na2SO4 in the solvent mixture but also the type of solid salt which precipitates (sodium sulfate or sodium sulfate decahydrate). Figure 1 shows some experimental results for the solubility of sodium sulfate in mixtures of water + methanol at 298.15 K in comparison with the correlation. Full symbols characterize the deposition of sodium sulfate decahydrate, whichsat that temperaturesprecipitates at methanol concentrations up to about x˜ M ≈ 0.166. Empty symbols characterize the deposition of sodium sulfate. The short bold line in that figure marks the transition from Na2SO4‚10H2O to Na2SO4, as calculated from the model. As was already mentioned before, in some of the models for describing the thermodynamic properties of liquid mixed-solvent mixtures containing strong electrolytes, which are based on an equation for the Gibbs energy (e.g., the electrolyte NRTL model; cf. Austgen et al.11), the Gibbs energy of transfer of those electrolytes (from one pure solvent to the solvent mixture) is taken into account by means of the Born term.25,26 However, as it is explained in Appendix III, this can only be regarded as an approximation. Figure 2 illustrates the difference between

458


Table 2. Comparison of Experimental Results with Model Correlations/Predictions for the System H2O + CH3OH + Na2SO4 T (K)

average deviation

m j Na2SO4,sat.

283.15-328.15 283.15-328.15 298.15-328.15 348.95-376.95 313.15 293-308

0.013 mol‚kg-1 0.035 mol‚kg-1 0.040 mol‚kg-1 0.070 mol‚kg-1 0.014 mol‚kg-1 0.035 mol‚kg-1

m j Na2SO4,sat.

283.15-328.15 341.05-345.75 313.15 341.05-376.95 313.8-394.8

p, yM

data points

ref

measurement

Correlated Data 39a 17 7 8a 13a 15

Emons and Ro¨ser15 Emons and Ro¨ser16 Emons et al.17 Meranda and Furter3 Fleischmann and Mersmann18 Zhang et al.19

salt solubility salt solubility salt solubility salt solubility salt solubility salt solubility

Predicted Data 0.00057 mol‚kg-1 8b c 8b 0.00051 mol‚kg-1 7b 10% 1.7%d 16 2.2% 5.9% 24

Emons and Ro¨ser15 Meranda and Furter3 Fleischmann and Mersmann18 Meranda and Furter3 this work

salt solubility salt solubility salt solubility VLE VLE

a Only the data at methanol concentrations below x ˜ M ) 0.55 were considered in the correlation. b The data at methanol concentrations above x˜ M ) 0.55 were predicted. c The concentration of the salt in the saturated liquid was very small and not reported. d One single data point at x˜ M ) 0.001 was not considered in that average deviation calculation.

Figure 2. Gibbs energy of transfer of Na2SO4 from pure water to the solvent mixture H2O + CH3OH (on the molality scale) at 298.15 K: (s) correlation, this work; (- -) prediction from Born’s equation25,26 (cf. eq 38).

Prediction of the Vapor-Liquid Equilibrium Figure 1. Solubility of Na2SO4 in H2O + CH3OH at 298.15 K (precipitated salt: filled symbols Na2SO4‚10H2O, empty symbols Na2SO4): ([, ]) Emons and Ro¨ser;15 (9, 0) Emons and Ro¨ser;16 (b, O) Emons et al.;17 (2, 4) Zhang et al.;19 (s) correlation, this work; (- -) prediction from Born’s equation25,26 (cf. eq 38).

the correlated Gibbs energy of transfer at 298.15 K and a prediction resulting from Born’s equation alone (cf. also ref 9):

[ ][

2

m ∆tGNa (T,x˜ M) ) 2SO4,Born

] ()

NAe 1 4 2 1 + + 8π0 W rNa+ rSO24 3RT ln

d (38) dW

The ionic radii were adopted from ref 27 (rNa+ ) 102 pm, rSO24 ) 230 pm). A rather large disagreement is observed between m Born’s equation and the correlation; for example, ∆tGNa2SO4,Born m (T ) 298.15 K, x˜ M ) 0.5) and ∆tGNa (T ) 298.15 K, x˜ M 2SO4,Born ) 1) deviate from the correlation by about -46% and -10%, respectively. If the Gibbs energy of transfer is estimated from Born’s equation, the prediction results for the salt solubility in the mixed-solvent are far off from the experimental data (cf. Figure 1, for salt-free solvent mixture concentrations up to x˜ M ≈ 0.06). Furthermore, the results are unreasonable as an increase of the solubility of the salt with increasing concentration of the organic solvent component is predicted. Neglecting the Gibbs energy of transfer at all results in similar bad predictions for the salt solubility in those mixtures.

The vapor-liquid equilibrium of the system (water + methanol + sodium sulfate) is described by means of the extended Raoult’s law applied to both volatile components water and methanol

psi φsi exp

[

]

(p - psi )

(di/Mi)RT

ai ) yipφi (i ) W, M)

(39)

where psi , φsi , di, and Mi are the vapor pressure, fugacity coefficient of the saturated vapor, specific liquid density, and molar mass, respectively, of the pure solvent component i. The term p is the total pressure, R is the universal gas constant, and yi and φi are the mole fraction and the fugacity coefficient of component i in the vapor phase. The vapor pressure of pure water (psW), as well as the density of pure liquid water (dW), which was approximated by the density of the saturated liquid, were calculated from correlation equations given by Saul and Wagner.28 The correlation equation for the vapor pressure of pure methanol (psM) given by Reid et al.29 was adopted. The density of pure liquid methanol (dM) was approximated by the density of the saturated liquid. It was calculated from the correlation given by Hales and Ellender.30 The nonideality of the vapor phase is described by the virial equation of state, truncated after the second virial coefficient. Second virial coefficients Bii and Bij are calculated from a correlation by Hayden and O’Connell31 (for details, cf. ref 9). The equations for calculating the activities ai of the volatile solvent components water and methanol have already been given


Figure 3. Phase diagram (yM - x˜ M) for the system H2O + CH3OH + Na2SO4 at p ) 99.99 kPa (solutions are saturated with Na2SO4): (b) Meranda and Furter;3 (s) prediction, this work; (- -) no salt calculation, this work.

in the previous section. Furthermore, all parameters involved m (0) (1) in those equations, in particular ∆tGNa , βNa +,SO2-, βNa+,SO2-, 2SO4 4 4 φ and CNa2SO4, were fixed in that previous section, too. The capability of the model to predict the influence of sodium sulfate on the vapor-liquid equilibrium of the system (water + methanol) is tested by comparing prediction results with experimental results. However, only one publication containing experimental information on such data was found in the open literature. Meranda and Furter3 (Detherm Database, Dechema e.V.) reported 16 VLE data points of the ternary system at 99.99 kPa and for temperatures from about 341 to 377 K, the liquid phase always being saturated with the salt. The model is able to predict the concentration of methanol in the gaseous phase above the saturated solutions within the experimental uncertainty (cf. Figure 3 and Table 2). Predicted temperatures (at preset pressures) systematically deviate from the experimental results by about -2.7 K. Predicted pressures (at preset temperatures) systematically deviate from the experimental results by about +10%. When the water content in the binary solvent mixture is sufficiently high, sodium sulfate dissolves in that liquid in considerable amounts. Increasing the sodium sulfate concentration in the liquid results in an increase of the methanol concentration in the vapor (see also Figure 3). This well-known salting-out effect is often attributed to the hydration of the salt in the liquid phase which “reduces” the concentration of “free” water. However, as already demonstrated in ref 9, without considering any kind of chemical interactions, the model is able to quantitatively predict that behavior. To further test the predictive character of the model, the VLE of that ternary system was experimentally investigated. Experimental Investigation of the Vapor-Liquid Equilibrium Apparatus. A special thin-film evaporator apparatus was used for the vapor-liquid equilibrium experiments. That apparatus has been described before by Hasse,32 Albert et al.,33 and Jo¨decke et al.34; therefore, only an outline is repeated here. The thin-film evaporator consists of a rotating coil (made up of glass-fiber reinforced Teflon) in a stainless steel tube. That

coil spreads the thermostated liquid feed on the inner surface of the tube. The tube is heated from the outside by a liquid which is thermostated to a few kelvin above the temperature of the liquid feed. The heating results in a partial evaporation of the liquid feed at a nearly constant pressure. That pressure is supplied by a backpressure regulator. For measurements above ambient pressure, the evaporator chamber is pressurized from a storage tank (filled with nitrogen). For measurements at lower pressures, the storage tank is replaced by a buffer tank which is connected to a vacuum pump and a nitrogen flask. After passing through the heated section of the tube, the coexisting and equilibrated phases are separated. The liquid phase is subcooled. The gaseous phase is completely condensed. Both phases are separately collected in vials. The temperature was measured (with an accuracy of (0.1 K) with a calibrated platinum resistance thermometer. The pressure was measured with two absolute pressure transducers (WIKA, Klingenberg, Germany) with ranges up to 0.25 and 0.6 MPa with an accuracy of (0.1% of each transducer’s full range. Before and after a series of measurements, the transducers were calibrated against a high precision pressure gauge (Desgranges & Huot, Aubervilliers, France) and against a mercury manometer for pressures above and below ambient pressure, respectively. The salt molality in the liquid phase was determined gravimetrically by slowly heating the sample in an oven and thereby completely evaporating the solvent. Each sample was analyzed three times and relative deviations between the single results were below 2% in most cases. The (salt-free) composition of the coexisting phases was determined by gas chromatography (GC) using commercial GC equipment (HewlettPackard, GC model HP-5890, autosampler model HP-7673; column packing Porapak T from Alltech, Unterhaching, Germany; thermal conductivity detector). Generally, for calibration (for analysis), each sample was analyzed at least five (three) times. Absolute deviations between the single results in the (saltfree) molar fraction of methanol were below (0.007. Substances and Sample Pretreatment. Methanol (g99.8 mol %, Honeywell Specialty Chemicals Seelze, Seelze, Germany) was degassed under vacuum. Deionized water was degassed by vacuum distillation. Sodium sulfate (g99 mass %) was purchased from Merck GmbH, Darmstadt, Germany and was degassed and dried under vacuum. The liquid feed mixtures were gravimetrically prepared. New Experimental Results and Comparison with Model Predictions. The vapor-liquid equilibrium of the system H2O + CH3OH + Na2SO4 was investigated at temperatures of about 314, 354, and 394 K. The liquid feed solutions contained up to about 23 mol % methanol in the salt-free solvent mixture. The maximum salt concentration was 1.25 mol‚(kg of solvent mixture)-1, but it was always below the solubility limit of the salt. The total pressure was below 0.39 MPa. The experimental results are given in Table 3. In Figures 4-6, the experimental results for the composition of the gaseous phase and for the total pressure are shown as full symbols and plotted versus the molality of the saltsat about a constant temperature and a constant salt-free liquid mole fraction of methanol. As expected, with increasing salt concentration in the liquid phase, the concentration of methanol in the gaseous phase increases, i.e., methanol is salted-out. Along with the experimental results, prediction results for the composition of the gaseous phase and for the total pressure above liquid solutions of water + methanol + sodium sulfate, calculated for a preset temperature and liquid phase composition, are also given in Table 3 (cf., also, Table 4) and shown in

460


Figure 4. Mole fraction of methanol in the vapor phase (upper diagram) and total pressure (lower diagram) versus the molality of Na2SO4 in aqueous solutions of methanol at T ≈ 314 K (filled symbols experimental data, this work; empty symbols prediction results, this work): (9, 0) x˜ M ≈ 0.036; (2, 4) x˜ M ≈ 0.08; (b, O) x˜ M ≈ 0.21.


Table 3. Experimental and Prediction Results for the Vapor-Liquid Equilibrium of the System H2O + CH3OH + Na2SO4 (Calculation for Preset Temperature and Liquid Phase Composition) T (K)

x˜ M

m j Na2SO4 (mol‚kg-1)

yM,exp

yM,pred

pexp (kPa)

ppred (kPa)

316.4 314.4 314.0 313.8 315.8 314.9 314.4 313.8 314.2 314.1 314.0 354.7 354.4 354.1 353.8 354.6 354.0 353.1 352.1 353.9 353.2 352.9 394.4 394.3 394.0 393.8 394.8 394.3 393.7 393.1 394.1 393.1 392.8

0.0368 0.0368 0.0361 0.0360 0.0762 0.0777 0.0813 0.0804 0.214 0.209 0.210 0.0414 0.0388 0.0388 0.0377 0.0820 0.0822 0.0849 0.0856 0.228 0.215 0.218 0.0398 0.0394 0.0360 0.0341 0.0746 0.0743 0.0761 0.0815 0.215 0.214 0.212

0 0.362 0.712 1.13 0 0.196 0.475 0.721 0 0.104 0.135 0 0.361 0.780 1.11 0 0.271 0.610 1.13 0 0.0876 0.175 0 0.374 0.797 1.25 0 0.284 0.639 1.07 0 0.0734 0.209

0.212 0.244 0.264 0.283 0.383 0.405 0.428 0.444 0.633 0.651 0.649 0.229 0.249 0.270 0.288 0.370 0.384 0.418 0.452 0.592 0.601 0.614 0.203 0.229 0.224 0.234 0.324 0.353 0.362 0.384 0.542 0.556 0.559

0.240 0.273 0.300 0.336 0.391 0.419 0.460 0.483 0.637 0.643 0.647 0.239 0.257 0.288 0.306 0.377 0.407 0.447 0.495 0.614 0.612 0.623 0.205 0.229 0.237 0.250 0.320 0.345 0.378 0.423 0.558 0.567 0.578

10.0 10.0 10.0 10.0 13.0 13.0 13.0 13.0 18.0 18.0 18.0 64.0 64.0 64.0 64.0 76.0 76.0 76.0 76.0 102.0 102.0 102.0 256.0 256.0 256.0 256.0 300.0 300.0 300.0 300.0 390.0 390.0 390.0

11.11 10.28 10.24 10.50 12.92 12.73 13.02 12.97 17.43 17.44 17.50 63.62 63.35 63.98 63.91 74.52 74.94 75.48 76.71 101.8 98.34 98.78 250.4 252.4 249.4 247.8 287.3 288.1 291.0 299.7 379.3 371.0 373.0

Figures 4-6. The aforementioned salt-effects are quantitatively predicted by the model, i.e., the differences between experi-


mental data and predictions lie within the experimental uncertainty. The average relative deviations between the new experimental data for the salt-free system and the calculation results for the

Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 461 Table 4. Comparison of the New Experimental Results for the Vapor-Liquid Equilibrium of the System H2O + CH3OH + Na2SO4 with the Prediction Results ≈T (K)

≈x˜ M

100|∆yM/yM,exp|a

100|∆p/pexp|a

|∆T|b (K)

314.0

0.036 0.08 0.21 0.039 0.082 0.22 0.037 0.076 0.21

14.7 6.6 0.8 5.3 7.5 1.6 4.2 5.6 2.7

3.4 0.8 2.9 0.4 1.0 3.4 2.4 2.3 4.6

0.6 0.2 0.6 0.1 0.3 0.9 0.8 0.8 1.6

354.0 394.0

a Values of y and p were calculated at experimental temperature and M N liquid phase composition. |∆Z/Zexp| ) 1/N∑i)1 |(Zi,exp - Zi,pred)/Zi,exp|. b T was calculated at experimental pressure and liquid phase composition. N |∆Z| ) 1/N∑i)1 |Zi,exp - Zi,pred|.

pressure and the mole fraction of methanol in the vapor phase (for a preset temperature and liquid phase composition) amount to (3.0 % and (3.4 %, respectively. When the pressure is preset (instead of the temperature), the average deviation in the temperature amounts to (0.7 K. As already discussed before,9 the binary UNIQUAC interaction parameters between water and methanol were adjusted to the VLE data for that binary system found in the literature. The average relative deviations between the literature data and the correlation results for the pressure and the mole fraction of methanol in the vapor phase (for a preset temperature and liquid phase composition) amount to (1.6% and (2.5%, respectively. For the salt-containing system, the average relative deviations between the experimental and prediction results for the pressure and the mole fraction of methanol in the vapor phase (for a preset temperature and liquid phase composition) amount to (2.2% and (5.9%, respectively. If the pressure is preset instead of the temperature, the average deviation in the temperature amounts to (0.6 K. These deviations are similar to those for the salt-free system. If Born’s equation (eq 38) is used to estimate the Gibbs energy of transfer of sodium sulfate (from water to the solvent mixture of water + methanol), the model predictions for the VLE of the ternary system are rather good as well. The average relative deviations between the experimental and calculation results for the pressure and the mole fraction of methanol in

the vapor phase (for a preset temperature and liquid phase composition) then amount to (5.1% and (4.3%, respectively. This quite satisfactory prediction is only due to the relatively low solubility of that particular salt in that particular solvent mixture (cf. eqs 14 and 15). If the Gibbs energy of transfer is even neglected in the calculations, the average relative deviations between the experimental and calculation results for the pressure and the mole fraction of methanol in the vapor phase (for a preset temperature and liquid phase composition) increase to (9% and (15%, respectively. As was shown before, it is very important to use correct numerical values for the Gibbs energy of transfer of the salt (from one pure solvent to a solvent mixture), if the solubility of that salt in that solvent mixture is to be described correctly. That Gibbs energy of transfer also has a large influence on the VLE of the ternary system, but only when the solubility of the salt in the binary solvent mixture is sufficiently high. Explanation of the Salt-Effect For a better understanding of the salting-out and salting-in effects, the different contributions to the activities of the solvent components are considered here (cf. eqs 14 and 15). A similar discussion is available for the system (water + methanol + sodium chloride).9 It is adapted here for the system (water + methanol + sodium sulfate). Figure 7 shows the contributions to the activities of methanol and water at T ) 298.15 K and x˜ M ) 0.1 plotted versus the stoichiometric molality of sodium sulfate in the liquid solvent mixture (up to the solubility limit of the salt). Obviously, the dilution term (ln xi) has the largest influence of all contributions to the activity of the solvent components. As expected, the UNIQUAC term (ln γi,UNIQUAC), which describes the interactions between those solvent components in the salt-free solvent mixture, is positive and does not change when the salt is added. A significant contribution results from the (linear) term involving the partial derivative of the (molar) Gibbs energy of transfer of Na2SO4 (from one pure solvent to the solvent mixture) with respect to x˜ M (called the “salt-effect term” in Figure 7). This contribution results in an increase of the activity of methanol with the rising salt concentration (methanol is salted-out) and in a decrease of the activity of water (water is salted-in).

Figure 7. Contributions to the activities of the solvent components methanol (left) and water (right) (cf. eqs 14 and 15) plotted versus the molality of Na2SO4 in the liquid solvent mixture (up to the solubility limit of the salt) at T ) 298.15 K and x˜ M ) 0.1.

462


Obviously, that term is most responsible for the salt-effect. The contribution resulting from the Pitzer term (ln γi,Pitzer) to some minor extent reduces that salt effect, and the contribution resulting from the conversion term (ln γi,conv) is nearly negligible here.

The stoichiometric molality of the salt can be rearranged to

m j CA njCA x/CA 1 1 ) ) / m° M* M* (nW + nM) (1 - xCA)

(47)

Derivation of Furter’s Equation Furter and co-workers1,2 empirically established an equation describing the salt-effect on the VLE of volatile solvent mixtures. For a single salt Cν+Aν- in a binary mixed-solvent (e.g., water, W, and methanol, M) at a fixed salt-free solvent mixture composition (x˜ M) and temperature (T), the so-called Furter’s equation relates the relative volatility in the saltcontaining mixture,

R

(CA)

)

y(CA) ˜M M /x

Therefore,

ln

( )

( ) ( )

x x/CA R(CA) 1 ∂∆tGCA ) R (1 - x/CA) RT ∂x˜ M

T

(48)

At very low to moderate salt concentrations (x/CA , 1),

(40)

y(CA) ˜W W /x

γM,Pitzer γW,Pitzer

+ ln

x/CA 1 - x/CA

≈ x/CA

(49)

to the relative volatility in the salt-free mixture,

R)

yM/x˜ M yW/x˜ W

(41)

In many cases (depending on the salt and the solvent mixture, see below), the second contribution on the right-hand side of eq 46 can be neglected in comparison with the first contribution:

as follows:

( )

ln

R(CA) ) k(T,x˜ M)x/CA log R

(42)

(cf. also, e.g., Prausnitz et al.6). The term k(T,x˜ M) is the salteffect parameter (which remains constant at moderate salt concentrations) and x/CA is the mole fraction of the salt in the liquid phase:

x/CA )

njCA nW + nM + njCA

( ) ( )

x 1 ∂∆tGCA / R(CA) ) x ln R RT ∂x˜ M T CA

(44)

W

( ) ( ) a(CA) R(CA) M /aM ) ln (CA) R aW /aW

(45)

Calculating the activities of the solvent components in the saltfree and salt-containing solutions from eqs 14 and 15 leads to (the dilution and conversion terms vanish and the contributions resulting from the UNIQUAC equation cancel each other)

( ) ( )( )( ) ( ) m j CA M* ∂∆tGxCA R(CA) ) R m° RT ∂x˜ M

+ ln

T

(51)

The salt-effect parameter in Furter’s equation, then, is

Applying Raoult’s law to both volatile solvent components, i.e., neglecting the influence of the pressure on the fugacity of the liquid components, as well as all interactions in the gaseous phases, results in the following:

ln

(50)

(43)

(CA) R(CA) yM /yM ) (CA) R y /y

ln

)

γM,Pitzer ≈0 γW,Pitzer

Equation 46 then results in

( )

Since the salt is nonvolatile, the gaseous phase only contains water and methanol. To facilitate direct comparison, the definition of R(CA) used by Furter uses liquid compositions on a salt-free basis (x˜ i); therefore,

W

(

γM,Pitzer γW,Pitzer

(46)

∂∆tGxCA 1 k(T,x˜ M) ) (ln 10) RT ∂x˜ M

(52)

T

The partial derivative (with respect to the solute-free mole fraction of one of the solvent components) of the Gibbs energy of transfer of a general salt CA (from one pure solvent to the binary solvent mixture) can easily be converted from the molality scale to the mole fraction scale (cf. eq 18). Here,

( ) ( ) ∂∆tGxCA ∂x˜ M

)

T

∂∆tGm CA ∂x˜ M

(

- (ν+ + ν-)RT T

)

M/M - M/W M*

(53)

Obviously, as was already discussed before,9 the salt-effect may directly be attributed to that Gibbs energy of transfer. It is, therefore, also closely related to the solubility product of the salt (or any solid hydration form of that salt) in the liquid solvent mixture. It might be worthwhile to note that the last term on the righthand side of eq 46 {ln(γM,Pitzer/γW,Pitzer)} might not always be negligible even at very low salt concentrations, i.e., at very low ionic strengths. For the system under consideration in the present work (cf. eqs 24 and 25),

( )


[

Cν+Aν- f ν+Cz+ + ν-Az-

γM,Pitzer ln ) -(M/M - M/W) (If′ - f) + γW,Pitzer m j Na2SO4 2 (0) (1) {βNa 4 +,SO2- + βNa+,SO2- exp(-2xI)} + 4 4 m° m j Na2SO4 3 ∂f φ 4x2 CNa + M* + 2SO4 m° ∂x˜ M I

(

4

) ( ) ( ){

(0) m j Na2SO4 2 ∂βNa +,SO24

m°

∂x˜ M

] [(

+

)

(1) ∂βNa +,SO24

∂x˜ M

(

2x2

}

g(2xI) +

)

]

φ m j Na2SO4 3∂CNa 2SO4

m°

∂x˜ M

(A1)

and in equilibrium with a solid phase Cν+Aν-‚(H2O)νW, the phase equilibrium condition yields m ν- νW (T,p,x˜ M) ) aν+ Ksp,C C aA aW ν+Aν-‚(H2O)νW

(A2)

m Ksp,C (T,p,x˜ M) is the solubility product for the forν+Aν-‚(H2O)νW mation of solid Cν+Aν-‚(H2O)νW (on the molality scale):

m RT ln Ksp,C (T,p,x˜ M) ) µCν+Aν-‚(H2O)νW(T,p) ν+Aν-‚(H2O)νW

µ∞,m ˜ M) - νWµpure CA (T,p,x W (T,p) (A3)

(54)

Both terms (If ′ - f) and (∂f/∂x˜ M)I are proportional to the DebyeHu¨ckel parameter Aφ (cf. eqs 8, 26, and 27). Aφ depends on the density d and the relative dielectric constant of the salt-free solvent mixture (cf. eq 10). In particular, if that dielectric constant is low, Aφ can become very large, and neglecting ln(γM,Pitzer/γW,Pitzer) can result in a large error even at very low ionic strengths. Very small amounts of salts may then have a large influence on the relative volatility R(CA). Conclusions New experimental data for the vapor-liquid equilibrium of the system (water + methanol + sodium sulfate), measured with a thin-film evaporator technique at temperatures (and pressures) ranging from about 314 to 394 K (and 10 to 390 kPa) and for sodium sulfate concentrations below the solubility limit, are presented. The recently published9 extension of Pitzer’s Gibbs excess energy equation for aqueous electrolyte solutions7,8 to mixedsolvent electrolyte solutions is successfully applied to describe the phase equilibrium data. The model requires information on the solubility of the salt in pure water (which was taken from the literature), on the vapor-liquid equilibrium of the salt-free binary system (which was also taken from the literature), and on the Gibbs energy of transfer of sodium sulfate from pure water to the mixtures of water and methanol. That Gibbs energy of transfer was determined from literature data for the solubility of sodium sulfate in mixtures (water + methanol). With only three parameters (for the aforementioned temperature and solvent mixture composition dependent Gibbs energy of transfer), the model is able to describe the solubility limit of the salt in the mixed solvent over the ranges of experimental temperature and solvent concentration values and predicts the salt precipitating (either sodium sulfate or sodium sulfate decahydrate). In particular, that new model is able to quantitatively predict the new experimental results for the vapor-liquid equilibrium of that ternary system within experimental uncertainty. Acknowledgment The authors express their gratitude to the German Government and to BASF AG, Bayer AG, Degussa AG, Lurgi Oel Gas Chemie GmbH, Lurgi Energie und Entsorgung GmbH, and Siemens-Axiva GmbH & Co.KG for supporting the experimental part of this investigation (BMBF Grant No. 01/RK9808/8). Appendix I For a salt Cν+Aν-, which is fully dissociated in the liquid solvent mixture of water + methanol

µCν+Aν-‚(H2O)νW(T,p) is the chemical potential of the pure solid phase, µpure W (T,p) is the chemical potential of the pure liquid water, and

˜ M) ) ν+µ∞,m ˜ M) + ν-µ∞,m ˜ M) µ∞,m CA (T,p,x C (T,p,x A (T,p,x

(A4)

where µ∞,m ˜ M) is the chemical potential of the ionic k (T,p,x species k in its reference state, where one mole of that ionic species is dissolved in one kilogram of the solVent mixture of water + methanol and it experiences interactions as infinitely diluted in the solVent mixture. The molar Gibbs energy of transfer of solute species k from pure water to an aqueous solution of, e.g., methanol, is defined (on the molality scale) as ∞,m ˜ M) ) µ∞,m ˜ M) - µk,W (T,p) ∆tGm k (T,p,x k (T,p,x

(A5)

It is the change in the reference state of the chemical potential of solute species k encountered in the isothermal and isobaric transfer from pure water to the solvent mixture. Therefore

˜ M) ) µ∞,m ˜ M) - µ∞,m ∆tGm CA(T,p,x CA (T,p,x CA,W(T,p)

(A6)

Combining eqs A3 and A6 results in m (T,p,x˜ M) ) Ksp,C ν+Aν-‚(H2O)νW

[

]

∆tGm ˜ M) CA(T,p,x (A7) RT

m (T,p) exp Ksp,C ν+Aν-‚(H2O)νW,W

Appendix II Pitzer proposed the following (molality scale based) equation for describing the Gibbs excess energy of a pure solvent electrolyte containing solution:7,8

GEPitzer M/s nsRT

) f(I) +

m i mj

∑ ∑ m° m°λij(I) +

i,j*s

mi mj mk

∑ ∑∑ m°m° m°µijk i,j,k*s

(A8)

where i, j, and k stand for the ionic solute species, i.e., for / example, for Na+ and SO24 , if sodium sulfate is added. Ms is the relative molar mass of the solvent component divided by 1000, and ns is the amount of substance of the solvent component. The function f(I) is Pitzer’s modification of the Debye-Hu¨ckel term. The terms λij(I) and µijk are osmotic virial

464


coefficients for interactions between solute species in the particular solvent. They are symmetric:

λij ) λji

(A9)

µijk ) µikj ) µjik ) µjki ) µkij ) µkji

(A10)

It is usual practice to neglect all parameters describing interactions between ionic species carrying charges of the same sign. The coefficient λij is expressed by means of (1) (1) (2) (2) λij ) β(0) ij + βij g(Rij xI) + βij g(Rij xI)

species k in a reference state, where 1 mol of that ionic species is dissolved in 1 L of a liquid solution of water + that ionic species and where it experiences interactions as infinitely diluted in pure water. The term µ∞,c k,W(T,p) is split up into ∞,c ∞,c ∞,c µk,W (T,p) ) µk,W (T,p)|uncharged + ∆µk,W (T,p)|electrical charge (A17)

where

(A11)

∞,c (T,p)|electrical charge ) ∆µk,W

where

g(x) ) β(0) ij ,

β(1) ij ,

2 [1 - (1 + x) exp(-x)] x2

β(2) ij ,

R(1) ij ,

(A12)

R(2) ij

The terms and are binary parameters. The terms µijk are ternary parameters. For binary (chemically nonreacting) systems (water + salt Cν+Aν-), it is not possible to separate the influence of µC,C,A from that of µC,A,A. Therefore, both ternary parameters are usually summarized in CφCA:

CφCA )

3

[ν+µC,C,A + ν-µC,A,A]

xν+ν-

(A13)

(1) (2) φ It is common practice to report β(0) C,A, βC,A, βC,A, and CCA (or to set either µC,C,A or µC,A,A to zero and report either µC,A,A or (2) µC,C,A). The numerical values for R(1) CA and RCA are usually fixed. For example, for a 2-1 type electrolyte (like Na2SO4), (2) R(1) CA is usually set to 2, and because βC,A is not required for that type of electrolyte, the numerical value for R(2) CA is meaningless.

Appendix III The molar Gibbs energy of transfer of solute species k from pure water to an aqueous solution of, e.g., methanol, (based on the molarity scale) is ∞,c ˜ M) - µk,W (T,p) ∆tGck(T,p,x˜ M) ) µ∞,c k (T,p,x

(A14)

˜ M) is the chemical potential of the The expression µ∞,c k (T,p,x ionic species k in a reference state, where 1 mol of that ionic species is dissolved in 1 L of a liquid solution of water + methanol + that ionic species and where it experiences interactions as infinitely diluted in the solVent mixture (of water ˜ M) may be split up in + methanol). The expression µ∞,c k (T,p,x two contributions:

˜ M) ) µ∞,c ˜ M)|uncharged + µ∞,c k (T,p,x k (T,p,x ˜ M)|electrical charge (A15) ∆µ∞,c k (T,p,x The first contribution µ∞,c ˜ M)|uncharged represents the k (T,p,x chemical potential of the uncharged species k in the reference state described before, and the second contribution ∆µ∞,c ˜ M)|electrical charge represents the change in that chemik (T,p,x cal potential which is only due to the charging process. This second term is described by Born:25

˜ M)|electrical charge ) ∆µ∞,c k (T,p,x

NAe2 zk2 8π0 rk

(A16)

By analogy, µ∞,c k,W(T,p) is the chemical potential of the ionic

NAe2 zk2 8π0W rk

(A18)

Combining eqs A14-A18 results in

∆tGck(T,p,x˜ M) ) ∆tGck(T,p,x˜ M)|uncharged + ∆tGck(T,p,x˜ M)|Born (A19) where

˜ M)|uncharged ∆tGck(T,p,x˜ M)|uncharged ) µ∞,c k (T,p,x ∞,c (T,p)|uncharged (A20) µk,W

and

∆tGck(T,p,x˜ M)|Born ) ∆µ∞,c ˜ M)|electrical charge k (T,p,x ∞,c (T,p)|electrical charge ) ∆µk,W

[

]

2 N Ae 2 1 1 zk (A21) 8π0 W rk

As it is exemplarily shown in the present publication (cf. also ref 9), if the Gibbs energy of transfer of a salt (from pure water to the mixed-solvent) is approximated by the Born term alone, i.e., if the first contribution on the right-hand side of eq A19 is discarded, a large disagreement is observed between the predicted salt solubility in the mixed-solvent and the experimental data. Nomenclature ai ) activity of species i A ) anion AF,n ) Redlich-Kister parameter (for describing property F; n ) 0, 1, 2, ...) Aφ ) Debye-Hu¨ckel parameter b ) parameter in Pitzer’s modified Debye-Hu¨ckel term B ) second virial coefficient C ) cation Cφ ) third osmotic virial coefficient in Pitzer’s GE equation d ) specific density e ) electron charge f ) function g ) function G ) Gibbs energy (or molar Gibbs energy) GE ) Gibbs excess energy h ) function I ) ionic strength (on the molality scale) k ) salt-effect parameter kB ) Boltzmann constant Km sp solubility product (on the molality scale) m j CA ) stoichiometric molality of salt CA m ˜ i ) mass of solvent component i m° ) reference molality (m° ) 1 mol‚kg-1)


Mi ) molar mass of solvent component i M/i ) relative molar mass of solvent component i divided by 1000 M* ) (mean) relative molar mass of the solute-free solvent mixture divided by 1000 ni ) amount of substance of species i njCA ) stoichiometric amount of substance of salt CA N ) number of experimental points NA ) Avogadro number p ) total pressure psi ) vapor pressure of component i qi ) UNIQUAC parameter of pure component i (depending on external surface area) ri ) UNIQUAC parameter of pure component i (depending on molecular size) ri ) ionic radius of species i R ) universal gas constant S ) molar entropy T ) absolute temperature T° ) reference temperature (here, the standard temperature, T° ) 298.15 K) Vi ) (partial) molar volume of component i x ) variable xi ) true mole fraction of species i x˜ i ) mole fraction of component i in the solute-free solvent mixture x/CA ) mole fraction of salt CA in the liquid phase (cf. eq 43) yi ) vapor phase mole fraction of component i z ) coordination number (z ) 10) z+, z- ) number of charges on the cation and anion in electrolyte CA Z ) property Z ) yM, T, or p ∆Z ) average difference between the experimental and calculated numerical values for property Z (see footnote of Table 4) Greek Letters R(CA), R ) relative volatilities with and without salt, respectively (cf. eqs 40 and 41) E R(k) ij ) binary parameters in Pitzer’s G equation (k ) 1, 2) (k) E βij ) parameters in Pitzer’s G equation (k ) 0, 1, 2) describing binary interactions (between solute species i and j in the solvent mixture) γi ) activity coefficient of species i γ( ) mean ionic activity coefficient ∆t ) transfer property ) relative dielectric constant 0 ) permittivity of vacuum θ˜ i ) area fraction of component i in the solute-free solvent mixture λij ) second virial coefficient in Pitzer’s GE equation µi ) chemical potential of species i µijk ) third virial coefficient in Pitzer’s GE equation, parameter describing ternary interactions (between solute species i, j, and k in the solvent mixture) ν+, ν- ) number of cations and anions in electrolyte CA νW ) hydration number φi ) fugacity coefficient of component i φ˜ i ) segment fraction of component i in the solute-free solvent mixture ψij ) UNIQUAC parameter for interactions between solvent components i and j

Subscripts conv ) conversion exp ) experimental ijk ) species i, j, k M ) methanol pred ) predicted s ) solvent component W ) water Superscripts c ) on the molarity scale m ) on the molality scale x ) on the mole fraction scale o ) in a reference state s ) saturated vapor property ∞ ) infinite dilution Literature Cited (1) Furter, W. F. Ph.D. Thesis, University of Toronto, Toronto, Ontario, 1958. (2) Johnson, A. I.; Furter, W. F. Can. J. Chem. Eng. 1960, 38, 78-87. (3) Meranda, D.; Furter, W. F. AIChE J. 1974, 20, 103-108. (4) Ramalho, R. S.; James, W.; Carnaham, J. F. J. Chem. Eng. Data 1964, 9, 215-217. (5) Iliuta, M. C.; Thyrion, F. C. Fluid Phase Equilib. 1995, 103, 257284. (6) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall: New York, 1999. (7) Pitzer, K. S. J. Phys. Chem. 1973, 77, 268-277. (8) Pitzer, K. S. ActiVity Coefficients in Electrolyte Solutions; CRC: Boca Raton, FL, 1991; pp. 75-153. (9) Pe´rez-Salado Kamps, A Ä . Ind. Eng. Chem. Res. 2005, 44, 201-225. (10) Iliuta, M. C.; Thomsen, K.; Rasmussen, P. Chem. Eng. Sci. 2000, 55, 2673-2686. (11) Austgen, D. M.; Rochelle, G. T.; Peng, X.; Chen, Ch.-Ch. Ind. Eng. Chem. Res. 1989, 28, 1060-1073. (12) Kalidas, C.; Hefter, G.; Marcus, Y. Chem. ReV. 2000, 100, 819852. (13) Hefter, G.; Marcus, Y.; Waghorne, W. E. Chem. ReV. 2002, 102, 2773-2836. (14) Pabalan, R. T.; Pitzer, K. S. Geochim. Cosmochim. Acta 1987, 51, 2429-2443. (15) Emons, H.-H.; Ro¨ser, H. Z. Anorg. Allg. Chem. 1966, 346, 225333. (16) Emons, H.-H.; Ro¨ser, H. Z. Anorg. Allg. Chem. 1967, 353, 135147. (17) Emons, H.-H.; Ro¨ser, H.; Roschke, E. Z. Anorg. Allg. Chem. 1970, 375, 281-290. (18) Fleischmann, W.; Mersmann, A. J. Chem. Eng. Data 1984, 29, 452-456. (19) Zhang, D.; Okada, S.; Yazawa, A. Shigen to Sozai 1989, 105, 481486. (20) Rumpf, B.; Maurer, G. Ind. Eng. Chem. Res. 1993, 32, 17801789. (21) Abrams, D. S.; Prausnitz, J. M. AIChE J. 1975, 21, 116-128. (22) Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Glasses; John Wiley & Sons: New York, London, Sydney, 1968. (23) Rogers, P. S. Z.; Pitzer, K. S. J. Phys. Chem. 1981, 85, 28862895. (24) Okorafor, O. C. J. Chem. Eng. Data 1999, 44, 488-490. (25) Born, M. Z. Phys. 1920, 1, 45-48. (26) Bjerrum, N.; Larsson, E. Z. Phys. Chem. 1927, 127, 358-384. (27) Marcus, Y. Ion Properties; Dekker: New York, 1997. (28) Saul, A.; Wagner, W. J. Phys. Chem. Ref. Data 1987, 16, 893901. (29) Reid, C. R.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (30) Hales, J. L.; Ellender, J. H. J. Chem. Thermodyn. 1976, 8, 11771184. (31) Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem. Process Des. DeV. 1975, 14, 209-216.

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(32) Hasse, H. Ph.D. Thesis, University of Kaiserslautern, Germany, 1990. (33) Albert, M.; Hahnenstein, I.; Hasse, H.; Maurer, G. AIChE J. 1987, 42, 1741-1752. (34) Jo¨decke, M.; Pe´rez-Salado Kamps, A Ä .; Maurer, G. J. Chem. Eng. Data 2005, 50, 138-141.

ReceiVed for reView July 12, 2005 ReVised manuscript receiVed October 14, 2005 Accepted November 7, 2005 IE0508177

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