1722
J. Phys. Chem. 1982, 86, 1722-1729
Thermodynamic Properties of Liquld Mixtures of Argon 4- Krypton Susana F. Barrelros, Jorge C. 0. Calado,' Paulette Clancy,t Manuel Nunes da Ponte, and Wllllam B. Streettt Centro de Ouimlca Estrutural, Complex0 I, 1096 Lisboa. Portugal (Received: October 1, 1981; I n Final Form: December 1. 1961)
The equations of state of liquid argon, liquid krypton, and of a mixture of mole fraction composition 0.485 Ar + 0.515 Kr have been measured at 129.32,134.32,142.68,and 147.08 K, from just above the vapor pressure to the freezing pressure of krypton. Similar studies have been made for three other mixtures, of mole fraction in argon x = 0.277,0.698, and 0.787, at 134.32 K. The results show that the excess volume p is negative at low pressures but increases rapidly with pressure, becoming almost zero at high pressures. The curve of VE against x , which is rather asymmetric at low pressures, becomes more symmetric as the pressure is raised to approximately 20 MPa, reverting to an asymmetric shape with a tendency toward S-shaped curves at higher pressures. The changes in the excess functions GE, HE,and TSEwith rising pressure have been calculated; HE and TSEprove to be extremely sensitive to pressure changes especially at the higher temperatures, increasing very steeply when the first 20 MPa are applied to the mixtures; GE is much less affected by pressure. The results are well reproduced by conformal solution theory (van der Waals-1), using either the argon or the Lennard-Jones equation of state for the reference fluid.
Introduction
The measurements reported in this paper were obtained in the course of a program to investigate the effect of pressure on the thermodynamic properties of liquid mixtures of condensed gases. Mixtures of argon and krypton consist of simple monatomic molecules, hence their properties should be accurately predicted by any successful theory. Extensive theoretical studies of these mixtures have been performed, either by computer simulation,' using the Lennard-Jones 12-6 intermolecular potential, or by perturbation theory.2 Moreover, the argon + krypton liquid system can be used as a convenient reference fluid to study the effect of anisotropic forces on the properties of mixtures, as has been done by Gubbins, Gray, and coworker~.~-~ The available experimental data for the thermodynamic properties of this system have been recently summarized by Lewis, Lobo, and Staveley.6 They stress the need for more experimental information on the equation of state, to determine the influence of pressure on such thermodynamic excess properties as the excess Gibbs energy GE, the excess enthalpy HE,the excess entropy SE,and the excess volume P, adding a new dimension to comparisons of the theory and experiment. Most of the experimental data published so far have been obtained at low saturation vapor pressure (virtually zero pressure). The only exceptions are the work of Blagoi and Sorokin,' who report measurements of the molar volume of a mixture of approximately equimolar composition at three temperatures and pressures to 50 MPa, and the work of Schouten, Deerenberg, and Trappeniers? who performed vapor-liquid equilibrium experiments, at eight temperatures, between 138 and 193 K. Our results cover wider pressure and composition ranges and temperatures closer to the critical temperature of argon (150.86 K) than the measurements of Blagoi and Sorokin. We performed p , V,T measurements on liquid argon, liquid krypton, and four mixtures of these substances, with mole fractions of Ar approximately x = 0.3, 0.5, 0.7, and 0.8 at pressures up to the freezing pressure of krypton at each temperature. The study of these four mixtures provides information about the effect of pressure School of Chemical Engineering, Cornel1 University, Olin Hall, Ithaca. NY 14853-0294. 0022-3654/82/2086-1722$01.25/0
on the asymmetry of the VE vs. composition curve. The calculation of the effect of pressure on HEand SErequires an accurate knowledge of the derivative ( d P / d T ) , and measurements were therefore performed at four diierent temperatures, on the pure components and on the mixture of x cz 0.5. The studies of M c D ~ n a l d ' *and ~ J ~Singer and Singer" have shown that one of the most successful, but also one of the simplest, theories available to predict the thermodynamic properties of simple liquid mixtures is the conformal solution theory in the van der Waals-1 form (vdw-1) developed by Leland, Rowlinson, and Sather.12 We used our results to test the applicability of this theory, employing two different equations of state to calculate the properties of the reference fluid: the Gosman et al. equation of state for =god3 and the equation of state of the Lennard-Jones fluid, recently proposed by Nicolas et a1.14
Experimental Section
The apparatus used for the p , V , T measurements presented here is very similar to the one described by Nunes da Ponte, Streett, and Sta~eley.'~The main difference (1)I. R. McDonald, Mol. Phys., 23, 41 (1972). (2)J. R. Lee, D. Henderson, and J. A. Barker, Mol. Phys., 29, 429 (1975). (3)C. H. Twu, K. E. Gubbins, and C. G. Gray, Mol. Phys., 29, 713 (1975). (4)M.Flytzani-Stephanopoulos,K.E. Gubbins, and C. G. Gray, Mol. Phys., 30, 1649 (1975). (5)C. H. Twu, K. E. Gubbins, and C. G. Gray, J. Chem. Phys., 64, 5186 (1976). (6)K.L. Lewis, L. Q.Lobo, and L. A. K. Staveley, J. Chem. Thermodyn., 10,351 (1978). (7)Y. P. Blagoi and V. A. Sorokin, Trans. Phys. Techn. Inst. Low Temp. Acad. Sci. Ukr. SSR,Phys. Condensed State, Part U,5 (1969). (8)J. A. Schouten, A. Deerenberg, and N. J. Trappeneiers, Physica A, A_l -. ,1.51 _ _(197.5). ~
(9)I. R. McDonald, Mol. Phys., 24, 391 (1972). (10)G. Jacucci and I. R. McDonald, Physica A, 80, 607 (1975). (11)J. V. L. Singer and K. Singer, Mol.~Phys.,24, 357 (1972). (12)T.W.Leland, J. S. Rowlinson, and G. A. Sather, Trans. Faraday SOC.,64, 1447 (1968). (13)A. L. Gosman, R. D. McCarty, and J. G. Hust, Natl. Stand. Ref. Data Ser. Natl. Bur. Stand., No.27 (1969). (14)J. J. Nicolas, K. E. Gubbins, W. B. Streett, and D. J. Tildesley, Mol. Phys., 37, 1429 (1979). (15)M.Nunes da Ponte, W. B. Streett, and L. A. K. Staveley, J. Chem. Thermodyn., 10, 151 (1978).
0 1982 American Chemical Society
Roperties of Liquid Ar
+ Kr Mixtures
The Journal of Physical Chemisby, Vol. 86, No. 9, 1982 1729
is that the pressure is measured by a Ruska dead weight gauge with a precision of a t least 10.01 MPa. Temperature was measured on the IPTS-68 by a platinum resistance thermometer, coupled to an ASL automatic resistance bridge, with an estimated precision of fO.O1 K. The accuracy of the calibration of the thermometer was checked, as described by Albuquerque et al.,18 by measuring the vapor pressure of the argon at several temperatures. A new measurement performed on krypton at 147.08 K yielded a vapor pressure of 0.566 MPa, which compared with 0.5663 MPa given by Michels et al." The argon and krypton used were supplied by Air Liquide with purities guaranteed to be 99.998 and 99.95 mol %, respectively. Purity checks on the argon used were described by Albuquerque et al.18 The mixtures were prepared in small steel cylinders at pressures of approximately 10 MPa, and the composition measured by weighing in a Mettler balance of 1 kg capacity and f O . l mg precision. The accuracy in composition is estimated to be fO.OO1 mole fraction.
Results In our experimental method, temperature, pressure, and amount of substance inside the cell are measured directly. In order to calculate molar volumes, we need to know the volume of the cell from an independent calibration. This calibration and some comparisons with other authors' values were described by Albuquerque et al.18 We need only state here that we used the molar volume of orthobaric liquid krypton, given by Terry et al.19 at 129.32 K, divided by 1.004,18as our reference value. As a further test of our calibration, we performed p,V,T measurements on methane at 120.00 K. Agreement with values due to Nunes da Ponte et al.16 at the same temperature is very good. The molar volumes reported here, as well as those in ref 15,16, and 18, are estimated to be accurate to +0.1%. The experimental p,V,T results for argon, krypton, and four binary mixtures are given in Table I. For krypton, the p, V isotherms terminate at the freezing pressure. When freezing intervenes, the p,V,T apparatus acts as a blocked capillary, and, since no more substance enters the cell when pressure is applied, we obtain a change of slope in the isotherm and a horizontal line at higher pressures. This behavior is shown in Figure 1, where the 142.68 K isotherms for the pure components and for the equimolar mixtures are represented. The interaction of this horizontal line with the liquid isotherm gives the freezing pressure and the volume of the liquid in equilibrium with its solid. These results for krypton have been reported by Calado and Nunes da Ponte.20 Table I also records values of AV, the difference between the experimental molar volumes and calculated ones, obtained from a fitting of each isotherm to the Tait equation V = V , + A In [ ( B + p , ) / ( B + P)I (1) where p s is the saturation vapor pressure, V , is the saturation molar volume, A and B are constants (for each substance at each temperature), p is the pressure, and V the molar volume. This equation was used for its simplicity and because it is explicit in V , which makes it (16) G.M. N.Albuquerque, J. C. G. Calado, M. Nunea da Ponte, and A. M. F. Palavra, Cryogenics, 20,601 (1980). (17) A. Michels, T. Waesenaar, and T. Zwietering, Physica, 18, 63 (1952). (18) G.M. N.Albuquerque, J. C. G. Calado, M. Nunes da Ponte, and L. A. K.Staveley, Cryogenics, 20,416 (1980). (19) M. J. Terry, J. T. Lynch, M. Bunclark, K. R. Mansell, and L. A. K. Staveley, J. Chem. Thermodyn. 1, 413 (1969). (20) J. C. G.Calado and M. Nun- da Ponte, Rev. Port. Quim.,21,M (1979).
36
k p/MPa
Flgwe 1. Experimental p ,V resub for argon, krypton, and a mixture 0.485 Ar 0.515 Kr, at 142.68 K. The open circles represent points obtained above the freezing pressure of krypton.
+
particularly suitable for the calculation of the molar volumes VE, at rounded values of pressure
P(p,T,X) = V,b,T,X) - XVl(p,T) - (1 - X)VZ(p,T) (2) where subscripts 1,2, and m refer to component 1 (argon), component 2, and the mixture, respectively, and 3c is the argon mole fraction. Equation 1 gave a good fit for all isotherms and the differences, AV, are within the estimated precision of the data. However, for argon at the two higher temperatures (142.68 and 147.08 K)this equation was not suitable below the vapor pressure. At these temperatures, argon is close to its critical temperature (150.86 K)and, with the corresponding low densities, the slope of the V,p isotherm is rather large and negative, as shown in Figure 1. To account for this, the parameter B in eq 1 becomes negative and the logarithmic term in this equation becomes infiiite at any positive value of the pressure (p = 4).To avoid these difficulties, excess volumes a t 142.68 K were calculated from eq 2, using the smoothed values given by eq 1 for both krypton and the mixture, and using the experimental, unsmoothed results for argon. We therefore obtained a series of excess volumes, which were found to be fitted by the equation In
(-P) =
4.53818 - 2.03658 In (p
+ 3) + 0.097148 ln2 (p + 3)
(3)
with P in cm3 mol-' and p in MPa. At 147.08 K, it was not possible to obtain a good fit with an equation of this type. The excess volumes at the lower pressure, below the vapor pressure of argon ( - 4 MPa), were thus calculated by extrapolation, at constant pressure, of the results obtained for the mixture 0.485 Ar + 0.515 Kr at the other three temperatures. The following equation was found to give a good fit to the data: In (-P) = al + a,y + a o 2 (4) where y = exp(0.04T), T being the temperature in K, and a', a2, and as constants a t each pressure. The VE value at zero pressure and 115.76 K obtained by Davies et aL2'
1724
The Journal of Physical Chemjstty, Vol. 86, No. 9, 7982
Barreiros et al.
TABLE I : Experimental Values of the Molar Volume V and of the Difference A V between Experimental and Calculated Molar Volumes, at Pressure p and Temperature T, for Liquid Argon, Liquid Krypton, and Four Mixtures plMPa
VI Avl (em3mol-') (cm3mol-')
plMPa
VI AVl (cm3mol-') (cm3mol-')
p/MPa
VI
AV/
(cm3mol-') (em3mol-')
11.32 15.77 22.91
34.249 33.460 32.462
-0.009 0.009 0.007
29.60 35.94
Ar T = 129.32 K 31.746 31.155
0.020 0.001
44.07 48.82
30.531 30.208
-0.005 -0.013
7.77 8.14 8.45 11.07 16.29 21.80
36.399 36.276 36.234 35.421 34.287 33.370
-0.031 -0.026 0.035
24.27 30.94 39.28 44.42 53.35
T = 134.32 K 33.019 0.020 32.224 0.018 31.425 0.012 31.003 0.003 30.374 -0.006
60.49 66.21 71.16 77.02 81.84
29.936 29.621 29.366 29.088 28.903
-0.016 -0.022 -0.030 -0.035 -0.011
6.65 9.66 11.77 14.42 18.15 23.95 28.54 31.62 37.41
40.282 38.475 37.516 36.641 35.644 34.477 33.754 33.372 32.636
-0.022 0.030 -0.014 0.018 0.018 0.018 0.020 0.057 0.003
42.31 44.58 53.69 61.03 66.36 72.97 80.76 86.12 88.73
0.004 0.022 -0.002 -0.005 -0.009 -0.005 -0.013 -0.006 -0.011
95.90 98.91 100.04 104.92 112.86 122.23 124.99 142.23
28.881 28.757 28.744 28.560 28.274 27.983 27.881 27.391
-0.015 -0.018 0.014 0.016 0.015 0.036 0.021 0.034
5.56 7.84 10.27 11.63 17.70 24.02 28.38 35.32
44.721 41.685 39.838 29.126 36.844 35.338 34.554 33.547
-0.064 0.052 0.039 0.076 0.061 0.043 0.036 0.020
39.74 43.31 49.03 49.03 56.82 64.47 70.95 78.25
0.006 0.024 0.006 -0.008 -0.005 0.001 -0.011 -0.016
84.18 84.87 91.35 99.48 105.34 112.03 114.51 118.37
29.764 29.731 29.433 29.063 28.840 28.591 28.497 28.368
- 0.024
2.20 8.00 14.57
35.532 35.129 34.712
- 0.001
23.68 29.47
32.141 31.949 31.183 30.671 30.336 29.965 29.558 29.312 29.190 T = 147.08 K 33.009 32.649 32.091 32.077 31.446 30.913 30.496 30.078 Kr T = 129.32 K 34.210 33.919
0.008 0.006
39.25 46.29
33.919 33.195
0.007 0.010
2.37 2.99 7.72 16.00
36.132 36.079 35.673 35.119
0.007 0.007 -0.021 0.003
21.35 28.29 36.36
T = 134.32 K 34.779 -0.009 34.414 0.011 34.012 0.009
43.17 50.77 57.69
33.697 33.385 33.116
0.000 0.000 -0.007
1.31 2.02 5.07 7.73 12.04 17.55 21.74 29.47
37.373 37.295 36.982 36.730 36.333 35.913 35.631 35.106
0.003 0.002 0.002 0.003 -0.017 - 0.004 0.014 - 0.004
30.56 35.83 41.05 46.50 51.25 60.72 64.74
T = 142.68 K 35.060 0.005 34.766 0.011 34.486 0.006 34.203 -0.009 34.006 0.013 33.567 -0.024 33.445 0.013
72.55 73.04 79.10 81.23 85.81 92.19 95.49
33.152 33.119 32.917 32.854 32.675 32.502 32.397
0.010 -0.005 0.003 0.011 -0.020 0.005 -0.002
1.04 1.77 3.02 3.12 3.12 5.65 5.65 11.39 14.39
38.069 37.977 37.841 37.814 37.828 37.527 37.537 36.940 36.680
-0.005 -0.006 0.008 -0.007 0.007 -0.008 0.007 -0.014 -0.002
22.77 28.78 36.76 43.80 50.49 51.66 57.73 65.44 72.75
77.71 85.92 95.50 98.40 104.61 108.05 108.33 114.59 114.67
33.261 32.970 32.666 32.574 32.367 32.279 32.313 32.109 32.105
0.011 0.004 0.007 0.003 -0.022 -0.012 0.030 - 0.004 -0.005
3.98 5.43 7.80 15.13
34.977 34.782 34.507 33.770
0.012 -0.005 -0.006 -0.009
36.84 42.83 51.79
32.234 31.888 31.420
0.017 0.004 -0.018
3.30 7.68 10.41 12.41
35.950 35.314 34.959 34.214
0.004 0.002 -0.009 -0.001
52.65 59.95 65.37
31.831 31.488 31.252
0.001 -0.001 -0.003
-0.008
0.018 0.025
T = 142.68 K
0.006 0.003
T = 147.08 K 36.005 - 0.006 35.598 0.005 35.107 0.008 34.703 -0.008 34.410 0.034 34.325 0.005 34.003 -0.040 33.722 0.005 33.425 -0.007 0.485 Ar t 0.515 Kr T = 129.32 K 15.13 33.769 -0.010 21.47 33.250 0.002 30.43 32.624 0.013
24.21 30.57 37.25 46.90
T = 134.32 K 33.614 0.003 33.133 0.115 32.680 -0.002 32.128 0.003
-0.022 -0.013 -0.028 -0.013 -0.007 -0.010 -0.002
Properties of Liquid Ar
+ Kr Mixtures
The Journal of Fhyslcal Chemktv, Vol. 86,No. 9, 1982 1725
TABLE I (Continued)
plMPa
VI Av/ (cm3mol-') (cm' mol-')
pIMPa
VI Av/ (cm3mol-') (cm3mol-')
p/MPa
VI Av/ (cm3mol-') (cm3mol-')
T = 142.68 K 4.49 7.80 14.52 22.01 28.53 29.12
37.426 36.831 35.735 34.857 34.236 34.152
-0.016 0.036 -0.018 -0.003 0.009 -0.023
36.98 43.62 50.55 58.13 64.75 71.23
6.11 8.22 13.58 16.71 17.03 24.23 30.63
38.101 37.594 36.626 36.147 36.103 35.223 34.565
0.026 -0.015 -0.005 -0.014 -0.012 0.002 -0.004
35.94 44.76 52.07 57.37 67.65 71.51
4.62 8.74 15.66 24.03
35.681 35.249 34.592 33.988
0.007 0.006 - 0.028 -0.003
5.52 8.35 16.43 21.82
35.881 35.343 34.179 33.559
0.005 -0.008 0.004 -0.003
3.66 6.24 10.58 10.58
36.635 35.982 35.084 35.068
0.002 0.008 0.001 -0.015
33.552 0.010 33.084 0.003 -0.010 32.644 0.002 32.240 31.928 0.018 31.621 0.006 T = 147.08 K 34.116 0.009 33.447 0.005 32.980 0.012 32.661 0.003 32.130 0.010 31.945 0.009
79.43 84.60 91.50 105.97 128.04 150.65
31.276 31.073 30.815 30.330 29.694 29.141
78.67 94.25 104.46 109.42 114.25 120.45
31.623 31.002 30.650 30.484 30.333 30.145
0.277Ar + 0.723 Kr T = 134.32K 28.36 33.715 0.010 36.29 33.241 0.005 44.70 32.810 0.012
50.97 59.31 66.97
32.510 32.146 31.842
0.005 -0.003 -0.010
58.42 64.48 68.07
30.959 30.665 30.508
-0.004 - 0.004 0.002
50.69 58.95 66.68 76.60
31.134 30.667 30.281 29.843
0.005 -0.002 -0.006 - 0.008
0.698 Ar + 0.302 Kr T = 134.32 K 29.61 32.827 -0.002 36.43 32.291 0.000 46.28 31.639 0.005 50.56 31.387 0.004 0.787 Ar + 0.213 Kr T = 134.32 K 19.40 33.752 -0.011 27.39 32.883 -0.001 35.80 32.158 0.012 42.42 31.672 0.015
was also included in the fitting. Calculation of the change of the excess enthalpy HE with pressure required the knowledge of the derivative ( d P / dT)P AHE = HE(T,x,p)- HE(T,x,p=O)=
J p [ P- T ( d P / d T ) , l d p (5) Therefore the excess volumes obtained for the mixture 0.485 Ar + 0.515 Kr were cross fitted, at a given pressure (up to 8 MPa), to a function of T, using eq 4. At higher pressures (p > 8 MPa) where this equation yields too steep a variation of P with T , a simple polynominal was used instead
P = bl + bzT + b3TL
(6) where bl, b2, and b3 are constants for each isobar. The values of Ve shown in Table I1 for that mixture are therefore those obtained from eq 4 or 6. Values of P for the other three mixtures a t 134.32 K, recorded in Table 111, were obtained from eq 1 and 2. We estimate the precision of the P values to be f0.03 cm3 mol-', except at 142.68and 147.08K,below the vapor pressure of argon, where the extrapolation procedures employed may have introduced errors which increase with decreasing pressure. The change in the excess Gibbs energy between pressures 0 and p , AGE,can be easily calculated from AGE = GE(T,x,p)- GE(T,x,p=O)=
$,P P d p
0.004 0.002 -0.005 -0.012
0.006
- 0.002
0.000 -0.006 -0.007 -0.010
and TASE,the change in excess entropy with pressure, can be obtained from the well-known equation
- AGE (8) Values of @,AGE,TASE,and T ( a P / d T ) ,are listed in TASE =
AHE
Table 11. The imprecisions in both AHE and TASE are certainly larger than AGE, as their evaluation involves differentiation followed by integration. We estimate it to be f2% for AGE and f10% for AHEand TASE. The excess molar volumes at zero pressure are especially important for the calculation of excess Gibbs energies GE from vapor-liquid equilibrium data, as shown by Lewis et al.6 The constants used in eq 4 for the zero pressure isobar were ul = -1.5961,uz = 6.8594 X and u3 = 2.1485 X 10". This equation should give accurate P values from the triple-point temperature of krypton (115.76K) to close to the critical temperature of argon. Comparison with Other Experiments Our p,V,T values for pure argon and krypton have been compared to the data of Streett and Staveley for argonzz and krypton.= The agreement is better than 0.1% when one takes into account the difference in calibration of the cell volume, as discussed elsewhere.'* For the p,V,T measurements on the mixtures, there are only three sets of results available for comparison. The values of P reported by Davies et al.zl were obtained at the triplet point temperature of krypton (115.7K) and at
(7)
(21)R. H.Davies, A. G. Duncan, G. Saville, and L. A. K. Staveley, Trans. Faraday SOC.,63,855 (1967).
(22) W. B. Streett and L. A. K. Staveley, J . Chem. Phys., 60, 2302 (1969). (23) W. B. Streett and L. A. K. Staveley, J. Chem. Phys., 66, 2495 (1971).
1726
Barreiros et al.
The Journal of physicai chemisby, Vol. 86, No. 9, 1982
TABLE 11: Values of the Excess Molar Volume VE, Its Temperature Derivative -T(aVEkT),, and the Change in the Molar Excess Enthalpy A HE, Molar Excess Gibbs Energy AGE ,and Molar Excess Entropy T A S over Their Values at Zero Pressure for the Mixture XI,= 0.485 at Four Temrmatures (T=129.32.134.32. 142.68.and 147.08 KI
0 2 5 10 15 20 30 40 50
- 1.326
0 2 5 10 15 20 30 40 50 60 70
-2.409 -1.594 -1.017 -0.606 -0.410 -0.296 -0.184 -0.130 -0.096 -0.068 -0.058
0 2 5 10 15 20 30 40 50 60 70 80 100
-11.19 -4.530 -2.041 -0.954 -0.580 -0.398 -0.233 -0.156 -0.112 -0.082 -0.064 -0.046 -0.022
0 2 5 10 15 20 30 40 50 60 70 80 100 120
-37.9 -9.86 -3.421 -1.223 -0.708 -0.479 -0.274 -0.180 -0.127 -0.095 -0.074 -0.057 -0.032 -0.025
-1.031 -0.759 -0.499 -0.355 -0.259 -0.173 -0.126 -0.094
T = 129.32K 17.5 0 10.2 24.7 5.03 44.1 57.7 1.77 1.00 62.4 0.63 64.8 0.10 66.3 0.03 65.4 0.00 64.3 T = 134.32K 44.9 0 21.2 59.2 9.05 98.0 3.89 123 1.94 135 1.22 141 0.48 148 0.22 149 0.12 150 0.10 150 0.04 150 T = 142.68 K 381 0 101 405 30.2 567 7.76 637 3.73 662 2.31 674 1.17 689 0.66 694 0.41 697 0.35 700 0.23 702 T = 147.08 K 1783 0 290 1614 67 2032 10.0 2178 4.76 2210 2.92 2226 1.57 2245 0.90 2255 0.58 2261 0.48 2265 0.29 2268
0 -2.3 -5.0 -8.1 - 10.2 -11.7 -13.9 - 15.4 -16.5
0 27.0 49.1 65.8 72.6 76.5 80.2 80.8 80.8
0 -3.9 -7.8 -11.7 -14.2 -16.0 -18.4 -20.0 -21.1 -21.9 -22.5
0 63.1 105.8 135 149 157 176 169 171 172 173
0 -14.5 -23.7 -30.6 - 34.4 -36.9 -40.0 -42.0 -43.3 -44.2 -45.0 -45.6 -46.2
0 420 591 668 696 711 729 736 740 744 747
0 -41 -59 - 69 - 74 -76 - 80 -83 - 84 -85 - 86 - 87 - 88 -88
0 1655 2091 2247 2284 2302 2325 2338 2345 2350 2354
TABLE 111: Values of the Excess Molar Volume VE and the Increase of the Molar Excess Gibbs Energy A G E over the Value at Zero Pressure, at 134.32 K and Round Values of Pressure, for Three Mixtures of Liquid Argon and Krypton ( X A ~= 0.277,0.698,0.787) XA* = 0.277 X A ~ = 0.698 XA, = 0.787 p/MPa VE/(cm3mol-') aGE/(J mol-') VE/(cm3mol-') a G E / ( J mol-') VE/(cm3mol-') aGE/(J mol-') -2.631 0 -1.562 0 -2.816 0 0 -1.593 -4.5 -4.1 -1.777 - 1.082 - 2.6 2 -0.935 -7.8 -8.7 - 1.082 -0.733 5 -5.3 -0.491 -11.2 -0.586 -0.459 -12.6 10 -8.2 -0.299 -13.1 -0.362 -0.322 - 15.0 15 -10.1 -0.197 - 14.4 -0.239 -0.242 - 16.5 20 -11.5 -0.097 -15.9 -0.156 -0.118 - 18.3 30 -13.5 -16.6 -0.112 - 0.064 - 0.053 -19.2 40 - 14.9 -'17 .O -0.087 -0.036 -0.030 -19.7 50 -15.9 -0.017 -17.3 -0.071 -20.0 -0.020 -16.6 60 - 0.009 -17.4 -0.060 -20.2 -0.020 -17.3 70
Properties of Liquid Ar -t Kr Mixtures
The Journal of Physical Chemistry, Voi. 86, No. 9, 7982
1727
C
1-
E
mL - I \
- I 511 0
,
>
,
40
80
p/MPa
Figure 2. Experlmental excess volumes VE for a mixture 0.485 Ar 0.515 Kr, as a hncbkn of presswe, at fwr temperawes: (a) 129.31 K; (b) 134.32 K (c) 142.68 K (d) 147.08 K; (0)Blagol and Sorokin's data at 129.31 K, for a mixture 0.491 Ar 0.509 Kr.
+
+
-2
05
virtually zero pressure. They agree with the values given a t the same temperature and pressure by Chui and Canfield." For a mixture of composition 0.485 Ar + 0.515 Kr, Davies et al. give p = -0.512 cm3 mol-', a value that fits smoothly on a curve drawn through our own Ve points at zero pressure. Blagoi and Sorokin' report p,V,T values for the mixture 0.491 AI + 0.509Kr, at three temperatures, 119.98,129.31, and 141.58 K, and at pressures to 50 MPa. The excess volumes reported here for the mixture 0.485 Ar 0.515 Kr are represented as a function of pressure, at four temperatures, in Figure 2. The shape of these curves is similar to the corresponding curves given by Blagoi et al.7 in their paper. The large, negative excess molar volume at zero pressure, the sharp increase with pressure at low pressures, and the approach to a limiting value at high pressures (which in the case of these mixtures is 0 cm3mol-l) found here is emerging as typical behavior of binary mixtures of condensed gases. The mixture studied by Blagoi et al. has a very similar composition to one of ours, but unfortunately they report excess volumes in graphical form only. From these graphs we conclude that agreement of their values at 129.31 K with ours at 129.32 K is good at low pressures, but worsens as the pressure is raised, Blagoi's results remaining more negative. However, when we recalculated the VE values, using their molar volumes for the mixture, but our own for the two pure components at 129.32 K, we obtained a much improved agreement with our VE results, as shown in Figure 2. Figure 3 shows the dependence of VE on composition and pressure a t a temperature of 134.32 K. The asymmetry of the VE vs. composition curves is very pronounced at low pressures, the minimum in the curve occurring at a composition richer in the more volatile component. This was also the case for nitrogen methane mixtures, reported by Nunes da Ponte et a1.16 Those curves become more symmetric as the pressure is increased to about 20 MPa. Above this pressure, however, the asymmetry reappears, with a tendency toward producing S-shaped curves. In Figure 4, AHEand AGE are plotted as a function of pressure, for the mixture 0.485 Ar + 0.515 Kr, at two temperatures, 134.32 and 142.68 K. It is obvious from this figure that the first 20 MPa of applied pressure exert a dramatic effect on the excess enthalpy, while the effect on
XA,
6ool /
+
+
(24) C. H. Chui and F. B. Canfield, Trans. Faraday SOC.,67, 2933 (1971).
+
Figure 3. Experimental excess volumes VE for argon krypton Hquid mixtures, as a function of compositlon, at several pressures and at 134.32 K.
J mol-'
4oot
I
0
2 0 -
40 '
60
A G ~ -100-
Figure 4. Changes of the excess enthalpy AHEand the excess Gbbs energy AGE with applied pressure, for the mixture 0.485 Ar 0.515 Kr, at two temperatures: (a) 134.32 K; (b) 142.68 K (-, experiment; , theory).
+
____
the excess Gibbs energy is comparatively small. It is also apparent that the effect of pressure on HE is much stronger at higher temperatures, while for GE the influence of temperature is comparatively much smaller. These effects are a consequence of very large negative values of T( a p / d T & ,which is the dominant term in eq 5, at the lower pressures. Lewis et a1.6 have summarized the available experimental information on the excess functions a t zero pressure. From their paper we estimate GE ( x = 0.5) at zero pressure to be 87,89, and 104 J mol-l, at 129.32, 134.32, and 142.68 K, respectively. It is not possible to extrapolate their data to 147.08 K within reasonable limits of accuracy, but the value at this temperature should be significantly higher than at the other three. This fact would be consistent with a sharp decrease of HE ( x = 0.5) with rising temperature, a kind of behavior that, as Lewis et al. pointed out, is by no means improbable. However, the only available measurements of HE which can lay claim to high
1728
The Journal of Physical Chemistry, Vol. 86, No. 9, 1982
Baneiros et al.
accuracy are those of Lewis et al.eat 116.9 K. They give
0
HE= 43.6 J mol-' for a mixture of xAr = 0.485. Comparisons with Conformal Solution Theory It is well-known that the 12-6 Lennard-Jones intermolecular potential is a good effective pair potential for simple liquids. Extensive calculations have been reported by several authors on this subject (for a review see Barker and Henderson26),either using perturbation theories or computer simulations, comparison being often carried out with experimental properties of liquid argon. Recently, Nicolas et al.I4 published a 33-constant equation, fitted to a large number of computer simulation values for the equation of state of the 12-6 Lennard-Jones fluid, covering a wide range of temperatures and densities. We have used this equation to examine the adequacy of the 12-6 Lennard-Jones potential to describe the behavior of argon + krypton mixtures. The procedure was as follows: first, the equation was fitted to our experimental p , V , T results for pure krypton and pure argon, incorporating into the argon data the results obtained by Nunes da Ponte et and also results recently obtained in this laboratory a t 119.32 K. The values obtained for the intermolecular parameters were the following: t / k = 117.5 K and a = 0.3398 nm, for argon; t / k = 165.8 K and a = 0.3633 nm, for krypton. The argon parameters agree very well with those obtained by McDonald and Singer in the fitting of their Monte Carlo calculations using a 12-6 potential to experimental density data2' ( t / k = 117.2 f 1.4 K and a = 0.3405 f 0.013 nm). As for krypton, while the agreement is excellent for a (a = 0.3634 nm), it is less good for e ( t / k = 163.1 K), possibly due to the fact that p,V,T data for krypton covering only a narrow region of low reduced temperatures were used.27 Secondly, p,V,T values for the mixture 0.485 Ar + 0.515 Kr were fitted to the same equation of state by using conformal solution theory,12yielding t, and a,, the energy and length parameters for the intermolecular potential of a hypothetical pure fluid. The usual van der Waals mixing rules :a
= Ex,x,a,;
(9)
aB
€a ,:
=
EX,X,t,,a,;
(10)
aB
may then be used to calculate the cross interaction parameters t,, and ,,a, and subsequently to calculate the deviations k,, and j,, from the geometric and arithmetic mean combining rules, respectively, as given by ea, = (1 - kug)(ea,$3,)1'2
(11)
Flgure 5. Comparison between experimental (points) and calculated (lines) excess mder vokrmes at 134.32 K -, argon equation of state (2, 5, 20, 40 MPa); ----, Lennarddones equation of state (5, 20, 40 MPa).
computer simulation (on which the equation of state of Nicolas et al. is based) cannot yield realistic values for the critical region, because it does not allow for the large density fluctuations characteristic of real fluids in that region. For argon, with the parameters used here, the equation gives a critical temperature 10 K higher than the real value. In Figure 5 we present a comparison between experimental and calculated values of VE. At low pressures, the inability of the theory to reproduce the experimental results is simply a consequence of the bad fit of the Lennard-Jones equation to the experimental p,V,T results of argon, mentioned above. Nevertheless, the theoretical results exhibit the correct asymmetry with composition. At pressures above 20 MPa for a mixture 0.5 Ar + 0.5 Kr, the theory agrees with the experimental results within their estimated precision, but does not predict any S-shaped curves at the highest pressures. Better agreement between experiment and conformal solution theory is obtained if the properties of the reference fluid are calculated with the Gosman et al. equation of state for The equation was used in reduced form, with the commonly assumed values of the parameters t / k = 119.8 K and a = 0.3405 nm for argon, and c / k = 167.0 K and u = 0.3633 nm for krypton.% The value of k,, was also fixed a priori, k,, = 0.0109, as obtained from Monte Carlo simulation data," but j d was adjusted to reproduce the composition dependence of VE at 134.25 K and a pressure of 2 MPa, since this property is known to be extremely sensitive to the value of jab chosen. A value of ' of 0.005 was obtained by this method. Subsequently HE,and VE were predicted at pressures from zero to 70 MPa. The results for VE are presented in Figure 5, showing good reproduction of the composition dependence at lower pressures which worsens as the pressure is raised. At P = 40 MPa the predicted curve has the wrong shape. This is probably due to the density ( p $ = 0.85 for x = 1) approaching the upper limit of the reference equation of state. The variations of GEand HE with pressure are very well reproduced, as shown in Figure 4, providing a severe test of the theory. Finally, conformal solution theory was used to evaluate HEas a function of temperature, pressure, and composition. At 116.9 K, Lewis et ala6obtained HE(x = 0.5) = 43 J mol-l, while the Lennard-Jones equation of state gives
$&, where the subscripts a and @ refer to molecules of species CY and @, respectively. We obtained k,, = 0.029 and j = -0.0002. The very weak deviation from the arithmetic mean rule for ad is confirmed by Monte Carlo calculations of Singer and Singer.l' The experimental p,V,T results are reproduced within experimental error, except for argon near its vapor pressure and above 129.32 K. In this region, as argon approaches its critical point, the experimental V,p isotherms have increasingly large, negative slopes. On the other hand, (25) J. A. Barker and D. Henderson, Reu. Mod. Phys., 48,587(1976). (26)M. Nunes da Ponte, W. B. Streett, R. C. Miller, and L. A. K. Staveley, J. Chem. Thermodyn. 13, 767 (1981). (27) I. R. McDonald and K. Singer, Mol. Phys., 23, 25 (1972).
(28)I. R.McDonald in 'Statistical Mechanics", Vol. 1, K. Singer,Ed., Chemical Society, London, 1973.
1729
J. Phys. Chem. 1982, 86, 1729-1734
//;,/-----------...\ /,/'
0.021
/,,' _ _ < / - - - - - - -
'\
-----.~ '.'.
''\
1
ever, restored by using the argon equation of state for the reference fluid (maximum at x h = 0.37), although qualitatively the agreement is less good than that obtained with the Lennard-Jones equation of state. In Figure 6, we plot results of our calculation of HEas a function of composition and pressure, at a temperature of 134.32 K using both the Lennard-Jones and Gosman et al. equations of state with k12 = 0.0109. Both equations seem to lead to the same qualitative pattern, and although quantitatively the results are different, several conclusions can be drawn from this figure. (a) HE (x = 0.5) is negative at low pressures. Although no experimental results exist for comparison at this or similar temperatures, HEcan be estimated from a plot of GE/Tas a function of 1/T. Lewis et al.6give a plot of this type,which suggests that HE should become negative above 130 K.
Figure 6. Mdar excess enthalpies HE,as a function of composition, at several pressures and 134.32 K, calculated for argon krypton llquid "s, uslng conformal solution theory (-, argon equation of state: ----, Lennarddones equatlon of state): k , 2 = 0.0109.
+
40 J mol-', with k12 = 0.109. It should be noted, however,
that the calculations predict a fairly symmetric curve for
HE,whereas the experimental results of Lewis et ala6show a markedly asymmetric curve of HE as a function of com-
position, with a maximum at xAr = 0.35 and HE= 47.3 J mol-'. This seems to be a common feature of several theoretical2and ComputeP studies of liquid Ar Kr, which led McDonalds to point out that 'the variation of GE and HE with composition is rather uninteresting, both functions are nearly symmetrical about x1 = x2 = 0.5". This behavior is not borne out by experiment. The asymmetry is, how-
+
(b) The dependence of HEon composition seems to be much more interesting than has been suggested previously. The curves at lower pressures, in Figure 6, exhibit marked asymmetry with higher (less negative) values on the krypton-rich side, as in the experimental results of Lewis et al. However, small changes in pressure (and in the density of the mixtures) can result in dramatic changes in the composition dependence of HE. We may conclude by saying that, although the present work supplies part of the information on argon krypton liquid mixtures, which, following Lewis et a1.,6 was necessary for a complete thermodynamic knowledge of these mixtures, there still remains the need for calorimetric measurements of HEat high temperatures and over wide ranges of composition and pressure. Acknowledgment. This work was partly financed by research grants from NATO and Junta Nacional de Investigacgo Cientifica e TecnolBgica, and in part by a grant from the donor of the Petroleum Research Fund, administered by the American Chemical Society.
+
Temperature Dependence of Molecular Motlon in Smectic Liquid Crystals of Hydrated Sodium 4-( 1'-Heptylnonyl) benzenesulfonate Frank D. Blumt and Wllmer 0. Mlller" Department of Chemkby, Unhwsky of Minnesota, Minneapolis, Minnesota 55455 (Received: February 3, 1981; I n Flnal Form: October 2, 1981)
The surfactant sodium 4-(1'-heptylnony1)benzenesulfonate (SHBS), upon hydration to form a smectic liquid crystalline phase, exhibits a thermal transition of 1.62 f 0.22 kcal per mole of SHBS centered at -70 "C. From measurements of the temperature dependence of the carbon-13NMR spectra and from the calorimetric studies, transition was identified as the freezing-in of motion of the aliphatic chains and can thus be labeled as the so-called gel-liquid crystal transition observed in phospholipids. The enthalpy and the entropy associated with the transition correspond to a change of approximatelyone trans-gauche rotation per hydrocarbon tail. A more intense, broad thermal transition with a heat of 15 f 3 cal per gram of smectic phase is observed from ca. -10 to -50 "C and shown to come primarily from the freezing-in of bilayer water. At ca. -23 "C water associated with the ionic groups, as well as the motion of the SHBS head, becomes frozen-in. Several comparisonsbetween SHBS and similar studies on synthetic phosphatidylcholine are made. Introduction Carbon-13 nuclear magnetic resonance spectroscopy (NMR) and differential scanning calorimetry (DSC) have
been important tools in the study of biological membranes. Proton-decoupled, natural-abundance 13C NMR spectra have been reported for naturally occurring and model membrane Spectral line widths can be cor-
?Departmentof Chemistry, Drexel University, Philadelphia, PA 19104.
(1) E. Oldfield and D. Chapman, Biochem. Biophys. Res. Commun., 43,949 (1971).
0022-3654/82/2086-1729$01.25/0
0 1982 American Chemical Society
