a new group contribution method based on equation of state parameters to evaluate the critical...

VOLUME 84, AUGUST 2006 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 431

INTRODUCTION

Supercritical fl uid extraction has experienced important developments during the last decades, especially with the use of carbon dioxide as an extracting fl uid of biomolecules. The subject has become then an interesting source for chemical, thermodynamic and process research (Brennecke and Eckert, 1989; McHugh and Krukonis, 1994). Modelling and simulation of supercritical fl uid separation processes have also received some attention and several equations of state models have been proposed. These nowadays commonly used methods require of the knowledge of the critical properties of the fl uids involved in the supercritical mixtures and of the acentric factor. However, in many cases these properties cannot be evaluated in experimental form, but still the values are required for the application of equations of state to the calculation of properties of pure fl uids and mixtures (Valderrama, 2003). The so-called group contribution methods have been commonly used to estimate the critical properties of those substances for which these properties are not available.

A New Group Contribution Method based on Equation of State Parameters to Evaluate the Critical Properties of Simple and Complex Molecules

Jos O. Valderrama1, 2* and Vctor H. Alvarez2

1. Faculty of Engineering, Mechanical Engineering Department, University of La Serena, Casilla 554, La Serena, Chile

2. Centro de Informacin Tecnolgica, Casilla 724, La Serena, Chile

Among the several proposals presented in the literature, the method developed by Lydersen (1955) is perhaps the most widely used group contribution method to estimate critical properties. Similar proposals were presented by Ambrose (1978) and Klincewicz and Reid (1984). Later, Joback and Reid (1987) developed a method that is frequently mentioned in the literature and used in several applications. In all these methods, the property of a compound is calculated by summing up the contributions of certain defi ned groups of atoms, considering at the same time the number frequency of each group occurring in the molecule. Although all these methods have been questioned in the literature (Poling et al., 2001), they have the advantage of quick estimates without requiring sophisticated computational calculations.

Lydersen (1955) defi ned 43 structural groups and proposed the following model equations for the critical properties:

A new group contribution method to evaluate the critical properties (temperature, pressure and volume) is presented and applied to estimate the critical properties of biomolecules. Similar to other group contribution methods, the one proposed here divides the molecule into conven-iently defi ned groups and evaluates the properties as the sum of the different contributions according to a specifi ed model equation for each of the properties. The proposed method consists of a one-step calculation that uses simple model equations and does not require additional data besides the knowledge of the structure of the molecule, except for isomers. For these substances the normal boiling temperature, the molecular mass and the number of atoms in the molecule are used to distinguish among isomers. The method is applicable to high molecular weight compounds, as most biomolecules and large molecules present in natural products.

On prsente une nouvelle mthode de contribution de groupe pour valuer les proprits critiques (temprature, pression et volume) de biomolcules. Comme dans le cas dautres mthodes de contribution de groupe, celle quon prsente ici divise la molcule en groupes dfi nis de manire pratique et value les proprits comme la somme des diffrentes contributions selon une quation de modle spcifi que pour chacune des proprits. La mthode propose consiste en un calcul en une tape qui utilise des quations de modle simple et, except pour les isomres, ne requiert pas de donnes supplmentaires hormis la structure de la molcule. Pour ces substances, on utilise la temprature dbullition normale, la masse molculaire et le nombre datomes dans la molcule pour distinguer entre les isomres. La mthode est applicable des composs de poids molculaire lev, comme la plupart des biomolcules et des molcules larges prsentes dans les produits naturels.

Keywords: group contribution, critical properties, molecular structure, van der Waals

* Author to whom correspondence may be addressed.E-mail address: [email protected]

432 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 84, AUGUST 2006

TT

A n nc

b

L i T i TL L

=

+ ( ) 2 (1)

PM

C nc

L i PL

=

+ 2 (2)

V E nc L i VL= + (3)In these equations ni is the number of times that a group appears in the molecule, Tb is the normal boiling temperature, TL is the contribution to the critical temperature, PL is the contribution to the critical pressure, VL is the contribution to the critical volume, M is the molecular mass, and AL, CL and EL are constants. The values of these constants are: AL=0.567, CL=0.34 and EL=40.

Joback and Reid (1987) defi ned 41 structural groups and proposed the following model equations for the critical properties :

TT

A B n nc

b

J J i T i TJ J

=

+ ( ) 2 (4)

PC D N n

c

J J i PJ

=

+ 1

2

(5)

V E nc J i VJ= + (6)In these equations ni is the number of times that a group appears in the molecule, N is the number of atoms in the molecule, Tb is the normal boiling temperature, TJ is the contribution to the critical temperature, PJ is the contribution to the critical pressure, VJ is the contribution to the critical volume, and AJ, BJ, CJ, DJ and EJ are constants. The values of these constants are: AJ=0.584, BJ=0.965, CJ=0.113, DJ=0.0032 and EJ=17.5.

Constantinou et al. (1993), proposed a complex estimation technique, which is based on conjugate forms (alternative formal arrangements of valence electrons). This technique provides reasonable estimations of several properties of pure compounds and allows capturing the differences among isomers. However, the generation of conjugate forms is a nontrivial issue and requires a symbolic computing environment. Another somewhat complex method has been proposed by Constantinou and Gani (1994). The method considers the molecular structure of the molecule and estimates a given property at two levels. The primary level uses contributions from simple groups that allow describing a wide variety of organic compounds while the higher level involves polyfunctional and structural groups that provide more information about molecular fragments whose description through fi rst-order groups is not possible. Joback (2003) has shown that the method of Constantinou and Gani (1994) gives errors a little higher than Lydersens method. Marrero and Pardillo (1999), proposed another technique that considers the contributions of interactions between bonding groups instead of the contributions of simple groups, method that allows distinguishing among isomers.

Group Contribution Methods have also been proposed for mixture properties, being the most recent one that proposed by

Jaubert and coworkers to estimate the binary interaction parameter kij for the Peng-Robinson equation of state (Jaubert and Mutelet, 2004; Jaubert et al., 2005; Vitu et al., 2006). To the best of our knowledge the most recent Group Contribution Method proposed in the literature to estimate the critical proper-ties is that of Skander and Chitour (2003). The authors consid-ered the estimation of critical properties, heat of vaporization, refractive index parameter and molar volume of hydrocarbons and petroleum fractions. In another development, Kolsk (2005) presented a group contribution method to estimate the enthalpy and entropy of vaporization of pure organic compounds. Stefanis et al. (2004) have extended the method of Constantinou and Gani (1994), and applied it to the calculation of the octanol-water partition coeffi cient, the solubility parameters at 25C, and the fl ash point.

As described above, group contribution methods have been commonly used to estimate critical properties. However, most applications of equations of state also require the knowledge of the acentric factor , which can be calculated from the vapour pressure and the defi nition of the acentric factor: =-Log[Pvap/Pc] at T/Tc=0.7-1 (Prausnitz et al., 1999). If vapour pressure data are not available to calculate the acentric factor, a good estimation can be done using the critical properties and the normal boiling temperature, assuming that the vapour pressure and the temperature follow the classical relation Log Pvap=A-B/T. Applying this equation to both the normal boiling point and the critical point, and using the defi nition of the acentric factor, this can be estimated as: =(3/7)TbxLog[Pc/Pb]/(Tc-Tb) - 1. Here, Pc is in MPa, Pb=0.101325 MPa, Tb and Tc are in Kelvin.

In this paper, the 180 substances shown in Table 1, for which the critical properties are known, have been considered to propose a new simple and accurate group contribution method. Most of the values are those indicated as experimental data in the DIPPR compilation of Daubert et al. (1996). The substances designated with (1) in Table 1 were obtained from the Korean Data Base available on Internet (KBD, 2003).

The proposed one-step method does not require the normal boiling temperature (except to distinguish among isomers), and uses simple expressions for the critical temperature, the critical pressure and the critical volume. The model equations are based on an idea of group contribution applied to estimate the force and the volume constants of the van der Waals equation of state presented by Vukalovich and Novikov (1948). Large molecules have been considered in the proposed method to determine the contributions for each of the defi ned groups so the technique could be applied to large complex substances such as those present in many natural products.

THE NEW METHODVukalovich and Novikov (1948) presented an extensive analysis of the van der Waals equation of state. In that analysis the authors discussed the characteristic of the equation of state constants a and b and related them to atoms and groups that form a given molecule. From a theoretical analysis, they concluded that the volume constant b could be determined by summing up the values of certain defi ned groups bi while for a the contributions are for a . That is:

a n ai i= (7)b n bi i= (8)


that affect the critical properties of isomers are the normal boiling point and the predicted critical properties (Tc, Pc and Vc), calculated using Equations (11) to (13). To distinguish between groups of isomers the variables to consider are the molecular mass (M) and the number of atoms in the molecule (N).

Thus, the critical properties for isomers are estimated as for any other compound and then correction functions Tc, Pc and Vc are added. These correction functions include the properties Tc, Pc and Vc, calculated using Equations (11) to (13).

T a a T T a M a Nc T T b c T T= + + +1 2 3 41 ( / ) (14)P a a T T a M a Nc P P b c P P= + + +1 2 3 41 ( / ) (15)

V a a T T a M a Nc V V b c V V= + + +1 2 3 41 ( / ) (16)It should be noticed that the terms containing the molecular mass (M) and the number of atoms in the molecule (N) allow distinguishing between groups of isomers (for instance the group of dimethyl hexanes from the group of xylenes). The term containing the normal boiling temperature allows distinguishing among a group of isomers of the same substance (for instance between o-xylene, m-xylene and p-xylene). It should be also noticed that for isomers the method loses its predictive capabili-ties since the normal boiling temperature is needed.

Finally, the critical properties for the isomers are calculated as:

T T Tciso

c c= + (17)

P P Pciso

c c= + (18)

V V Vciso

c c= + (19)

The parameters aTi, aPi, and aVi in Equations (14) to (16) were determined using data for 40 isomers, already included among the 180 substances shown in Table 1.

Results and DiscussionA computer program to evaluate these contributions and constants was specially developed. The program considers the use of a Modifi ed Marquardt method (Reilly, 1972) as a basic numerical algorithm, and explores for multiple acceptable solutions to choose as a fi nal optimum solution that which give the lowest deviation of an established objective function, F. In the proposed method the objective function considers the Ns=180 compounds listed in Table 1 and is defi ned as:

F Q QcalNs

= ( ) exp 21

(20)

It should be observed that the variables Q used for the regres-sion analysis are T Pc c/ , Tc/Pc and Vc. However, to analyze the results obtained using the new method, a percent deviation has been defi ned for each of the critical properties, as follows:

%exp

expTT T

Tc

c ccal

c i

=

(21)

%exp

expPP P

Pc

c ccal

c i

=

(22)

Here, ni is the number of times that the atom or group i appears in the molecule, ai and bi are the contributions to a and b. These contributions are given by Vukalovich and Novikov for atoms and groups of atoms (25 for a and 35 for b).

The relations (7) and (8) are based on the van Laar concepts about the van der Waals constants for the case of mixtures: b=xibi and a x ai i= , being xi the mole fraction of component i in the mixtures. Vukalovich and Novikov consid-ered xi as the number of groups in the molecule.

The expressions for a and b for the van der Waals equation of state written in terms of the critical properties and the ideal gas constant R, are (Prausnitz et al., 1999):

aR TP

c

c=

2764

2 2 (9)

bRTPc

c=

8 (10)

Replacing Equations (9) and (10) into Equations (7) and (8) and cancelling the constants, it can be seen that the groups to be defi ned in a group contribution method are T Pc c/ , [Tc/Pc] and [Vc].

It should be mentioned here that the van der Waals equation is not directly used but only the relation between the constants a and b with the critical properties. The relation between the a and b parameters with the critical properties is the same for all equations of state derived from van der Waals equation: a= aR2Tc2/Pc; b=bRTc/Pc (being a and b different for each equation of state). Thus, there is no need of using more complex or accurate equations of state.

As any group contribution method, the one proposed here not only requires the values of the contributions T Pc c/ , [Tc/Pc] and [Vc], but also a model to add in some way those contribu-tions. After exploring several models, the following equations for these groups are defi ned:

T

Pn

T

Pc

ci

c

c i

= +

1

1

(11)

TP

nTP

c

ci

c

c i

= +

2

2

(12)

For the critical volume, a similar expression is used:

V n Vc i c i= + [ ]

3

3

(13)

In these equations, ni is the number of times that a group i appears in a molecule, i and i are constants, and T Pc c/ i, [Tc/Pc]i and [Vc]i are the contributions to T Pc c/ , [Tc/Pc]and [Vc], respectively. These contributions and the constants i and i are determined from regression analysis of experimental data of Tc, Pc and Vc. Once the groups (T Pc c/ )i and (Tc/Pc)i are known, the critical temperature and the critical pressure are directly obtained. Data for the 180 substances shown in Table 1 were used in the regression analysis and 39 groups were defi ned as shown in Table 2.

ISOMER CORRECTIONSThe proposed method has been extended to predict critical properties of isomers. We have found that the main properties


critical temperature, reason that explains the low average deviation in the prediction of this property. For Pc and Vc, however, the proposed method does not use the normal boiling temperature as input data, and give lower deviations.

The deviations shown in Table 5 depend not only on the type of substances but also on the so-called occurrence, that is the number of substances in which a given defi ned group appears in the list of 180 substances considered in the analysis. When solving the system of equations those groups having higher occurrence have greater infl uence in determining the values of the group contributions. Apart from this there is no other numerical or convergence problem when determining the values of the 117 groups (3 values for each of the 39 groups).

Finally, as an additional test, eighteen substances, mostly biomolecules such as -pinene, cholesterol, capsaicin or astaxan-tin, not used in the calculation of the 39 groups and the six parameters i and i, have been used to evaluate the applicabil-ity of the proposed method. The results are shown in Table 8. Of the substances shown in the Table, only three of them have experimental values in the literature. For the other substances, true experimental values are not available, so the values predicted by the Joback-Reid method were used for comparison only. Deviations for these cases are not given since the experi-mental values are unknown.

CONCLUSIONSAccording to the results, the following conclusions are drawn: (i) a new simple method has been proposed to estimate the critical properties of biomolecules; (ii) the method can be applied to any organic substance that contain any of the 39 groups defi ned by the method; (iii) the method requires only the molecular mass and the structure of the molecule, and does not need the normal boiling temperature, except if corrections for isomers are introduced; (iv) the average deviations are below 0.3% for Tc, Pc and Vc while the average absolute deviations are 2.2% for Tc, 4.5% for Pc and 2.6% for Vc; and (v) it is shown that the proposed method gives results similar to other more sophisticated methods that use different levels of calculations and properties not readily available in the literature.

%exp

expVV V

Vc

c ccal

c i

=

(23)

Using the designed program, the contributions shown in Table 3 for the groups T Pc c i/

, [Tc/Pc]i and [Vc]i and for the parameters i and i were calculated. The parameters are: 1 = 38.91, 1 = 0.88, 2 = 5.84, 2 = 1.27, 3 = 26.86 and 3 = 1.06. A summary of the method is given in Table 4, while Table 5 gives the average deviations and the average absolute deviations obtained for the 180 substances using the model summarized in Table 4.

For the 40 isomers included in the list of 180 substances considered in this study the results are also separately shown in Table 6. In this Table, Tc, Pc, and Vc are the values calculated without the correction for isomers while Tc

iso, Pciso, and Vc

iso are the values calculated with the corrections of Equations (14) to (16). It can be observed that overall deviations are lower for the three properties when values are compared with the overall deviations given by the method without isomer corrections. The deviations are also lower than those found for the 180 compounds given in Table 4. More important, however, is that the corrected values are different for the different isomers, as experimentally found.

The results shown in Table 5 indicate that for most substances deviations between experimental and calculated values are lower than 5% for all properties (%S10% in the Table 5), are 2.2% for Tc, 10.6% for Pc and 3.9% for Vc. The average deviations % are 0.1% for Tc, 0.3% for Pc and 0.2% for Vc. The absolute average deviations %|| are 2.2% for Tc, 4.5% for Pc and 2.6% for Vc. These results are similar to those found using more sophisticated methods as shown in Table 7. The Lydersen and Joback-Reid results shown in Table 7 were also evaluated in this work for the 180 selected substances presented in Table 1. For the Constantinou-Gani and the Marrero-Pardillo methods the results shown in the Table were taken from Poling et al. (2000). It should be noticed that the Lydersen and the Joback-Reid methods employ the normal boiling temperature to estimate the

Table 1. Critical properties of the 180 substances chosen to evaluate the 39 group contributions and the model parameters. In the Table, the substances designated with (1) were obtained from the Korean Data Base, available on the Internet (KBD, 2003).

NS Substances Formula M (g/gmol) Tc (K) Pc (bar) Vc (cc/mol) Tb

1 Methyl_hydrazine (1) CH6N2 46.072 567.00 82.400 271.00 360.60

2 Nitromethane CH3NO2 61.040 588.15 63.126 173.41 374.35

3 Carbon_Tetrafl uoride CF4 88.005 227.50 37.400 140.00 145.09

4 Chloroform CHCl3 119.377 536.40 54.720 239.00 334.33

5 Trichlorofl uoromethane CCl3F 137.368 471.20 44.076 248.00 296.97

6 Acetonitrile C2H3N 41.052 545.50 48.500 173.00 354.80

7 Ketene (1) C2H2O 42.037 380.00 65.000 145.00 232.00

8 Ethylene_oxide C2H4O 44.053 469.15 71.900 140.30 283.60

9 Acetaldehyde (1) C2H4O 44.053 466.00 55.700 154.00 293.20

10 Dimethylamine (1) C2H7N 45.084 437.22 53.400 182.50 280.03

11 Ethanol C2H6O 46.069 513.92 61.480 167.00 351.44

12 Methyl_Isocyanate (1) C2H3NO 57.052 491.00 55.700 196.98 312.00



13 Acetic_acid C2H4O2 60.053 591.95 57.860 179.70 391.05

14 Methyl_formate C2H4O2 60.053 487.20 59.984 172.00 304.90

15 Ethylenediamine (1) C2H8N2 60.099 593.00 62.800 206.00 390.40

16 Monoethanolamine (1) C2H7NO 61.083 614.00 44.500 196.00 443.50

17 1_1-Difl uoroethane C2H4F2 66.051 386.60 44.988 181.00 247.35

18 Nitroethane (1) C2H5NO2 75.067 595.00 48.500 269.00 387.20

19 Acetyl_chloride (1) C2H3ClO 78.498 508.00 58.700 204.00 323.90

20 Bromoethane C2H5Br 108.966 503.80 62.315 214.92 311.50

21 1_2-Diiodoethane C2H4I2 281.863 749.90 47.300 323.50 473.20

22 Methylacetylene C3H4 40.065 402.39 56.276 164.00 249.94

23 Acrylonitrile (1) C3H3N 53.063 536.00 45.600 210.00 350.50

24 Propionitrile C3H5N 55.079 564.40 41.847 229.00 370.50

25 Acetone C3H6O 58.080 508.20 47.015 209.00 329.44

26 1_2-Propylene_oxide C3H6O 58.080 482.25 49.244 186.00 307.05

27 Allyl_alcohol (1) C3H6O 58.080 545.00 53.100 203.50 370.23

28 Trimethylamine C3H9N 59.111 433.25 40.733 254.00 276.02

29 Isopropylamine C3H9N 59.111 471.85 45.400 221.00 304.90

30 Isopropanol C3H8O 60.096 508.31 47.643 220.13 355.41

31 Methyl_ethyl_ether C3H8O 60.096 437.80 44.000 221.00 280.50

32 1-Propanol C3H8O 60.096 536.78 51.750 219.00 370.35

33 Acrylic_acid (1) C3H4O2 72.063 615.00 56.700 210.00 414.00

34 Propionic_acid C3H6O2 74.079 600.81 46.170 233.00 414.32

35 Ethyl_formate C3H6O2 74.079 508.40 47.420 229.00 327.46

36 Methyl_acetate C3H6O2 74.079 506.55 47.500 228.00 330.09

37 Methylal C3H8O2 76.095 480.60 39.517 213.00 315.00

38 1_2-Propylene_glycol (1) C3H8O2 76.095 625.00 60.700 237.00 460.50

39 1-Nitropropane (1) C3H7NO2 89.094 606.00 40.000 328.66 404.70

40 2-Nitropropane (1) C3H7NO2 89.094 597.00 41.500 316.10 393.30

41 Glycerol (1) C3H8O3 92.094 726.00 66.800 255.00 563.00

42 Hexafl uoroacetone C3F6O 166.023 357.14 28.371 329.00 245.88

43 2-Iodopropane (1) C3H7I 169.990 574.60 43.300 285.50 362.60

44 1_2-Butadiene (1) C4H6 54.091 443.70 44.900 219.00 284.00

45 Isobutene C4H8 56.107 417.90 39.990 238.88 266.25

46 Furan C4H4O 68.075 490.15 55.000 218.00 304.50

47 Isobutyronitrile (1) C4H7N 69.106 566.00 38.200 279.50 377.00

48 Pyrrolidine C4H9N 71.122 568.55 56.134 248.68 359.72

49 Methyl_ethyl_ketone C4H8O 72.107 535.50 41.543 267.00 352.79

50 Diethyl_ether C4H10O 74.123 466.70 36.400 280.00 307.58

51 2-Butanol C4H10O 74.123 536.05 41.790 269.00 372.70

52 2-Methyl-2-propanol C4H10O 74.123 506.21 39.730 275.00 355.57

53 2-Methyl-1-propanol C4H10O 74.123 547.78 43.000 273.00 380.81

Table 1 (continued)



54 Tetrahydro-furan C4H8O 72.107 540.15 51.878 223.93 338.00

55 Tert-butylamine C4H11N 73.138 483.90 38.400 293.00 317.55

56 Diethylamine C4H11N 73.138 496.60 37.085 301.00 328.60

57 Vinyl_acetate (1) C4H6O2 86.090 525.00 43.500 265.00 346.00

58 Methyl_acrylate (1) C4H6O2 86.090 536.00 43.000 265.00 353.50

59 Morpholine (1) C4H9NO 87.121 618.00 54.700 253.00 401.40

60 Ethyl_acetate C4H8O2 88.106 523.30 38.800 286.00 350.21

61 Methyl_propionate C4H8O2 88.106 530.60 40.040 282.00 352.60

62 N-propyl_formate C4H8O2 88.106 538.00 40.631 285.00 353.97

63 1_4-Dioxane C4H8O2 88.106 587.00 52.081 238.00 374.47

64 1_2-Dimethoxyethane C4H10O2 90.122 536.15 38.706 270.64 357.20

65 1_3-Butanediol C4H10O2 90.122 676.00 40.200 305.00 481.38

66 1-Nitrobutane (1) C4H9NO2 103.121 624.00 38.000 347.94 426.10

67 2-Nitrobutane (1) C4H9NO2 103.121 615.00 36.000 372.76 412.90

68 2-Bromobutane (1) C4H9Br 137.019 558.70 43.000 315.50 364.40

69 3-Methyl-1-butyne (1) C5H8 68.118 476.00 42.100 271.50 299.50

70 3-Methyl-1_2-butadiene (1) C5H8 68.118 496.00 41.100 267.00 314.00

71 1_2-Pentadiene (1) C5H8 68.118 503.00 40.700 276.00 318.00

72 2-methyl-1-butene C5H10 70.134 465.00 34.000 292.00 304.30

73 2-methyl-2-butene C5H10 70.134 471.00 34.000 292.00 311.71

74 n-Methylpyrrole C5H7N 81.117 596.00 48.500 271.90 385.80

75 Cyclopentanone (1) C5H8O 84.118 624.50 46.000 268.00 403.72

76 Piperidine C5H11N 85.149 594.10 49.400 288.00 379.37

77 1-Pentanal C5H10O 86.134 566.10 39.700 313.00 376.15

78 Diethyl_ketone C5H10O 86.134 560.95 37.389 336.00 375.14

79 2-Pentanone C5H10O 86.134 561.08 36.943 301.00 375.46

80 Pyridine (1) C5H5N 79.101 620.00 56.700 243.00 388.38

81 1-Pentanal C5H10O 86.134 566.10 39.700 313.00 376.15

82 2-Methyl_tetrahydrofuran (1) C5H10O 86.134 537.00 37.600 267.00 351.00

83 n-Methyl-2-Pyrrolidone C5H9NO 99.133 721.60 45.200 310.00 477.42

84 Ethyl_acrylate (1) C5H8O2 100.117 552.00 37.400 320.00 373.00

85 Ethyl_propionate C5H10O2 102.133 546.00 33.620 345.00 372.25

86 Methyl_n-butyrate C5H10O2 102.133 554.50 34.730 340.00 375.90

87 Benzene C6H6 78.114 562.05 48.950 256.00 353.24

88 1-Hexene C6H12 84.161 504.03 31.400 354.00 336.63

89 Aniline C6H7N 93.128 699.00 53.094 270.00 457.60

90 2-Methylpyridine C6H7N 93.128 621.00 46.000 335.00 402.55

91 4-Methylpyridine C6H7N 93.128 646.15 46.610 325.62 418.50

92 Phenol C6H6O 94.113 694.25 61.300 229.00 454.99

93 Fluorobenzene C6H5F 96.104 560.09 45.505 269.00 357.88

94 Cyclohexanone (1) C6H10O 98.144 653.00 40.000 346.99 428.58

95 2-Hexanone C6H12O 100.161 587.61 32.870 378.00 400.70

Table 1 (continued)



96 3-Hexanone C6H12O 100.161 582.82 33.200 378.00 396.65

97 Triethylamine C6H15N 101.192 535.15 30.398 390.00 361.92

98 Diisopropyl_ether C6H14O 102.177 500.05 28.800 386.00 341.45

99 1-Hexanol C6H14O 102.177 611.35 35.100 381.00 430.55

100 Monochlorobenzene C6H5Cl 112.558 632.35 45.191 308.00 404.87

101 Isobutyl_acetate (1) C6H12O2 116.160 561.00 31.600 414.00 389.70

102 Hexanoic_acid C6H12O2 116.160 659.10 33.080 377.20 478.85

103 2-Butoxy_ethanol C6H14O2 118.176 633.90 32.700 423.57 444.47

104 Bromobenzene C6H5Br 157.010 670.15 45.191 324.00 429.24

105 Iodobenzene C6H5I 204.010 721.15 45.191 351.00 461.60

106 1-Methylcyclohexene (1) C7H12 96.172 584.20 35.690 353.60 383.20

107 1-Heptene C7H14 98.188 537.29 28.300 413.00 366.79

108 1_1-Dimethylcyclopentane C7H14 98.188 547.00 34.450 360.00 361.00

109 Benzonitrile C7H5N 103.123 699.35 42.150 313.20 464.15

110 p-Toluidine C7H9N 107.155 693.15 40.000 346.00 473.57

111 m-Cresol C7H8O 108.140 705.85 45.600 312.00 475.43

112 o-Cresol C7H8O 108.140 697.55 50.055 282.00 464.15

113 p-Cresol C7H8O 108.140 704.65 51.500 277.00 475.13

114 2-Heptanone C7H14O 114.188 611.40 29.400 434.00 424.18

115 1-Heptanol C7H16O 116.203 631.90 31.500 435.00 449.45

116 Benzoic_acid (1) C7H6O2 122.123 752.00 45.600 341.00 523.00

117 Heptanoic_acid C7H14O2 130.187 676.84 30.430 429.70 496.15

118 Diethylene_Glycol_monopropyl_ether C7H16O3 148.202 679.80 30.020 489.00 487.99

119 Perfl uoromethylcyclohexane (1) C7F14 350.055 485.91 20.190 570.00 349.50

120 Perfl uoroheptane (1) C7F16 388.051 474.80 16.200 664.00 355.66

121 m-Xylene C8H10 106.167 617.00 35.410 375.00 412.27

122 o-Xylene C8H10 106.167 630.30 37.320 370.00 417.58

123 p-Xylene C8H10 106.167 616.20 35.110 378.00 411.51

124 1-octene C8H16 112.215 566.60 25.500 472.00 394.44

125 1_1-Dimethylcyclohexane C8H16 112.215 591.15 29.384 450.00 392.70

126 n-Propylcyclopentane (1) C8H16 112.215 603.00 30.000 425.00 404.10

127 2_2-Dimethylhexane C8H18 114.231 549.80 25.300 478.00 379.99




131 Acetophenone (1) C8H8O 120.151 709.50 40.600 386.00 475.00

132 n_n-Dimethylaniline (1) C8H11N 121.182 687.00 36.300 406.17 467.30

133 1-octanol C8H18O 130.230 652.50 28.600 490.00 468.35

134 Methyl_benzoate (1) C8H8O2 136.150 692.00 36.400 396.00 472.20

135 Octanoic_acid C8H16O2 144.214 694.26 27.790 499.00 512.85

136 Phthalic_anhydride (1) C8H4O3 148.118 810.00 47.600 368.00 560.00

137 n-propylbenzene C9H12 120.194 638.35 32.000 440.00 432.39

Table 1 (continued)



138 Cumene C9H12 120.194 631.00 32.090 434.00 425.56

139 Mesitylene C9H12 120.194 637.30 31.270 433.00 437.89

140 n-Propylcyclohexane C9H18 126.242 639.15 28.067 477.00 429.90

141 2_2_4_4-Tetramethylpentane C9H20 128.258 571.35 23.609 504.00 395.44

142 2_2_3_3-Tetramethylpentane C9H20 128.258 610.85 27.358 478.00 413.44

143 2_2_5-trimethylhexane C9H20 128.258 568.05 23.305 519.00 397.24

144 Isoquinoline (1) C9H7N 129.161 803.00 51.000 374.00 516.37

145 Quinoline (1) C9H7N 129.161 782.00 48.600 437.00 510.31

146 5-Nonanone C9H18O 142.241 640.00 23.200 560.00 461.60

147 1-Nonanal (1) C9H18O 142.241 640.00 24.800 556.50 461.60

148 Naphthalene C10H8 128.171 748.40 40.500 407.00 491.10

149 1_2_3_4-Tetrahydronaphthalene C10H12 132.205 720.00 36.500 408.00 480.77

150 n-butylbenzene C10H14 134.221 660.50 28.900 497.00 456.46

151 Isobutylbenzene (1) C10H14 134.221 650.00 30.500 480.00 445.94

152 p-cymene C10H14 134.221 652.00 28.000 497.00 450.28

153 cis-Decahydronaphthalene C10H18 138.253 702.25 32.424 480.00 468.97

154 trans-Decahydronaphthalene C10H18 138.253 687.05 28.371 480.00 460.46

155 1-decene C10H20 140.269 617.05 21.684 585.00 443.75

156 1-Decanal (1) C10H20O 156.267 674.20 22.600 612.50 481.60

157 n-Pentylbenzene C11H16 148.248 679.90 26.040 550.00 478.61

158 n-Butyl_benzoate (1) C11H14O2 178.230 723.00 26.000 561.00 523.00

159 Biphenyl C12H10 154.211 773.00 33.800 497.00 528.15

160 1_2-Dimethylnaphthalene (1) C12H12 156.227 775.30 30.100 521.50 539.50








168 Dibenzopyrrole (1) C12H9N 167.210 901.80 39.300 502.00 627.85

169 Di-n-hexyl_ether (1) C12H26O 186.337 657.00 18.200 720.00 499.60

170 Diphenylmethane C13H12 168.238 760.00 27.100 563.00 537.42

171 n-Nonylbenzene C15H24 204.356 741.00 18.950 790.00 555.20

172 Pyrene (1) C16H10 202.255 938.20 27.210 626.00 666.00

173 1-Hexadecanol (1) C16H34O 242.444 770.00 16.100 950.50 607.00

174 Undecylbenzene C17H28 232.409 764.00 16.720 910.00 586.40

175 m-Terphenyl C18H14 230.309 883.00 24.800 724.00 648.15

176 o-Terphenyl C18H14 230.309 857.00 29.900 731.00 609.15

177 p-Terphenyl C18H14 230.309 908.00 29.900 729.00 655.15

178 Triphenylene (1) C18H12 228.293 1013.20 25.110 700.00 698.00

179 n-Tridecylbenzene (1) C19H32 260.462 784.00 15.000 1040.00 614.40

180 1-Nonadecanol (1) C19H40O 284.524 775.30 11.500 1118.50 631.00

Table 1 (continued)


Table 2. Defi ned groups for the proposed method

Non-ring groups

1 -CH3

2 -CH2-

3 >CH-

4 >CN-

22 =N-

23 -CN

24 -NO2

25 -F

26 -Cl

27 -Br

28 -I

Ring groups

29 -CH2-

30 >CH-

31 =CH-

32 >CN-

39 =N-

Table 3. Values for the contributions to T Pc c/ , Tc/Pc and Vc for the 39 groups and values of the constants in the model Equations (11) to (13)

Non-ring groups T Pc c/ Tc/Pc Vc

1 -CH3 8.26 0.8446 45.42

2 -CH2- 20.07 1.2910 37.60

3 >CH- 27.11 1.4806 30.00

4 >C< 35.81 1.6350 13.28

5 =CH2 3.27 0.4418 41.36

6 =CH- 18.81 1.1606 33.48

7 =C< 26.64 1.4472 21.93

8 =C= 15.40 0.6119 21.56

9 =(-)CH 6.46 0.2734 28.60

10 =(-)C- 6.46 0.2734 28.60

11 -OH 11.21 0.0881 20.75

12 -O- 9.93 0.3628 10.81

13 >C=O 34.18 1.5195 47.41

14 -CHO 21.98 0.7953 52.80

15 -COOH 50.52 1.9871 60.10

16 -COO- 40.07 1.9804 56.69

17 -HCOO- 28.83 1.3779 65.87

18 =O(Any_other) -7.64 -0.9769 26.61

19 -NH2 13.70 0.4922 32.98

20 -NH- 15.00 0.2946 51.67

21 >N- 24.66 0.9936 17.34

22 =N- 26.42 1.8782 32.62

23 -CN 49.30 2.5229 58.84

24 -NO2 48.83 2.0970 83.38

25 -F -7.25 0.0210 21.46

26 -Cl 8.28 0.5375 41.72

27 -Br 16.07 0.5006 51.66

28 -I 41.78 1.7649 68.45

Ring groups

29 -CH2- 15.09 0.9240 34.69

30 >CH- 21.92 1.0535 20.42

31 =CH- 11.23 0.6178 28.49

32 >C< 31.39 1.3134 11.00

33 =C< 27.40 1.2796 20.63

34 -O- 2.13 -0.0201 10.59

35 -OH(Phenols) 1.17 -0.4922 -11.19

36 >C=O 27.48 1.0798 40.67

37 -NH- 4.28 -0.4429 19.04

38 >N- 33.80 1.2695 16.14

39 =N- 9.23 -0.0791 29.24

1 = 38.91 2 = 5.84 3 = 26.86

1 = 0.88 2 = 1.27 3 = 1.06


Table 4. Summary of the equations that describe the proposed group contribution method

All compounds

T

Pg n

T

Pc

ci

c

c i

= = +

1 11

TP

g nTP

c

ci

c

c i

= = +

2 22

Pc = (g1 / g2)2

Tc = Pc x g2

V n Vc i c i= + [ ]

3

3

1 = 38.91 2 = 5.84 3 = 26.86

1 = 0.88 2 = 1.27 3 = 1.06Isomers

T T T P P P V V Vciso

c c ciso

c c ciso

c c= + = + = +

T a a T T a M a Nc T T b c T T= + + +1 2 3 41 ( / )*

P a a T T a M a Nc P P b c P P= + + +1 2 3 41 ( / )*

V a a T T a M a Nc V V b c V V= + + +1 2 3 41 ( / )*

aT1 = - 281.22 aP1 = - 20.47 aV1 = +105.82

aT2 = - 1.587 aP2 = - 1.460 aV2 = -1.505

aT3 = - 0.1931 aP3 = 0.0106 aV3 = 0.0856

aT4 = 0.0531 aP4 = - 0.0464 aV4 = - 0.4402

Table 5. Deviations for the calculated critical properties using the new method for the 180 compounds. In the table, %S>10% is the number of substances with average absolute deviation greater than 10%, %S


Components %Tc %Pc %Vc42 Hexafl uoroacetone 22.5 15.1 -6.3

43 2-Iodopropane 1.7 0.4 1.1

44 1-2-Butadiene 3.3 2.3 0.2

45 Isobutene 1.7 -2.5 -0.7

46 Furan 6.4 5.8 -10.7

47 Isobutyronitrile 1.0 0.1 -1.8

48 Pyrrolidine 3.3 1.4 -2.5

49 Methyl ethyl ketone -2.6 -0.4 0.7

50 Diethyl ether 3.7 3.4 -3.5

51 2-Butanol 3.7 2.9 0.5

52 2-Methyl-2-propanol 2.7 1.1 -1.8

53 2-Methyl-1-propanol -0.4 -0.1 0.3

54 Tetrahydro-furan -1.6 -7.5 2.9

55 Tert-butylamine 10.0 7.4 -4.9

56 Diethylamine 5.2 10.7 9.8

57 Vinyl acetate -0.6 -4.1 3.1

58 Methyl acrylate -1.5 -2.0 2.3

59 Morpholine -1.9 7.8 2.0

60 Ethyl acetate 0.3 -0.4 -1.0

61 Methyl propionate -0.7 -3.1 0.2

62 N-propyl formate 0.5 0.9 -0.2

63 1 4-Dioxane -6.5 -4.5 3.2

64 1-2-Dimethoxyethane -2.8 -0.8 5.7

65 1 3-Butanediol -2.9 19.0 -4.1

66 1-Nitrobutane 1.4 -3.8 5.2

67 2-Nitrobutane 1.5 3.5 -1.8

68 2-Bromobutane -2.5 -6.1 1.2

69 3-Methyl-1-butyne 1.9 3.5 0.2

70 3-Methyl-1-2-butadiene -2.2 -1.8 0.6

71 1-2-Pentadiene -0.7 -0.3 -0.7

72 2-methyl-1-butene 0.4 3.1 0.0

73 2-methyl-2-butene 0.1 0.1 0.0

74 n-Methylpyrrole 5.2 1.2 -1.3

75 Cyclopentanone -1.5 1.8 2.3

76 Piperidine 3.1 3.4 1.8

77 1-Pentanal 0.7 -1.6 2.4

78 Diethyl ketone -0.5 -2.6 -4.2

79 2-Pentanone -0.5 -1.4 6.9

80 Pyridine -0.2 10.3 8.1

81 1-Pentanal 0.7 -1.6 2.4

82 2-Methyl tetrahydrofuran 3.4 14.6 3.2

83 n-Methyl-2-Pyrrolidone -4.5 -1.8 1.1

84 Ethyl acrylate 0.4 -1.3 1.8

85 Ethyl propionate 1.3 2.0 -2.1

86 Methyl n-butyrate 0.3 -0.7 -1.0

Components %Tc %Pc %Vc87 Benzene 0.6 3.8 2.2

88 1-Hexene 1.6 0.4 -0.2

89 Aniline -3.6 -4.6 10.5

90 2-Methylpyridine 1.5 8.8 -2.3

91 4-Methylpyridine -0.8 8.9 -0.6

92 Phenol -1.8 -2.7 2.2

93 Fluorobenzene 1.6 0.9 4.7

94 Cyclohexanone -2.8 6.0 -6.3

95 2-Hexanone -0.1 -0.4 -0.8

96 3-Hexanone 0.2 -2.0 -0.5

97 Triethylamine 2.3 -3.0 3.3

98 Diisopropyl ether 5.0 6.8 -1.0

99 1-Hexanol -1.2 -7.1 1.0

100 Monochlorobenzene -0.2 4.3 1.1

101 Isobutyl acetate 1.3 -1.8 -4.8

102 Hexanoic acid 1.3 1.5 2.7

103 2-Butoxy ethanol 0.0 0.2 -5.4

104 Bromobenzene 2.9 14.7 0.6

105 Iodobenzene 3.0 3.6 0.0

106 1-Methylcyclohexene 3.0 2.2 0.0

107 1-Heptene 0.4 0.2 -0.9

108 1-1-Dimethylcyclopentane 7.5 0.1 1.2

109 Benzonitrile 3.1 -2.0 7.5

110 p-Toluidine 0.6 6.6 2.3

111 m-Cresol -0.1 8.2 -7.2

112 o-Cresol 0.1 -2.4 3.6

113 p-Cresol 0.1 -4.2 4.5

114 2-Heptanone -2.3 -1.9 0.5

115 1-Heptanol -1.3 -7.6 1.4

116 Benzoic acid 3.6 4.7 -0.7

117 Heptanoic acid 0.8 -1.7 3.3

118 Diethylene Glycol monopropyl ether -0.4 -2.3 -3.1

119 Perfl uoromethylcyclohexane -1.6 -1.5 0.8

120 Perfl uoroheptane -6.6 -5.0 -0.3

121 m-Xylene 1.0 2.6 0.2

122 o-Xylene -0.5 -1.9 1.2

123 p-Xylene 1.1 3.4 -0.5

124 1-octene -0.6 0.7 -1.3

125 1-1-Dimethylcyclohexane 2.8 8.5 -7.5

126 n-Propylcyclopentane 1.0 0.1 -0.5

127 2-2-Dimethylhexane 1.2 0.4 -2.7

128 2 3-Dimethylhexane -0.3 -2.4 0.3

129 2 4-Dimethylhexane 0.6 -1.1 -0.1

130 2 5-Dimethylhexane 1.2 1.7 -2.1

Table 5 (continued)


Components %Tc %Pc %Vc131 Acetophenone -2.4 -0.4 0.3

132 n n-Dimethylaniline -4.2 1.8 1.0

133 1-octanol -1.7 -8.2 1.6

134 Methyl benzoate -0.4 4.5 1.3

135 Octanoic acid 0.3 -3.2 0.3

136 Phthalic anhydride 4.6 -1.5 1.7

137 n-propylbenzene 0.7 3.4 -2.5

138 Cumene 0.6 4.9 -1.1

139 Mesitylene 3.5 3.4 -1.0

140 n-Propylcyclohexane -2.0 -0.3 -0.3

141 2-2-4-4-Tetramethylpentane 0.9 3.5 -0.2

142 2-2-3-3-Tetramethylpentane -3.4 -7.3 4.2

143 2-2-5-trimethylhexane 0.4 0.7 0.7

144 Isoquinoline -3.2 -10.9 10.6

145 Quinoline -1.0 -7.0 -5.1

146 5-Nonanone -0.8 2.2 -1.7

147 1-Nonanal 1.5 0.3 -1.8

148 Naphthalene 0.3 2.4 0.8

149 1-2-3 4-Tetrahydronaphthalene -1.1 -2.7 0.0

150 n-butylbenzene 1.2 -7.3 8.9

151 isobutylbenzene -0.2 2.1 -2.3

152 p-cymene 0.3 -1.8 1.3

153 cis-Decahydronaphthalene 1.3 5.9 -2.2

154 trans-Decahydronaphthalene -1.7 -8.4 -0.2

155 1-decene -0.4 3.4 0.2

156 1-Decanal -2.0 -0.7 -0.8

157 n-Pentylbenzene -1.3 0.6 -1.3

158 n-Butyl benzoate -0.8 2.1 -1.3

Components %Tc %Pc %Vc159 Biphenyl 0.5 3.3 2.0

160 1-2-Dimethylnaphthalene -0.8 5.3 -0.8





165 1-7-Dimethylnaphthalene 0.2 2.5 -0.6

166 2-3-Dimethylnaphthalene 0.2 2.5 -0.6


168 Dibenzopyrrole -0.3 -3.1 -0.8

169 Di-n-hexyl ether 0.6 -1.8 0.8

170 Diphenylmethane 2.0 16.4 -2.3

171 n-Nonylbenzene -1.5 0.0 -1.9

172 Pyrene -0.3 0.9 -1.1

173 1-Hexadecanol -1.8 -9.2 1.5

174 Undecylbenzene -1.2 -0.8 -1.9

175 m-Terphenyl 1.4 15.3 0.5

176 o-Terphenyl 2.2 -8.7 0.5

177 p-Terphenyl -1.1 -3.6 -0.3

178 Triphenylene -6.1 1.6 1.0

179 n-Tridecylbenzene -0.6 -1.7 -2.7

180 1-Nonadecanol 2.2 9.2 2.3

Average Deviation, % 0.1 0.3 0.2

Absolute Average Deviation, %|| 2.2 4.5 2.6

%S10% 2.2 10.6 3.9

Table 5 (continued)


Table 6. Predicted critical properties for the 40 isomers included in the list of 180 substances considered in this study. Here, Tc*, Pc*, and Vc* are the values without isomer correction and Tc, Pc, and Vc are the values calculated with isomer correction.

Substance Tc* Tc Pc

* Pc Vc* Vc

1 2-Butanol 545.58 555.72 42.98 43.02 273.69 270.44

2 2-Methyl-1-propanol 545.58 562.35 42.98 43.46 273.69 268.07

3 Vinyl_acetate 524.76 521.82 42.11 41.70 270.41 273.29

4 Methyl_acrylate 524.76 528.20 42.11 42.13 270.41 271.02

5 Ethyl_acetate 524.54 525.02 38.88 38.65 282.39 283.24

6 Methyl_propionate 524.54 527.05 38.88 38.79 282.39 282.51

7 Diethyl_ketone 551.17 557.92 36.37 36.42 324.09 321.82

8 2-Pentanone 551.17 558.18 36.37 36.43 324.09 321.73

9 Ethyl_propionate 552.88 553.32 34.32 34.28 337.82 337.65

10 Methyl_n-butyrate 552.88 556.27 34.32 34.47 337.82 336.59

11 2-Methylpyridine 653.85 630.15 51.76 50.04 317.86 327.42

12 4-Methylpyridine 653.85 641.04 51.76 50.77 317.86 323.53

13 2-Hexanone 575.93 586.88 32.23 32.74 379.96 375.17

14 3-Hexanone 575.93 583.74 32.23 32.53 379.96 376.29

15 m-Cresol 706.16 705.38 49.28 49.34 288.71 289.50

16 o-Cresol 706.16 698.25 49.28 48.87 288.71 292.04

17 p-Cresol 706.16 705.19 49.28 49.33 288.71 289.57

18 m-Xylene 633.90 623.39 37.08 36.35 372.49 375.88

19 o-Xylene 633.90 627.13 37.08 36.60 372.49 374.55

20 p-Xylene 633.90 622.86 37.08 36.31 372.49 376.07

21 2_3-Dimethylhexane 546.17 561.94 24.85 25.67 479.01 469.78

22 2_4-Dimethylhexane 546.17 556.89 24.85 25.33 479.01 471.58

23 2_5-Dimethylhexane 546.17 556.63 24.85 25.31 479.01 471.68

24 2_2_4_4-Tetramethylpentane 572.29 576.22 24.23 24.43 509.07 503.04

25 2_2_3_3-Tetramethylpentane 572.29 590.26 24.23 25.37 509.07 498.03

26 Isoquinoline 791.45 777.37 45.81 45.43 408.17 413.64

27 Quinoline 791.45 773.95 45.81 45.20 408.17 414.86

28 cis-Decahydronaphthalene 695.50 690.00 29.83 29.69 481.27 479.19

29 trans-Decahydronaphthalene 695.50 684.54 29.83 29.32 481.27 481.14

30 1_2-Dimethylnaphthalene 771.79 773.65 30.01 30.98 520.61 517.89








38 m-Terphenyl 896.06 894.89 26.48 28.59 731.60 727.83

39 o-Terphenyl 896.06 875.47 26.48 27.29 731.60 734.76

40 p-Terphenyl 896.06 898.38 26.48 28.82 731.60 726.59

Average Deviation (%) -0.03 0.04 -0.80 0.20 0.30 0.10

Absolute Average Deviation (%) 1.40 0.90 3.90 4.00 2.00 1.90


Table 7. Overall comparison between the proposed methods and others presented in the literature. The results for the methods Constantinou-Gani and Marrero-Pardillo have been taken from Poling et al. (2000).

Property Method %S>10% %S


ACKNOWLEDGEMENTThe authors thank the support of the Direction of Research of the University of La Serena for the permanent support of the Center for Technological Information of La Serena-Chile and of the National Council for Scientifi c and Technological Research (CONICYT), through the research grant FONDECYT 1040285.

NOMENCLATUREa, b force and volume parameters in van der Waals

equation of stateai, bi contributions of atoms and groups for the van der

Waals constantsaTi, aPi, aVi coeffi cients in the model for the isomersAL, CL, EL coeffi cients in Lydersens methodAJ, BJ, CJ, DJ, EJ coeffi cients in Joback-Reids methodF objective functiong1, g2 auxiliary variables in Table 4M molecular mass ni number of times that an atom or group i appears

in a moleculeN number of atoms in a molecule (Equations (5) and

(14) to (16))Ns number of substances considered in this study

(Equation (20))Pc critical pressurePc

* critical pressure calculated with the proposed method Pc

iso calculated critical pressure for isomersPc

exp experimental critical pressure for the isomersPc

Lit literature value for critical pressureQ variable in the objective function F (Equation (20))R ideal gas constant%S percent of substances (%S>10 and %S


Skander, N. and C. E. Chitour, Prediction of Physical Properties of Hydrocarbons and Petroleum Fractions by a New Group-Contribution Method, Petrol. Coal 45(34), 168173 (2003)

Stefanis, E., L. Constantinou and P. Costas, A Group-Contribution Method for Predicting Pure Component Properties of Biochemical and Safety Interest, Ind. Eng. Chem. Res. 43(19), 62536261 (2004).

Valderrama J. O., The State of the Cubic Equations of State, Ind. Eng. Chem. Res. 42(7), 16031618 (2003).

Vitu, S., J. N. Jaubert and F. Mutelet, Extension of the PPR78 Model (Predictive 1978, PengRobinson EOS with Temperature Dependent kij Calculated Through a Group Contribution Method) to Systems Containing Naphthenic Compounds, Fluid Phase Equilibria 243, 928 (2006).

Vukalovich, M. P. and I. I. Novikov, Equation of State of Real Gases (in Russian), State Energy Publishers, Moscow, U.R.S.S. (1948).

Manuscript received February 11, 2006; revised manuscript received May 10, 2006; accepted for publication May 10, 2006.

a new group contribution method based on equation of state parameters to evaluate the critical...

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