The Second Virial Coefficient and the Redlich−Kwong Equation

The Second Virial Coefficient and the Redlich−Kwong Equation. Paul M. Mathias*. Aspen Technology, Inc., 10 Canal Park, Cambridge, Massachusetts 0214...
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Ind. Eng. Chem. Res. 2003, 42, 7037-7044

7037

The Second Virial Coefficient and the Redlich-Kwong Equation Paul M. Mathias* Aspen Technology, Inc., 10 Canal Park, Cambridge, Massachusetts 02141-2201

A two-parameter corresponding-states correlation, based upon the Redlich-Kwong equation, is proposed for the second virial coefficient at relatively high temperature (reduced temperature above 0.8). This remarkably simple correlation suggests an improvement to the venerable Redlich-Kwong equation of state, gives guidance for the high-temperature extrapolation of R-functions, and provides an independent tool to evaluate theoretical and empirical correlations for the second virial coefficient. RTc b ) 0.08664035 Pc

(2)

R2Tc2 ac ) 0.42748025 Pc

(3)

Introduction Redlich and Kwong were concerned about the limiting behavior of their equation of state (EoS).1 They sought an accurate gas-phase model at low density and high density and hence paid special attention to the second virial coefficient. This emphasis on the second virial coefficient was distracted by the success of the proposal of Wilson2 and Soave,3 which recommended fitting the EoS directly to vapor pressures without attention to the second virial coefficient, PVT properties, or other absolute thermodynamic properties. The focus on the second virial coefficient was partially revived by the densitydependent EoS mixing rules of Mollerup4 and Whiting and Prausnitz,5 and the Wong-Sandler6 mixing rules; these researchers developed EoS models that followed the theoretically correct quadratic mole fraction dependence of the second virial coefficient, but they paid little attention to the quantitative accuracy of the second virial coefficients. In this work, we revisit the approach of Redlich and Kwong1 by seeking a simple model that provides an accurate description of the second virial coefficient in the high-temperature region. It turns out that a twoparameter corresponding-states model provides an accurate prediction of the second virial coefficients of nearly all substances, from nonpolar alkanes to polar molecules to hydrogen-bonding substances such as water and methanol, as long as the range of the model is limited to relatively high temperatures (reduced temperatures above 0.8). This simple model suggests an improved Redlich-Kwong EoS, provides guidance for the high-temperature extrapolation of R-functions, and provides an independent tool to evaluate experimental data and theoretical and empirical correlations for the second virial coefficient. The Redlich-Kwong EoS and the Second Virial Coefficient The Redlich-Kwong EoS relates the pressure P of a fluid to its molar volume v and the absolute temperature T.

P)

acR(T) RT v - b v(v + b)

(1)

* Tel: (617) 949-1727. Fax: (617) 949-1030. E-mail: Paul. [email protected].

R(T) )

1 TeR

(4)

R is the gas constant, Tc and Pc are the critical temperature and critical pressure of the fluid, respectively, and TR is the reduced temperature (TR ) T/Tc). The R-function (eq 4) for the Redlich-Kwong equation has been generalized by introducing the exponent e. In the original RK EoS, e ) 0.5, but in this work, we show that a better value for most substances is e ) 1.0. The virial equation is a power series for the compressibility factor in the reciprocal molar volume, 1/v.

z)

B 2 B3 Pv )1+ + 2 + ... RT v v

(5)

z is the compressibility factor, B2 is the second virial coefficient, and B3 is the third virial coefficient. Following Pitzer and Curl7 and Tsonopoulos,8 we develop a correlation for the reduced second virial coefficient in terms of Tc and Pc, which, according to the RK EoS, is as follows:

B2Pc 0.42748025 ) 0.08664035 RTc T 1+e

(6)

R

According to the RK EoS, the reduced second virial coefficient at the critical temperature has a universal value for all substances, which is equal to -0.34084. The purpose of this paper is to report that eq 6 provides a surprisingly accurate correlation for B2 of most substances for reduced temperatures above about 0.8 if e is changed from the original RK value of 0.5 to the value of 1.0. As a historical note, we point out that the model presented by Clausius9 used an equivalent exponent of e ) 1. Another historical note is that a correlation quantitatively equivalent to eq 6 was proposed over 100 years ago by Berthelot.10

B2Pc 54/128 ) 9/128 RTc T 2

10.1021/ie0340037 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/12/2003

R

(7)

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Figure 1. Comparison between DIPPR correlations12 for the reduced second virial coefficient of various substances and the predictions of RK EoS (e ) 0.5), - - -; RK EoS (e ) 1.0), s; and PR EoS (e ) 1), - -. The error bars for the RK EoS (e ) 1.0) show (0.05 in the reduced second virial coefficient.

Figure 2. Comparison between DIPPR correlations12 for the reduced second virial coefficient of n-alkanes and the predictions of RK EoS (e ) 1.0), s; SRK (ω ) 0.0), - -; and SRK (ω ) 1.0), - - -.

The modified RK EoS (e ) 1.0) is a better choice than the original model if the EoS is used as the vapor-phase EoS together with activity-coefficient models. In addition, we demonstrate that the simple correlation provides guidance for the high-temperature extrapolation of R-functions and is reliable enough to question or analyze correlations that significantly disagree with it, thus serving as an independent tool for theoretical and correlation analysis. In the remaining sections of this paper, the proposed correlation is first evaluated broadly against data for various classes of compounds, and is then compared in detail with data for three classes of compounds: nonpolar, polar, and hydrogen-bonding polar. The experimental data and empirical correlations for B2 have been taken from the DIPPR 801 Project,11 and specifically, the data and correlations of DIADEM Version 1.512 have been used.

Optimum Value for e Figure 1 presents comparisons between data for the reduced second virial coefficients of a wide variety of substances and two forms of the RK EoS, with e ) 0.5 (original proposal1) and e ) 1.0. The data points are the evaluated DIPPR correlations presented in DIADEM Version 1.5.12 Examination of Figure 1 shows that the reduced second virial coefficient at the critical temperature (TR ) 1) is, on average, equal to about -0.34, which is the prediction of the RK EoS. In addition, while the RK EoS with e ) 0.5 provides a reasonable prediction of the reduced second virial coefficient over the temperature range of interest, the RK EoS with e ) 1.0 provides an improved prediction for a broad range of species. The equation analogous to eq 6 for the PengRobinson13 EoS is as follows:

Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 7039

Figure 3. Second virial coefficient of helium. Comparison of the RK EoS (e ) 0.0), Soave (ω ) -0.-39003) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

Figure 5. Second virial coefficient of neon. Comparison of the RK EoS (e ) 0.5), Soave (ω ) 0.0) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

beyond the critical temperature or to use an extrapolation function that smoothly goes to zero at high reduced temperatures.15 Past practices have ignored the useful insight and guidance of the second virial coefficient. Evaluation for Nonpolar Substances Figure 2 evaluates the predictions of the RK EoS (e ) 1.0) against the DIPPR correlations (DIADEM 1.512) for the reduced second virial coefficients of the n-alkanes (methane to n-decane) as representative nonpolar compounds. This figure also evaluates the Soave3 RK (SRK) model with values of the acentric factor (ω) equal to 0.0 (slightly less than methane, ω ) 0.01155) and equal to 0.5 (slightly greater than n-decane, ω ) 0.49233). In the formalism of eqs 1-3, the R-function of the SRK model is as follows.

Figure 4. Second virial coefficient of hydrogen. Comparison of the RK EoS (e ) 0.2), Soave (ω ) -0.-21599) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

B2Pc 0.45724 ) 0.07780 RTc T 1+e

(8)

R

While the PR EoS is usually not written in the form of the original RK EoS, it is instructive to directly compare these two popular models in this way. Evaluation of the predictions of the PR EoS in Figure 1 suggests that it is slightly worse than the RK EoS for the prediction of the high-temperature second virial coefficient, and according to this criterion, the RK EoS is the preferred of these two popular models. Lundgaard and Mollerup14 have demonstrated that the RK EoS gives more accurate high-pressure gasphase fugacities than the PR EoS, and this is another related reason that the RK EoS is the preferred of these two popular models. Figure 1 suggests that eq 4 should be used to guide the extrapolation of R-functions for TR > 1. Previous practice has been to simply use the subcritical model

R(TR) ) [1 + m(1 - TR0.5)]2

(9)

m ) 0.480 + 1.57ω - 0.176ω2

(10)

Analysis of Figure 2 indicates that the RK EoS (e ) 1.0) provides, on average, the best prediction of the hightemperature second virial coefficient. More importantly, it provides guidance for the optimum extrapolation of the R-function to high reduced temperatures. The Wilson2 and Soave3 approach offers no guidance on how to extrapolate the R-function to temperatures above the critical temperature, and we see that the extrapolation of the R-function for substances such as n-decane yields a second virial coefficient that is too positive. We recommend that RK EoS with e ) 1 should be used as this guide. It is interesting that while the subcritical R-function depends on additional parameters such as the acentric factor, the high-temperature R-function seems to be a universal function of the reduced temperature alone. Figures 3-5 (see also Table 1) compare predictions of the RK EoS, the Soave R-function,3 and the DIPPR correlation12 to experimental data for the quantum fluids, helium, hydrogen, and neon. The acentric factors used in the Soave R-function3 are those reported by DIPPR.12 Chueh and Prausnitz16 proposed a special correlation using effective critical constants to capture

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Figure 6. Comparison between DIPPR correlations12 for the reduced second virial coefficient of polar compounds and the predictions of RK EoS (e ) 1.0)

Figure 7. Second virial coefficient of ethyl ether. Comparison of the RK EoS (e ) 1.0) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

the quantum effects of these low-molecular-weight fluids. Figures 3-5 demonstrate that these quantum fluids also follow the universal reduced second virial coefficient at the critical temperature, and that the quantum character is demonstrated by componentspecific values of the exponent e. Helium, with the strongest quantum effects, is best described by e ) 0, while the optimum exponents for hydrogen and neon are 0.2 and 0.5, respectively. The model proposed in this work provides an objective way to extrapolate the R-functions of the quantum fluids to high reduced temperatures. In the case of neon where the lowest temperature data are at TR ) 1.13, the proposed method provides a reliable way to extrapolate the data down to about TR ) 0.8. Evaluation for Polar Substances Figure 6 presents comparisons between the predictions of the RK EoS (e ) 1.0) and the DIPPR correlations

Figure 8. Second virial coefficient of acetonitrile. Comparison of the RK EoS (e ) 1.0) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

(DIADEM 1.512) for the reduced second virial coefficients of representative polar compounds. It is remarkable that the accuracy of the RK EoS (e ) 1.0) is as good for polar compounds as it is for nonpolar compounds (Figure 2). The average reduced second virial coefficient at the critical temperature remains about -0.34. There is a tendency of the predicted second virial coefficient to be biased high as the temperature is decreased, and this is to be expected because the specific interactions of polar compounds will become dominant at lower temperatures. Figure 7 provides an example (ethyl ether) where the simple model proposed here (RK EoS with e ) 1) provides an essentially quantitative prediction of the data. Figure 8 presents a counter example (acetonitrile) where the simple model proposed here (RK EoS with e ) 1) does not capture the experimental data. The model predictions for B2 are significantly higher than the data from several sources, and the agreement among the data sets is good. Prausnitz and Carter17 indicate that

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Figure 9. Comparison between DIPPR correlations12 for the reduced second virial coefficient of hydrogen-bonding polar compounds and the predictions of RK EoS (e ) 1.0). Table 1. References for Experimental Data in Figures 3-5, 7, 8, 10, and 11 substance

DIPPR ref no.

ethyl ether (Figure 7)

1

TRC21

phenol (Figure 10)

2 4 5 6 7 9 10 12 1

TRC21 Lambert et al.22 Strein et al.23 Zaalishvili et al.24 Zaalishvili et al.24 Stryjek26 Knoebel and Edmister27 Olf et al.28 Kudchadker et al.30

2 1 4 1

Opel32 TRC21 Dymond and Smith29 TRC21

2 4

McCarty35 Kestin et al.36

hydrogen (Figure 4) helium (Figure 3)

ref

acetonitrile forms dimers in the vapor phase, and this is probably the reason the proposed simple model fails. One value of the proposed simple model is to identify species where vapor-phase association is significant. Evaluation of Hydrogen-Bonding Substances Figure 9 presents comparisons between the predictions of the RK EoS (e ) 1.0) and DIPPR correlations (DIADEM 1.512) for the reduced second virial coefficients of representative hydrogen-bonding polar compounds. It is even more remarkable that the accuracy of the RK EoS (e ) 1.0) is maintained for hydrogen-bonding polar compounds as it is for nonpolar and polar compounds. In this case, the average reduced second virial coefficient at the critical temperature is slightly above -0.34, but -0.34 is still a good estimate. Two substances show significant differences from the proposed correlation, and we discuss these below. Figure 9 suggests that the model predictions for phenol at high temperatures (TR > 1) are significantly lower than the DIPPR correlation. In Figure 10, the

substance

DIPPR ref no.

acetonitrile (Figure 8)

1

TRC21

2 3 4 6 7 1 2 3 4

TRC21 Lambert et al.22 Prausnitz and Carter17 Olf et al.28 Demiriz et al.25 TRC21 TRC21 Dymond and Smith29 Kretschmer and Wiebe31

5 6 7 2

Knoebel and Edmister27 Wilson et al.33 Smith and Srivastava34 Dymond and Smith29

4 5 6 7

Dymond and Smith29 Dymond and Smith29 Dymond and Smith29 Kestin et al.36

ethanol (Figure 11)

Neon (Figure 5)

ref

proposed correlation and the DIPPR correlation are compared against experimental data. There are no data above the critical temperature of phenol, so both correlations are extrapolations, but the RK-based model should be preferred since it has been validated by agreement with a broad range of substances. Figure 9 suggests that the model predictions for ethanol are low in the region of the critical temperature and high for reduced temperatures above 1.8. Figure 11 provides an assessment of this observation. The DIPPR correlation for ethanol12 is based upon several compilations among which there is some disagreement above TR ) 0.9. There is at least one of these compilations34 that agrees with the prediction of the RK EoS (e ) 1.0), and examination of Figure 9 shows that ethanol does not follow the pattern of the other normal alcohols. We conclude that the DIPPR correlation for the second virial coefficient of ethanol needs to be reassessed. Figures 10 and 11 clearly show that the two-parameter corresponding-states correlation presented here will

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Figure 10. Second virial coefficient of phenol. Comparison of the RK EoS (e ) 1.0) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

Figure 11. Second virial coefficient of ethanol. Comparison of the RK EoS (e ) 1.0) and the DIPPR correlation to experimental data. See Table 1 for details on data references.

be biased high as the temperature decreases. This bias will remain significant at higher reduced temperatures as the polarity and hydrogen-bonding nature of the substance increase. But it is also true that essentially all substances, including polar and hydrogen-bonding molecules, follow the two-parameter correspondingstates correlation at relatively high temperatures (TR > 0.8), and this finding is useful for the advancement of applied thermodynamics. Evaluation of Predictive Methods Figure 12 presents a comparison between the proposed correlation and the DIPPR correlation,12 and the predictions of the McCann18 group-contribution method for butyronitrile. The DIPPR correlation agrees very well with the McCann18 predictions because it was fit on this basis; no experimental data are available for butyronitrile. The McCann group-contribution method predicts that the reduced second virial coefficient of butyronitrile at its critical temperature is -0.68. This is far more negative than the value found for most other

Figure 12. Second virial coefficient of butyronitrile. Comparison of the RK EoS (e ) 1.0) and the DIPPR correlation to groupcontribution predictive method of McCann.18

Figure 13. Second virial coefficient of ethanol. Comparison of the RK EoS (e ) 1.0), the DIPPR correlation, and the Twu R-function20 to experimental data. See Table 1 for details on data references.

substances and is thus subject to doubt. But we should also point out that B2(Tc)Pc/(RTc) for acetonitrile and propionitrile are -0.56 and -0.70, respectively. It would be useful and interesting if new experimental measurements were made on the nitrile family to determine if their reduced second virial coefficients at the critical temperature are indeed so strongly negative. If the experimental studies confirm the existing data, theoretical studies to determine why the specific interactions of this family persist up such high reduced temperatures would also be useful. Evaluation of High-Temperature r-Functions Many researchers have proposed R-functions to improve the correlation of vapor pressures.19 Perhaps the most effective of these correlations is the one proposed by Twu et al.,20 which is shown here.

R(TR) ) TN(M-1) exp{L(1 - TNM R R )}

(11)

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Figure 13 compares the R-function of eq 6 and that of the Twu correlation to experimental data for the second virial coefficient of ethanol. Figure 13 indicates that there is very little observable difference between eqs 6 and 11 for the subcritical R-function (of course, vapor pressure is extremely sensitive to the R-function, so small differences are significant), but the extrapolation from the R-function model of Twu et al.20 gives a poor second virial coefficient at supercritical temperatures. The conclusion of this work is that new R-functions, which will provide both good vaporpressure predictions and accurate correlations for the supercritical second virial coefficients, need to be invented. On the basis of the analysis in this paper, we propose a new model for EoS R-functions

R(TR) )

a1 b1 c1 1 + + + (12) N 2 3 TR a2 + TR b2 + TR c2 + T4R

where N equals 1 for most fluids, and will be less than for quantum fluids; a2, b2, and c2 are positive adjustable constants smaller than unity; and a1, b1, and c1 are adjustable constants subject to the constraint that R(TR) ) 1 when TR ) 1. Limited investigation of eq 12 indicates that it provides an accurate prediction of purecomponent vapor pressures and an accurate description of the supercritical second virial coefficient. Conclusion This work presents a two-parameter correspondingstates correlation for the second virial coefficient that has the following benefits for applied thermodynamics: (i) It suggests an improved R-function for the RedlichKwong EoS, which will be useful when this EoS is used as the vapor-phase EoS together with activity-coefficient models. (ii) The reduced second virial coefficient predicted by the Redlich-Kwong EoS at the critical temperature (-0.341) agrees with data for most substances. This suggests that the RK EoS should be preferred over other popular cubic EoSs such as the Peng-Robinson13 model. (iii) It offers guidance for the high-temperature extrapolation of R-functions based upon the Wilson2 and Soave3 proposal. (iv) It offers a useful independent tool to evaluate and interpret data and models for the second virial coefficient. Literature Cited (1) Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev. 1949, 44, 233. (2) Wilson, G. M. Vapor-Liquid Equilibria Correlated by Means of a Modified Redlich-Kwong Equation of State. Adv. Cryog. Eng. 1964, 9, 168. (3) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (4) Mollerup, J. A Note on the Excess Gibbs Energy Models, Equations of State and Local Composition Concept. Fluid Phase Equilib. 1981, 7, 121. (5) Whiting, W. B.; Prausnitz, J. M. Equations of State for Strongly Non-Ideal Fluid Mixtures: Application of Local Composition Towards Density dependent Mixing Rules. Fluid Phase Equilib. 1982, 9, 119. (6) Wong, D. S.; Sandler, S. I. Theoretically Correct Mixing Rules for Cubic Equations of State. AIChE J. 1992, 38, 671.

(7) Pitzer, K. S.; Curl, R. F., Jr. Empirical Equation for the Second Virial Coefficient. J. Am. Chem. Soc. 1957, 79, 2369. (8) Tsonopoulos, C. An Empirical Correlation of Second Virial Coefficients. AIChE J. 1974, 20, 263. (9) Clausius, R. Ann. Phys. 1880, 9, 337. See description: Partington, J. R. Fundamental Principles and Properties of Gases. An Advanced Treatise on Physical Chemistry; Longmans Publishers: Harlow, U.K., 1949; Vol. 1. (10) See description: Hirschfelder, J. O.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1964; p252. (11) DIPPR Project 801, Design Institute for Physical Properties, http://www.aiche.org/dippr/projects/801.htm. (12) DIADEM Version 1.5, DIPPR Information and Data Evaluation Manager, Copyright BYU-TPL 2000. (13) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (14) Lundgaard, L.; Mollerup, J. M. The Influence of Gas-Phase Fugacity and Solubility on Correlation of Gas-Hydrate Formation Pressure. Fluid Phase Equilib. 1991, 70, 199. (15) Boston, J. F.; Mathias, P. M. Proceedings of the 2nd International Conference, Berlin, Germany, Knapp, H., Sandler, S. I., Eds.; EFCF Publication Series No. 11; DECHEMA: Frankfurt, Germany, 1980. (16) Chueh, P. L.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1967, 6, 492. (17) Prausnitz, J. M.; Carter, W. B. Virial Coefficients of the Acetonitrile-Acetaldehyde System. AIChE J. 1960, 6, 611. (18) McCann, D. W. A Group-Contribution Method for Second Virial Coefficients. M.S. Thesis, The Pennsylvania State University, University Park, Pennsylvania, 1982. (19) Valderrama, J. O. The State of the Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 1603. (20) Twu, C. H.; Bluck, D.; Cunningham, J. R.; Coon, J. E. A Cubic Equation of State with a New Alpha Function and a New Mixing Rule. Fluid Phase Equilib. 1991, 69, 33. (21) Selected Values of the Properties of chemical Compounds, Data Project, Thermodynamics Research Center, Texas A&M University, College Station, Texas (1980-extant); loose-leaf data sheets. (22) Lambert, J. D.; Roberts, G. A. H.; Rowlinson, J. S.; Wilkinson, V. J. Heat Capacity of Methanol Vapors. Proc. R. Soc., Ser. A 1949, 196, 113. (23) Strein, V. K.; Lichtenthaler, R. N.; Schramm, B. Messwerte des ZweitenVirialkoeffizienten eniger gesa¨ttiger Kohlenwasstrstoffe von 300-500 K. Ber. Bunsen-Ges. Phys. Chem. 1971, 17, 1308. (24) Zaalishvili, Sh. D.; Kolysko, L. E. The Second Virial Coefficients of Vapors and Their Mixtures. I. The Ethyl EtherAcetone System. Russ. J. Phys. Chem. 1960, 34, 1223. (25) Demiriz, A. M.; Koopman, C.; Moeller, D.; Sauermann, P.; Igselias-Silva, G. A.; Kohler, F. The Virial Coefficients and Equation of State Behavior of Polar Components Chlorodifluoromethane, Fluoromethane and Ethanenitrile. Fluid Phase Equil. 1993, 85, 313. (26) Stryjek, R. The Second Virial Coefficients of n-Alkyl Formates. Bull. Acad. Pol. Sci., Ser. Sci. Chim. 1966, 14, 307. (27) Knoebel, D. H.; Edmister, J. C. Second Virial Coefficients of Binary Mixtures of Benzene with Methanol, Ethanol, Acetone and Diethyl Ether. J. Chem. Eng. Data. 1968, 13, 312. (28) Olf, G.; Schnitzler, A.; Gaube, J. Virial Coefficients of Binary Mixtures Composed of Polar Substances. Fluid Phase Equilib. 1989, 49, 49. (29) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases, A Critical Compilation; Clarendon Press: Oxford, 1969. (30) Kudchadker, A. P.; Kudchadker, S. A.; Wilhoit, R. C. Key Chemicals Data BookssPhenol; Thermodynamics Research Center, Texas A&M University: College Station, Texas, 1977. (31) Kretschmer, C. B.; Wiebe, R. Pressure-Volume-Temperature Relationships of Alcohol Vapors. J. Am. Chem.. Soc. 1954, 76, 2579. (32) Opel, G. Zweite Virialkoeffizienten von Phenol zwischen 205 und 350 C, Chem. Tech. (Leipzig) 1969, 21, 776.

7044 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 (33) Wilson, K. S.; Lindley, D. D.; Kay, W. B.; Hershey, H. C. Virial Coefficients of Ethanol from 373.07 to 473.15. J. Chem. Eng. Data 1984, 29, 243. (34) Smith, B. D.; Srivastava, R. Data for Pure Compounds. Part B. Halogenated Hydrocarbons and Alcohols; Elsevier: Amsterdam, 1986. (35) McCarty, R. D. Thermodynamic Properties of helium-4 from 2 to 1, 500 K at pressures to 1.0E+08 Pa. J. Phys. Chem. Ref. Data 1973, 2, 923.

(36) Kestin, J.; Knierim, E. A.; Mason, B.; Majafi, S. T.; Ro, S. T.; Wadman, M. Equilibrium and Transport Properties of Noble Gases and Their mixtures at Low Density. J. Phys. Chem. Ref. Data 1984, 13, 229.

Received for review July 15, 2003 Revised manuscript received October 6, 2003 Accepted October 10, 2003 IE0340037