Bulk and surface thermodynamic properties in mixtures of small rigid

Apr 7, 1987 - ... from the surface tensions and chemical potentials, the interface being richer in CC14 than the bulk liquid, while the gas phase is p...
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J . Phys. Chem. 1988, 92, 228-234

228

Bulk and Surface Thermodynamic Properties in Mixtures of Small Rigid Molecules: The CCI4 CS2 System

+

Gustavo Luengo, Javier Aracil, Ramdn G. Rubio,* and M. Diaz Peiia Departamento de Quimica Fisica, Facultad de Quimicas, Universidad Complutense. 28040 Madrid, Spain (Received: April 7 , 1987; In Final Form: July 22, 1987) The surface tension of CCI4 + CS2 has been measured at six temperatures between 293.15 and 31 3.15 K. The excess surface tension, surface energy, and entropy have been found to be temperature independent within the experimental uncertainties. The vapor pressure of the mixture has been measured at five temperatures between 298.15 and 31 3.15 K. The calculated GE slightly decreases with increasing temperature. The relative surface adsorption has been calculated from the surface tensions and chemical potentials, the interface being richer in CC14 than the bulk liquid, while the gas phase is poorer. The concentration-concentration correlation function has been calculated from the chemical potentials, indicating that there is a certain tendency to homocoordination in this mixture. The bulk excess properties have been correlated in terms of generalized van der Waals and king-like models. All of them predict excess volumes which are lower than the experimental ones. While some of the models predict too large excess enthalpies, others predict too low ones. The surface tension data have been correlated in terms of two corresponding states models, a squared-gradient type model and the discontinuous interface model. Predictions range from poor to fair.

Introduction It seems unnecessary to state the importance of surface tension and adsorption in understanding the behavior of fluid mixtures.’ Even though there is a considerable degree of phenomenological knowledge on surface properties, the study of surface-induced phase transitions,* micro emulsion^,^ monolayer^,^ etc., demands a deeper knowledge at a molecular leveL4 In the past decade, the vapor-liquid interface of simple fluids has been extensively ~ t u d i e d .In ~ more complex systems other phenomena arise, like preferential orientations in the interfacial region due to shape and/or interaction ani~otropy.~Several perturbation theories for the vapor-liquid interface of molecular fluids have been developed and predict that kind of effects.6 However, when the molecules present a high degree of shape anisotropy, these theories are hardly applicable, even to homogeneous fluids.’ Very recently, perturbation theories for multicenter Lennard-Jones fluids have been published: their extension to inhomogeneous fluids being possible.6 Even though bulk and surface experimental thermodynamic data are available for mixtures of simple or complex data for mixtures formed by rigid, nonpolar, and small polyatomic molecules are quite scarce. Thus, we found it interesting to obtain experimental information for these types of mixtures. We have chosen the system CCI4 + CS2 since both pure components have been extensively studied,1° and some data are already available in the literature.”-’3 ~~~~

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( I ) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.;Wiley: New York, 1982. (2) Sullivan, D.; Telo da Gama, M. M. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: New York, 1986. Moldover, M. R.; Schmidt, J. W. Physica D 1984,12, 351. Douillard, J. M.; Bennes, R.; Privat, M.; Tenebre, L. J . Colloid Interface Sei. 1985,106, 146. (3) Kalweit, D. F.; Mitchell, D. J.; Ninham, B. W. J . Phys. Chem. 1986, 90, 2817. Rushforth, D. S.; Sanchez-Rubio, M.; Santos-Vidals, L. M.; Wormuth, K. R.; Kaler, E. W.; Cuervas, R.; Puig, J. E. J . Phys. Chem. 1986, 90, 6668. Lichterfeld, F.;Schmeling, T.; Strey, R. J . Phys. Chem. 1986,90, 5162. (4) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; University Press: Oxford, 1982. (5) Beaglehole, D. Mol. Cryst. Liq. Cryst. 1982,89, 319. AIS Nielsen, J. Z . Phys. B: Condens. Matter 1985,61,41 I. Thurtell, J. H.; Telo da Gama, M. M.; Gubbins, K. E. Mol. Phys. 1985,54, 231. (6). Dickinson, E.;Lal, M. Mol. Relax. Interact. Proc. 1980, 17, 1 . Gubbins, K. E. In Fluid Interfacial Phenomena: Croxton, C. A., Ed.; Wiley: New York, 1982. (7) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Oxford Universitv Press: London. 1984. (8) Bdhm, M., Lustig, R.; Fischer, J Fluid Phase Equilib 1986,25, 251 Lustis?. R. Mol. Phvs. , 1986. 59. 173 (9j’Rowlinson. J. S.; Swinton, F. L. Liquids and Liquid Mixlures; Butterworths: London, 1982. (IO) Narten, A. H. J . Chem. Phys. 1976,65, 573. Montague, D. G.; Chowdhurry, M. R.; Dore, J. C.; Reed, J. Mol.Phys. 1983,50,1. Van Tricht, J. B.: Jansen, G. H.; Davis, G. J. J . Mol. Liq. 1985,31,91. (1 1) Siddiqi, M. A.; Lucas, K. J . Chem. Thermodyn. 1983, 15, 1181. ~

I

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0022-3654/88/2092-0228$01.50/0

In the present work, surface tension has been measured as a function of temperature and composition. In order to calculate the surface adsorption, the chemical potentials are needed as a function of composition; they have been obtained from vapor pressure measurements using standard procedures.

Experimental Section The surface tension of the mixture has been obtained, under saturation conditions, by using a differential capillary-rise technique, similar to the one used by McLure’s group,I4 using three different capillaries. Care has been taken to ensure that the contact angle was zero. To this end all the measurements were made with receding menisci and by going from high to low temperatures. The vapor pressures were measured by using a static technique described previously.15 The mixtures have been prepared by weight from degassed substances, the uncertainty in mole fraction being estimated to be not more than fO.OOO1. The heights of the menisci in the capillaries and in the manometers were measured within f5 pm. The temperature was controlled to within f 2 mK in both techniques, and the temperature scale agrees with the IPTS-68 within f0.02 K. The experimental uncertainty in the vapor pressures is 1 8 Pa. The Sugden parameter’ S was calculated from the differences of heights by the method described by Gielen et a1.I6 For the calculation of the surface tension y from S, the densities of the liquid and the vapor phases are required. The densities of the liquid phase were calculated from the pure component ones and the excess volumes measured by Vigil.I2 These were fitted to a Redlich-Kister type equation whose coefficients are given in Table I1 for 298.15 and 308.15 K. seems to be temperature independent for the present mixture; thus, the same curve has been assumed between 293.15 and 3 18.15 K. For the densities of the vapor phase the virial equation of state has been used. In spite of the high precision in the readings of height, and due to the high densities of CCll and CS2,the uncertainty in y is f0.12 mN/m. The pure components were the same as for the excess volume measurements.’* CC14 was Carlo Erba RS grade with a minimum purity of 99.95%; its density at 293.15 K was 1.594 12 g/cm3 which compares favorably with the recommended value of 1.59404;” (12) Vigil, M. R., unpublished results. (13) Shipp, W. E. J . Chem. Eng. Data 1970,15, 308. Chaudhri, M. N.; Katti, P. K.; Baliga, M. N. Trans. Faraday SOC.1959,55, 2013. (14) Soares, V. A. M.; McLure, I.: Calado, J. C. G. Fluid Phase Equilib. 1978,2 , 99. McLure, I.; Edmons, B.; Lal, M. J. Colloid InferfaceSci. 1983, 91,361. (1 5) Rubio, R. G.; Renuncio, J. A. R.; Diaz PeRa, M . J. Chem. Thermodyn. 1982,14, 983. (16) Gielen, H. L.; Verbeke, 0. B.; Thoen, J. J. Chem. Phys. 1984,81, 6154.

0 1988 American Chemical Society

The CC14

+ CS2 System

The Journal of Physical Chemistry, Vol. 92, No. 1, 1988 229

32kh ,-

i 2 9 3 15 K 29815 K

TABLE I: Experimental Values of the Sugden Parameter S, Surface Tension y , Excess Surface Tension yE, and the Relative Surface Adsorption of CS2 with Respect to CCle r2,1 for the Mixture at Six Temperatures X,

S. cm2

0.0000 0.1003 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.8881 1.0000

0.05221 0.048 48 0.045 56 0.04406 0.041 65 0.039 90 0.037 55 0.036 13 0.035 55 0.03462

0.0000 0.1003 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.8881 1 .OOOO

0.051 36 0.047 62 0.045 13 0.043 28 0.041 06 0.039 19 0.036 77 0.035 49 0.03481 0.03404

0.0000 0.1003 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.8881 1.0000

0.050 66 0.046 94 0.04422 0.042 35 0.040 24 0.038 42 0.036 03 0.034 75 0.034 17 0.033 45

0.0000 0.1003 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.8881 1.0000

0.049 75 0.045 97 0.043 26 0.041 59 0.039 60 0.037 60 0.035 55 0.034 10 0.033 53 0.032 87

0.0000 0.1003 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.8881 1 .OOOO

0.048 86 0.045 15 0.042 58 0.041 10 0.038 73 0.036 97 0.034 75 0.033 46 0.03295 0.032 27

0.0000 0.1003 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.888 1 1.OOOO

0.048 00 0.044 28 0.041 64 0.040 24 0.037 89 0.036 22 0.034 11 0.032 64 0.032 23 0.03 1 67

Y.

mN/m

YE,

mN/m

r2.11

mol/m2 X lo6

T = 293.15 K

2L

0

1

I

02

QL

06

08

1

X

Figure 1. Surface tension of xCC14 + (1 - x)CS2 at the temperatures indicated. The values for the pure components agree with those given in ref 17.

CS2 was Fluka puriss with a minimum purity of 99.9%, and its density was 1.263 11 g/cm3 at 298.15 K which also compares well with the value of 1.263 20 recommended in the 1 i t e r a t ~ r e . l ~

Results Table I shows the Sudgen parameter S and the surface tension y for the mixture at six temperatures between 293.15 and 313.15 K. Figure 1 shows the values of y as a function of composition. The values of y for the pure components agree with those recommended by Jasperi7within the experimental error. The surface Shipp13has tension of CC14 CS2 has been studied previo~sly.'~ measured y for this mixture at 298.15 K. Our results do not agree very well with his. This is not surprising since he reported y(CS2) = 32.75 m N / m while the value recommended by JasperI7 is y(CS2) = 31.58 mN/m, a difference which is well beyond the experimental error claimed by Shipp. For CC14 Shipp reported a value 26.86 m N / m larger than that recommended by Jasper, 26.43 mN/m. Therefore, the y vs x curve given by Shipp has too negative a slope when compared with the curve in Figure 1. Chaudhri et give the parameters of Guggenheim's quasicrystalline model for this mixture at 305.05 K. We have interpolated y at the same temperature and found that they are lower than those of Chaudhri et al.13 The value of y reported by those authors for the CS2 agree quite well with that given by Jasperi7 while the value for CC1, is 0.3 mN/m lower than that of Jasper.I7 Except for this, the difference between both sets of data for the mixture is well beyond the combined uncertainties, and we have not found an explanation for them. Unlike most bulk properties, there is not a well-based statement of surface ideality. We have calculated the excess surface tension yE as yE = 7 - X l Y l - X2Y2 (1)

+

the values of which are shown in Table I. We have fitted yE to a Redlich-Kister type equation 3

YE = XlX2CAI(X, - x2)'

(2)

I=o

The parameters A iand the standard deviations of the fits are shown in Table 11. As can be observed in Figure 2, yE is temperature independent within the experimental error in y. The surface entropy SY and energy uY have been calculated from the temperature dependence of y.

The temperature dependence of the data in Table I can be described, for a fixed composition, by straight lines within the experimental error. This means that both s7 and MY are temperature (17) Jasper, J. J. J . Phys. Chem. ReJ Data 1972, I , 841.

32.32 31.13 30.05 29.75 28.91 28.39 27.65 27.39 27.31 27.04

T = 298.15 31.61 30.40 29.60 29.05 28.32 27.73 26.90 26.74 26.59 26.43

-0.66 -1.30 -1.18 -1.47 -1.43 -1.29 -0.67 -0.32

K -0.69 -1.06 -1.20 -1.39 -1.42 -1.40 -0.69 -0.42

-3.589 -2.745 -2.243 -1.689 -1.136 -0.114 -0.071 -0.037

T = 303.15 K 30.99 29.78 28.85 28.25 27.60 27.01 26.20 26.03 25.94 25.81

T = 308.15 30.25 28.99 28.03 27.57 26.98 26.28 25.70 25.39 25.29 25.21

-0.69 -1.19 -1.38 -1.49 -1.52 -1.48 -0.78 -0.45

-3.801 -2.965 -2.358 -1.658 -1.042 -0.373 -0.072 -0.034

K -0.75 -1.29 -1.36 -1.42 -1.58 -1.33 -0.79 -0.48

-3.927 -2.808 -2.075 -1.414 -0.937 -0.405 -0.099 -0.035

T = 313.15 K 29.52 28.29 27.41 27.08 26.23 25.67 24.96 24.75 24.70 24.59

T = 318.15 28.81 27.56 26.63 26.34 25.46 24.99 24.35 24.00 24.01 23.98

-0.74 -1.20 -1.14 -1.43 -1.51 -1.41 -0.79 -0.44

-3.562 -2.530 -1.977 -1.503 -1.072 -0.403 -0.032 -0.009

K -0.76 -1.29 -1.20 -1.57 -1.53 -1.37 -0.9 1 -0.5 1

-3.585 -2.552 -1.945 -1.405 -0.967 -0.391 -0.053 -0.005

independent. Table I11 shows the vapor pressure of the mixture at five temperatures between 298.15 and 313.15 K. The data are plotted in Figure 3 together with previous data reported by

230

The Journal of Physical Chemistry, Vol. 92, No. 1, 1988

TABLE 11: Parameters A, of the Fits of Y~ and VE to a Redlich-Kister Tvw Eauation and the Standard Deviations (I T, K A0 AI ’42 A3 293.15 298.15 303.15 308.15 313.15 318.15

-5.914 -5.840 -6.363 -6.040 -6.069 -6.165

7,mN/m 0.556 1.760 0.073 0.338 1.172 0.291 -1.120 1.688 -0.366 0.129 0.658 -1.267

Luengo et al. I

I

I

I

I OL

I

06

1 08

64

90

1.847 3.230 1.635 1.159 3.872 2.502

0.12 0.06 0.07 0.08 0.08 0.10

80

VE,cm’/mol 298.15 308.15

1.4083 1.3982

-0.0472 -0.2753

0.0487 0.1545

-0.3732 0.1170

0.0009 0.0009

70

“The units of u are the same as those of the excess function TABLE III: Calculated Values of the Surface Entropy s 7 and Energy u’, Their Estimated Uncertainties, and the Molar Surface Entropy ST and Enerev UT at 293.15 K 104s7, 104b, l o w , I O ~ A ~ , s7, cn, xi J / ( m 2 K ) J/(m2 K) J / m 2 J / m 2 J/(mol K ) kJ/mol 0.0000 1.40 0.02 7.348 0.05 18.2 9.6 0.1007 0.1837 0.2627 0.3674 0.4743 0.6397 0.8069 0.8881 1.oooo

1.43 1.40 1.36 1.38 1.37 1.30 1.35 1.30 1.22

0.02 0.05 0.04 0.03 0.02 0.03 0.01 0.02 0.002

7.297 7.120 6.930 6.941 6.844 6.582 6.687 6.550 6.292

. L

9.9 10.0 10.0 10.4 10.6 10.7 11.3 11.3 11.2

19.3 19.5 19.5 20.5 21.1 21.0 22.8 22.5 21.7

0.05 0.15 0.13 0.09 0.05 0.10 0.05 0.05 0.01

60 0

L Y

50

LO

-zow 30

-

20

‘E

.

-1.0

I

E

02

X

W

Figure 3. Vapor pressures of the mixture xCCI4 + (1 - x)CS2 at the Curves are pretemperatures indicated: 0, this work; A, H1a~aty.I~ dictions from the regression of the data according to a modified Barker’s

>

-318.15K

method. 0 0

02

06

OL

08

1

X

Figure 2. Excess surface tension for xCCI4 + (1 - x ) C S 2 at the temperatures indicated as calculated from eq 2 with the parameters of Table 11.

Hlavaty;I8 the agreement between both sets of data is very good, although the scattering of the data in Table IV is much smaller than that of Hlavaty’s data. As discussed below, this will turn out to be important for the calculation of the relative surface adsorption. A thermodynamic consistency test of the data has been carried out by a modified Barker’s method described prev i o ~ s l y . The ~ ~ reduced molar excess Gibbs energy, G E / R T ,has been assumed to be given by eq 2; the parameters of the fits and the standard deviations of the variables are given in Table V. In comparing Hlavaty’s work,I9one can find that the maximum value of the standard deviation of vapor pressures u ( p ) in Table IV is half the minimum value of Hlavaty’s data. Table IV also gives the residuals of the variables, Ax and Ap, the GE,the activity coefficients, y l and yz,and the vapor-phase mole fraction of CCl,, y , . As can be observed, GE shows a very weak temperature dependence; this is especially clear if one considers that the uncertainty of the GEvalues at the maximum of the GEvs xIcurves is A2 J/mol. The Gibbs-Helmholtz equation represents a very (18) Hlavaty, K. Collect. Czech. Chem. Commun. 1970, 35, 2878. (19) Rubio, R. G.; Renuncio, J. A. R.; Diaz Pefia, M. Fluid Phase Equilib. 1983, 12, 217.

sensitive test for the vapor-liquid equilibrium data. Figure 4 shows G E / Tvs 1/T for the equimolecular mixture. The slope of the straight line represents the experimental value of HE obtained by Siddiqi and Lucas at 303.15 K;” they also observed that HE very slightly decreases as T is increased. As can be observed, the agreement between both sets of data is remarkable. It can be concluded that for the CC14 + CS2 system the excess properties F,l2P,” GE,and yE show a very weak temperature dependence in the temperature interval studied in this paper.

Discussion Besides the y, u y , and SY data, another fundamental piece of information is the surface adsorption. The relative adsorption l?z,l was calculated via the Gibbs isotherm in the form

-(&( g)T

r2,1=

(dy/dx2), was calculated from eq 1, and from the GE composition dependence through

(4) was calculated

(5)

and

The CCI,

+ CS2 System

The Journal of Physical Chemistry, Vol. 92, No. I , 1988 231

TABLE I V Experimental Vapor Pressures p and Calculated Values of the Excess Molar Gibbs Energy CE,Activity Coefficients y, and y2, and the Mole Fraction of CCI, in the Vapor Phase Y I 4 fl, GE, AXl AP, GE, xI ( X IO5) P,kPa Pa J/mol y, 72 Yl xI (X IO5) P , kPa Pa J/mol y, 7 2 Yl T = 298.15 K 0.0509 0.1047 0.1964 0.2879 0.3729 0.4437 0.5127

2.7 0.9 -1.4 1.2 3.5 4.0 2.3

46.673 24 45.142 7 42.639 -14 40.186 13 37.945 -29 35.873 27 33.772 16

29 61 112 149 169 174 172

1.2720 0.9995 1.2710 0.9997 1.2259 1.0065 1.1632 1.0236 1.1110 1.047 1 1.0777 1.0692 1.0547 1.0904

0.0215 0.0457 0.0874 0.1287 0.1686 0.2053 0.2463

0.5169 0.6729 0.7485 0.7932 0.8210 0.8419 0.8559 0.8655

-3.9 3.1 -5.3 2.9 -3.0 -3.0 10.3 -4.3

33.684 28.275 25.541 23.813 22.799 22.001 21.382 21.097

-27 19 -34 18 -18 -18 60 -26

172 145 125 111 101 93 87 83

1.0536 1.0277 1.0208 1.0168 1.0142 1.0121 1.0107 1.0097

1.0916 1.1310 1.1500 1.1654 1.1782 1.1901 1.1995 1.2066

0.2489 0.3747 0.4584 0.5172 0.5581 0.5916 0.6155 0.6326

0.0509 0.1047 0.1964 0.2879 0.3729 0.4437 0.5127

6.7 4.2 -1.7 -3.0 -2.1 0.4 3.1

56.245 46 54.465 29 51.545 -12 48.665 -22 45.916 -15 43.483 3 40.945 19

32 65 1I4 150 168 172 169

1.2828 1.2692 1.2161 1.1538 1.1035 1.0714 1.0488

0.9999 1.0009 1.0088 1.0259 1.0487 1.0703 1.0913

T = 303.15 K 0.0225 0.5169 0.0471 0.6729 0.0893 0.7485 0.1314 0.7932 0.1722 0.8210 0.2097 0.8419 0.2514 0.8559 0.8655

1.1 1.5 -4.6 -1.4 -0.3 -4.5 4.7 3.6

40.799 6 34.413 8 31.101 -24 -7 29.066 27.788 -1 26.846 -22 26.150 23 25.702 18

168 139 117 102 92 84 78 74

1.0477 1.0215 1.0152 1.0119 1.0099 1.0084 1.0074 1.0067

1.0925 1.1325 1.1500 1.1625 1.1722 1.1808 1.1875 1.1926

0.2541 0.3809 0.4653 0.5249 0.5666 0.6006 0.6249 0.6425

0.0509 0.1047 0.1964 0.2879 0.3729 0.4437 0.5127

-0.8 1.9 -1.8 1.2 -2.4 4.6

67.529 -5 65.535 1 62.083 11 58.582 -10 55.194 7 52.318 -12 49.344 25

40 75 122 153 168 173 168

1.3274 1.0013 1.2695 1.005 1 1.1956 1.0157 1.1404 1.0311 1.0988 1.0502 1.0694 1.0700 1.0450 1.0929

T = 308.15 K 0.0239 0.5169 0.0484 0.6729 0.0901 0.7485 0.1332 0.7932 0.1761 0.8210 0.2148 0.8419 0.2566 0.8559 0.8655

-3.9 4.5 -4.2 -2.5 1.1 -2.6 3.9 1.2

49.211 41.847 37.873 35.345 33.707 32.484 31.623 31.053

-21 20 -17 -11 4 -10 16 4

168 131 101 82 69 60 53 49

1.0437 1.0065 0.9980 0.9958 0.9954 0.9955 0.9957 0.9959

1.0943 1.1532 1.1772 1.1859 1.1880 1.1876 1.1861 1.1845

0.2592 0.3807 0.4633 0.5239 0.5676 0.6038 0.6298 0.6488

0.0509 0.1047 0.1964 0.2879 0.3729 0.4437 0.5127

-2.5 4.5 -0.2 -1.4 1.7 -5.2 2.7

80.283 -15 77.886 23 73.846 -3 69.704 -5 65.744 9 62.398 -25 58.930 12

38 72 117 148 164 169 165

1.3042 1.2544 1.1886 1.1369 1.0964 1.0672 1.0428

1.0011 1.0044 1.0140 1.0285 1.0470 1.0667 1.0895

T = 313.15 K 0.0242 0.5169 0.0493 0.6729 0.0923 0.7485 0.1368 0.7932 0.1807 0.8210 0.2202 0.8419 0.2626 0.8559 0.8655

-1.4 12.7 -9.9 -11.1 8.0 -7.1 4.3 6.4

58.736 -6 50.105 49 45.487 -34 42.531 -38 40.550 26 39.158 -24 38.140 14 37.450 21

164 126 97 78 65 56 50 46

1.0415 1.0046 0.9965 0.9946 0.9940 0.9946 0.9950 0.9952

1.0910 1.1494 1.1720 1.1794 1.1805 1.1791 1.1770 1.1748

0.2653 0.3883 0.4714 0.5324 0.5764 0.6 124 0.6384 0.6574

0.1047 0.1964 0.2879 0.3729 0.4437 0.5169

6.3 -2.2 -6.0 7.4 -0.1 -4.6

92.034 28 87.355 -10 82.620 -26 78.014 31 74.102 -0.4 69.814 -18

70 116 148 164 168 163

1.2524 1.1910 1.1358 1.0924 1.0630 1.0392

1.0031 1.0122 1.0277 1.0476 1.0674 1.0900

T = 318.15 K 0.0507 0.6729 0.0951 0.7485 0.7932 0.1402 0.1845 0.8210 0.8419 0.2245 0.2708 0.8559 0.8655

7.1 -3.3 -1.5 -8.1 -7.2 12.8 2.8

59.575 24 54.130 -10 50.744 -5 48.617 -24 46.979 -21 45.811 37 45.082 8

127 100 83 72 63 57 53

1.0085 1.0020 1.0001 0.9995 0.9992 0.9991 0.9991

1.1378 1.1558 1.1633 1.1663 1.1678 1.1682 1.1683

0.3986 0.4836 0.5445 0.5874 0.6226 0.6479 0.6659

0

TABLE V Parameters Aiof the Fits of GEto a Redlich-Kister Type Equation and the Standard Deviations of the Vapor Pressure a ( p ) and Concentration u ( x ) T, K A0 AI A2 A3

298.15

303.15

308.15

313.15

318.15

0.2795 -0.0528 0.0034 0.1183

0.2698 -0.0657 -0.0054 0.0882

0.2654 -0.0735 -0.0522 -0.0482

0.2552 -0.0715 -0.0635 -0.0467

0.2489 -0.0763 -0.0407 0.0049

19 3

13 3

22 6

20 6

.@)/Pa 25 . ( X ) x 105 4

The double differentiation in eq 6 places a heavy strain on the GEdata; thus, very precise vapor pressure data are necessary. This was the reason why Hlavaty's datals were not suitable for the calculation of r2,1, and we decided to remeasure the vapor pressures of this system. Matteoli and LeporiZ0and Rubio et aL2I have discussed the main sources of error in calculating the second derivatives of GE, and we have followed the same procedure as in ref 21. Table I shows the calculated l?2,1;the values at 293.15 (20) Matteoli, E.; Lepori, L. J. Chem. Phys. 1984, 80, 2856. (21) Rubio, R. G.; Prolongo, M. G.; Diaz Pefia, M.; Renuncio, J. A. R.

J . Phys. Chem. 1987, 91, 1177.

K have not been reported since the vapor pressure results at that temperature lead to a GE curve inconsistent with the temperature dependence of GE shown in Figure 4 and with the H E data of Siddiqi and Lucas;" thus, we decided to discard those data. This turned out not to be a problem since, as observed in Table I, the r2,1 values can be considered to be temperature independent. They are negative over the whole composition range, indicating that it is CC1, which is preferentially adsorbed at the vapor-liquid interface. This is an expected result since CC1, has the smallest y. Taking into account that the results of Table IV indicate that the vapor phase is poorer in CC1, than the liquid phase, the r2,, results mean that the concentration profile of CCll is not monotonic in going from the bulk liquid to the bulk vapor, but increases as the interface is approached from the liquid side and then decreases as the bulk vapor is entered up to a value smaller than that of the bulk liquid. The opposite holds for CS2. It has become common to think of surface energy as a fraction of vaporization to this end it is interesting to express the surface properties per mole of substance. The area per molecule has been calculated by using standard approximate and the molar surface energy and entropy at 293.15 K are given in Table 111. Since the molar volume is involved in (22) Moelwyn-Hughes, E. A. Physical Chemistry; Pergamon: London, 1957.

232 The Journal of Physical Chemistry, Vol. 92, No. 1, 1988 0 6:

I

I

I

I

Luengo et al.

I I

I

0 60

I

.Y. I

T 7

0 9

.

1

0

+

02

01

06

08

1

X1

0

Figure 5. Long-wavelength limit of the concentration-concentration correlation function, S,--(O), for the xCCI, + (1 - x)CS2 system: (-), at 298.15 K; (---), at 308.15 K; ideal random mixture. Bar indicates the estimated uncertainty.

W

0 0

(-e-),

0 5t

1

0 LE

I

I

I

3 2

I

33

(

io3/ T

3

I K-'

Figure 4. Gibbs-Helmholtz test of the vapor-liquid equilibrium data for the equimolecular mixture. The slope of the straight line represents the experimental HEfor x = 0.5 at 303.15 K."

the calculation of the area occupied by a molecule, these molar quantities are temperature dependent, increasing very slightly with T, which is in agreement with the results of This behavior is compatible with Eotvos rule'*22 yw3 =

K(T, - T )

(7)

The present data are described very well by eq 7, with correlation coefficients of at least 0.996. Moreover, the calculated T,'s increase smoothly from 552 K (pure CS2) to 557 K (pure CClJ, while the literature values of both T,'s are 552 and 556 K, respectively. It must be observed that the u Y and SY values given in Table 111 decrease with increasing concentration of CCI,, while the corresponding molar quantities follow the opposite trend, which is the same for the vaporization energies. The second term in eq 6 is directly related to the long-wavelength limit of the concentration-concentration correlation function Scc(0) introduced by Bathia and T h ~ r n t o n .Scc(0) ~~ is directly related to the partial structure factors of the mixture aij(ij) ai,(;)

= 1 + Evj mo[ g , , ( r ) - 11 exp(iq'+) dr

with gij(r) being the radial distribution function and wavenumber

(8)

G

proper tie^.^,' Theoretical Calculations Bulk Properties. Most of the frequently used models for mixtures of complex molecular fluids are generalized van der Waals (GVDW)30or Ising-like models.3' The partition function of a GVDW models for a pure r-mer fluid can be written as

the

+ 2 x l S N C ( G ) + SCC(G) - x l x 2 x ~ ~ ~ =~ x~ ~ (~ SGN N) ( G-) 2X2SNC(& + SdG) - ~ 1 x 2(9) x1x2a12(& = XIX2SNN(G) + (x2 - xl)SNC(G) - S C C ( i ) + x1x2 x12all(s') =

The calculation of Scc(0) from experimental data on excess properties has been discussed in detail previously.21 Figure 5 shows Scc(0) vs x1 at 298.15 K for the bulk liquid phase. The curves at the other temperatures agree with the one shown within the estimated uncertainty, which at the maximum of the curve is f O . O 1 . The Scc curve corresponding to a mixture in which the molecules are randomly distributed is given by S$(O) = xlx2and is also plotted in Figure 5. Values of Scc(0) > S&(O) indicate that molecules of type i have a tendency to be preferentially surrounded by molecules of type i , and the opposite holds for Scc(0) < S&(0).21*25From the results in Figure 5 one can conclude that the CCl, and CS2 molecules have an almost random distribution, though a small tendency to homocoordination seems to be present in the temperature range studied in this paper. This is fortunate for testing perturbation theories with the present data, since these kind of theories assume that the structure of the fluid is mainly due to packing effects and that attractive forces play ~ ' ~ ~ ~ effects tend a minor role.26 As discussed p r e v i ~ u s l y ,packing to prevent concentration fluctuations, thus leading to Scc(0) < S',dc(0),while addition of attractive forces, even in a mean field approach, can yield Scc(0) > S$(0).27 Figure 5 indicates that in the present system the attractive forces must play a nonnegligible role; however, the almost ideal behavior of Scc(0) allows one to expect that perturbation theories will deal with these effects in this system. Moreover, Lebowitz et al.28929have shown that fluids with molecules of not very high anisotropy can be studied using a pseudospherical reference system, which would simplify the study of these kinds of systems both for bulk and interfacial

x12SNN(G)

S"(q), S ~ c ( g )and , Scc(q') are the number-number, numberconcentration, and concentration-concentration correlation functions, respectively. The long-wavelength limit lim, Scc(q) is the quantity of interest here. (23) Luck, W. A. P. Angew. Chem., Int. Ed. Engl. 1979, 18, 350. (24) Bathia, A. B.; Thornton, D. E. Phys. Reu. E : Solid State 1970, 2 , 3004.

where V, is the free volume, 3c the external degrees of freedom per mer, N the number of molecules, A the de Broglie wavelength, k T the thermal energy, qr,vthe internal contribution, and @ / 2 the (25) Gallego, L. J.; Silbert, M. Chem. Phys. Lett. 1986, 125, 80. (26) Hoheisel, C.; Kohler, F. Fluid Phase Equilib. 1984, 16, 13. (27) Joarder, R. N.; Silbert, M. Chem. Phys. 1985, 95, 357. (28) Lebowitz, J. L.; Percus, J. K. J. Chem. Phys. 1983, 79, 443. Williams, G. 0.;Lebowitz, J. L.; Percus, J. K. J . Chem. Phys. 1984, 81, 2070. (29) Johnson, J. D.; Shaw, M. S . J. Chem. Phys. 1985, 83, 1271. (30) Vera, J. H.; Prausnitz, J. M. Chem. Eng. J. 1972, 3, 1 . Sandler, S. I . Fluid Phase Equilib. 1985, 19, 233. (31). Lacombe, R. H.; Sinchez, I. C. J . Phys. Chem. 1976, 80, 2586. Cabrerizo, U.; Rubio, R. G.; MenduiRa, C.; Renuncio, J. A. R. J. Phys. Chem. 1986, 90, 889.

The CCI,

+ CS2System

The Journal of Physical Chemistry, Vol. 92, No. 1, 1988 233 I

TABLE VI: Characteristic Parameters of the Pure Components at 298.15 K for the Different Theoretical Models substance model I T,O K E'," J.cm3 P,cm3/mol CCI4

PFP FCS

LS

cs2

CSC PFP FCS

LS CSC

9.06 9.17 5.94 5.97

4698 6786 503 517 4738 7654 536 552

569 662 447 460 638 708 495 508

75.07 69.56 84.69 85.60 47.04 44.66 53.41 53.93

I

I

I

150

"The parameters for the FPV model are the same as those for the PFP one.

TABLE VII: Binary Parameters, Mean Standard Deviations, and Predictions for the Equimolar VE and HEat 298.15 K for the Different Models

-I

2 loo

. 7

a(Ge),

model PFP FPV FCS

t

LS

0.9827 f 0.0001 0 9800 f 0.0002 1 0087 f 0.0002 0.9838 f 0.0002

CSC

0.9807 & 0.0002

J/mol 6.4 6.9 8.4 9.8 10.3

VE(x=0.5)," cm3/mol 0.2412 0.1945 0.1648 0.2323 0.2593

@(~=0.5),~ J/mol 328 266 244 325 332

"Experimental value p ( x = 0 . 5 ) = 0.3521 cm3/mol. bExperimental value @ ( x = 0 . 5 ) = 311 J/mol.

potential of the mean force. Extension of eq 9 to mixtures is straightforward once some mixing rules are assumed for the interaction parameters, the external degrees of freedom, and the core volume of the mers, v*.30 Flory's choices32for the mixing rules of c and I"c have been used in this paper as in a previous work.33 For the energy of the mixture, both Flory's random assumption32and a nonrandom one based on the quasi-chemical hypothesis and due to Panayiotou and Vera3, have been used combined with Flory's free volume term (models PFP and FPV, respectively); also, the free volume arising from the Carnahan-Starling hard-sphere equation was used coupled with the random assumption (FCS model). An Ising-like model previously p r o p ~ s e dhas ~ ~also , ~ ~been used in correlating and predicting the thermodynamic properties of the mixtures. Two different values for the coordination number of the lattice, z = 10 and z = -, have been used; for the latter value the model reduces to the one proposed by Lacombe and SBr~chez.~' In all these models there is an adjustable binary parameter 5 which accounts for the departure of the unlike interactions from the Lorentz-Bertheiot combination rule. Table VI shows the characteristic parameters of the pure components at 298.15 K for the different models, calculated from the density, thermal expansion coefficient, and the isothermal compressibility at 298.15 K.3',32 Table VI1 shows the binary parameter 5 obtained by fitting GE at 298.15 K, the standard deviation of the fits, and the predictions for p and p at x = 0.5. Figure 6 shows the GEvalues calculated from the Redlich-Kister equation and from the theoretical models. As can be observed, the PFP and the FPV models lead to the best fits, especially in the CC14-rich concentration range, although they underestimate the maximum of the curve. All the models underestimate the values, the PFP and the CSC models leading to the smallest errors, though they are of 30%. The same can be said about HE;however, in this case the values predicted by the PFP, LS, and CSC models are 6% larger than the experimental value, while for the FCS and FPV they are 21% and 14% smaller, respectively. It is surprising than the PFP model leads t o better results than t h e FCS one since the latter uses for Vf the expression given by Carnahan and Starling,36while the PFP ~ ~ clearly model uses the expression of Eyring and H i r ~ f e l d e r .This (32) Flory, P. J. J . A m . Chem. SOC.1965, 87, 1833. (33) Saez, C.; Compostizo, A,; Rubio, R. G.; Cresp Colin, A,; Diaz Pefia, M. J . Chem. Soc., Faraday Trans. I 1986, 82, 1839. (34) Panayiotou, C.; Vera, J. H. Fluid Phase Equilib. 1980, 5, 55. (35) Costas, M.; Sanctuary, B. C. J . Phys. Chem. 1981, 85, 3153. (36) Carnahan, N. F.; Starling, K. E. J . Chem. Phys. 1969, 51, 635.

w

(3

50

0

I

I

I

I

02

OL

06

08

X

Figure 6. Excess Gibbs energy of xCC14 + (1 - x)CS2 at 298.15 K. 0 represents values from fitting to eq 2 and the parameters of Table IV. Predictions from theoretical models with the parameters of Table VI1 (see text): (-*-), PFP and FPV; (---), FCS; (-), LS; (---), CSC.

indicates that there exists a compensation of errors between the approximate expressions of V, and of a. The use of a V, suitable for nonspherical particles could improve the results of the GVDW models.37 It is interesting to note that the Ising-like model with z = 10 leads to predictions that are similar to those of the GVDW models. This is fortunate, since Ising-like models can accommodate polar interactions and hydrogen bonds in a more natural way than GVDW models, and these type of interactions are present in many mixtures in which surface effects are the most interesting (microemulsions, vesicles, etc.). The same conclusions are valid at the other temperatures; thus, we will not describe them for the sake of brevity. Surface Properties. Surface tension has been frequently studied within the framework of corresponding states formulations. Several expressions for y have appeared in the literature. Rastogi and Patterson38 extended the PFP model of the previous section to nonuniform fluids. Figure 7 shows the values predicted at 298.15 K by using the binary parameter 5 obtained from GE.Since the predictions for pure CCI, and CS, are so poor, no attempt has been made to optimize f using the y values or to predict y at other temperatures. Rice and Teja39have proposed a model which predicts y using the pure component data and the critical parameters of the pure components and the pseudocritical parameters of the mixtures. There is a binary adjustable parameter f defined by T, = E[ T,, Tc2]l/, where T,,is the critical temperature of component i. Figure 7 shows the results of this model for f = 0.977, the agreement being very satisfactory; it also shows the predictions (37) Siddiqi, M. A,; Svejda, P.; Kohler, F. Ber. Bunsen-Ges. Phys. Chem. 1983,87, 1176.

(38) Patterson, D.; Rastogi, A. K. J . Phys. Chem. 1970, 74, 1067. (39) Rice, P.; Teja, A. J . Colloid Interface Sci. 1982, 86, 158.

234

The Journal of Physical Chemistry, Vol. 92, No. 1. 1988

Luengo et al.

li 0

02

0 4

06

08

1

X1

Figure 8. I*, defined by eq 15, for the vapor-liquid interface at 298.15 K. -

i

I

I

I

I

0

02

04

06

08

in this case it is necessary to calculate the density and concentration profiles through the interface.6 Therefore, an empirical extension of eq 11 has been proposed44

X

Figure 7. Excess surface tension: (-), from best fits to eq 2; (-), Rice and Teja’s model; (---), discontinuous interface model; (-.-), modified Sgnchez’s model.

at 308.15 K using the same value of 6. As can be observed, the calculated values agree with the experimental ones within the estimated uncertainty. Unfortunately, using this model offers no possibility of predicting the surface properties from bulk properties. As already stated, the statistical mechanical theories of liquids have been extended to inhomogeneous fluids. One of the most simple models is the so-called discontinuous interface,40 in which the density and concentration profiles are approximated by step functions. As described by Davis and S ~ r i v e n ,under ~’ suitable approximations for mixtures the surface tension is given by

where nio and :y are the bulk number density and the surface tension for pure liquid i, respectively, and n1(l) is the number density of i in the mixture. The results are shown in Figure 7; it can be observed that the predictions are not as good as those of Rice and Teja’s model, and the predictions become worse as the temperature increases, although at the highest temperature the maximum error in y is 3.2%, which is of the same order of the results found by Winterfeld et al. using the same Perhaps the most promising models for the interfacial properties of complex fluids are the GVDW with gradient-squared corrections for the Helmholtz free energy d e n ~ i t y . ~Recently, .~ Sgnche~ has ~~ proposed a universal correlation for y of pure fluids based on a simplification of one of such models y ( ~ ~ / p ) ’= / ’ A 0 ‘1’

(12) with A , being a universal constant, p the denisty, and KT the isothermal compresibility. Using the recommended value for A,, the predictions of y both for the pure components and the mixture are very poor; thus, we have not plotted the results in Figure 7. It is not possible to find a similar correlation for mixtures, since (40) Fowler, R. H. Proc. R. Soc. London, A 1937, 159, 229. Kirkwood, J. G.; Buff, F. P.J . Chem. Phys. 1949, 17, 3 3 8 . Gray, C. G.; Gubbins, K. E. Mol. Phys. 1975, 30, 179. (41) Davis, H. T.; Scriven, L. E. In Advances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; Interscience: New York, 1982; Vol. XLIX. (42) Winterfeld, P. H.; Scriven, L. E.; Davis, H. T. AIChE J . 1978, 24, 1010. (43) SBnchez, I. C . J . Chem. Phys. 1983, 79, 405.

Equation 13 is equivalent to the assumption that the mixtures and that there behave like a pure component with KT = CX~KT, is no excess volume. It is known that KT of the mixture is better approximated by taking into account the density of the mixture calculated from the experimental excess volumes and using KT = c p , K ~ , .In ~ ~addition, it was found necessary to use a composition-dependent A. A0‘12 = 91Ao,I1l2 92A0,2‘I2 t 14)

+

Figure 7 shows the results obtained with this modified Sgnchez’s method. The predictions are worse than those of Rice and Teja’s and of the discontinuous interface model; also, the predictions get poorer as the temperature increases. The density and concentration profiles allow one to estimate the characteristic length of the interface, 1. Using a gradientsquared GVDW model, Bathia and March have related I to y,

where 6 = ( u l - u z ) / [xu1+ (1 - x ) u 2 ] ,u, being the partial molar volumes of component i. Using the present data and KT = &,KT,, we have calculated I*, the results being plotted in Figure 8, where one can see that I* is of the same order as the values obtained for simple fluids4’ and liquid metals.46 I* shows a maximum in the midconcentration range, which is probably associated with the maximum in the relative adsorption of CC14 in excess volume and in Scc(0), the latter playing the most important role in eq 15. One has to recall that according to the gradient-squared theory the width of the interface is directly related to the stability of the density and concentration fluctuations:’ which in turn are directly related to Scc(0).

Acknowledgment. This work was supported in part by the USA-Spain Corcmittee for Scientific Cooperation under Grant CCB84-003. Registry No. CCI4, 56-23-5; CS2, 75-15-0. (44) Acree, W. E. J . Colloid Interface Sci. 1984, 101, 515. Ballard, R. E.; Jones, J.; Read, D. Chem. Phys. Lett. 1985, 122, 161. (45) Aicart, E.; Tardajos, G.; Diaz Peiia, M . J . Chem. Thermodyn. 1980, 12. -, 10x5. ~ - - -

(46) Bathia, A. B.; March, N. H . J . Chem. Phys. 1978, 68, 4651. (47) Egelstaff, P. A,; Widom, B. J . Chem. Phys. 1970, 53, 2667.