Correlations for Second and Third Virial Coefficients

Correlations for second and third virial coefficients of pure fluids

Contents

Abstract Introduction Second virial coefficient Third virial coefficient Discussion Conclusions Acknowledgements References

Abstract

The new correlation Tsonopoulos to the second virial coefficient was developed based on the corresponding-states principle.

The present correlation is more accurate, reliable and satisfactory for nonpolar compounds.

The new correlation Weber can accurately represent the literature data for the third virial coefficients of polar fluids.

Both correlations for the second and third virial coefficient need additional parameters as the critical temperature, critical pressure, acentric factor and low dipole moment.

Introduction

The virial coefficients are basic thermodynamic properties that represent the nonideal behavior of real gases.

The second virial coefficient represents the departure from ideality due to interactions between pairs of molecules.

The third virial coefficient gives the effects of interactions of molecular triplets.

The fourth virial coefficient usually contribute little to the densities of gases and have relatively large uncertainties.

Introduction

Virial coefficients are usually derived from experimental measurements such as :

PVT measurements.

Speed of sound measurements.

Joule–Thomson measurements.

Refractive index and relative permittivity measurements.

Vapor pressure and enthalpy of vaporization measurements.

Introduction

The new correlation was developed to enhance the prediction accuracy and reliability based on the corresponding-states form, which accounts for nonspherical effects and dipole moment effects, while neglecting chemical associations and quantum effects.

For the third virial coefficients reliable data is scarce and for accurate results, require corrections to the assumption of pairwise additivity.

In this work we have taken the limited amount of experimental third virial coefficients now available and correlated them as best as we can.

Second virial coefficient

For the gaseous second virial coefficient data, Tsonopoulos modified the Pitzer–Curl equation to give the following widely used expression:

= = f (0)() + ωf (1)() + f (2)() (1)

where,

f (0) = 0.1445 − − − − (2a)

f (1) = 0.0637 + − − (2b)

f (2) =

(3)


Eq. (3) is very important at reduced temperatures less than unity for representing the behavior of polar fluids.

The parameter a is proposed to be a function of the reduced dipole moment μr defined as follows:

μr = (4)

The second virial coefficients of polar compounds are more negative than those of nonpolar compounds (for the same Tr and ω). Thus, a should be negative. However, a usually becomes slightly positive for μr < 100.


For the halogenated methanes and ethanes of interest in the refrigeration industry, Tsonopoulos and Weber gave the following expressions:

a Tsonopoulos = −2.188 × μr4 − 7.831 × μr

8 (5)

a Weber = −9 × μr2 (6)

Tsonopoulos also gave

a Tsonopoulos = −2.140 × μr − 4.308 × μr8 (7)

for other nonhydrogen bonding polar compounds.


Eq. (1) is also used in this work.

For nonpolar fluids, Eqs. (2a) and (2b) are redefined here as

f (0) = 0.13356 − − − − (8)

f (1) = 0.17404 − + − − (9)


Fig. 1. Br = f (0)(Tr): (- - -) Tsonopoulos [3]; (· · ·) Pitzer–Curl [1]; (—)present work.

Fig.2. Deviations of measured second virial coefficient data for argon (ω =−0.0022) from the present correlation: () Gilgen et al. [21]; (O) Estrada-Alexanders and Trusler [22]; (- - -) Tsonopoulos [3]; (· · ·) Pitzer–Curl [1]; (-·-·-) Weber [7].


Fig. 3. Deviations of measured second virial coefficients for methane from the present correlation: () Roe [25]; () Haendel et al. [26]; () Trusler [27]; () Michels et al. [28]; () Trappeniers et al. [29]; () Hou et al. [30]; () Douslin et al. [31]; () Holleran [32]; (- - -) Tsonopoulos [3]; (· · ·) Pitzer–Curl [1]; (-·-·-) Weber [7].

Fig. 4. Deviations of measured second virial coefficients for ethane from the present correlation: () Estrada-Alexanders and Trusler [33]; () Douslin and Harrison [34]; () Pompe and Spurling [35]; () Jaeschke [36]; (- - -) Tsonopoulos [3]; (· · ·) Pitzer–Curl [1]; (-·-·-) Weber [7].


Fig. 5. Deviations of measured second virial coefficients for butane from the present correlation: () Gupta and Eubank [24]; (- - -) Tsonopoulos [3]; (· · ·) Pitzer–Curl [1]; (-·-·-) Weber [7].

Fig. 6. Deviations of measured second virial coefficients for benzene from the present correlation: () Sherwood and Prausnitz [37]; () Waelbroeck [38]; () Bich et al. [39]; () Bich et al. [40]; () Connolly and Kandalic [41]; (- - -) Tsonopoulos [3]; (· · ·) Pitzer–Curl [1]; (-·-·-) Weber [7].

Substance μr Optimum a RMSD (cm3 mol-1) Present work Tsonopoulos Weber

R11 3.97 0.00614 33.4 35.3 28.2 R12 7.16 0.00171 10.2 16.9 8.9 R13 10.92 0.00856 4.7 6.0 4.2 R22 76.76 −0.00469 8.3 11.9 14.2 R23 144.77 −0.01469 4.9 3.7 3.6 R32 180.95 −0.02586 7.8 5.4 7.9 R40 136.80 −0.01053 10.7 10.2 13.0 R41 198.08 −0.05129 5.7 6.3 8.9

R114 7.93 0.00264 23.3 35.4 18.3 R115 6.68 0.01404 11.0 13.5 12.2

R141b 77.50 −0.00132 54.8 37.0 7.7 R142b 109.29 −0.00452 22.6 15.3 21.4 R123 31.84 −0.00091 6.5 22.8 76.2 R124 49.36 −0.00069 10.8 5.7 21.3 R125 75.82 0.00069 7.7 4.5 4.5

R134a 121.17 −0.00740 3.0 6.2 3.7 R143a 169.91 −0.01703 6.4 14.9 11.4 R152a 152.76 −0.01661 22.0 9.9 7.3

R227ea 43.58 0.00245 12.5 7.3 7.1 R236ea 25.90 −0.00078 21.6 25.7 28.1

Propanone 149.03 −0.03410 38.1 35.3 2-Butanone 111.29 −0.02313 65.5 74.1

2-Pentanone 84.42 −0.01803 163.6 142.8 3-Pentanone 92.84 −0.01308 65.9 76.9

Dimethyl ether 55.98 −0.01752 64.2 54.2 Diethyl ether 21.80 −0.00449 63.8 44.7

Diisopropyl ether 14.26 −0.00277 68.7 71.0 Ethanol 191.44 −0.04482 38.0 36.7

Acetonitrile 249.48 −0.12116 175.5 178.5


Fig. 7. Dependence of the polar parameter a on the reduced dipole moment μr: (—) Eq. (11); (- - -) Eq. (12); () optimum values for haloalkanes; () optimum values for other nonhydrogen bonding polar fluids.

Fig. 8. Deviations of measured second virial coefficients for R22 from the present correlation: () Zander [42]; () Lisal et al. [43]; () Schramm and Weber [44]; () Natour et al. [45]; () Schramm et al. [46]; () Demiriz et al. [47]; () Haendel et al. [48]; () Haworth and Sutton [49]; () Esper et al. [50]; (- - -) Tsonopoulos [3]; (-·-·-) Weber [7].


Fig. 9. Deviations of measured second virial coefficients for 32 from the present correlation: () Qian et al. [51]; () Sato et al. [52]; () Defibaugh et al. [53]; () Sun et al. [54]; ()Weber and Goodwin [55]; (- - -) Tsonopoulos [3]; (-·-·-) Weber [7].

Fig. 10. Deviations of measured second virial coefficients for R134a from the present correlation: () Schramm et al. [46]; () Tillner-Roth and Baehr [56]; () Qian et al. [57]; () Bignell and Dunlop [58]; () Goodwin and Moldover [59]; () Beckermann and Kohler [60]; () Weber [61]; (- - -) Tsonopoulos [3]; (-·-·-) Weber [7].


Fig. 11. Deviations of measured second virial coefficients for R123 from the present correlation: () Schramm and Weber [44]; () Goodwin and Moldover [62]; (—) Weber’s equation [63]; (- - -) Tsonopoulos [3]; (-·-·-) Weber [7].

The results for 28 kinds of polar molecules listed in Table 1 show that in general, all three correlations are roughly equivalent for polar haloalkanes.

And both this work and Tsonopoulos correlation have large errors in calculating the data for other nonhydrogen bonding polar molecules.

So, this work does not give significantly improved predictions for the second virial coefficients of polar fluids due to the weakness of Eq. (3).

Third virial coefficient

The Van Nhu model is often considered to be better than most for correlating the experimental data for the third virial coefficients.

Weber successfully simplified the model using the critical volume as a parameter:

C = + ϑ () (13)

where = b, = 0.625, b = 0.36, is a function of and ϑ () is a strong functions of , and is the critical volume.


As virial equation may be truncated after the second or after the third virial coefficient, and are substituted into Eq. (13) and both sides of the equation are multiplied by (, so:

= 0.081 + ϑ () (14) where, = B ( = C ( = /R In Eq. (14), = 0.26. Thus, the critical volume parameter is not

needed.


Therefore, the new correlation representing the third virial coefficients for nonpolar gases and haloalkanes is

Cr = c0 + (Br − c1)2 [f0 (Tr) + f1 (Tr)] (15)

where c0 = 5.476 × 10−3, c1 = 0.0936 ;f0 (Tr) = 1094.051 − + − (16)

f1 (Tr) = ( 2.0243 − ) × 10−10 (17)


The first term, f0 (Tr), was determined by fitting the C data for argon, carbon dioxide, methane, nitrogen and benzene, which have zero reduced dipole

moments.

Noted that f0 (Tr) is equal to 0.174 when the reduced temperature is equal to unity.



Weber suggested that the polar term should be correlated using the cube of the reduced dipole moment, however, we got better results with the quartic of the reduced dipole moment which is similar to the case of Van Nhu.


f1 (Tr) was determined by fitting the data for some haloalkanes, such as R134a, R143a, R152a, R32 and R23.

Although f1 (Tr) was somewhat arbitrary, the results show that it accurately represents the experimental data of haloalkanes.





Discussion

The importance of the virial coefficients lies in the fact that they are very useful for representing the PVT behavior of real gases at low densities.

The virial equation of state, truncated after the third virial coefficient, can provide a very good fit to precise PVT data for densities up to about 0.5 ρc for nonpolar gases.For polar gases, this maximum density decreases to about 0.25 ρc or even lower.

The accuracies of the present correlation and the Weber correlation for the nonpolar gases argon and nitrogen and polar gas R134a are shown in Figs. 24–26.

Discussion

Discussion

Discussion

Discussion

For argon, nitrogen and R134a the present correlation is better than Weber’s in representing the gas-phase densities over the whole range.

Estimates for the density error for the nonpolar gases methane, ethane, propane, butane, carbon dioxide and oxygen and the polar gases R143a, R125a, R32 and R22 are shown in Table 2.

Discussion

For most nonpolar fluids and polar haloalkanes, the present correlation is roughly equivalent to the Weber correlation in predicting the gas-phase nonideality.

The Weber prediction is often slightly better than this work near the critical temperature and worse at supercritical temperatures.

Conclusions

A modified correlation was developed for the second and third virial coefficients of nonpolar and polar fluids.

For the second virial coefficients, the simple spherical term f (0) and the nonpolar term f (1) very successfully represent the best available data within the experimental imprecision for Tr = 0.5–6 or higher.

For the third virial coefficients, Eq. (13) gives an implicit dependence on ω through the second virial coefficient, so the reasonable form of the empirical correlation should be

= f (0)(Tr ) + f (1) (Tr) + f (2) (Tr , μr ) (18)

Eq. (18) accurately represents nonpolar fluids; however it does notwork well for polar fluids, which was the same as the correlation for B.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 50225622) and the Fok Ying Tung Education Foundation (No. 81051).

References

[1] K.S. Pitzer, R.F. Curl Jr., J. Am. Chem. Soc. 79 (1957) 2369–2370.

[2] J.P. O’Connell, J.M. Prausnitz, Ind. Eng. Chem. Process. Des. Dev. 6 (1967) 245–306.

[3] C. Tsonopoulos, AIChE J. 20 (1974) 263–272. [4] C. Tsonopoulos, AIChE J. 21 (1975) 827–829. [5] R.R. Tarakad, R.P. Danner, AIChE J. 23 (1977) 685–

695. [6] H.A. Orbey, Chem. Eng. Commun. 65 (1988) 1–19. [7] L.A. Weber, Int. J. Thermophys. 15 (1994) 461–

482. [8] J.G. Hayden, J.P. O’Connell, Ind. Eng. Chem. Proc.

Des. Dev. 14 (1975) 209–216. [9] J.H. Dymond, K.N. Marsh, R.C. Wilhoit, K.C. Wong,

Virial coefficients of pure gases and mixtures, vol. A, in: Virial Coefficients of Pure Gases, Landolt-B¨ornstein, 2002.

[10] J.S. Rowlinson, J. Chem. Phys. 19 (1951) 827–831.

[11] W.H. Stockmayer, J. Chem. Phys. 9 (1941) 398–402.

[12] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory ofGases and Liquids, Wiley, New York, 1954.

[13] P.L. Chueh, J.M. Prausnitz, AIChE J. 12 (1967) 896–902.

[14] R.D. Santis, B. Grande, AIChE J. 25 (1979) 931–937.

[15] A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses, Wiley, New York, 1968.

[16] G.A. Pope, P.S. Chappelear, R. Kobayashi, J. Chem. Phys. 59

(1973) 423–434. [17] H. Orbey, J.H. Vera, AIChE J. 29 (1983) 107–113. [18] N. Van Nhu, G.A. Iglesias-Silva, F. Kohler, Ber.

Bunsenges. Phys. Chem. 93 (1989) 526–531. [19] M.A. Bosse, R. Reich, Chem. Eng. Commun. 66

(1988) 83–99. [20] E.M. Besher, J. Lielmezs, Thermochim. Acta 200

(1992) 1–13. [21] R. Gilgen, R. Kleinrahm, W. Wagner, J. Chem.

Thermodyn. 26 (1994) 383–398. [22] A.F. Estrada-Alexanders, J.P.M. Trusler, J. Chem.

Thermodyn. 27 (1995) 1075–1089. [23] R.A. Aziz, M.J. Slaman, J. Chem. Phys. 92 (1990)

1030–1035. [24] D. Gupta, P.T. Eubank, J. Chem. Eng. Data 42

(1997) 961–970. [25] D.R. Roe, Thermodynamic properties of gases

and gas mixtures at low temperatures and high pressures. Ph.D. Thesis,

University of London, London, England, 1972. [26] G. Haendel, R. Kleinrahm, W. Wagner, J. Chem.

Thermodyn. 24 (1992) 685–695. [27] J.P.M. Trusler, J. Chem. Thermodyn. 26 (1994)

751–763. [28] J.P.J. Michels, J.A. Schouten, M. Jaeschke, Int. J.

Thermophys. 9 (1988) 985–992. [29] N.J. Trappeniers, T. Wassenaar, J.C. Abels,

Physica A 98 (1979) 289–297. [30] H. Hou, J.C. Holste, K.R. Hall, K.N. Marsh, B.E.

Gammon, J. Chem. Eng. Data 41 (1996) 344–353.

References

[31] D.R. Douslin, R.H. Harrison, R.T. Moore, J.P. McCullough, J. Chem. Eng. Data 9 (1964) 358–363. [32] E.M. Holleran, J. Chem. Thermodyn. 2 (1970) 779–786. [33] A.F. Estrada-Alexanders, J.P.M. Trusler, J. Chem.

Thermodyn. 29 (1997) 991–1015. [34] D.R. Douslin, R.H. Harrison, J. Chem. Thermodyn. 5 (1973) 491–512. [35] A. Pompe, T.H. Spurling, Commonwealth Scientific and

Indust. Res. Org. Div. of App. Organic Chemistry Technical Paper No.

1, CSIRO, Melbourne, Australia, 1974. [36] M. Jaeschke, Int. J. Thermophys. 8 (1987) 81–95. [37] A.E. Sherwood, J.M. Prausnitz, J. Chem. Phys. 41 (1964)

429– 437. [38] F.G. Waelbroeck, J. Chem. Phys. 23 (1955) 749–750. [39] E. Bich, H. Hendl, T. Lober, J. Millat, Fluid Phase Equilib.

76 (1992) 199–211. [40] E. Bich, G. Opel, R. Pietsch, E. Vogel, Z. Phys. Chem.

(Leipzig) 260 (1979) 1145–1159. [41] J.F. Connolly, G.A. Kandalic, Phys. Fluids 3 (1960) 463–

467. [42] M. Zander, Proc. Symp. Thermophys. Prop., vol. 4, ASME,

New York, 1968, pp. 114–123. [43] M. Lisal, K. Watanabe, V. Vacek, Int. J. Thermophys. 17

(1996) 1269–1280. [44] B. Schramm, C.H. Weber, J. Chem. Thermodyn. 23 (1991) 281–292. [45] G. Natour, H. Schuhmacher, B. Schramm, Fluid Phase

Equilib. 49 (1989) 67–74. [46] B. Schramm, J. Hauck, L. Kern, Ber. Bunsenges. Phys.

Chem. 96 (1992) 745–750. [47] A.M. Demiriz, R. Kohlen, C. Koopmann, D. Moeller, P.

Sauermann, G.A. Iglesias-Silva, F. Kohler, Fluid Phase Equilib. 84 (1993) 313–333. [48] G. Haendel, R. Kleinrahm, W. Wagner, J. Chem.

Thermodyn. 24 (1992) 697–713. [49] W.S. Haworth, L.E. Sutton, Trans. Faraday. Soc. 67 (1971) 2907–2914.

[50] G. Esper, W. Lemming, W. Beckermann, F. Kohler, Fluid Phase

Equilib. 105 (1995) 173–192. [51] Z.Y. Qian, A. Nishimura, H. Sato, K. Watanabe, JSME Int. J.

36 (1993) 665–670. [52] T. Sato, H. Sato, K. Watanabe, J. Chem. Eng. Data 39

(1994) 851–854. [53] D.R. Defibaugh, G. Morrison, L.A. Weber, J. Chem. Eng.

Data 39 (1994) 333–340. [54] L.Q. Sun, Y.-Y. Duan, L. Shi, M.S. Zhu, L.Z. Han, J. Chem.

Eng. Data 42 (1997) 795–799. [55] L.A. Weber, A.R.H. Goodwin, J. Chem. Eng. Data 38 (1993) 254–256. [56] R. Tillner-Roth, H.D. Baehr, J. Chem. Thermodyn. 24

(1992) 413–424. [57] Z.Y. Qian, H. Sato, K. Watanabe, Fluid Phase Equilib. 78

(1992) 323–329. [58] C.M. Bignell, P.J. Dunlop, J. Chem. Phys. 98 (1993) 4889–

4891. [59] A.R.H. Goodwin, M.R. Moldover, J. Chem. Phys. 93 (1990) 2741–2753. [60] W. Beckermann, F. Kohler, Int. J. Thermophys. 16 (1995)

455–464. [61] F. Weber, PTB-Mitt 106 (1996) 17–23. [62] A.R.H. Goodwin, M.R. Moldover, J. Chem. Phys. 95 (1991) 5236–5242. [63] L.A. Weber, J. Chem. Eng. Data 35 (1990) 237–240. [64] L. Holborn, J. Otto, Z. Phys. 33 (1925) 1–11. [65] A. Michels, H. Wijker, H.K. Wijker, Physica 15 (1949) 627–

633. [66] E. Whalley, Y. Lupien, W.G. Schneider, Can. J. Chem. 31

(1953) 722–733. [67] N.K. Kalfoglou, J.G. Miller, J. Phys. Chem. 71 (1967) 1256–

1264. [68] C.C. Tanner, I. Masson, Proc. R. Soc. London, Ser. A 126

(1930) 268–288. [69] R.W. Crain, R.E. Sonntag, Adv. Cryog. Eng. 11 (1966) 379–

390. [70] A. Michels, J.M.H.L. Sengers, W. de Graaff, Physica 24

(1958) 659–671.

References

[71] A.F. Estrada-Alexanders, J.P.M. Trusler, Int. J. Thermophys. 17

(1996) 1325–1347. [72] W. Zhang, J.A. Schouten, H.M. Hinze, M. Jaeschke, J.

Chem. Eng. Data 37 (1992) 114–119. [73] A. Michels, H. Wouters, J. de Boer, Physica 1 (1934)

587–594. [74] A. Michels, R.J. Lunbeck, G.J. Wolkers, Physica 17

(1951) 801–816. [75] F.B. Canfield, T.W. Leland, R. Kobayashi, Adv. Cryog.

Eng. 8 (1963) 146–157. [76] A.E. Hoover, F.B. Canfield, R. Kobayashi, T.W.

Leland, J. Chem. Eng. Data 9 (1964) 568–573. [77] W. Duschek, R. Kleinrahm, W. Wagner, M. Jaeschke,

J. Chem. Thermodyn. 20 (1988) 1069–1077. [78] J. Otto, A. Michels, H. Wouters, Phys. Z. 35 (1934)

97–100. [79] M.R. Patel, L.L. Joffrion, P.T. Eubank, AIChE J. 34

(1988) 1229–1232. [80] M.P. Vukalovich, Y.F. Masalov, Teploenergetika

(Moscow) 13 (1966) 58–62. [81] S. Glowka, Pol. J. Chem. 64 (1990) 699–709. [82] W.C. Pfefferle, J.A. Goff, J.G. Miller, J. Chem. Phys. 23

(1955) 509–513. [83] J.C. Holste, M.Q. Watson, M.T. Bellomy, P.T. Eubank,

K.R. Hall, AIChE J. 26 (1980) 954–964. [84] E.G. Butcher, R.S. Dadson, Proc. R. Soc. London,

Ser. A 277 (1964) 448–467. [85] W. Duschek, R. Kleinrahm, W. Wagner, J. Chem.

Thermodyn. 22 (1990) 827–840. [86] T. Katayama, K. Ohgaki, H. Ohmori, J. Chem. Eng.

Jpn. 13 (1980) 257–262. [87] A. Michels, C. Michels, Proc. R. Soc. London, Ser. A

153 (1935) 201–224. [88] J.C. Holste, K.R. Hall, P.T. Eubank, G. Esper, M.Q.

Watson, W. Warowny, D.M. Bailey, J.G. Young, M.T. Bellomy, J. Chem.

Thermodyn. 19 (1987) 1233–1250. [89] R. Kleinrahm, W. Duschek, W. Wagner, M. Jaeschke,

J. Chem. Thermodyn. 20 (1988) 621–631. [90] A. Michels, G.W. Nederbragt, Physica 3 (1936) 569–

577. [91] J.P.M. Trusler, W.A. Wakeham, M.P. Zarari, Int. J.

Thermophys. 17 (1996) 35–42. [92] A.E. Hoover, I. Nagata, T.W. Leland, R. Kobayashi, J.

Chem. Phys. 48 (1968) 2633–2647. [93] H.W. Schamp, E.A. Mason, A.C.B. Richardson, A.

Altman, Phys. Flu ids 1 (1958) 329–337. [94] N.I. Timoshenko, E.P. Kholodov, A.L. Yamnov, T.A.

Tatarinova, in: V.A. Rabinovich (Ed.), Teplofiz. Svoistva Veshchestv

Mater., vol. 8, Standards Publ., Moscow, 1975, pp. 17–39. [95] D.S. Rasskazov, E.K. Petrov, G.A. Spiridonov, E.R.

Ushmajkin, in: V.A. Rabinovich (Ed.), Teplofiz. Svoistva Veshchestv

Mater., vol. 8, Standards Publ., Moscow, 1975, pp. 4–16. [96] H.B. Lange, F.P. Stein, J. Chem. Eng. Data 15 (1970)

56–61. [97] A. Yokozeki, H. Sato, K. Watanabe, Int. J.

Thermophys. 19 (1998) 89–127. [98] A.P. Kuznetsov, L.V. Los, Kholod. Tekh. Tekhnol. 14

(1972) 64– 66.

References

[99] H.L. Zhang, H. Sato, K. Watanabe, J. Chem. Eng. Data 41 (1996)

1401–1408. [100] Y.D. Fu, L.Z. Han, M.S. Zhu, Fluid

Phase Equilib. 111 (1995) 273–286. [101] L.A. Weber, D.R. Defibaugh, J.

Chem. Eng. Data 41 (1996) 1477–1480. [102] S. Nakamura, K. Fujiwara, M.

Noguchi, J. Chem. Eng. Data 42 (1997) 334–338. [103] K.A. Gillis, Int. J. Thermophys. 18

(1997) 73–135. [104] T. Tamatsu, T. Sato, H. Sato, K.

Watanabe, Int. J. Thermophys. 13 (1992) 985–997. [105] F. Ye, H. Sato, K. Watanabe, J.

Chem. Eng. Data 40 (1995) 148–152. [106] S.J. Boyes, L.A. Weber, J. Chem.

Thermodyn. 27 (1995) 163–174. [107] H.A. Duarte-Garza, C.E. Stouffer,

K.R. Hall, J.C. Holste, K.N. Marsh, B.E. Gammon, J. Chem. Eng.

Data 42 (1997) 745–753. [108] S.J. Boyes, L.A. Weber, Int. J.

Thermophys. 15 (1994) 443–460. [109] G. Adam, B. Schramm, Ber.

Bunsenges. Phys. Chem. 81 (1977) 442–444. [110] Ch. Tegeler, R. Span, W. Wagner, J.

Phys. Chem. Ref. Data 28 (1999) 779–850. [111] R. Span, E.W. Lemmon, R.T.

Jacobsen, W. Wagner, A. Yokozeki, J. Phys. Chem. Ref. Data 29 (2000)

1361–1433. [112] R. Tillner-Roth, H.D. Baehr, J. Phys.

Chem. Ref. Data 23 (1994) 657–729.

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Correlations for Second and Third Virial Coefficients