or Correlation of Second Virial

Mar 15, 2001 - We present a simple expression that satisfies known limits for correlating and/or predicting the second virial coefficient. For predict...
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1968

Ind. Eng. Chem. Res. 2001, 40, 1968-1974

GENERAL RESEARCH An Equation for Prediction and/or Correlation of Second Virial Coefficients Gustavo A. Iglesias-Silva Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya, Celaya, Guanajuato CP 38010, Me´ xico

Kenneth R. Hall* Chemical Engineering Department, Texas A&M University, College Station, Texas 77843

We present a simple expression that satisfies known limits for correlating and/or predicting the second virial coefficient. For prediction, we utilize a generalized function that can provide the second virial coefficient of nonpolar and polar substances, as well as for molecules with association and quantum effects. The expression does not require additional terms to encompass these effects. Introduction The virial equation of state (VEOS) can predict accurate thermodynamic properties in the gas phase for many practical applications within the chemical industry, and it has a precise basis in statistical mechanics that allows its use in theoretical developments. Hall et al.1 and Luongo-Ortiz and Starling2 showed that mixing rules for virial coefficients are quadratic under certain assumptions. McGregor et al.3 provided virial solution models. Hall and Iglesias-Silva4 derived the general expression for the fugacity coefficient of a component in a mixture using the virial equation of state. The virial equation of state truncated after the second virial coefficient is

Z)

P ) 1 + BF FRT

(1)

where Z is the compression factor, P is the pressure, T is the temperature, F is the molar density, R is the gas constant (8.31451 J mol-1 K-1), and B is the second virial coefficient. This equation has been used extensively to calculate phase equilibria and thermodynamic properties, and it appears to provide reasonably accurate results up to about one-half of the critical density. For pure substances, the second virial coefficient is a function of temperature, but for mixtures, it is also a function of composition. Statistical mechanics provides this composition dependence involving pure and cross virial coefficients that must be determined experimentally or predicted from correlations. Experimental determination of pure and cross second virial coefficients is time-consuming and expensive. Many correlations for pure and cross second virial coefficients have appeared in the literature. Some use an equation of state principle,5,6 while others use the corresponding states principle with the acentric factor * Corresponding author. Phone: 979 845 3357. Fax: 979 845 6446. E-mail: [email protected].

and the critical properties as characteristic parameters.7-9 The latter group uses the reduced dipole moment and group constants, together with different temperature functions to describe polar and complex substances. The existing correlations predict fairly well pure and cross second virial coefficients; however, they have trouble with low temperatures and strongly polar and/ or associating substances. The Thermodynamics Research Center (TRC has relocated to NIST in Boulder, CO) has compiled an exhaustive database of second virial coefficients and requires a correlation/predictor to fit these data. Of course, TRC could use a different function for each substance, but it is vastly preferable to have a single function for all substances. Unfortunately, none of the existing formulations could serve as this function within the accuracy required. From time to time, TRC also receives requests for values in regions where data do not exist (e.g., very high-temperature requests from astrophysicists). As a result, we have developed an equation that can predict or correlate second virial coefficients within TRC requirements. This equation is a function that uses the Boyle temperature as a normalizing parameter and handles complex molecules within experimental error using the same equation as for simple molecules. Development In corresponding states theory, the reduced second virial coefficient is a summation of several contributions including a term for nonpolar, spherical molecules; another term for nonpolar, nonspherical molecules; another term for polar molecules; plus additional terms as needed. In practical correlations, the critical properties (TC, the critical temperature, usually in Kelvin; and PC, the critical pressure, usually in bar), the acentric factor (ω), and the reduced dipole moment (105 µPC/TC2) appear as input parameters. Most authors employ the usual dimensionless variables; we use the Boyle temperature (the temperature at which the second virial coefficient becomes zero) as the normalizing variable for

10.1021/ie0006817 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/15/2001

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1969

temperature.

b1 )

B ) f(T) ) 0 when T ) TB

(2)

The gas behaves more nearly like an ideal gas because

Z ) 1 + O(F2)

(3)

and the contributions of higher terms are often negligible at high temperatures (such as those approaching the Boyle temperature). Wagner et al.,10 Holleran,11 and Ben-Amotz and Herschbach12 have presented applications for the Boyle temperature. In addition to eq 2, second virial coefficients must satisfy the following constraints:

B f -∞ when T f 0

(4)

B ) max at T ) Tmax (Tmax > TB)

(5)

B ) 0 (or possibly finite) when T f ∞

(6)

Equation 6 reflects one possible interpretation of the nature of the intermolecular potential function as temperatures reach extremely high levels and go beyond the decomposition value. One side holds that the potential function continues to sample unavailable states and asymptotically approaches infinity at zero separation, while the other side believes that the potential effectively becomes infinite at the separation associated with the decomposition temperature. We do not propose to solve this disagreement, but we choose to use the model that works best for the problem at hand, eq 6. Using the Boyle temperature instead of the critical temperature as the normalizing variable allows us to propose a simple equation that satisfies eqs 2-6

B/b0 ) θm(1 - θl) exp(blθn)

(7)

where θ ) TB/T and b0, m, l, b1, and n are adjustable parameters. This equation has the characteristics

B/b0 ) 0 when θ ) 1 or T ) TB

(8)

B/b0 ) -∞ when θ ) ∞ or T ) 0

(9)

B/b0 ) 0 when θ ) 0 or T ) ∞

(10)

To reduce the number of characteristic parameters and to ensure qualitatively correct representation at high temperatures (those beyond which data do not exist), we have chosen the m and l values to approximate the high-temperature second virial coefficients obtained from a Lennard-Jones (12-6) potential. This selection provides a means for extrapolating into the hightemperature regions that lack data and for approximating eq 5. The values of m and l that offer a reasonable compromise between accuracy and simplicity are m ) 0.2 and l ) 0.8. In addition to fitting the limit equations, it is desirable for the equation to fit the second virial coefficient at the critical temperature. This constraint follows from writing eq 7 at the critical temperature

( ) [ ( ) ] [ ( )]

TB BC ) b0 TC

0.2

1-

TB TC

0.8

exp b1

TB TC

n

Upon rearrangement of eq 11, it follows that

(11)

() { TC TB

n

ln

b0[(TB/TC)

}

BC 0.2

- (TB/TC)]

Therefore, the equation for the second virial coefficient becomes

()[

TB B ) b0 T

)

0.2

1-

()[ TB T

0.2

1-

{

( ) TB T

0.8

( ) TB T

0.8

]

]

exp

[

{( ) TC T

n

ln

b0[(TB/TC)

BC

BC 0.2

- (TB/TC)]

]

]}

(Tc/T)n

b0[(TB/TC)0.2 - (TB/TC)]

(12)

In this work, we have used eq 12 to correlate the second virial coefficient of nonpolar, polar, and complex molecules. This equation has three characteristic parameters, b0, BC, and n, that can be found from fitting experimental second virial coefficients. We suggest the following generalized expressions for BC, b0, and n based upon our fits to the data sample considered here.

BC/VC ) -1.1747 - 0.3668ω - 0.00061µR (13) in which BC is the second virial coefficient at the critical temperature and VC is the critical volume. In theory, the acentric factor also accounts for polar effects. However, in the case of second virial coefficients, the accepted acentric factors do not adequately reflect the total polar contribution. As a result, we include the reduced dipole moment in this equation. The coefficients multiplying the acentric factor and the reduced dipole moment adjust the contributions and permit use of published property values. Whereas most correlations use the approximation

BCPC/RTC ) -0.34 ( 0.01 we find that using this expression in eq 12 causes severe degeneration of data fits. The exponent n correlates as

n ) 1.4187 + 1.2058ω

(14)

for both polar and nonpolar fluids, while

b0/VC ) 0.1368 - 0.4791ω + 13.81(TB/TC)2 exp[-1.95(TB/TC)] (15) Boyle Temperatures Because Boyle temperatures are usually high (∼2.5TC), it is rare to have actual measurements for them. We have used three techniques to establish values for this paper. Tellez-Morales13 has suggested an accurateinterpolation/reasonable-extrapolation method in which he plots (T - TB)/BTC against (T/TC)1.1. Figure 1 illustrates this interpolation technique for nitrogen. The upper portion of the figure uses the correct Boyle temperature, while the lower portion uses a temperature that is 1 K removed from the correct value. Clearly, a pole exists in the lower portion (because B ) 0 at the

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Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001

Table 1. Physical Properties for Included Substances substance

ω

ω (ref)

TC (K)

TC (ref)

VC (cm3 mol-1)

VC (ref)

helium hydrogen neon argon krypton xenon methane oxygen nitrogen carbon monoxide ethene ethane propene propane R-11 R-13 n-butane R-12 benzene R-22 carbon dioxide methyl ethyl ether n-pentane R-152a toluene acetone R-134a water n-octane methanol 1-propanol ethanol

-0.3900 -0.2150 -0.0414 0.0000 0.0000 0.0000 0.0120 0.0218 0.0403 0.0663 0.0852 0.0990 0.1424 0.1518 0.189 0.198 0.1993 0.204 0.2108 0.2191 0.2276 0.244 0.2486 0.256 0.263 0.304 0.3328 0.3449 0.398 0.5656 0.623 0.644

18 17 18 18 18 18 18 18 18 17 18 17 18 18 15 15 18 15 18 33 18 15 18 15 15 15 33 18 15 18 15 15

5.19 33.2 44.4 150.687 209.4 289.734 190.551 154.581 126.192 132.85 282.35 305.33 364.9 369.85 471.15 301.92 425.16 385.01 562.161 369.3 304.128 438.03 469.7 386.44 591.8 508.1 374.18 647.09 568.8 512.6 536.7 513.9

17 17 17 19 17 20 23 10 21 17 24 25 17 27 15 15 28 15 31 33 22 15 29 15 15 15 34 35 15 17 15 15

57.400 65.100 41.600 74.586 91.200 118.000 98.630 73.367 89.414 92.170 130.945 147.060 181.000 203.000 247.96 180.1 255.100 212.88 257.120 165.600 94.120 220.78 314.790 179.09 316.64 215.91 200.850 55.950 492.4 116.279 218.53 166.92

17 17 17 19 17 18 23 10 21 17 24 26 17 17 15 15 28 15 32 33 22 15 30 15 15 15 34 35 15 17 15 15

µ (Debye)

µ (ref)

0.1

17

0.49 0.48

17 17

0.54

17

1.41

17

1.2

17

2.53 0.4 2.9 1.3 1.7 0 1.8 1.7 1.7

17 17 17 17 17 17 17 17 17

TB (K)

TBa

22.5 107.3 122.6 412.0 580.0 775.0 508.7 406.5 326.8 345.5 681.3 757.1 891.4 891.7 1119.9 714.5 1003.5 908.6 1347.0 888.3 702.1 1015.6 1081.6 891.6 1361.6 1151.7 867.7 1589.0 1254.1 1281.5 1137.8 1086.9

TI TI TI TI TI TI TI TI TI TI TE TE TE TE E E TE E TE ZL TE E TE E E E ZL ZL E CS E E

a TI ) Tellez-Morales interpolation; TE ) Tellez-Morales extrapolation; ZL ) unit compression factor extrapolation; CS ) 2.5T estimate; C E ) eq 19.

Figure 2. Example of TB determination by extrapolating the unit Z line.

Figure 1. Example of TB determination using the Tellez-Morales method.

true Boyle temperature, while T - TB is not zero unless the assumed temperature is actually the correct value) that is easy to see at even smaller differences in temperature. When the pole disappears, the Boyle temperature is correct. The extrapolation technique uses the observation that the Tellez-Morales plots are straight lines and the lines are extrapolated to B ) 0. It is also possible to extrapolate the line of unit compression factor to zero density to establish TB. Given that the virial equation is

Z ) 1 + BF + CF2 + ...

(16)

then, when Z is unity at nonzero density, the equation becomes

0 ) BF + CF2 + ... ) B + CF + ...

(17)

and in the limit as F approaches zero, eq 17 reduces to B ) 0, which is the definition of the Boyle temperature. Figure 2 illustrates this method for R-22. It is another extrapolation; this time of the straight line to zero density. If we even lack the data for the latter technique, we can resort to a data fit. Here, we use

TB ) 2.5TC

(18)

as the initial guess for TB (for methanol in Table 1, this is the proper value for the Boyle temperature). Judging from the values in Table 1, we suggest the correlation for TB

TB/TC ) 2.0525 + 0.6428 exp(-3.6167ω) (19) Results Equation 12 employs the accepted limits for the second virial coefficient and should extrapolate well in temperature. Figure 3 shows deviations for the second

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Figure 3. Percentage deviation of second virial coefficient calcu;ated using eqs 12-16 from that calculated using a LennardJones potential.

Figure 4. Second virial coefficient for Ar from prediction using eqs 12-16 (O), and from correlation using eq 13 (0). The region between the dashed lines has a linear scale, whereas the regions outside the dashed lines have a logarithmic scale.

Table 2. Average Absolute Deviation of Second Virial Coefficients Using Tsonopoulos, Hayden/O’Connell, the New Equation as a Predictor, and the New Equation as a Correlation substance helium hydrogen neon argon krypton xenon methane oxygen nitrogen carbon monoxide carbon dioxide ethene ethane propene propane Freon 11 Freon 13 n-butane Freon 12 benzene R-22 methyl ethyl ether n-pentane Freon 152a toluene acetone R-134a water n-octane methanol 1-propanol ethanol

∆B (cm3 mol-1) number Tsonopoulos O’Connell eqs 12-16 eq 12 of refsa 56.9 33 1 2.1 1.6 2.4 0.7 5.8 2.3 1.6 1.6 1.9 2.5 3.1 5.3 63.6 6.23 5.3 23.5 44.2 11.2 141 29.4 82.6 38.1 74.4 13.5 168.5 16.3 39 23.8 79.9 54.2

25.5 23.6 0.7 2.1 3.7 4.8 1.7 3.9 3.4 4.2 12 2.4 1.6 3.7 1.5 70.8 6.6 11.3 21.8 14.6 5.5 16.4 25.9 55 26.1 -

2 4.1 0.4 1.8 1.5 2.4 0.8 3.6 3.2 2.2 0.9 1.1 2 4.6 4.4 53 8.3 9 21 21.9 5.9 98.5 13.1 28.5 78.3 297 9.7 20.1

0.6 0.7 0.2 0.4 0.9 2.2 0.2 4.1 0.1 1.3 0.5 0.2 0.5 2.3 1 16.5b 4.0b 4.4 18.0b 12.2 2.9 c 12 5.8b 14.0b 75.5b 1.6 5.9

63 29 11 46 20 16 68 16 41 17 45 46 45 20 33 4 8 30 11 39 10 1 19 9 8 17 8 22

40.9 -

48.4 108.5 146 66.2

24.6b 14.3 44.1b 43.8b

6 26 5 7

a References are available from the corresponding author upon request. b Boyle temperature from the correlation. c Not enough points to fit a curve.

virial coefficient calculated from eqs 12-16 compared to the second virial coefficient calculated from a Lennard-Jones (12-6) potential. Although the agreement is not perfect, the deviation remains small over a very large temperature range and provides an indication that the temperature dependence is at least qualitatively correct. We have assembled a varied group of substances to test the validity of eqs 12-16. Table 1 presents the physical properties of the substances used in this work, and Table 2 contains a comparison among the five-term Tsonopoulos9 prediction, the Hayden and O’Connell14 prediction, eqs 12-16 used as a prediction, and eq 12 used as a correlation. Table 2 also indicates the number of references from which we used data for each substance to develop the correlation (564 different references). Rather than including such a large number of citations in this paper, interested readers can obtain a listing from the corresponding author. These data reside in the TRC Source Database,15 along with their esti-

Figure 5. Second virial coefficient for He from eqs 12-16 (b), Lennard-Jones 6-12 (0), and Tsonopoulos (4). The region between the dashed lines has a linear scale, whereas the regions outside the dashed lines have a logarithmic scale.

mated uncertainties, and we used the uncertainties to weight the data fits. The overall absolute average deviations for the three correlations are reasonably close, except for those for helium and hydrogen (which Tsonopoulos and Hayden and O’Connell did not use in their fits) and the lower alcohols (for which Tsonopoulos specifically inserted a term). The advantage of eq 12 is that it retains the same form for molecules of any complexity, and it is much easier to apply than Hayden/ O’Connell. Figure 4 presents deviations from the second virial coefficient of Ar with eq 12 used as a correlation (O) and with eqs 12-16 used for prediction (0). The deviation axis is a log-linear plot, as suggested in Holste et al.16 In the figure, the interval -1 to 1 has a linear scale, whereas the ranges outside -1 to 1 have logarithmic scales. This presentation is especially useful for plots such as this with many small deviations and a few large ones. From this figure, it appears that eq 12 as correlation and eqs 12-16 as prediction are essentially indistinguishable above about 300 K for Ar. At the lower temperatures, the correlation behaves better, as would be expected. Figure 5 contains the deviations from data for He using eqs 12-16 (b), Lennard-Jones (12-6) (0), and Tsonopoulos (4). In the remaining figures, we have compared only with Tsonopoulos because that correlation is more commonly used and easier to apply than Hayden/O’Connell. Again, this is a log-linear plot with the linear region lying between -1 and 1. The former

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Figure 6. Comparison between eqs 12-16 (b) and Tsonopoulos (4) for various substances.

two representations fit the data quite well over an enormous temperature range. The latter appears to have an almost constant displacement. The problem is that Tsonopoulos does not include substances that have negative acentric factors. Figure 6 compares Tsonopoulos to eqs 12-16 for several substances (short- and longer-chain hydrocarbons, non-hydrocarbons, and polar molecules). In general, the two predictions are comparable at higher temperatures, and for the smaller molecules, eqs 1216 appear to work better for the chain hydrocarbons.

For water, Tsonopoulos presents two equations, one of which (three-parameter) fits poorly and the other of which (one-parameter) provides an excellent fit. Figure 7 illustrates the behavior of Tsonopoulos (dashed line) and eqs 12-16 (solid line) over a very wide temperature range. Equations 12-16 reach a maximum at a reduced temperature of about 25 (as would be expected), whereas Tsonopoulos appears to have plateaued before returning to zero. This would indicate that eqs 12-16 provide a better high-temperature extrapolation.

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1973

Figure 7. Second virial coefficient of Ar predicted from eqs 1216 (s) and from Tsonopoulos (- - -).

Conclusions We have developed a new predictor/correlator equation for the second virial coefficients of pure substances. The new equation differs from existing ones in that it does not have a polynomial form. Also, the new equation compares favorably with the Tsonopoulos and Hayden/ O’Connell correlations, but it does not require extra terms to account for hydrogen bonding, quantum effects or polarity. The new expression should provide better extrapolations than existing expressions. Acknowledgment The Texas Engineering Experiment Station and the Instituto Tecnolo´gico de Celaya have provided financial support for this work. Nomenclature b0 ) reducing parameter for the second virial coefficient b1 ) fit parameter in eq 8 (cm3 mol-1) B ) second virial coefficient (cm3 mol-1) l ) fit parameter in eq 8 m ) fit parameter in eq 8 n ) fit parameter in eq 8 P ) pressure (bar) R ) gas constant (8.31451 J mol-1 K-1) T ) temperature (K) Z ) compression factor ()P/FRT) θ ) Boyle temperature divided by temperature µ ) dipole moment ω ) acentric factor Subscripts B ) Boyle temperature C ) critical condition max ) temperature at which B becomes a maximum

Literature Cited (1) Hall, K. R.; Iglesias-Silva, G. A.; Mansoori, G. A. Quadratic mixing rules for equations of state. Origins in and relationships to the virial expansion. Fluid Phase Equilib. 1993, 91, 67. (2) Luongo-Ortiz, J. F.; Starling, K. E. A new combining rule for the mixture equations of state: Higher-order composition dependencies reduce to quadratic composition dependence. Fluid Phase Equilib. 1997, 132, 159. (3) McGregor, D. R.; Holste, J. C.; Eubank, P. T.; Marsh, K. N.; Hall, K. R. Simple solution models applied to virial coefficients. AIChE J. 1986, 32, 1221. (4) Hall, K. R.; Iglesias-Silva, G. A. Generalized derivations for mixtures using the virial equation: Application to fugacity. Chem. Eng. Commun. 1995, 137, 211.

(5) Black, C. Vapor-phase imperfections in vapor-liquid equilibria. Ind. Eng. Chem. 1958, 50, 391. (6) Nothnagel, K. H.; Abrams, D. S.; Prausnitz, J. M. Generalized correlation for fugacity coefficients in mixtures at moderate pressures. Ind. Eng. Chem. Process Des. Dev. 1973, 12, 25. (7) Pitzer, K. S.; Curl, R. F. The volumetric and thermodynamic properties of fluids. III. Empirical equation for the second virial coefficient. J. Am. Chem. Soc. 1957, 79, 2369. (8) O’Connell, J. P.; Prausnitz, J. M. Empirical correlation of second virial coefficients for vapor-liquid equilibrium calculations. Ind. Eng. Chem. Process Des. Dev. 1967, 6, 245. (9) Tsonopoulos, C. An empirical correlation of second virial coefficients. AIChE J. 1974, 20, 263. (10) Wagner, W.; Ewers, J.; Schmidt, R. An equation of state for oxygen vaporssecond and third virial coefficients. Cryogenics 1997, 37. (11) Holleran, E. M. J. Chem. Phys. 1967, 47, 5318. (12) Ben-Amotz, D.; Herschenbach, A. Correlation of Zeno line for supercritical fluids with vapor-liquid rectilinear diameters. Israel J. Chem. 1990, 30, 59. (13) Tellez-Morales, R. Correlations for the second and third virial coefficients. M.S. Thesis, Instituto Tecnolo´gico de Celaya, Celaya, Guanajuato, Mexico, 1998. (14) Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 209. (15) TRC Databases for Chemistry and EngineeringsSource Database; Thermodynamics Research Center, Texas A&M University: College Station, TX, 1999. (16) Holste, J. C.; Hall, K. R.; Iglesias-Silva, G. A. Log-linear plots for data representation. AIChE J. 1996, 42, 296. (17) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (18) Daubert, T. E.; Danner, R. P. Physical and thermodynamic properties of pure chemicals data compilation. Design Institute for Physical Property Data, American Institute of Chemical Engineers: New York, 1991; Parts 1-4. (19) Gielgen, R.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (p, F, T) relation of argon. J. Chem. Thermodyn. 1994, 26, 383. (20) Sifner, O.; Klomfar, J. Thermodynamic properties of xenon from the triple point to 800 K with pressures up to 350 MPa. J. Phys. Chem. Ref. Data 1994, 23 (1), 63. (21) Nowak, P.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (p, F, T) relation of nitrogen. J. Chem. Thermodyn. 1997, 29, 1137. (22) Duschek, W.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (p, F, T) relation of carbon dioxide. J. Chem. Thermodyn. 1990, 22, 827. (23) Kleinrahm, R.; Duschek, W.; Wagner, W. (p, F, T) measurements in the critical region of methane. J. Chem. Thermodyn. 1986, 18, 1103. (24) Nowak, P.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (p, F, T) relation of ethylene. J. Chem. Thermodyn. 1996, 28, 1423. (25) Douslin, D. R.; Harrison, R. H. Pressure, volume, temperature relations of ethane. J. Chem. Thermodyn. 1973, 5, 491. (26) Goodwin, R. D.; Roder, H. M.; Straty, G. C. Thermophysical properties of ethane from 90 to 600 K at pressures to 70 MPa; NBS Technical Note 684; National Bureau of Standards: Washington, D.C., 1982. (27) Goodwin, R. D.; Haynes, W. M. Thermophysical properties of propane from 85 to 700 K at pressures to 70 MPa; NBS Monograph 170; National Bureau of Standards: Washington, D.C., 1976. (28) Haynes, W. M.; Goodwin, R. D. Thermophysical properties of n-butane from 135 to 700 K at pressures to 70 MPa; NBS Monograph 169; National Bureau of Standards: Washington, D.C., 1982. (29) TRC Databases for Chemistry and Engineerings Thermodynamic Tables Hydrocarbons; Thermodynamics Research Center, Texas A&M University: College Station, TX, 1999. (30) Holcomb, C. D.; Magee, J. W.; Haynes, W. M. Density measurements on natural gas liquids. GPA RR-147, 1994. (31) Ambrose, D. Vapor pressures for some aromatic hydrocarbons. J. Chem. Thermodyn. 1987, 19, 1007.

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(32) Chirico, R. D.; Steele, W. V. Reconciliation of calorimetrically and spectoscopically derived methylbenzene. The importance of the third virial coefficient. Ind. Eng. Chem. Res. 1989, 33, 157. (33) TRC Databases for Chemistry and EngineeringsFreons Database; Thermodynamics Research Center, Texas A&M University: College Station, TX, 1999. (34) Tillner-Roth, R.; Baehr, H. D. An international formulation for the thermodynamic properties of 1,1,1,2-tetrafluoroethane (HFC-134a) for temperatures from 170 to 455 K and pressures up to 70 MPa. J. Phys. Chem. Ref. Data 1994, 23 (5), 657.

(35) Saul, A.; Wagner, W. A fundamental equation for water covering the range from the melting line to 1273 K at pressures up to 25000 MPa. J. Phys. Chem. Ref. Data 1989, 18 (4), 1537.

Received for review July 20, 2000 Revised manuscript received January 23, 2001 Accepted January 24, 2001 IE0006817