Liquid mixtures involving triangular molecules: (vapour + liquid) equilibria of (xenon + trimethylboron)

J. Chem. Thermodynamics 1998, 30, 1543]1553 Article No. ct980429

Liquid mixtures involving triangular molecules: ( vapour H liquid) equilibria of ( xenon H trimethylboron) Eduardo J. M. Filipe, Ulrich K. Deiters, a and Jorge C. G. Calado Centro de Quımica Estrutural, Instituto Superior Tecnico, 1096 Lisboa, ´ ´ Portugal

The total vapour pressure of  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. has been measured at T s 161.39 K and T s 182.33 K Žthe triple points of xenon and nitrous oxide, respectively.. The excess molar Gibbs energy GmE has been calculated, as a function of composition, from the vapour pressure data. The molar volumes of the mixtures were also measured at T s 182.33 K, and the corresponding excess molar volumes VmE calculated. The results were interpreted using the Deiters equation of state DEOS, and Monte Carlo simulation. q 1998 Academic Press KEYWORDS: Žvapour q liquid. equilibria; excess properties; xenon; trimethylboron

1. Introduction Molecular shape is a crucial factor to the understanding of the thermodynamic properties of liquid mixtures of non-spherical molecules at high densities. This is because for such systems the liquid structure is mainly determined by the repulsive portions of the intermolecular potentials. With this in mind, we have recently started a systematic study of mixtures involving triangular molecules. Several types of triangular molecules have been considered Žfigure 1.: open and flexible Žinternal rotation. such as propane; cyclic and rigid such as cyclopropane, or ethylene oxide; and radial, like the boron compounds. Xenon was chosen as the second component since it is spherical Žstructureless in molecular terms., with a high polarizability Žwhich enhances dispersion forces. and a liquid range suitable for mixing, in the liquid state, with a wide variety of other substances. For each type of triangular molecule, several binary mixtures have already been studied and reported: Ža. Žxenon q propane.Ž1. and Žxenon q dimethylether.;Ž2. Žb. Žxenon q cyclopropane.Ž3. and Žxenon q ethylene oxide.;Ž4. Žc. Žxenon q boron trifluoride.Ž5. and Žxenon q boron trichloride..Ž5. The present work is concerned with this latter type. a Permanent address: Universitat ¨ zu Koln, ¨ Institut fur ¨ Physikalische Chemie, Luxemburger Str. 116, D-50939 Koln, ¨ F.R.G.

0021]9614r98r121543 q 11 $30.00r0

q 1998 Academic Press

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E. J. M. Filipe, U. K. Deiters, and J. C. G. Calado

FIGURE 1. Different types of triangular molecules: a, open; b, cyclic; c, radial.

Trimethylboron, like the boron halides boron trifluoride ŽBF3 . and boron trichloride ŽBCl 3 ., is a planar molecule with a considerable quadrupole. Trimethylboron has, in addition, some interesting features which make the study of systems involving this molecule worthwhile. Ži. The nature of the B C bond in trialkylboranes is not fully understood. Experimental evidence has been accumulated indicating the possibility of some double bond character in these compounds. For example, BŽCH 3 . 3 does not dimerize, whereas AlŽCH 3 . 3 forms stable dimers. One possible explanation for this is that the empty p-orbital on boron in BŽCH 3 . 3 is at least partially satisfied by intramolecular hyperconjugation. Extended Huckel calculations suggest that the p-bond character of BŽCH 3 . 3 is ¨ about nine times that of AlŽCH 3 . 3 .Ž6. However, ab initio calculations have found hyperconjugation in BŽCH 3 . 3 to be unimportant.Ž7. Žii. The compound BŽCH 3 . 3 is a typical Lewis acid due to the existence of the empty p-orbital on boron. Therefore, this molecule could show some form of incipient association with xenon, which can be considered an extremely weak Lewis base. In reality, trimethylboron was found to be less acidic than might be expected, another argument in favour of the p-bond character hypothesis.Ž8. The goal of this work, therefore, is the experimental investigation of the thermodynamic properties of the Žxenon q trimethylboron. system, followed by an interpretation of the results by means of an equation of state and computer simulation. As far as we are aware, no other experimental or theoretical work has been done on the Žxenon q trimethylboron. system.

2. Experimental The experimental techniques used to measure the vapour pressures and densities of the mixtures have already been described.Ž9. The vapour pressures were measured using a quartz-spiral gauge ŽTexas Instruments, model 145. which had been calibrated against mercury manometers. The pyknometer used for the molar volume determinations was calibrated by performing the measurements of the density of ethane, and checking the value against the results of Haynes and Hiza.Ž10. The mixtures were prepared by condensing known amounts of each component into the pyknometer, and the measurements carried out at T s 161.39 K

VLE of Žxenon q trimethylboron.

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and T s 182.33 K, the triple point temperatures of xenon and nitrous oxide, respectively. Xenon and nitrous oxide from Air Liquide Žmole fraction: 0.99995 and 0.9999, respectively. were further purified by fractionation in a low temperature column and the final purity checked by the constancy of the triple point pressure during melting. The measured triple point pressure of xenon was p s 81.669 kPa, comparable with the literature value Ž81.674 " 0.011. kPa.Ž11. The experimental triple point pressure of nitrous oxide, p s 87.815 kPa, also compares favourably with the literature value, p s Ž87.865 " 0.012. kPa.Ž11. A further check on the purity of the gases is provided by the values of the vapour pressure and molar volume of the mixture components at the working temperatures. In the case of xenon, the vapour pressure was found to be p s 247.67 kPa  compared with p s Ž247.55 " 0.21. kPa4 ,Ž12. while the average value of seven determinations obtained for the molar volume was Ž46.468 " 0.006. cm3 . moly1 ,  compared with Ž46.485 " 0.049. cm3 . moly1 ..Ž12. Trimethylboron was synthesized following the procedure of Nesmeyanov,Ž13. by reacting a solution of boron trifluoride ŽMatheson, mole fraction: 0.995. in di-nbutylether with a methyl Grignard reagent solution. This was prepared by reacting methyl iodide ŽAldrich, mole fraction: 0.99. with magnesium in dried di-nbutylether. The reactants and products were handled in an inert gas atmosphere in order to prevent contamination and also because trimethylboron ignites spontaneously on contact with air. The crude trimethylboron formed in the reaction was purified by fractional distillation. The triple point pressure of trimethylboron is too low to be measured with sufficient accuracy in our apparatus. The purity was checked by ion cyclotron resonance spectroscopy ŽFinnigan spectrometer, model FTMS 2001.. The obtained spectrum compared favourably with the characteristic fragmentation pattern of trimethylboron,Ž14. and no impurity peaks were detected. The vapour pressure and molar volume of pure trimethylboron could not be reliably measured at the working temperatures with our technique because the substance condenses easily in the pyknometer inlet tube, splitting into droplets and bubbles. The sources of ancillary data needed in the calculation of the excess functions have already been reported for xenon.Ž2. Since there are no experimental values for the second virial coefficients of trimethylboron, they were estimated using a corresponding states method.Ž15. However, this method requires the knowledge of the critical constants which are also not known in the case of BŽCH 3 . 3 . We thus estimated the critical constants of BŽCH 3 . 3 by using the Deiters equation of state, relying on the literature vapour pressure at the working temperatures Ž16. and our own extrapolated molar volume at T s 182.33 K. We obtained Tc s 417.46 K, pc s 39.63 bar, and Vc s 0.254 dm3 . moly1 . The estimated second virial coefficients are: y647 cm3 . moly1 at T s 298.15 K, y3242 cm3 . moly1 at T s 182.33 K, and y6170 cm3 . moly1 at T s 161.39 K. A similar procedure applied to fluids of known critical constants led to values for the second virial coefficients that differ by 20 per cent from the ones calculated using the experimental critical constants. The cross

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E. J. M. Filipe, U. K. Deiters, and J. C. G. Calado

TABLE 1. Total vapour pressure p, and excess molar Gibbs energy GmE of  x Xe q Ž1 y x . BŽCH 3 . 3 4Žl. at T s 161.39 K and T s 182.33 K. y is the xenon mole fraction in the gas phase; R p are the pressure residuals defined as R p s pexpt y pcalc x

y

prkPa

GmE rŽJ . moly1 .

R prkPa

T s 161.39 K 0 0.31034 0.49647 0.62690 0.70628 1

0 0.99292 0.99647 0.99773 0.99829 1

0.228 25.67 41.31 51.90 58.32 81.67

0

y0.009 0.045 y0.107 0.076

84 108 113 113 0

T s 182.33 K 0 0.10713 0.28338 0.44703 0.62309 0.71715 0.81187 0.93111 1

0 0.94340 0.97967 0.98930 0.99439 0.99618 0.99764 0.99918 1

1.638 27.24 67.41 107.44 152.09 175.75 200.10 230.69 247.78

0 7.7 3.4 10.7 y0.2 y0.8 3.6 7.2 0

0.09 y0.22 0.20 0.09 y0.32 0.04 0.63

virial coefficient for the mixtures was assumed to be the arithmetic mean of the virial coefficients of the pure components.

3. Results The vapour pressure p of  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. at T s 161.39 K and T s 182.33 K is given in table 1, and plotted in figure 2 as a function of the liquid mole fraction of xenon. The excess molar Gibbs energy of each mixture GmE , and the mole fraction of xenon in the vapour phase y were calculated from the total vapour pressure measurements using the Redlich]Kister model, and the method of Barker Ž17. for the minimization of the pressure residuals d p s p y pcalc : 2

GmE rRT s x 1 x 2  A q B Ž x 1 y x 2 . q C Ž x 1 y x 2 . 4 .

Ž 1.

The values of d p, which are also given in table 1, are a good indication of the self-consistency of the experimental data. The value of pcalc is the calculated vapour pressure from equation Ž1. and Barker’s method. The corresponding Redlich]Kister expansion coefficients, as well as the values of GmE for the equimolar mixture, are recorded in table 2. The GmE results are plotted in figure 3 as a function of x. It should be noted that at T s 182.33 K the mixtures are almost ideal and consequently the values of GmE lie within experimental error, showing a much larger scatter than usual. The description of the vapour phase imperfection becomes thus very important in this case, and a check was made on the influence of the second virial coefficient of BŽCH 3 . 3 on the values of GmE . It was found that a

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FIGURE 2. Vapour pressure p of  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. s  y Xe q Ž1 y y .BŽCH 3 . 3 4Žg. plotted against x, or against y at: l, T s 161.39 K; v, T s 182.33 K; }}}, DEOS.

TABLE 2. Coefficients for equation Ž1., and excess molar Gibbs energy values for the equimolar E  Ž . Ž . 4Ž . mixture G1r 2 of x Xe q 1 y x B CH 3 3 l at T s 161.39 K and T s 182.33 K

v,

TrK

A

B

C

E y1 . Ž . G1r 2 r J mol

161.39 182.33

0.3221 " 0.0290 y0.0028 " 0.0092

y0.1334 " 0.0382 0.0137 " 0.0123

0.1334 " 0.0419 0.0662 " 0.0155

108.0 " 9.8 y1.1 " 3.5

FIGURE 3. Excess molar Gibbs energy GmE for  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. at: l, T s 161.39 K; T s 182.33 K.

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E. J. M. Filipe, U. K. Deiters, and J. C. G. Calado

TABLE 3. Molar volumes Vm , and excess molar volumes VmE of  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. at T s 182.33 K and under the saturation vapour pressure. R V are the volume residuals, defined as R V s VmE y VmE equation Ž3.4 x 0 0.32135 0.34819 0.63726 0.75853 0.84259 1

Vm rŽcm3 . moly1 . 87.559 74.724 73.631 61.542 56.452 52.948 46.468

VmE rŽcm3 . moly1 . 0 0.369 0.379 0.169 0.062 0.011 0

R V rŽcm3 . moly1 . 0 y0.007 y0.007 0.002 y0.001 0.002 0

change of 50 per cent from the estimated value of B causes GmE Ž x s 0.5. to change between y3.17 J . moly1 and 0.98 J . moly1 , which is well within experimental error. The excess molar enthalpy HmE was estimated within the limits of uncertainty of EŽ Gm x s 0.5. at both temperatures. For the equimolar composition a value of HmE s 900 J . moly1 was found. The orthobaric molar volumes Vm of  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. mixtures at T s 182.33 K are recorded in table 3. Due to the low pressures involved, the corrections for the conversion of VmE to zero pressure are negligible. As mentioned before, the molar volume of pure trimethylboron Vm BŽCH 3 . 3 4 could not be reliably measured due to condensation problems in the inlet tube. It was estimated through the extrapolation, lim

x ŽXe .ª0

 Vm y x Ž Xe. Vm Ž Xe. 4 rx  B Ž CH 3 . 3 4 ,

Ž 2.

which led to a value of Ž87.559 " 0.012. cm3 . moly1 . A similar procedure applied to xenon leads to a molar volume of Ž46.456 " 0.021. cm3 . moly1 , in agreement with the measured value Ž46.485 " 0.049. cm3 . moly1 . The excess molar volume results were fitted to a Redlich]Kister type equation: VmE s x 1 x 2  D q E Ž x 1 y x 2 . 4 ,

Ž 3.

yielding D s yŽ1.161 " 0.026. cm3 . moly1 and E s Ž1.575 " 0.028. cm3 . moly1 . The molar volume residuals d V s VmE y VmE Žfit. are also given in table 3. The experimental and fitted VmE results are presented in figure 4. For x s 0.5, VmE s Ž0.290 " 0.007. cm3 . moly1 . The VmE data are positive and the VmE Žx. curve is highly asymmetric.

4. Discussion The phase equilibrium data for this system has been interpreted using the Deiters Ž18.

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1549

FIGURE 4. Excess molar volume VmE for  x Xe q Ž1 y x .BŽCH 3 . 3 4Žl. at T s 182.33 K.

equation of state DEOS. This is a semi-empirical equation based on the square-well potential model: p s Ž RTrVm . 1 q cc0  a1 j q a2 Ž 3a2 y a1 q 2 . j 2 y a2 j 3 4 r Ž 1 y j .

ž

 Ž RT*b . rVm2 4 T˜Ž e1r T˜ y 1. I1Ž j . ,

3

/y Ž 4.

with T˜ s Ž cTrT * q lj . 1ryŽ j .4 , and T * s «rk. In this equation Vm denotes the molar volume; b is a volumetric parameter Žcovolume, b s NA s 3r2 1r2 .; j is the reduced density Ž j s 2 1r2p br6V .; c is a shape parameter which corrects for the non-sphericity of the molecules Ž c s 1 for spherical molecules.; c 0 s 0.6887 is a universal constant which accounts for the deviation of the real pair potential from the rigid core model; l s y0.09333 is another universal constant which accounts for the influence of three-body interactions; and I1Ž j . and y Ž j . are complicated functions of density Žand generally of c . derived from statistical mechanics.Ž18. The coefficients a1 and a 2 are from Mansoori]Leland] Carnahan]Starling theory Žan extension of Carnahan]Starling theory towards mixtures., and are given by: a1 s 3rsr¨ ,

a2 s Ž s 3r¨ 2 . y 1,

Ž 5.

with, rs

Ý x i ri ,

ss

Ý x i ri2 ,

¨s

Ý x i ri3 ,

and, ri s bii1r3 . The composition dependence of the mixture parameters was given by the following

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E. J. M. Filipe, U. K. Deiters, and J. C. G. Calado

mixing theory. A density dependent relation for the averaged attraction parameter was used:

«s g s

Ý Ý x i x k « i k sigk , i

Ž 6.

k

with,

g s 3 Ž1 y j 2 . , where g is the structure exponent of the square-well potential. Furthermore, quantum and quasi-chemical corrections were added to these equations which are explained in detail in reference 19. The binary attraction parameter « 12 was defined as:

« 12 s u Ž « 11 « 22 .

1r2

,

Ž 7.

where u is an adjustable interaction parameter, and was calculated from the experimental results for the mixture. The size parameter ratio is given by: 2

s 2rs 1 s c Ž c11rc22 . Ž b 22rb11 .

1r3

,

Ž 8.

where c is a second adjustable interaction parameter. The composition dependence of b and c is given by: b s x 12 b11 q 2 x 1 x 2 b12 q x 22 b 22 ,

Ž 9.

c s x 1 c11 q x 2 c 22 ,

Ž 10 .

b12 s Ž b11 q b 22 . r2.

Ž 11 .

with,

The parameters for the pure components were determined from critical and vapour pressure data and are listed in table 4. The binary interaction parameters were calculated from two experimental VLE data points for the mixtures at T s 182.33 K. No experimental molar volumes for the mixtures were included. All parameters are listed in table 4. The calculated curves are plotted in figure 2. As can be observed, the phase diagrams are well reproduced at both temperatures. The predicted values of GmE at TABLE 4. Parameters of DEOS for xenon and trimethylboron, and binary interaction parameters for Žxenon q trimethylboron. mixture arK Xe BŽCH 3 . 3 Xe q BŽCH 3 . 3 4

199.11 314.57

brŽdm3 . moly1 . 0.02611 0.05619

c

u

c

0.9569

1.2679

1.000 1.043

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TABLE 5. Excess molar functions of some xenon mixtures at equimolar composition GmE rŽJ . moly1 .

System T: Žxenon q trimethylboron. Žxenon q methyl propane. Žxenon q propane. Žxenon q ethylene .

161.39 K

182.33 K

108

y1

y5 145

y30

195.48 K y43 y44

HmE rŽJ . moly1 .

VmE rŽcm3 . moly1 .

900

q0.29 y0.33 y0.31 q0.35

138 285

equimolar composition are 49.8 J . moly1 at T s 161.39 K and y27.0 J . moly1 at T s 182.33 K, lower than the experimental values, 108.0 J . moly1 at T s 161.39 K and y1.1 J . moly1 at T s 182.33 K. However, the prediction for VmE can be considered good, since the calculated value is VmE s 0.361 cm3 . moly1 , as opposed to the experimental one, VmE s 0.290 cm3 . moly1 . No experimental molar volumes for the mixtures were used in the calculation of the binary interaction parameters. The predicted value of HmE is 627 J . moly1 , in reasonable agreement with the indirect experimental value of 900 J . moly1 . Owing to the chemical nature of its substituents one might expect trimethylboron to behave much like an alkane when mixed with xenon. For this purpose, methyl propane Ž i-butane. should be a good compound for comparison since the shape of both molecules is very similar. Although trimethylboron is said to be flat and methyl propane pyramidal, due to the large volume occupied by the methyl groups, the two molecules look very much alike, except for the presence of an extra hydrogen atom on the central carbon in methyl propane. This can be readily seen using ‘‘space filled’’ molecular models with realistic radii. In table 5 the results for the present system are compared with those of the Žxenon q methyl propane. mixture.Ž20. The excess functions for the latter system follow the pattern found for mixtures of xenon with alkanes Žnegative GmE and VmE , and positive HmE ., and are very similar to those of Žxenon q propane.,Ž1. also included in table 5. The obvious conclusion is that the excess functions of Žxenon q trimethylboron. seem to exhibit a different type of behaviour from that observed for the other mixtures: GmE is positive at the lower temperature, HmE is much higher, and VmE is now positive. Interestingly enough, this is the kind of behaviour found in mixtures of xenon with unsaturated compounds, of which Žxenon q ethylene .Ž9. is a typical example Žtable 5.. This result might be an indication of the double bond character in trimethylboron, although we realize that the assumption is highly speculative. Alternatively, the positive VmE found for this system might be attributed  as has been in the case of Žxenon q ethylene .4 , to the shape of the trimethylboron molecule. Flat molecules are believed to have a tendency to form more structurally layered liquids than those involving spherical or quasi-spherical molecules. In other words, a liquid of flat molecules is more anisotropic, and the introduction of xenon atoms would disrupt this structure resulting in larger positive excess volumes. In order to check this hypothesis we performed Monte Carlo computer

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E. J. M. Filipe, U. K. Deiters, and J. C. G. Calado

TABLE 6. Simulation results for Žxenon q trimethylboron. and Žxenon q methyl propane. at equimolar composition Vm rŽcm3 . moly1 .

System

˜p

xenon

0.05 0.10 0.20

65.33 53.17 44.87

2-methylpropane

0.05 0.10 0.20

122.07 102.56 88.46

Žxenon q methyl propane.

0.05 0.10 0.20

93.21 77.90 66.66

trimethylboron

0.05 0.10 0.20

122.96 103.65 89.78

Žxenon q trimethylboron.

0.05 0.10 0.20

93.79 78.38 67.52

E V1r2 rŽcm3 . moly1 .

y0.49 0.03 y0.01

y0.37 y0.04 0.14

simulations of the mixture and its pure constituents. The molecules were modelled as fused hard sphere particles, i.e. each atom was taken to be a hard sphere. The simulations were made for NpT ensembles with 256 molecules, and the pressure was set to a sufficiently high value so that liquid-like densities resulted for the simulation ensembles. Comparison of simulation results for the mixture and for pure fluids allowed us to obtain the excess volumes, or, more specifically, the contribution of molecular shape and packing to the excess volumes. The simulation technique has been described elsewhere.Ž21. In addition, similar simulations have been made for the system Žxenon q methylpropane.. For the hard sphere radii we used the van der Waals radii published by Ewsley, reduced by a factor of 0.96 in order to give better agreement with liquid molar volumes. The simulation results are shown in table 6. It should be noted that for hard-body fluids the pressure is strictly proportional to temperature. The dimensionless pressure shown in the table is defined as:

˜p s p . u 3r Ž k B T . ,

Ž 12 .

where u is the unit length Ž1 . 10y1 0 m. of our calculations. The resulting pressures  at T s Ž160 to 200. K4 are therefore in the p s 100 MPa range, which is also typical of repulsive pressures in liquids. As can be seen from table 6, the experimental excess volume of Žxenon q methylpropane., y0.33 cm3 moly1 , is approximately reproduced by the simulations. The simulation results for the VmE of Žxenon q trimethylboron. are even slightly more negative. This, however, is in contradiction to the experimental observations. We must, therefore, conclude that the positive VmE for this system is probably not due to a shape effect. There remains the possibility that the positive VmE and the rather large positive HmE are

VLE of Žxenon q trimethylboron.

1553

caused either by a quantum chemical peculiarity of the trimethylboron molecule Ždimer formation, interactions involving the free p-orbital., or by electrostatic interactions: trimethylboron has a strong quadrupole, and the boron atom carries a significant positive partial charge. These hypotheses will be investigated in subsequent publications. We are grateful to Prof. A. Griesbeck, University at Cologne, for providing us with structural data of the molecules studied in this work. REFERENCES 1. Calado, J. C. G.; Filipe, E. J. M.; Gomes de Azevedo, E. J. S.; Martins, L. F. G.; Jackson, G.; Soares V. A. M. J. Phys. Chem. Žsubmitted for publication.. 2. Calado, J. C. G.; Rebelo, L. P. N.; Streett W. B.; Zollweg, J. A. J. Chem. Thermodynamics 1986, 18, 931]938. 3. Calado, J. C. G.; Filipe, E. J. M.; Lopes, J. N. C.; Lucio, J. M. R.; Martins J. F.; Martins L. F. G. ´ J. Phys. Chem. B 1997, 101, 7135]7138. 4. Calado, J. C. G.; Deiters, U. K.; Filipe, E. J. M. J. Chem. Thermodynamics 1996, 28, 201]207. 5. Calado, J. C. G.; Filipe, E. J. M. J. Chem. Soc. Faraday Trans. 1996, 92Ž2., 215]218. 6. Ohkubo, K.; Shimada, H.; Okada, M. Bull. Chem. Soc. Jpn. 1971, 44, 2025. 7. Guest, M. F.; Hillier, I. H.; Saunders, V. R. J. Organomet. Chem. 1972, 44, 59]68. 8. Murphy, M. K.; Beauchamp, J. L. Inorg. Chem. 1977, 16, 2437]2443. 9. Calado, J. C. G.; Gomes de Azevedo, E. J. S.; Soares, V. A. M. Chem. Eng. Commun. 1980, 5, 149]163. 10. Haynes, W. M.; Hiza, M. J. J. Chem. Thermodynamics 1977, 9, 179]187. 11. Staveley, L. A. K; Lobo L. Q.; Calado, J. C. G. Cryogenics 1981, 21, 131]144. 12. Machado, J. R. S.; Gubbins, K. E.; Lobo L. Q.; Staveley, L. A. K. J. Chem. Soc., Faraday Trans. 1 1980, 76, 2496]2506. 13. Nesmeyanov, A. N.; Sokolik, R. A. The Organic Compounds of Boron, Aluminum, Gallium, Indium and Thallium, Series: methods of elemento-organic chemistry, Vol. 1. Nesmeyanov, A. N.; Kocheshkov, K. A.: editors. North Holland P. C. 1967. 14. Murphy, M. K.; Beauchamp, J. L. J. Am. Chem. Soc. 1976, 98, 1433]1440. 15. Van Ness, H. C.; Abbott, M. M. Classical Thermodynamics of Nonelectrolyte Solutions. McGraw-Hill: New York. 1982. 16. Hughes, R. L.; Smith, I. C.; Lawless, E. W. Production of the Boranes and Related Research. Holzmann, R. T.: editor. Academic Press: New York. 1967. 17. Barker, J. A. Aust. J. Chem. 1953, 6, 207]210. 18. Deiters, U. Chem. Eng. Sci. 1981, 36, 1139]1146. 19. Deiters, U. K. Fluid Phase Equilibria 1987, 33, 267]293. 20. Martins, L. F. G.; Filipe, E. J. M.; Calado, J. C. G. J. Phys. Chem. Žsubmitted for publication.. 21. Klarner, F.-G.; Krawczyk, B.; Ruster, V.; Deiters, U. K. J. Am. Chem. Soc. 1994, 116, 7646]7657. ¨ 22. Ewsley, J. Die Elemente. de Gruyter: Berlin. 1994.

(Recei¨ ed 11 May 1998; in final form 17 Ausust 1998)

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