Prediction of vapor pressure for nonaqueous electrolyte solutions using an electrolyte equation of state

RglDPHASE [OUIUBRIA ELSEVIER

Fluid Phase Equilibria 138 (1997) 87-104

Prediction of vapor pressure for nonaqueous electrolyte solutions using an electrolyte equation of state You-Xiang Zuo, Walter Ftirst

*

Laboratoire Rdacteurs et Processus, ENSMP-ENSTA, 32 Bd. Victor, 75739 Paris, Cedex 15, France

Received 23 September 1996; accepted 21 April 1997

Abstract The aqueous electrolyte equation of state (EOS) developed by FiJrst and Renon [W. Fiirst, H. Renon, AIChE J. 39 (1993) 335-343] has been extended to nonaqueous electrolyte solutions. Binary interaction parameters for ion-solvent and ion-ion pairs are estimated from ionic Stokes and Pauling diameters. The temperature dependence of the interaction parameters involving ions has been neglected since the temperature range in question is from 278.15 to 348.25 K. The extended electrolyte EOS has been used to calculate vapor pressures and mean ionic activity coefficients of nonaqueous solutions of single electrolytes without any adjustable parameters. The predicted results are quite satisfactory, the overall average absolute deviation (AAD) for predicted vapor pressure being approximately 1%. In addition, the extended electrolyte EOS has been compared with the electrolyte NRTL model of Mock et al. [B. Mock, L.B. Evans, C.C. Chen, AIChE J. 32 (1986) 1655-1664], with the extended electrolyte UNIQUAC models of Sander et al. [B. Sander, Aa. Fredenslund, P. Rasmussen, Chem. Eng. Sci. 41 (1986) 1171-1183] and Macedo et al. [E.A. Macedo, V. Skovborg, P. Rasmussen, Chem. Eng. Sci. 45 (1990) 875-882] and also with the one and three adjustable parameter models of Pitzer [K.S. Pitzer, J. Phys. Chem. 77 (1973) 268-277]. The deviations obtained by the application of our electrolyte EOS to various nonaqueous systems are of the same order of magnitude as those obtained by the models of Mock et al., Sander et al. and Macedo et al. as well as by Pitzer's model with one adjustable parameter, although our model is a predictive one. © 1997 Elsevier Science B.V. Keywords: Theory; Equation of state; Electrolyte; Nonaqueous solutions; Alcohol; Vapor pressure; Modeling

1. I n t r o d u c t i o n

Phase equilibria in electrolyte systems are o f great importance in m a n y industries. In the past few decades, considerable progress has been m a d e in measuring and modeling t h e r m o d y n a m i c properties

* Corresponding author. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0378-3812(97)00145-3

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of electrolyte systems. However, most of the studies are restricted to aqueous systems. Much less attention has been given to nonaqueous electrolyte solutions. Although some measured property data are available in the literature, experimental data of nonaqueous electrolyte systems are still rather scarce compared to aqueous electrolyte systems. In engineering applications, therefore, a simple, reliable and predictive model is required for representing phase behavior of nonaqueous electrolyte systems. Up to now, very few models have been proposed for the representation of equilibrium properties in the case of nonaqueous electrolyte solutions (Mock et al. [1]; Sander et al. [2]; Macedo et al. [3]; Li et al. [4]). Pitzer's model [5] has also been extended to the representation of various data in nonaqueous electrolyte solutions (Tomasula et al. [6]; Barthel et al. [7-10]; Han and Pan [11]). However, all these models need adjustable parameters. It is also possible to consider nonaqueous electrolyte systems as ideal solutions, which result in a very simple predictive method. Obviously, this implies that excess functions are negligible. This point will be discussed later, and it will be shown that, although this problem highly depends on the type of data represented, the ideal solution hypothesis is inadequate in many cases. Furthermore, this is also the case of predictive models involving only a Debye-Hi.ickel term. Therefore, up to now, no thermodynamic models exist, which allow accurate calculations of equilibrium properties of nonaqueous electrolyte systems without parameter fitting. Ftirst and Renon [12] derived a new equation of state (EOS) for aqueous electrolyte systems from an expression of the Helmholtz energy containing a nonelectrolyte part and a part relative to ions. The nonelectrolyte part is taken from the modified SRK EOS of Schwartzentruber et al. [13]. The ionic part consists of an MSA long-range interaction term to account for electrostatic forces and a short-range interaction term to account for ionic solvation effects. Since all model parameters were correlated to cationic Stokes diameters and anionic Pauling diameters, this leads to a predictive method which has been successfully applied to aqueous electrolyte systems. In this work, we have tried to extend the aqueous electrolyte EOS proposed by Ftirst and Renon [12] to nonaqueous electrolyte systems, preserving the predictive characteristics of the model. The extended electrolyte EOS has been tested by predicting vapor pressures and mean ionic activity coefficients of single nonaqueous solvent-salt systems without any adjustable parameters. The predicted results are in good agreement with experimental data. In addition, the models of Mock et al. [1], Sander et al. [2], Macedo et al. [3], Pitzer [5] have been compared with our extended electrolyte EOS.

2. Equations of our electrolyte equation of state Fiirst and Renon [12] derived an aqueous electrolyte equation of state based on an expression of the Helmholtz free energy. The molar Helmholtz energy of an electrolyte mixture consists of four contributions:

The first two terms on the right-hand side of Eq. (1) are the repulsive force (RF) contribution and

E-X. Zuo, W. Fiirst / Fluid Phase Equilibria 138 (1997)87-104

89

the first attractive short-range (SR1) contribution, respectively. They are similar to the expression of the SRK equation of state:

RF

+

~

SR1

= Y'~xkln P0(-'v--b) k

+ RT(b+c)In

v + b + 2c

(2)

where the summation is over all components (including both ionic and molecular species). The energy parameters a sR and the volume translation parameters c of ions are set to zero, but the covolumes b of all the species are taken into account. The energy parameter and the covolume of pure molecular components are calculated in the same way as presented by Schwartzentruber et al. [13]. The classical van der Waals mixing rules are used for a sR, b and c. The third term on the right-hand side of Eq. (1) is the second short-range (SR2) contribution specific to interactions between a molecule and an ion or between two ions:

() Aa

~

x x,w , = E E u ( 1 - ~3) SR2 k /

(3)

where at least one of species k and l is an ion and ~3 is defined as: NTr s¢3=--E 6

xko -3 k

(4)

v

in which the summation is over all components. The last term on the right-hand side of Eq. (1) is the long-range (LR) interaction contribution. The simplified MSA model proposed by Ball et al. [14] is used:

"~

LR-

47r

. 1 + Fo"i + 3rr---N

(5)

where the shielding parameter F is given by:

'--g7 x

1 - ~ziF ~ , )2

(6)

and 2 __ OlLR

e2N

(7)

e oDRT

The dielectric constant of the solution takes into account the influence of the ions through Pottel's expression (Pottel [15]) and is expressed as:

D = 1 + ( D ~ - 1)

~3'

(8)

1+ T In this equation ~3' is similar to ~3 defined in Eq. (4) but the summation is only over ions. D S stands for the dielectric constant of the solvent.

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Y.-X. Zuo, W. Fiirst / Fluid Phase Equilibria 138 (1997) 87-104

As in the model, the following relation between ionic diameters and ionic covolumes is assumed:

~ 6b i °'i =

Nvr

(9)

The model contains four kinds of adjustable parameters which are anionic (b~) and cationic (be) covolumes and symmetrical interaction parameters between a cation and a solvent (Wcs) and between a cation and an anion (Wa~). The interaction parameters between ions with charges of the same sign and those between anions and solvents are set to zero because of the charge repulsive effect and the lower solvation of anions. Hence, the model contains a great number of adjustable parameters. However, all these parameters can be deduced from the properties related to solvation characteristics, as shown in a data treatment of numerous aqueous strong electrolytes solutions (Fiirst and Renon [121). The ionic covolumes are calculated by the following correlations: bc = Al(tycs)3 + A2

(10)

b a : A,(ozap)3 + A2

(ll)

and

where o-cs and o;ae are the cationic Stokes diameter and the anionic Pauling diameter, respectively. The use of crystalline diameter is also justified by a lower solvation of anions, compared to cations. Interaction parameters for cation-solvent pairs We, and for anion-cation pairs Wac are given by: Wcs = A 30-c s + A 4

(12)

Wac = a,(

(13)

and

s + op)4 + A6

Coefficients A~-A6 have been deduced from a data treatment of numerous experimental coefficients relative to halide and non-halide systems.

3. Extension of the model to n o n a q u e o u s solutions

The model takes into account molecular species as well as ions and since the solvent is treated as a molecular compound, the model equations can be applied without modifications to electrolyte solutions in various solvents. However, as we want to maintain the predictability of the model when it is applied to nonaqueous systems, some hypotheses have to be made for the evaluation of the various parameters, especially in the case of ionic parameters.

3.1. Discussion on data available and on their sensitivity to excess properties As said above, only few data sets are available to test the extension of our equations. Skabichevskii [16] reported osmotic coefficients of LiCI and LiBr solutions in methanol at 15-30°C from the

Y.-X. Zuo, W. Fiirst/ Fluid Phase Equilibria 138 (1997) 87 104

91

measurements of vapor pressures. Sada et al. [17] measured vapor pressure data of LiC1 and LiBr solutions in 2-propanol. Bixon et al. [18] presented vapor pressure data of methanol with seven salts. Tomasula et al. [6] reported vapor pressure data of 11 single electrolytes with methanol and successfully correlated the experimental data by use of two Bromley-type and two Pitzer-type models up to an ionic strength of 6 molalities. Mato and Cocero [19] measured vapor pressure data of ethanol solutions of salts. Recently, Barthel and co-workers ([7-10]) reported experimental vapor pressures over four nonaqueous electrolytes systems (methanol, ethanol, 2-propanol, acetonitrile). Vapor pressure data of normal alcohol and CaC12 solutions were presented by Hongo et al. [20]. More currently, Yamamoto et al. [21] reported vapor pressure data of ethanol with three single salts. Vapor pressure data of CuC12 and ethanol systems were measured by Oh and Campbell [22]. Han and Pan [I 1] reported mean ionic activity coefficients of NaBr in methanol and in ethanol at 298.15 K. Hence, data have been reported to systems with four nonaqueous solvents: methanol, ethanol, 2-propanol and acetonitrile. From the preceding list, it appears that most of the data available in the literature are equilibrium pressures. This is important because it is well known that this kind of data is not well adapted to the determination of parameters involved in expressions used for the calculation of excess properties. Even in the case of strong electrolyte aqueous solutions, the contribution of the activity coefficient to the activity of water (hence, also to the equilibrium pressure) is generally greater than 1% only at molalities greater than 1 m o l / k g , although depending on the salt considered. This is why, in the previous paper [12], we used osmotic coefficients (mean activity coefficients are also well adapted data). Hence, when the database used to adjust model parameters is only made of vapor pressure data, the resulting model may fail to represent accurately excess properties such as mean activity coefficients. In the case of our approach, the problem is different because it is a predictive model. What we want is to compare the predicted values to experimental ones and this can be done on both types of data: vapor pressures and mean activity coefficients (no direct experimental determination of osmotic coefficients being available in the literature). An important difference between aqueous solutions and most of nonaqueous systems is that the dipole moment of nonaqueous molecules is generally lower than that of water. This results in a significantly lower dielectric constant. Hence, the screening effect of electrostatic interactions between ions is less efficient than in water. This has to be related to the lower solubility of various salts in nonaqueous solvents, compared to the case of aqueous solutions, although the solubility limit may be high enough: for instance, the solubility limits of LiC1 in methanol and in ethanol at 25°C are 10.35 and 5.82 m o l / k g [23], respectively. However, the solvent influence on excess properties is not restricted to its screening effect (expressed through Debye-Hi~ckel type models). Another important interaction which has to be taken into account for the representation of excess properties is the ion-solvent interaction corresponding to solvation. If, for dilute solutions, electrostatic interactions between ions are responsible of the main part of excess properties, at higher ionic concentrations, solvation interactions become preponderant. In our database (see Table 1), the maximum molalities of most of the systems are higher than unity (up to 5.3 in the case of LiC1 in methanol) at 25°C. This explains why the ideal solution hypothesis does not represent vapor pressures over a whole molality range (see Fig. 1, for instance). This is more obvious in the case of mean activity coefficient data. Not only are the experimental values far from unity, but also the Debye-Hiickel model is clearly inadequate (as shown in Fig. 2). Therefore, a more accurate model is highly required.

Y.-X. Zuo, W. Fiirst/ Fluid Phase Equilibria 138 (1997) 87-104

92

Table 1 Average absolute deviations of the predicted vapor pressure for single nonaqueous solvent-salt systems by use of our model at 298.15 K Solvent + Salt

No. of data points

Methanol + Bu4NI Methanol + Am4NBr Methanol + Bu4NBr Methanol + Et4NBr Methanol + CaC1 z Methanol + CuC12 Methanol + KI Methanol + NaI Methanol + NaBr Methanol + LiCI Methanol + LiBr Methanol + CsI Methanol + RbI Methanol + KBr Methanol + NaCI Ethanol + CaC12 Ethanol + NH 41 Ethanol + CuCI 2 Ethanol + NaI Ethanol + LiC1 Ethanol + LiBr 2-Propanol + Na! Acetonitrile + Am4NBr Acetonitrile + Bu4NI Acetonitrile + Bu4NBr Acetonitrile + Bu4NCI Acetonitrile +Pr4NBr Acetonitrile+Et4NBr Acetonitrile + NaI Overall

43 29 35 30 16 11 55 66 37 27 18 41 46 23 33 13 5 6 36 12 12 22 32 22 27 26 31 18 26 798

AAD, % Method I

Method II

Mock et al.

0.57 1.08 0.95 0.45 2.08 1.50 0.26 0.78 0.32 1.65 2.03 0.21 0.22 0.19 0.18 0.35 0.96 0.30 0.93 1.75 0.88 1.66 1.29 1.66 0.65 0.58 0.45 0.11 0.36 0.75

0.49 0.94 0.77 0.39 1.68 2.64 0.29 0.97 0.41 3.28 2.24 0.21 (I.21 0.19 0.17 1.45 1.16 0.55 0.95 1.04 0.78 1.71 I. 11 1.24 0.54 0.55 0.33 0.11 0.38 0.78

0.38 0.24 0.25 0.29 1.41 0.41 0.25 0.84 0.22 2.90 3.17 0.16 0.24 0.12 0.19 0.49 0.18 0.84 1.19 1.87 1.92 2.26 0.85 0.63 0.81 0.86 0.86 0.82 0.78 0.69

Maximum molality

Source of data

0.907 1.560 1.651 1.874 2.635 3.396 1.122 4.520 1.556 5.669 4.467 0.130 0.436 0.134 0.219 1.879 1.519 2.840 2.610 2.340 2.330 1.465 2.237 2.401 1.957 2.513 1.573 0.367 1.542

b b b b e, e a, a, a, e, f a a a a g, g h c, h h c d d d d d d c

i e e, f e f, j

i

g, h

Source of data: (a) Barthel et al. [7]; (b) Barthel et al. [8]; (c) Barthel and Lauermann [9]; (d) Barthel and Kunz [10]; (e) Bixon et al. [18]; (f) Tomasula et al. [6]; (g) Yamamoto et al. [21]; (h) Mato and Cocero [19]; (i) Hongo et al. [20]; (j) Skabichevskii [16].

3.2. E v a l u a t i o n o f p u r e m o l e c u l a r c o m p o n e n t p a r a m e t e r s For pure molecular components, [24]. T h e p u r e c o m p o n e n t

c r i t i c a l p r o p e r t i e s a n d a c e n t r i c f a c t o r s a r e t a k e n f r o m R e i d et al.

parameters

are t h e v o l u m e

translation parameters

p a r a m e t e r s p~, P2 a n d P3 u s e d f o r t h e c a l c u l a t i o n o f p u r e c o m p o n e n t c o m p o u n d s aSR: SR

as

1 9(21/3-1)

c and the three polar

energy parameters of molecular

(RTc) 2

~(l+m(og)(1-~)-pl(1-Tr)(l+p2Tr+p3Tr2) rc

-

---

2

(14)

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Y.-X. Zuo, W. Fiirst / Fluid Phase Equilibria 138 (1997) 87-104

18,00

16,00 rl v

o

o '"'"-. o

14,00

•

""--....

o

......... °°°

i

o

•

'"+"

a. 12,00 >

o E*n.(Tomasulae~.[6])

10,00

"~"'to

o Exp.(Barthel et al., ['7]) - Predicted by Our Method I . . . . . . Ideal Solution

8,00 ........................ 0,00 1,00 2,00

o

3,00

4,00

5,00

Molality of Nal in Methanol

Fig. l. Use of our model to study the consistency of data sets relative to experimental vapor pressures for the methanol-NaI system at 298.15 K.

where:

m(w) =

0.48508 + 1.55191w - 0.15613o92

(15)

These parameters are determined by fitting experimental vapor pressure and density data of pure components (Boublik et al, [25] and Reid et al. [24]), respectively. The values of these parameters are given in Table 2.

1,00

z

0,80

"5 OJ

'6 0,60 0

o

1

o o --Our - - - - -

Exp. (Methanol, Han and Pan, [11]) Exp. (Ethanol, Han and Pan, [11]) Method I Our Method II Pitzer's Debye-H0ckel Model

.~ 0,40 "

0,20

'..!"-..... - . .

0,00 . . . . . . . . . . . . . . . . . . . 0,00 0,50 1,00 1,50

2,00

Molality of NaBr in Methanol/in Ethanol

Fig. 2. Prediction of mean activity coefficients relative to NaBr solutions in methanol/ethanol at 298.15 K.

Y.-X. Zuo, IV. F~rst/ Fluid Phase Equilibria 138 (1997) 87-104

94 Table 2

Parameters ( p t , P2, P3 and c) of pure solvents Solvent

Pl

P2

P3

c, × l0 -5 m o l / m 3

Methanol Ethanol 1-Propanol 2-Propanol Acetonitrile Water

-0.023387 - 0.383790 -0.0065221 - 0.00086683 - 0.022927 0.023175

- 15.960 - 3.7016 18.406 18.015 -15.529 4.6462

21.509 3.3217 - 28.659 5.0466 20.642 - 8.8079

1.29912 0.92059 1.13070 1.27565 0.00 0.60473

Pure solvent dielectric constants as a function of temperature are estimated by the following expression: d2

Ds = dl + - 7 + d3T + d4T2 + dST3

(16)

where coefficients dl-d 5 are solvent dependent, as listed in Table 3, which are fitted to experimental data (Akhadov [26]; Marcus [27]). For methanol, the expression presented by Maryott and Smith [28] is used: log D s = 1.514 - 0 . 0 0 2 6 4 ( T - 298.15)

(17)

In the SR2 term of the model, we have also to introduce some values for the molecular diameters. In the case of water, the chosen value (o-= 2.5 × 10 -1° m) is small compared to the published estimations of water diameters. The reason is that this parameter has some influence on the quality of the representation of excess properties. The smallest values give the best results. In the present study, it has been assumed that the same value could be given to the diameters of all nonaqueous solvents to facilitate the extension of the model to various solvents. To be consistent with the previous data treatment relative to aqueous solutions, the adopted value is: or= 2.5 × 10 -l° m.

3.3. Representation of equilibrium data using the directly extended model and evaluation of ionic parameters In this work, it is assumed that a salt is completely dissociated into ions in the liquid phase and not present in the vapor phase. The first hypothesis is not obvious and some studies have explained the variation of properties in nonaqueous solutions (e.g., conductance data) by partial ion pairing. However, if we take into account ion pairing, we have to include association constants which have to be adjusted. This results in a

Table 3 Temperature coefficients of dielectric constants in Eq. (16) for pure solvents Solvent

dt

d2

d3

d4, X 10 - 2

d 5, × l 0 -5

T-range, K

Water Ethanol l-Propanol 2-Propanol Acetonitrile

- 19.2905 175.72 173.19 178.22 35.9

29814.5 - 3.0699 - 151.07 31.152

- 0.019678 -0.35350 -0.36873 - 0.37237

0.013189 - 0.20285 -0.19387 -0.20296

- 0.031144 0.50644 0.49023 0.50106

288.15-403.15 288.15-328.15 293.15-328.15 298.15-328.15 298.15

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95

Table 4 Cationic Stokes diameters (in ,~) in various solvents at 298.15 K Ion

Water

Methanol

Li + Na + K+ Cs + Rb + NH~ Me 4 N + Et4 N+ Pr 4 N + Bu4N + Am 4 N + Ca 2+ Cu 2+

4.76 3.68 2.50 2.36 2.38 2.50 4.10 5.65 7.87 9.50 10.59 6.20 6.87

7.56 6.54 5.56 4.80 5.64 4.38 4.97 5.30 7.73 8.65 9.96 6.50

Ethanol 8.81 7.43 6.38 5.79 6.25 5.09 5.15 6.83 7.67 l 0.00 7.80

l -Propanol

2-Propanol

10.10 10.04 12.19

10.10 10.04

5.82 5.57 6.88 7.83 11.00

Acetonitrile 6.25 5.73 5.48

5.07 5.65 6.81 7.80 8.50 1 1.00

non-predictive model. Hence, to maintain the predictability of our model, total dissociation is assumed. This hypothesis is also made by most of the authors studying such systems (Mock et al. [1]; Sander et al. [2]; Macedo et al. [3]; Li et al. [4]). In fact the real justification of this hypothesis would be obtained from the capability of our equations to predict excess properties of nonaqueous systems. In the original equation of state proposed by Fiirst and Renon, coefficients A~-A4 in Eqs. (10)-(12) are universal constants for all the ions in aqueous electrolyte solutions: A~ = 0 . 1 0 6 8 8 X 10 - 6 , /~2 = 6.5665 X 10 - 6 , h 3 = 35.09 X 10 - 6 and h 4 = 6.004 X 10 - 6 , while h 5 and z~6 a r e anion dependent. For systems only containing halides (CI-, B r - and I - ) , h 5 = - 0 . 0 4 3 0 4 X 10 - 6 and h 6 = - 2 7 . 5 1 X 10 -6. In Eqs. (10)-(13), all the parameters are based on SI units but the ionic Stokes and Pauling diameters are expressed in Angstr6m (A). In the first step, a direct extension of the model has been tested. All the model parameters are calculated from the correlations mentioned above. Hence, the values of the coefficients h~-h 6 involved in Eqs. (10)-(13) were assumed to be the same as those determined by FLirst and Renon [ 12] in the case of aqueous electrolyte solutions at 298.15 K. Obviously, cationic volumes (b c) and size parameters (o-c) as well as binary interaction parameters (IV j) are solvent dependent and in this method which is hereafter referred to as method I, the influence of the solvents is simply expressed through the use of Stokes diameters measured in the corresponding solutions. By doing this, the model becomes purely predictive and does not need any adjustable parameters for the representation of equilibrium properties of nonaqueous electrolyte solutions. Table 4 gives the values of Stokes cationic diameters used in the calculations. Some are taken from Kay and Broadwater [29] and Hartley and Raikes [30], but most of them have been calculated from the limiting cationic conductance data published by Conway [31], using the Stokes law:

Izcl °'~s = 1.64 A0----~

(18)

where T/and A ° are the viscosity of a solvent and the limiting conductance of a cation in the solvent, respectively. Method I has been used to predict vapor pressure of single nonaqueous solvent electrolyte systems

96

E-X. Zuo, W. F~rst / Fluid Phase Equilibria 138 (1997) 87-104

at 298.15 K. The results are given in Table 1 (column 3). The overall average absolute deviation (AAD) is 0.75%. This deviation is remarkably good for an entirely predictive model. It should be noted that, in the case of Ca 2+ cation, there is only one published value for its Stokes diameter, which corresponds to the case of methanol (9.96 A). But when this value is used in the correlations, the interaction parameters between calcium and methanol are overestimated and the vapor pressures are underpredicted. In this case, due to the high cationic charge, the formation of partial ion pairing may be significant. Therefore, in order to calculate vapor pressures of such mixtures accurately, calcium Stokes diameters are fitted using experimental vapor pressure relative to calcium chloride in methanol and in ethanol at 25°C. The fitted calcium diameters in methanol and ethanol are 8.8 and 9.1 ,~, respectively. The results for methanol-CaC12 mixtures and for ethanolCaC12 mixtures shown in Table 1 are based on the fitted Stokes diameters. As mentioned above, it is much more difficult to represent mean ionic activity coefficients than to describe vapor pressure, but the representation of such excess properties may be important for several applications. Hence, method I has also been used to predict the mean activity coefficients of NaBr in methanol and in ethanol at 298.15 K. The results are shown in Fig. 2 (solid lines). For NaBr-methanol mixtures, the prediction is excellent while the mean activity coefficients are somewhat overpredicted for NaBr-ethanol mixtures. In addition, we have compared our results with the prediction obtained using a Debye-Hiickel model (Pitzer's Debye-Hiickel expression used by Han and Pan [11]). From Fig. 2, it can be seen that the long-range contribution is important only at very low salt concentrations, as pointed out by Zerres and Prausnitz [32]. It has to be noticed that, using our model without the SR2 term gives about the same results as what is obtained using Debye-Htickel expression.

3.4. Representation of equilibrium data using hypotheses based on the study of ethanol solutions In the second step, a new method which is referred to as method II has been developed. As the solvations properties vary a lot from water to propanol, it has been decided to use some informations taken from the experimental equilibrium properties of ethanol solutions (a 'middle' solvent). In this case, it is assumed that the values of cationic covolumes (b c) and size parameters (o-c) in nonaqueous electrolyte solutions are the same as in aqueous electrolyte solutions. This assumption has been made to simplify the model and because the complete model has too many adjustable parameters. On the contrary, binary interaction parameters (W,.j) are solvent dependent and estimated from Eqs. (12) and (13). In this case, however, coefficients A5 and A6 in Eq. (13) are determined by fitting only the vapor pressure data relative to salt solutions in ethanol at 298.15 K. The optimized values are A5 = - 0 . 0 1 2 0 8 X 10 -6 and /~6 = - - 3 3 8 . 0 0 × 10 -6. However, it has been found that these parameters are also suitable in the case of the other nonaqueous solvents (e.g., methanol, propanol and acetonitrile). Method II has been used to predict/correlate vapor pressure of single nonaqueous solvent electrolyte systems at 298.15 K. The results are given in Table 1 (column 4). The agreement with experimental data is associated to an overall AAD of 0.78%. Again, method II has been used to predict mean activity coefficients of NaBr in methanol and in ethanol at 298.15 K, as shown in Fig. 2 (dashed lines). The representation of mean activity coefficients of NaBr solutions in ethanol is excellent, while slight overpredictions at higher ionic strength and underpredictions at lower ionic strength are observed for NaBr-methanol mixtures.

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97

F r o m t h e r e s u l t s o b t a i n e d in the p r e v i o u s s e c t i o n s , w e c a n see that m e t h o d s I a n d II are c o m p a r a b l e in the r e p r e s e n t a t i o n o f b o t h v a p o r p r e s s u r e a n d m e a n i o n i c a c t i v i t y c o e f f i c i e n t data.

3.5. Representation o f equilibrium data at various temperatures A s w e k n o w , i o n i c S t o k e s d i a m e t e r s are t e m p e r a t u r e d e p e n d e n t . D u e to the l a c k o f e x p e r i m e n t a l d a t a f o r c a t i o n i c l i m i t i n g c o n d u c t a n c e s in s o l v e n t s at o t h e r t e m p e r a t u r e s , it is i m p o s s i b l e to d e t e r m i n e the t e m p e r a t u r e d e p e n d e n c e o f c a t i o n i c S t o k e s d i a m e t e r s . T h e r e f o r e , the S t o k e s d i a m e t e r s at 2 9 8 . 1 5 K h a v e b e e n d i r e c t l y a p p l i e d to p r e d i c t v a p o r p r e s s u r e s at o t h e r t e m p e r a t u r e s . T h e p r e d i c t e d r e s u l t s a r e t a b u l a t e d in T a b l e 5. A s it c a n b e s e e n f r o m T a b l e 5, the p r e d i c t i o n s at o t h e r t e m p e r a t u r e s are in g o o d a g r e e m e n t w i t h t h e m e a s u r e d data, e s p e c i a l l y w h e n it is t a k e n i n t o a c c o u n t that h i g h e r m a x i m u m c o n c e n t r a t i o n s o f salts a r e c o n s i d e r e d , c o m p a r e d to t h o s e at 2 9 8 . 1 5 K (as g i v e n in T a b l e 1). T h e p r e d i c t i o n s b y m e t h o d I are b e t t e r t h a n t h o s e b y m e t h o d II in m o s t cases. T o test the t e m p e r a t u r e d e p e n d e n c e o f the m o d e l p a r a m e t e r s further, the S t o k e s d i a m e t e r s at 2 9 8 . 1 5 K a r e u s e d to p r e d i c t the v a p o r p r e s s u r e o f a q u e o u s e l e c t r o l y t e s o l u t i o n s at o t h e r t e m p e r a t u r e s o f 3 0 3 . 1 5 - 3 4 3 . 1 5 K. T h e p r e d i c t e d r e s u l t s b y a p p l i c a t i o n o f o u r E O S a r e in g o o d a g r e e m e n t w i t h e x p e r i m e n t a l d a t a w i t h an o v e r a l l A A D o f 1.23%, as s h o w n in T a b l e 6. T h i s r e s u l t c a n b e e x p l a i n e d b y the f a c t that, at l e a s t o v e r the t e m p e r a t u r e r a n g e c o n s i d e r e d , t h e t e m p e r a t u r e e f f e c t o n the i o n i c p a r a m e t e r s is s m a l l c o m p a r e d to its e f f e c t o n the s o l v e n t p a r a m e t e r s . Table 5 Average absolute deviations of the predicted vapor pressure for single nonaqueous solvent-salt systems by use of our model at other temperatures Solvent + Salt

No. of data points AAD, %

Maximum molality T, K

Source of data

4.338 4.580 4.580 4.345 4.345 4.467 4.467 4.467 5.669 5.669 5.669 2.627 2.080 3.593 2.842 2.842 2.842 2.842 2.842 2.842

a a a a a d d d d d d c c b e e e e e e

Method I Method II Mock et al. Methanol + NaI 14 Methanol + LiC1 9 Methanol + LiCI 9 Methanol + LiBr 11 Methanol + LiBr 10 Methanol + LiBr 7 Methanol + LiBr 7 Methanol + LiBr 7 Methanol + LiCI 9 Methanol + LiC1 9 Methanol + LiCI 9 2-Propanol+LiCl 7 2-Propanol + LiBr 6 Ethanol + CuC12 8 Ethanol + NaI 8 Ethanol + NaI 8 Ethanol + NaI 8 Ethanol + NaI 8 Ethanol + NaI 8 Ethanol + NaI 8 Overall 170

1.46 1.46 1.53 2.12 2.71 1.39 1.70 1.39 0.96 1.46 1.55 2.60 2.69 0.94 0.63 0.87 1.69 2.49 2.48 2.86 1.72

2.06 3.08 2.88 2.20 2.48 1.28 1.30 1.38 3.49 3.13 3.22 2.94 2.93 0.94 0.53 0.76 0.94 1.52 1.74 2.16 2.08

2.38 2.77 2.98 3.15 3.49 1.85 1.89 1.93 3.05 2.95 2.83 0.79 0.30 0.70 0.42 0.53 0.70 0.80 1.00 1.23 1.90

313.15 308.15 318.15 308.15 313.15 293.15 288.15 303.15 303.15 293.15 288.15 348.25 348.15 303.15 318.15 308.15 293.15 288.15 283.15 278.15

Source of data: (a) Tomasula et al. [6]; (b) Oh and Campbell [22]; (c) Sada et al. [17]; (d) Skabichevskii [16]; (e) Subbotina and Mischchenko [33].

98

Y.-X. Zuo. W. F~rst / Fluid Phase Equilibria 138 (1997) 87-104

Table 6 Average absolute deviations of the predicted vapor pressures for aqueous solutions of single electrolytes in the temperature range of 303.15-343.15 K by use of our EOS Salt No. of data points Max. molality AAD (A P / P), % Source of data LiC1 LiBr LiI NaBr NaI KCI KBr KI RbCI CsCI CsBr CsI MgCI 2 CaCI 2 CaBr 2 CaI 2 SrCI 2 SrBr 2 SrI 2 BaC12 BaBr 2 Overall

19 20 20 30 35 30 30 40 30 40 30 25 25 20 40 25 40 40 35 25 40 639

5.998 7.026 4.987 7.981 6.693 4.286 4.349 5.648 6.949 8.590 5.888 2.595 3.167 3.887 4.596 2.915 3.203 3.340 3.517 1.338 3.398

2.18 1.44 1.68 1.51 0.93 0.57 0.63 0.81 0.85 0.81 1.33 0.45 2.24 1.14 1.06 3.92 0.99 0.72 1.87 0.27 1.62 1.23

Patil et al. [34] Patil et al. [34] Patil et al. [34] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patti et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patti et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patil et al. [35] Patti et al. [35] Patil et al. [35]

Therefore, it is c o n c l u d e d that the t e m p e r a t u r e d e p e n d e n c e o f the ionic m o d e l p a r a m e t e r s m a y be n e g l e c t e d b e t w e e n 278 and 350 K.

4. Discussion and conclusion Several m o d e l s for n o n a q u e o u s electrolyte s y s t e m s can be f o u n d in the literature ( M o c k et al. [1]; S a n d e r et al. [2]; T o m a s u l a et al. [6]; M a c e d o et al. [3]; Cisternas and L a m [36]; Li et al. [4]). H o w e v e r , all these m o d e l s contain parameters w h i c h are d e t e r m i n e d b y data reduction. Usually, the n u m b e r o f adjusted parameters ranges b e t w e e n 1 and 3 per salt for e a c h solvent. H e n c e , the results o f these p r e v i o u s studies can not be c o m p a r e d to our o w n results w h i c h are o b t a i n e d w i t h o u t p a r a m e t e r fitting (at least for m e t h o d I). H o w e v e r , w e have to verify that the precision o b t a i n e d in the present w o r k are o f the s a m e order o f m a g n i t u d e as w h a t has b e e n o b t a i n e d before. T o c h e c k this point, w e h a v e c o m p a r e d our results c o n c e r n i n g solutions in m e t h a n o l with T o m a s u l a et al.'s representations o f his o w n data. T o m a s u l a et al. [6] applied P i t z e r ' s m o d e l [5] to correlate v a p o r pressure o f m e t h a n o l electrolyte systems. In e a c h case, they h a v e reported the deviations associated to the d e t e r m i n a t i o n o f one or three adjustable p a r a m e t e r s per salt. A c o m p a r i s o n o f P i t z e r ' s m o d e l with our m o d e l is g i v e n in T a b l e 7, using the s a m e database: m e t h o d I is slightly better than P i t z e r ' s m o d e l with one adjustable parameter.

Y.-X. Zuo, W. Fiirst/ Fluid Phase Equilibria 138 (1997)87-104

99

Table 7 Comparison of our EOS with Pitzer's model (one and three adjustable parameters) for vapor pressure predictions in methanol-salt systems Salts

T, K

LiC1 LiCI LiC1 LiBr LiBr LiBr NaI Nal Overall

298.15 308.15 318.15 298.15 308.15 318.15 298.15 313.15

AAD ( A P / P ) , % Method I

Method II

Pitzer-3

Pitzer-1

1.7 1.5 1.5 2.4 2.1 2.7 1.2 1.5 1.8

3.6 3.1 2.9 2.8 2.2 2.5 2.1 2.1 2.5

0.9 1.2 1.3 1.1 1.0 1.1 0.4 0.6 0.9

3.4 2.7 2.7 3.8 3.5 3.5 0.7 0.7 2.5

Source of data: Tomasula et al. [6].

The electrolyte NRTL model of Mock et al. [1] has been used to correlate the vapor pressure of nonaqueous electrolyte solutions and compared with our model. In this case, the model of Mock et al. [1] has two adjustable parameters (rmxa, Zca,m) for a binary system at one temperature and a .... has been set to the optimized values obtained by Mock et al. The detailed comparison of our model with that of Mock et al. is also given in Tables 1 and 5. The model of Mock et al. and ours give comparable results. A more extensive comparison can also be made considering the data treatment of Sander et al. [2] and of Macedo et al. [3], although these models have been applied to a database including not only salt-alcohol binary systems but also ternary mixed solvent electrolyte systems. Both models combined Debye-Hiickel theory with an extended UNIQUAC term to calculate vapor-liquid equilibria in electrolyte solutions. Although the formulation of the Debye-Hiickel term in the Sander et al. model are incorrect, as pointed out by de M. Cardoso and O'Connell [37], the incorrect term has little influence on vapor pressure predictions (much greater influence on mean ionic activity coefficients). A comparison between our model and those of Sander et al. [2] and Macedo et al. [3] has been made using the same data containing only vapor pressures for various binary (salt and alcohols) systems. The result of this comparison are summarized in Table 8. It can be seen that our EOS is superior to the models of Sander et al. and Macedo et al. Fig. 3 compares the predicted vapor pressure of ethanol-LiCl system at 298.15 K by application of four methods. The predictions by method I are in good agreement with the experimental data. The model of Macedo et al. underestimates the salting effect over a whole LiC1 concentration range. Table 8 Comparison of our EOS with the models of Macedo et al. and Sander et al. for vapor pressure predictions in single nonaqueous solvent-salt systems A A D (diP~P), %

13 data sets, 210 data points 9 data sets, 138 data points

Method I

Method II

Macedo et al.

1.31 1.08

1.38 1.21

2.97

Sander et al. 1.67

100

Y.-X. Zuo, W. Fiirst / Fluid Phase Equilibria 138 (1997) 87-104

8,00 7,50 II. .x

7,00

g 6,50 6,00 Macedo's Method a.

5,5o

D.

=, 5,00

......

Sander's M e t h o d

....

O u r M e t h o d !1

- -

O u r Method I

o

"-,?o

Exp. (Mato and Cocero, [19])

4,50 4,00 0,00

0,50

1,00

1,50

2,00

2,50

Molality of LiCl in Ethanol

Fig. 3. Representation of equilibrium vapor pressure for ethanol-LiCl system at 298.15 K.

The predicted vapor pressure of methanol-LiC1 system at 298.15 K by use of four methods is shown in Fig. 4. The predictions by method I are in good agreement with the experimental data. The models of Macedo et al. and Sander et al. underpredict the salting effect at higher LiC1 concentrations. Fig. 5 illustrates the quality of the prediction in the case of acetonitrile and 2-propanol solvents. Using methods I and II, the predicted values of vapor pressures for the 2-propanol-NaI and acetonitrile-NaI systems at 298.15 K are in good agreement with the experimental data. A useful application of our model is to solve some problems of inconsistency between data sets. This could be illustrated by considering the data published by Tomasula et al. [6] and Bixon et al. [18]. From Fig. 4, it may be seen that the experimental data reported by Tomasula et al. and Bixon et al. are somewhat inconsistent. This is more obvious in the case of vapor pressure of methanol-NaI system at 298.15 K. Fig. 1 compares the predicted vapor pressure of methanol-NaI system at 298.15 K, using method I with the experimental data measured by Tomasula et al. [6], Bixon et al. [18] and Barthel et al. [7]. It can be seen that Bixon et al.'s experimental vapor pressures are significantly 18,00 16,00 m

n

14,00

.¢ 12,00 --i

a.

10,00 8,00 6,00

>

D

Exp,(Tomasula

et

- - -

Macedo's M e t h o d

4,00

......

Sander's Method

....

O u r M e t h o d II

2,00

- - O u r

0,00 0,00

,

,

al., [6])

Method I

i

2,00

,

,

,

,

,

,

,

4,00

6,00

Molality of I.iCl in Methanol

Fig. 4. Representation of equilibrium vapor pressure for methanol-LiCl system at 298.15 K.

E-X. Zuo, W. Fi~rst/ Fluid Phase Equilibria 138 (1997) 87-104

101

13,00 m

fit.

11,00

E -,

9,00

~.

7,00

~"

5,00

u} u)

o Exp.[Barthel& Lauermann, [9]) --Our Method I (Acetonitrile) - - - Our Method II (Acetonitrile) o Exp.[Barthel& Lauermann, [9]) --Our Method I (2-Propanol) - - --Our Method II (2-Prodanol)

3,00 0,00

0,40

0,80

1,20

1,60

Molality of Nal in 2-Propanol/in Acetonitrile Fig. 5. Representation of equilibrium vapor pressure for 2-propanol-NaI and acetonitrile-NaI systems at 298.15 K.

lower than those of Tomasula et al. as well as those of Barthel et al., although Barthel et al.'s data are available only over a limited molality range. From the predictions of our model, it may be concluded that the accuracy of Bixon et al.'s data is questionable. Another application concerns the calculations of mean activity coefficients. As mentioned above, Barthel and co-workers [7-10] reported experimental vapor pressures for a number of nonaqueous electrolyte systems at 298.15 K. To obtain the corresponding values of mean ionic activity coefficients, the experimental vapor pressures have been used to calculate osmotic coefficients. In the second step, the obtained osmotic coefficients have been fitted using Pitzer's model with three or four adjustable parameters per salt. Then the mean activity coefficients can be calculated using the corresponding expressions of Pitzer's model. The mean activity coefficients have been compared with those predicted by method I (see Fig. 6). From this figure, it appears that, at high concentrations, the mean activity coefficients calculated by Barthel et al. are systematically higher than those predicted by method I. In the case of NaBr solutions, experimental mean activity coefficients have been 0,60 0,55.

~..~,..

0,50. o

0,45.

~

...... Pitzer's Model (Nal) - - - Our Method I (Nal) - - - Pitzer's Model (KI) - - --Our Method I (KI) -Pitzer's Model (NaBr) --Our Method I [NaBr) N a B r ) ~ . . Exp. . .(Han . and . . Pan,, ]1[

._~ •

O ,< C

o,4o 0,35 0,30 0,00

0,40

0,80

1,20

1,60

Molality of Salt in Methanol Fig. 6. Prediction of mean activity coefficients of various salts in methanol: comparison with Barthel et al.'s results and experimental values.

E-X. Zuo, W. F~rst / Fluid Phase Equilibria 138 (1997) 87-104

102

published which were obtained using electromotive force measurements (experimental data of Han and Pan [ 11]). From Fig. 6, it appears that our model accurately represents experimental mean activity coefficients of NaBr in methanol. Hence, it may be concluded that the mean activity coefficients published by Barthel et al. are overestimated. The reason of this problem is probably connected to the fact that pressure determinations are not accurate enough to allow a good representation of mean activity coefficients. Hence, the aqueous electrolyte EOS proposed by Fiirst and Renon [12] has been extended to nonaqueous electrolyte systems, using experimental information on the ionic solvation in nonaqueous solvents. Doing this, we have obtained a predictive model which has been tested for predicting vapor pressures and mean ionic activity coefficients of single nonaqueous solvent-salt systems without any adjustable parameters, excepted in the case of calcium salts where ion pairing is probably significant. The precision of the extended EOS compares well with the results obtained by the models of Mock et al. [1], Sander et al. [2] and Macedo et al. [3] as well as the one and three adjustable parameter models of Pitzer. Therefore, our model can be used to predict equilibrium properties of nonaqueous electrolyte solutions when no data are available or to study inconsistencies between various data sets.

5. List of symbols a a sR

AAD, % b c cal D e exp N Np P p R T v W x Z

molar Helmholtz free energy short-range parameter defined in Eq. (2) (100/Np)Xjll - cal/explj covolume volume translation parameter calculated values dielectric constant protonic charge experimental values Avogadro number number of data points pressure polar parameters gas constant temperature molar volume interaction parameter mole fraction ionic charge

Greek letters

C~LR y+ F A

defined in Eq. (7) mean ionic activity coefficient defined in Eq. (6) property difference

Y.-X. Zuo, W. Fiirst / Fluid Phase Equilibria 138 (1997) 87-104

electric permittivity of free space viscosity parameters defined in Eqs. (10)-(13) limiting conductance of ions acentric factor defined in Eq. (4) diameter

6"(}

~7 A Ao

o-

Superscripts P S

Pauling diameter Stokes diameter

Subscripts anionic properties cationic properties ionic species ionic and molecular species long-range term repulsive forces nonelectrolyte short-range term ionic short-range term solvent properties standard properties

a c

i k,l LR RF SR1 SR2 s

0

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

B. Mock, L.B. Evans, C.C. Chen, AIChE J. 32 (1986) 1655-1664. B. Sander, Aa. Fredenslund, P. Rasmussen, Chem. Eng. Sci. 41 (1986) 1171-1183. E.A. Macedo, V. Skovborg, P. Rasmussen, Chem. Eng. Sci. 45 (1990) 875-882. Z. Li, H.M. Polka, J. Gmehling, Fluid Phase Equilibria 94 (1994) 89-114. K.S. Pitzer, J. Phys. Chem. 77 (1973) 268-277. P. Tomasula, G.J. Czerwienski, D. Tassios, Fluid Phase Equilibria 38 (1987) 129-153. J. Barthel, R. Neueder, G. Lauermann, J. Solution Chem. 14 (1986) 621-631. J. Barthel, G. Lauerrnann, R. Neueder, J. Solution Chem. 15 (1986) 851-867. J. Barthel, G. Lauermann, J. Solution Chem. 15 (1986) 869-877. J. Barthel, W. Kunz, J. Solution Chem. 17 (1988) 399-415. S. Han, H. Pan, Fluid Phase Equilibria 83 (1993) 261-270. W. Fiirst, H. Renon, AIChE J. 39 (1993) 335-343. J. Schwartzentruber, H. Renon, S. Watanasiri, Fluid Phase Equilibria 52 (1989) 127-134. F.-X. Ball, H. Planche, W. Ffirst, H. Renon, AIChE J. 31 (1985) 1233-1240. R. Pottel, Water a Comprehensive Treatise, Vol. 3, Plenum Press, New York, 1973. P.A. Skabichevskii, Russ. J. Phys. Chem. 43 (1969) 1432-1433. E. Sada, T. Morisu, K. Miyahara, J. Chem. Eng. Jpn. 8 (1975) 196-201. E. Bixon, R. Guerry, D. Tassios, J. Chem. Eng. Data 24 (1979) 9-11. F. Mato, M.J. Cocero, J. Chem. Eng. Data 33 (1988) 38-39. M. Hongo, M. Kusunoki, K. Mishima, Y. Arai, Kagaku Kogaku Ronbunshu 16 (1990) 1263-1265. H. Yamamoto, T. Terano, M. Yanagisawa, J. Tokunaga, Can. J. Chem. Eng. 73 (1995) 779-783.

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[22] S.K. Oh, S.W. Campbell, J. Chem. Eng. Data 40 (1995) 504-508. [23] W.F. Linke, A. Seidell, Solubilities of Inorganic and Metal-Organic Compounds, Vol. II, D. van Nostrand, Princeton, 1965. [24] R. Reid, J.M. Prausnitz, B. Poling, The Properties of Gases and Liquids, 4th edn., McGraw-Hill, New York, 1987. [25] T. Boublik, V. Fried, E. Hala, The Vapor Pressures of Pure Substances, Elsevier, Amsterdam, 1973. [26] Y.Y. Akhadov, Dielectric Properties of Binary Solutions, Pergamon, Oxford, 1981. [27] Y. Marcus, Ion Solvation, Wiley, Chichester, 1985. [28] A.A. Maryott, E.R. Smith, Table of Dielectric Constants of Pure Liquids, NBS Circular 514, U.S. Government Printing Office, WA, 1951. [29] R.L. Kay, T.L. Broadwater, J. Solution Chem. 5 (1976) 57-76. [30] H. Hartley, H.R. Raikes, Trans. Faraday Soc. 23 (1927) 393. [31] B.E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam, 1981. [32] H. Zerres, J.M. Prausnitz, AIChE J. 40 (1994) 676-691. [33] V.V. Subbotina, K.P. Mishchenko, Zh. Prikl. Khim. (Leningrad) 42 (1969) 204-206. [34] K.R. Patil, A.D. Tripathi, G. Pathak, S.S. Katti, J. Chem. Eng. Data 35 (1990) 166-168. [35] K.R. Patil, A.D. Tripathi, G. Pathak, S.S. Katti, J. Chem. Eng. Data 36 (1991) 225-230. [36] L.A. Cisternas, E.J. Lam, Fluid Phase Equilibria 62 (1991) 11-27. [37] M.J.E. de M. Cardoso, J.P. O'Connell, Fluid Phase Equilibria 33 (1987) 315-326.