Ind. Eng. Chem. Res. 1993,32, 759-761
759
Correlation of the Ideal Gas Properties of Five Aromatic Hydrocarbons Arno Laesecket Thermophysics Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80303-3328
The ideal gas thermodynamic properties cpo(77, so (T), and ho(0- h" (0) have been correlated for benzene, toluene, o-xylene, m-xylene, and p-xylene by a uniform, semiempirical function of temperature. The correlation is based on literature values calculated from statistical mechanics for temperatures up t o 1500 K for benzene and up to 3000 K for the other molecules. The temperature function chosen is more accurate than that used for previous correlations, and can be used in a wider temperature range.
Introduction Knowledge of the temperature dependence of the isobaric heat capacityc,O(T), entropysO(T),and enthalpy ho(T)- ho(0)for the ideal gas is a basic requirement to establish thermodynamic properties formulations for fluids. Furthermore, the specific heat capacity cUo is needed to calculate the contribution of internal degrees of freedom to the thermal conductivity XO(T )of a fluid in the ideal gas state (Laesecke et al., 1990). It is of particular interest to know these properties for aromatic hydrocarbons since they occur in many chemical production processes. The properties of many alternative fuels depend largely on their contents of aromatic hydrocarbons (Gilmutdinov et al., 1989). The thermodynamic properties of benzene and toluene were recently evaluated (Goodwin, 1988, 1989). The isobaric heat capacity c p o ( T ) was formulated for these fluids by two different empirical functions which were based on data calculated from statistical mechanics. The data which were used for benzene cover a temperature range from 50to 1500K (TRC, 1986). The data for toluene cover an extended temperature range from 50 to 3000 K (Chao et al., 1984). This reference presents also data for o-, m-,and p-xylene. The formulations of Goodwin require numerical integration to obtain the entropy s"(T) and enthalpy ho(T) - h"(0)from c,"(T). This is time consuming to program and run on personal computers. Goodwin's correlation for toluene appears of limited accuracy at extremely low temperatures since it yields a deviation from the data in ho(T) - ho(0) of as much as 17% at 50 K. It seemed worthwhile to develop more accurate formulations. The disadvantage of numerical integration should be overcome by selecting functional terms which could be integrated in closed form. Because data from statistical mechanics are available for o-, m-, and p-xylene, it appeared useful to include them in the present study with benzene and toluene to arrive eventually a t a uniform correlation of c p o ( T )for all five substances.
Selection of Functional Form The temperature dependence of the ideal gas heat capacity is often represented by simple polynomial functions (Reid et al., 1987). Such correlations, however, may be applied over only a limited temperature range (some hundreds of kelvin) and are not suitable for extrapolation. Better behaving correlations over temperature ranges of several thousand kelvin consist of polynomials with additional functional terms which are theoretically based. t
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Such an equation was proposed for molecular nitrogen (Barieau, 1965) and adopted in amended form in the IUPAC formulation for that fluid (Angus et al., 1979). It was also used in formulations of the thermodynamic properties of isobutane (Waxman and Gallagher, 1983) and the alkanes C1 to Cq (Younglove and Ely, 1987). This equation,
with u = a9/T and the universal gas constant R = 8.314 345 107 0 J.mol-l.K-l, was considered in this study as one of three alternatives. It includes nine adjustable parameters with only integral powers of the thermodynamic temperature T in the polynomial part. The exponential term was introduced as an approximation of the vibrational contribution to cpo. Goodwin (1971) used an equation with six adjustable parameters for methane which was retained in the recent properties formulation for that fluid (Friend et al., 1989). The exponential term is the same as in eq 1,but the polynomial part is reduced to four terms and includes nonintegral powers of T. An equation with seven adjustable parameters, originally developed for molecular oxygen (Wagner et al., 1982),was tested as a third alternative. The polynomial part of this expression is further reduced to three terms with one nonintegral power of T. An additional exponential term accounts for the effect of electronic excitation. It is generally desirable to reduce the number of polynomial terms since they lack physical significance. This can be accomplished by procedures to optimize the structure of empirical equations (Setzmann and Wagner, 1989). The equation of Wagner et al. (1982)resulted from such a structural optimization. Although this approach may yield more accurate correlations, it has the disadvantage of leading to varying forms of equations for different substances. Since the five aromatichydrocarbons are very similar molecules, a structural optimization was not attempted in this work.
The New Correlation The parameters in each of the three selected correlations were determined by nonlinear least-squaresregressionwith a Levenberg-Marquardt algorithm (Press et al., 1986). This algorithm can be conveniently used on personal computers. The final values, however, depend to some degree on the initial guesses. Several runs with varying initializations were made to ensure that an optimum set of parameters had been reached. Only cp" data were used for the fit. Equation 1yielded the best representation of the data for all five substances.
Published 1993 by the American Chemical Society
760 Ind. Eng. Chem. Res., Vol. 32, No. 4,1993 Table I. Parameter Values in Eq 1 for the Five Aromatic Hydrocarbons. toluene o-xylene benzene -2 077 000 al, K3 -1 968 000 -2 780 700 87 130 76 590 112 540 02, K2 03, K -864.5 -1411.9 -1285 9.0607 11.06 5.593 a4 2.0168 X 2.622 X 1.644 X le2 a5, K-' a6, K-2 -6.147 X 10-6 -7.1646 X 10-6 -9.322 X 10-6 1.171 X 8.199 X 9.0242 X a7, K3 11.92 12.06 as 11.32 1561.5 1730 as, K 1460 50-3000 50-1500 50-3000 temp range, K
m-xylene -2 162 000 91 650 -1265 9.882 2.561 X -9.074 X 10-6 1.141 X le9 13.61 1648 50-3000
p-xylene -2 738 000 113 600 -1497 10.56 2.492 X -8.825X 10-6 1.109x 10-9 13.71 1625 50-3000
The number of parameter digits as listed in this table was determined by rounding off a parameter set one digit at a time and recalculating the standard deviation of the correlation until it increased more than 0.1% . (I
Table 11. Representation of the Statistical Mechanical Data for the Five Aromatic Hydrocarbons by Eq 1. benzene toluene o-xylene rn-xylene p-xylene Representation of cpo(T) 0.03 0.9 X 0.3 X 10-2 0.7 X -0.5 X f0.08 f0.09 10.09 f0.02 10.08 0.17 0.16 0.18 0.021 0.14 -0.11 -0.08 -0.13 -0.12 -0.023 Representation of s0(T,po) -0.02 -0.02 -0.02 -0.02 -0.03 10.11 10.09 10.12 fO.10 f0.11 0.02 0.02 0.02 0.02 0.03 -0.45 -0.55 -0.60 -0.51 -0.48 Representation of ho(T)- ho(0) -0.16 -0.13 -0.15 -0.17 -0.17 10.8 fO.9 f0.7 f0.9 *0.8 0.09 0.06 0.19 0.11 0.15 -4.2 -4.3 -3.3 -3.5 -4.4 25 25 25 25 19
-
+ Goodwin (1988) -2.0
0
250
500
Reid et 81. (1987) Toubuklan, Makita (1970)
1000
750
1250
1500
T, K
Figure 1. Percent deviations between correlations and the cpo data from statistical mechanics for benzene.
a L V i sthe average deviation in percent; s is the standard deviation in percent; +Am and -Amar are the extreme deviations in percent.
A maximum deviation of 0.18% occurs for m-xylene at 1750 K. The tests of the two alternative equations did not yield deviations below 1%.The final parameters for eq 1 for each of the five aromatic hydrocarbons are compiled in Table I. Table I1 summarizes the data regression statistics. Chao et al. (1984) compared their calculated results for toluene and the three xylene isomers with experimental data and found deviations between 0.5% and 4 . 5 6 5% for temperatures up to 523.15 K. The key issue in the statistical mechanical calculation of these alkylbenzenes is the treatment of the rotation of the methyl groups. Chao et al. considered them as hindered rotors and found differences of 1.5% in cPoof p-xylene at 1000 K compared with earlier calculations in which the methyl groups were treated as free rotors. This indicates a higher uncertainty of the statistical mechanical data in the high-temperature range. There are only a limited number of other data sources which can be used to assess the absolute accuracy of the new correlation. Touloukian and Makita (1970) evaluated the available data for benzene and toluene and established correlations from 273 to 1500 K. They quote maximum deviations of 0.68% for benzene and 0.54% for toluene. Reid et al. (1987)present polynomials for all five substances but do not state their data base nor the range of applicability. Comparisons of these correlations, the previous equations of Goodwin, and the new correlation are shown in Figures 1and 2. No other data were found in reference books for the substances considered here (Gurvich, 1989). The deviations in Figure 1for benzene indicate that the new correlation is more accurate than the six-parameter
0.5
~
0.0 4.5
Toluene
-1.0 -1.5
t 0
I1
500
-i
\ 1000
1500
T,
2000
2500
3000
K
Figure 2. Percent deviations between correlations and the cpo data of Chao et al. (1984)for toluene.
equation based on the same data set (Goodwin, 1988). The polynomial by Reid et al. is not applicable below 273.15 K and is too low at higher temperatures. Values calculated from the correlation by Touloukian and Makita are generally too low with an extreme deviation of -1.5% at 500 K. A similar pattern is found for toluene, Figure 2. Although data were availableup to 3000 K, the formulation of Goodwin (1989) was based on a subset up to 1500 K. It is as accurate as the new correlation in that temperature range. Mainly negative deviations of the correlations of Touloukian and Makita and Reid et al. reflect probable differences in the data base between these and the present work. No information other than the polynomials of Reid et al. (1987) could be located for the three xylene isomers.
Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 761 Table 111. Reference Values for the Five Aromatic Hydrocarbons at TO= 298.15 K and DO = 0.101 325 MPa ~
benzene toluene o-xylene m-xylene p-xylene so (To,Po),
269.19 321.00 353.80 358.50 J/(mol.K) h"(T0)- h"(O), 14 233.0 18 170.0 23 390.0 22 120.0 J/mol
352.20 22 010.0
They seem to apply to temperatures from 273 to 1500 K although there are discrepancies of more than 1%.These exceed the differences with Chao et al. found between their calculations and the experimental data (0.5% and -0.56 % 1. The new correlation represents the data of Chao et al. for the xylene isomers well within these margins.
Calculation of Entropy and Enthalpy Equation 1for c,"(T) can be integrated in closed form to obtain the entropy so and the enthalpy ho in the ideal gas state. The following expressions yield the change of the ideal gas entropy and enthalpy from the reference state (T0,po)to the state (T,p):
and
hO(T0)-h"(O) R ( 1 K)
(3)
The values of the parameters al, ...,ag are listed in Table I for the five aromatic hydrocarbons. The reference values sO(T0,po)and ho(To)- ho(0)for benzene (Goodwin, 1988) and for the remaining molecules (Chao et al., 1984) are given in Table 111. The representation of the statistical mechanical data for the entropy so(T,po)and the enthalpy ho(T)- ho(0) is summarized in Table 11. Deviations in the entropy do not exceed -0.6%. The maximum enthalpy deviation occurs for all five substances a t 50 K and amounts to about -4.0%. This is a substantial improvement compared with the former correlation for toluene (Goodwin, 1989) which yielded deviations of 2.33 % in the entropy and 17 5% in the enthalpy a t that temperature. However, since eq 1is not correct in the limit as T 0, further improvements may require yet other functional forms.
-
Conclusion A uniform, semiempirical correlation has been established to represent the ideal gas heat capacity, entropy, and enthalpy of five aromatic hydrocarbons. It is more accurate than previous correlations and can be applied over a wider temperature range. It is also simpler to use, since entropy and enthalpy can be evaluated without numerical integration. The new correlation may be used
in conjunction with the equations of state for benzene and toluene (Goodwin, 1988, 1989) to calculate caloric data over wide ranges of the fluid region. Only small changes will occur compared to the tabulated data of Goodwin except for the properties of toluene at extremely low temperatures. Wide-ranging equations of state do not yet exist for the xylene isomers, but they can be developed using the new correlation.
Literature Cited Angus, S.; de Reuck, K. M.; Armstrong, B.; Jacobsen, R. T; Stewart, R. B. International Thermodynamic Tables of the Fluid State; Pergamon Press: Oxford, 1979; Vol. 6, Nitrogen. Barieau, R. E. Analytical Expressions for the Zero Pressure Thermodynamic Properties of Nitrogen Gas Including Corrections for the Latest Value of the Atomic Constanta and the New Carbon-12 Atomic Weight Scale. J. Phys. Chem. 1965,69,495. Chao,J.; Hall, K. R.; Yao, J.-M. ChemicalThermodynamic Properties of Toluene, 0-, m-, and p-Xylene. Thermochim. Acta 1984, 72, 323. Friend, D. G.; Ely, J. F.; Ingham, H. Thermophysical Properties of Methane. J. Phys. Chem. Ref. Data 1989,18, 583. Gilmutdinov, A. T.; Tanatarov, M. A.; Gilmutdinova, E. Sh. The Influence of Gasoline Hydrocarbon Composition on the Stability of the System Gasoline-Methanol-Water. Neft Gaz 1989,No. 6, 43. Goodwin, R. D. Thermophysical Properties of Methane: Orthobaric Densities and Some Thermal Properties. J.Res. Natl. Bur. Stand. (U.S.A.)1971, 75A, 15. Goodwin, R. D. Benzene Thermophysical Properties from 279 to 900 K at Pressures to 1000 Bar. J. Phys. Chem. Ref. Data 1988,17, 1541. Goodwin, R. D. Toluene Thermophysical Properties from 178to 800 K at Pressures to lo00 Bar. J. Phys. Chem. Ref. Data 1989,18, 1565. Gurvich, L. V. Reference Books and Data Banks on the Thermodynamic Properties of Individual Substances. Pure Appl. Chem. 1989,61, 1027. Laesecke, A.; Krauss, R.; Stephan, K.; Wagner, W. Transport Properties of Fluid Oxygen. J. Phys. Chem. Ref. Data 1990,19, 1089. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1986. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. Refer to Appendix A, Property Data Bank. Setzmann, U.; Wagner, W. A. New Method for Optimizing the Structure of Thermodynamic Correlation Equations. Int. J. Thermophys. 1989,10, 1103. Touloukian, Y. S.; Makita, T. Thermophysical Properties of Matter; Plenum Press: New York, 1970; Vol. 6. TRC Thermodynamic Tables-Hydrocarbons; Thermodynamics Research Center, Texas A&M University: College Station, April 1986. Wagner, W.; Ewers, J.; Schmidt, R. An Equation for the Ideal-Gas Heat Capacity of Molecular Oxygen for Temperatures from 30 K to 3000 K. Ber. Bunsen-Ges. Phys. Chem. 1982,86, 538. Waxman,M.; Gallagher,J. S.ThermodynamicProperties of Isobutane for Temperatures from 250 to 600 K and Pressures from 0.1 to 40 MPa. J. Chem. Eng. Data 1983,28,224. Younglove, B. A.; Ely, J. F. Thermophysical Properties of Fluids. 11. Methane, Ethane, Propane, Isobutane and Normal Butane. J. Phys. Chem. Ref. Data 1987,16, 577. Received for review July 6 , 1992 Revised manuscript received November 23, 1992 Accepted December 14, 1992
