Ind. Eng. Chem. Fundam. 1981, 20, 280-283
280
Table III. Comparison of Experimental and Predicted Excess Critical Roperties
system propane + nhexane propane t 2,2-dimethylbutane propane + 2-methylpentane propane t 3-methylpentane propane t 2,3-dimethylbutane
av abs maxabs avabs max abs error in error in error in error in TcE, K TcE, K PcE,bar PcE,bar 1.14 2.10 0.53 1.12 0.51
0.97
0.75
1.07
0.75
1.49
0.60
1.27
2.1 2
5.00
1.20
2.28
1.28
1.79
0.98
1.92
error for the five systems reported in this work.
Nomenclature A = constant P = pressure T = temperature V = volume x = composition (mole fraction) Subscripts c = critical i = component i max = maximum value
Literature Cited Beattle, J. A.; Kay, W. C.; Kaminsky, J. ' J . Am. Chem. Soc. 1937, 59,
with T,in K and P, in bar. Using eq 8-11 to obtain 4 E, and x,, we may then predict the excess critical property of these binary mixtures from a knowledge of the critical properties of the pure components. Predictions of the excess critical temperatures and pressures are summarized in Table I11 and shown in Figures 1 and 2. The excess critical volume could not be treated in the same way because ita magnitude was of the order of the experimental
1589. Etter, D. 0.;Kay, W. 8. J. Chem. Eng. Data 1961, 6, 409. Jones, A. E.; Kay, W. B. AIChE J . 1967, 13, 710. Kay, W. B.; Rambosek, (3. M. Ind. Eng. Chem. 1953, 45, 221. Pak, S. C.; Kay, W. B. Ind. Eng. Chem. Fundem. 1972, 11, 255. Rosslni, F. D. "Selected Va!ues of Physical and Thermodynamic Roperties of Hydrocarbons and Related Compounds"; Amerkan Petroleum Institute Project 44, Carnegie Press, PMsbwgh, Pa., 1953.
Receiued for reuiezu October 27, 1980 Accepted May 15,1981
Estimation of Vapor Pressures of Heavy LiquM Hydrocarbons Containing NItrogen or Sulfur by a Group-Contrtbutton Method D. R. Edwards" and J. M. Prausnltz Department of Chemical Engineering, Universi& of Callfornk Berkeky, California 94720
The groupcontribution method for vapor pressures of hydrocarbons, developed by Macknick et al. (1979), based on the kinetic theory of fluids, is extended to include groups containing nitrogen or sulfur. Flrst approxlmatlons of vapor pressures in the range 10-2000 ton are possible within a factor of 2. The method should be used only in the total absence of any data since accuracy can be improved dramatically by using only one vapor-pressure datum.
Introduction
Table I. Group Contributions t o Vapor Pressure Parameters 8 and E J R . Carbon Groupsa
Vapor-pressure data are generally available for lowmolecular-weight hydrocarbons. However, as molecular weight increases, data become more scarce. Moreover, vapor-pressure data for hydrocarbons containing heteroatoms are often unavailable for even lower-molecularweight compounds. With the development of alternate energy sources, it is important to estimate vapor pressures of these types of compounds to facilitate rational process design. It is usually not possible to determine all required data experimentally. Therefore, we often extend or extrapolate limited available data through various correlations. Many vapor-pressurecorrelations exist. Most are either empirical or are based in some way on integration of the Clapeyron equation which indicates that the vapor pressure is an exponential function of temperature. Due to the semilogarithmic nature of most vapor pressure correlations, 0196-4313/81/1020-0280$01.25/0
Vwi,
carbon type aliphatic CH, aliphatic CH, aliphatic CH aliphatic C aromatic Ar=C( Ar)H aromatic Ar= C( Ar)R condensed aromatic Ar=C(Ar)cond condensed aromatic Ar= C naphthenic >CH, naphthenic >CHR
si
e,ilR, K
2.359 0.479 -2.189 -4.318 1.175
1162.7 674.0 -372.9 -1127.1 939.5
cm3/ g-mol 13.67 10.23 6.78 3.33 8.06
-0.520
583.0
5.54
-0.774
432.5
4.74
0.321
623.5
4.74
1.188 -1.936
928.0 -431.0
9.47 9.47
From Macknick et al. (1979).
0 1981 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 281
In P = A
Table 11. Sample Calculations for Parameters in Eq 1. I. Carbazole
A = In R/Vw
H
+ B/T+ C In T + DT + E P
+ (s - y2) In (Eo/R) - In [(s-
vwi,
vi si aromatic Ar=C(Ar)H 8 1.175 condensed aromatic 4 -0.774 GOUP
Ar=C( Ar)cond heterocycloaromatic NH
1 s = &Si i
4.847
cm 3/ eoi/R,K g-mol
939.5 432.5
8.06 4.74
4253.0
8.08
i
= 91.52cm3/g-mol
i
Table 111. Sample Calculations. 11. 2,3-Dimethylthiophene
uwi,
group aliphatic CH, aromatic Ar=C(Ar)R aromatic Ar=C(Ar)H heterocycloaromatic (Ar),S
2 2 2 1
si eOi/R,K 2.359 1162.7 -0.520 583.0 1.175 939.5 2.272 1950.0
(S
+
l)!] In a (2)
B = -Eo/R
(3)
C=y..-s
(4)
- 3)(~ 1)/2(EO/R).
(6)
V, is the hard-core van der Waals volume; s is the number of equivalent oscillators per molecule; Eois the energy of vaporization of the hypothetical fluid at T = 0, and R is the gas constant (82.06 (cm3 atm/K g-mol)). When Eo/R and Tare in K and P is in atmospheres, the universal constant a equals 0.0966. The function (s - l)! found in eq 2 is the r function. A rapidly converging series to estimate this function is given by Abramowitz and Stegun (1964) and is shown in eq 7. (s - I)! = [exp(-s)][~(~.~)][(2*)~.~][1 + (1/12s) + (1/288s2)- (139/51840s3) - (571/2488320s4)] (7)
= 11.15
E,IR = Zvi(eoi/R)= 13499 K V , = &vWi
E=
(1)
cm3/ g-mol
13.67 5.54 8.06 11.10
s = Xvisi = 8.300 i
EJR = ?~i(e,i/R)= 7320 K I
V , = &y~,i = 65.64 cm3/g-mol
Group-Contribution Method for Hydrocarbons Abrams et al. (1974) and Macknick et al. (1977) have shown that eq 1 represents well the vapor pressures of heavy hydrocarbons in the range lo4 to 2 atm. Macknick et al. (1979) recognized that, because of the small number of parameters and their physical significance, a groupcontribution method may be applicable to the AMP equation. The group contributions determined by Macknick et al. (1979) are presented in Table I, and sample calculations are given in the Appendix and Tables I1 and III. Parameters s, Eo/R, and V, can be found by summing group contributions according to s = CUiSi I
i
correlating parameters must be known accurately to avoid large errors. Many correlating equations, based on the classical theories of corresponding states, use critical properties as reducing parameters. Heavy hydrocarbons, however, become unstable in the critical region; therefore, estimation of critical properties of heavy hydrocarbons is uncertain. A popular approach is to develop correlations for homologous series; this approach is a rudimentary groupcontribution method. Macknick et al. (1979) showed that a more general group-contribution method can be developed based on a theoretically derived relation with physically significant parameters. Macknick's correlation for hydrocarbons is extended here to compounds containing also nitrogen or sulfur. The method described below should be used only when no experimental data are available. The method often provides only an order-ofmagnitude estimate. Whenever just one vapor-pressure datum is available, accuracy of prediction is very much improved.
The AMP Equation Macknick's method is based on the AMP equation, as derived by Abrams et al. (1974), based on a suggestion by Moelwyn-Hughes (1961)
v, = Ci lJiU,i where si, tOi/R, and uWi are contributions from a group containing skeletal atom i; vi is the number of skeletal atoms of type i in the molecule. The group contributions to the hard-core van der Waals volume are determined from Bondi's (1968) correlation.
Extension to Heteroatom Groups The group parameters shown in Table IV for nitrogenand sulfur-containing groups were determined by using experimental data for the compounds listed in Table V. The parameters were fit by minimizing s u m s of squared relative errors using the Simplex (Nelder, 1965) regression routine. Bondi's (1968)correlation was used to determine uwi, the group van der Waals volume contributions. As mentioned by Macknick et al. (1979), a group-contribution analysis does not account for molecular fine structure; for instance, no distinction is made between 2-pentanethiol and 3-pentanethiol. However, the method can differentiate gross changes in structure, for instance, between butane-2,3-dithiol and 3,4-dithiahexane. To give some idea of the predictive capability of the groupcontribution method, Table VI shows estimated and
282 Ind. Eng. &em. Fundam., Vol. 20, No. 3, 1981
Table IV. Group Contributions to Vapor-Pressure Parameters s and EJR. Heteroatom Groups vapor-pressure data used to obtain group group si e,i/R, K uwi, cm3/g-mol parameters a h h a t i c SH 4.994 3489 14.81 alkyl thiols (C,-C,) arGmatic SH 2010 14.81 substituted thiophenols 1.759 aliphatic S 10.8 thiaalkanes (C,*,) 2.277 2627 thioanisole 10.8 1806 aliphatic RS aromatic 0.532 thionaphthenes ( C4-C6) 2416 heteronaphthenic S 10.8 1.928 1950 heterocycloaromatic S 11.1 substituted thiophenes 2.272 aliphatic SS 4527 dithiaalkanes (C,-C,) 3.740 22.7 aliphatic NH, 10.54 4229 primary amines 7.200 aromatic NH, aniline 10.54 3110 3.135 aliphatic NH dimethyl/diethylamine 8.08 3367 5.246 aliphatic N < trimethyl/triethylamine 874.7 4.33 1.049 heterocycloaromatic NH substituted pyrroles 4253 8.08 4.847 heterocycloaromatic N 5-member ring substituted thiazoles 5012 7.329 5.2 heterocycloaromatic N 6-member ring substituted pyridines 2184 2.201 5.2 Table V. Vapor-Pressure Data Used t o Determine Group Parameters range of compound data, torr sources thiophene 2-methylthiophene 3-methylthiophene 2,3-dithiabutane 3,4-dithiahexane 4,5-dithiaoctane 2-thiapropane 2-thiabutane 3-thiapentane thiacy clopentane thiacyclobutane thiacyclohexane methylamine 1-propylamine 2-propy lamine aniline dimethy lamine diethy lamine trimethylamine triethylamine 2-methyl-5-ethylpyridine 3-methylpyridine 4-methylpyridine 2-methylpyridine pyridine methanethiol ethanethiol l-propanethiol 2-propanethiol thiophenol 2-methylbenzenethiol 3-methylbenzenethiol 4-methylbenzenethiol thiazole methyl-2-thiazole l-methylpyrrole pyrrole 2,5-dimethylpyrrole thioanisole
10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 10-1500 4-760 300-2000 200-2000 50-780 5-760 300-900 6-7 60 200-900 20-760 70-2000 20-2000 150-2000 150-2000 10-1500 10-1500 10-1500 10-1 500 10-1500 10-1500 10-1500 10-1500 100-760 150-760 70-2000 70-2000 7 0-17 00
la
70-900
2
1 1 1 1 1 1 1 1 1 1 1 2b 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2
Wilhoit, R. C.; Zwolinski, B. J., “Handbook of Vapor Pressures and Heats of Vaporization of Hydrocarbons and Related Compounds”; ( APIRP 44) Thermodynamics Research Center, Department ofChemistry Texas P&M University, College Station, Texas, 1971. b’ Boubhk, T.; Fried, V.; Ha?a, E., “The Vapor Pressures of Pure Substances”; Elsevier Scientific Publishing Co. : Amsterdam, 1973.
experimental vapor pressures. All compounds shown in Table I11 were chosen at random from those shown in Table 11whose vapor-pressure data were used to determine the parameters. Errors range from 1% to 28% at 200 torr
and from 0% to 30% at 760 torr. These relatively good approximations are expected since the parameters were best fit to data for these compounds. Of more interest are comparisons using compounds not included in determination of group parameters. This provides a better test of potential unertaintiea. Examples I and II in the Appendix show calculations for carbazol and 2,3-dimethylthiophenee At the reported boiling point, calculated relative errors in vapor pressure are 31% and 0% ,respectively; these represent 13 K and 0 K errors in temperature, respectively. Estimations are not always this good, however. For example at the normal boiling point, the calculated error in vapor pressure for 2-methyl-2pentanethiol is 126%. In general one can expect estimates of vapor pressures to be within a factor of 2 although in many cases, estimates are likely to be much better. These examples serve to point out the desirability of obtaining at least some data for the compound of interest. While the method presented here provides a reasonable first estimate of the vapor pressure, it is pwible to increase remarkably the accuracy of one’s predictions by obtaining no more than one datum by using, for example, the SWAP method (Smith et al., 1976; Edwards et al., 1980). For instance, using SWAP at the reported normal boiling points of carbazol and 2-methyl-2-pentanethio1, the calculated relative errors in vapor pressure are 4% and lo%, respectively. Only in the absence of any experimental data should a group-contribution method by used, and then with caution. Acknowledgment For financial support the authors are grateful to the Fossil Energy Program, Assistant Secretary of Energy Technology, United States Department of Energy. Appendix I. Carbazole. Refer to Table I1 for calculations. Using eq 2,3,4,5, and 6 we obtain the constanb for eq 1 A = In (82.06/91.52) + (11.15 - 0.5) In (13499) In (5169264) + In (0.0966) = 83.38
B = -13499 C = 1.5 - 11.15 = -9.65
D = (11.15 - 1)/13499 = 7.52
X
lo-*
E = (11.15 - 3)(11.15 - 1)/(2)(13499) = 2.27
X
lo-’
At 627.64 K the calculated vapor pressure is 994.5 torr.
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 283 Table VI. Comparison of Experimental and Predicted Vapor Pressures at 200 and 760 torr compound Tm pcalcd % error T , 6 0 dimethylamineO triethylaminea pyridinea 2-propylamine a ethanethiol 3-methy lbenzenethiol
2-methylthiophene 3,4-dithiahexane 2-thiabutane thiacy clohexane thiazole“ 2,5-dimethylpyrrole
249.62 323.15 346.30 277.07 274.95 419.75 344.85 382.11 303.37 370.38 353.19 397.62
177.1 192.2 211.0 169.1 187.3 215.8 185.5 184.4 193.4 214.1 205.0 159.6
2.9 1.4 12.5 27.6 6.3 7.9 7.2 7.8 3.1 7.1 4.5 14.9
280.04 362.0 388.38 304.92 308.15 468.25 385.71 427.13 339.80 414.89 391.41 440.64
pcalcd
849.0 758.5 903.1 621.1 774.3 799.2 727.5 779.2 758.1 847.7 990.1 686.5
% error
11.7 0.2 18.8 18.3 1.9 5.2 4.3 2.5 0.2 11.5 30.2 9.7
Indicates compound for which datum at exactly 200 torr is not available, Available datum closest to 200 torr is used instead.
The reported vapor pressure is 760 torr; the error is 31% . 11. 2,3-Dimethylthiophene. Refer to Table I11 for calculations. Using eq 2,3,4,5, and 6 we obtain the constanta for eq 1 A = In (82.06/65.64)+ (8.300- 0.5) In (7320)In (9281)+ In (0.0966)= 58.16 B = -7320 C (1.5- 8.300)= -6.8 D = (8.300- 1)/7320 = 9.973 X lo4 E = (8.300- 3)(8.300 - 1)/(2)(7320)2= 3.610 X lo-’
At 141.6 OC the calculated vapor pressure is 758.7 torr. The reported vapor pressure is 760 torr; the error is 0%.
Abrams, D. S.; Massaldl, H. A.; Rausnitz, J. M. Ind. Eng. Chem. Fundem. i m , 13, 250; Errata w r r , 16, 392. Bondl, A. A. “Physical Ropertles of Molecular Crystals, Uquids, and Qiasses”; W y : New York, 1968. Boubk, T.; Fried, V.; Mia, E. “The Vapor Pressures of Pure Substances”; Elaevier: Amsterdam, 1073. Edwards. D. R.; Van de Rostyne, C. 0.;Wlnnick, J.; Rausnitz, J. M. Ind. €ng. Ch8m. Ruces.9 Dss. Dev. 1981, 20, 138. Macknlck, A. B.; Rausnlh, J. M. Ind. Eng. Chem. Fundem. 1979 18, 348. Macknick, A. B.; Winnlck, J.; Rausnitz, J. M. Ind. Eng. Chem. Fundem. 1877, 16, 302. Moelwyn-Hqbs, E. A. “Physicai Chemistry”, 2nd Ed.; Pergamon Press: New York, 1061. New,J. A.; Mead, R. Conput. J. 1 9 1 , 7, 308. Smlth, G.; Winnick, J.; A b ” , D. S.; Rausnitz, J. M. Can. J. Chem. Eng. 1878, 54, 337. WIyldt, R. C.; Zwollnskl. B. J. “Handbook of Vapor Pressures and Heats of Vaporization of Hydrocarbons and Related Compounds“, APIRP 44;Thecmodynamics Reeearch Center: Department of Chemistry, Texas A 8 M University, (%liege Station, Texas, 1071.
Literature Cited Abramowitz, M.; Stegun, I. A. Ed. “Handbook of Mathematical Functions”, 9th printing; Dover: New York, IOW p 257.
Received for review September 28, 1980 Accepted April 27, 1981
