3968
Ind. Eng. Chem. Res. 1997, 36, 3968-3973
A Group Contribution Estimation of the Thermodynamic Properties of Polymers† D. Boudouris, L. Constantinou, and C. Panayiotou* Department of Chemical Engineering, University of Thessaloniki, 540 06 Thessaloniki, Greece
A simple method is presented for the estimation of the thermodynamic properties of polymers over extended ranges of external pressure and temperature from a knowledge of the molecular structure of their repeating units. First-order and second-order groups are identified in the repeating unit which can capture even fine structural differences. A list of the most common such groups is presented along with their contributions to the scaling constants of the latticefluid equation-of-state model of polymers. Typical examples are given for the evaluation of these scaling constants of polymers. The accuracy of this calculation permits an estimation of the volumetric properties of polymers with an average absolute deviation from the experimental values of the order of 1%. The method may be applied to any thermodyamic property that the equation-of-state model can calculate. Introduction A rational design of polymer-processing equipment requires experimental information on, or reliable estimation of, the thermophysical behavior of the processed polymers. It is not always possible, however, to find reliable experimental information on these properties for the polymers of interest in the literature nor is it practical to measure these properties as the need arises. An estimation of the thermophysical properties is also essential in the design of new polymeric materials. Such a possibility would be a valuable guide for selecting the structures to be developed that would best suit the needs at hand. Such a screening depends very much on a fast, yet reliable evaluation of the key thermophysical properties for a large number of alternative polymeric structures. Therefore, in computer-aided process and product design, simple, efficient, and reliable methods for the estimation of properties of homopolymers and copolymers from their molecular structure are essential for the analysis and design of products and processes. Group contribution methods are the most widely used techniques for estimating or predicting the thermophysical properties of pure compounds and mixtures. We could divide the currently used methods into two classes. In the first class belong methods which estimate the property of a compound as a summation of the contributions of simple “first-order” groups, such as CH2 and OH, that can occur in the molecular structure. Representatives of this class are, among others, ASOG (Derr and Deal, 1969), UNIFAC (Fredenslund et al., 1977), DISQUAC (Kehiaian, 1984), and more recent group contribution methods (Joback and Reid, 1983; Reid et al., 1987; Lyman et al., 1990; Horvath, 1992; Smirnova and Victorov, 1987, just to name a few from a rather long list). The important advantage of these methods is the quick estimates they provide without requiring substantial computational resources. As, however, the molecular structure is oversimplified, these first-order groups are not able to capture fine structural differences such as proximity effects and isomer differences. As a consequence, many of these methods are * Author to whom correspondence should be addressed. FAX: +3031-996222. E-mail:
[email protected]. † Dedicated to Professor Juan H. Vera on the occasion of his 60th birthday. S0888-5885(97)00242-X CCC: $14.00
of questionable accuracy and utility (Tsonopoulos and Tan, 1993). The second class comprises group contribution methods that can capture fine stuctural differences by introducing in a consistent manner what one could call “higher-order” groups, besides the first-order groups. Representatives of this second class of methods are the connectivity index methods reviewed by Bicerano (1993) and the conjugation operators methods (Mavrovouniotis, 1990; Constantinou et al., 1993; Constantinou and Gani, 1994). These methods, by providing in a simple and methodological way enough information about the molecular structure of a compound, are able to offer significantly improved prediction of properties. In between the two classes belong methods like the ab initio quantum mechanical calculation method of Wu and Sandler (1991), which also attempts to capture fine structural differences in the molecules. On the other hand, equation-of-state models, such as the models proposed by Kurata and Isida (1955), Prigogine et al. (1967), Flory (1970), Patterson and Delmas (1970), Sanchez and Lacombe (1978), Panayiotou and Vera (1982) (again, just to name a few from a long list), remain one of the most widely used class of models for describing the thermodynamic behavior of polymer systems today. The thermodynamic behavior of each polymer is fully described by these models once the characteristic equation-of-state parameters or the scaling constants of the polymer are known. These scaling constants for high polymers are, in a sense, the alternatives of the critical constants used by the equations of state for small molecules. As an example, the latticefluid model (Sanchez and Lacombe, 1978) uses three scaling constants, T*, P*, and F*, for the temperature, pressure, and density, respectively, for each polymer. There are no methods available for an a priori estimation of these constants. Instead, experimental data over extended ranges of external conditions are usually needed in order to determine the scaling constants by a least-squares fitting procedure. The objective of this work is to propose and test a combination of the second class of group contribution methods mentioned previously with the equation-ofstate models of polymers. The combination will be examplified by using the conjugation operator method for group identification (Constantinou and Gani, 1994) in order to estimate the lattice-fluid (LF) scaling constants of polymers. These constants will subsequently © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3969
be used with the LF model in order to calculate the density of the polymers over extended ranges of temperature and pressure. The LF Equation of State According to the LF theory (Sanchez and Lacombe, 1978; Sanchez and Panayiotou, 1994), each fluid, in both pure state and in mixtures, is divided into r segments of hard-core volume v* and average intersegmental interaction energy *. The three parameters r, v*, and * may be replaced by the equivalent and more useful set of LF scaling constants T*, P*, and F* as follows. If M is the molecular weight of the fluid, its hard core volume V* can be obtained either by the product rv* or by the ratio M/F*. By equating the two alternative expressions for V*, we obtain for the scaling density
F* )
M rv*
(1)
On the other hand, * and v* are related with T* and P* through the equations
* ) RT* ) P*v*
(2)
where R is the gas constant. The reduced temperature, pressure, and density (or volume) of the fluid are given by the equations
T ˜ )
T P F 1 P ˜ ) F˜ ) ) T* P* F* v˜
(3)
respectively. With these definitions, the LF equation of state is
[
(
P ˜ + F˜ 2 + T ˜ ln(1 - F˜ ) + F˜ 1 -
1 )0 r
)]
(4)
For high polymers, this equation simplifies to
˜ [ln(1 - F˜ ) + F˜ ] ) 0 P ˜ + F˜ 2 + T
(5)
which can be written alternatively in the following convenient form:
[
F˜ ) 1 - exp -F˜ -
]
P ˜ F˜ 2 T ˜ T ˜
(6)
This equation can easily be solved numerically for a given T and P (or T ˜ and P ˜ ) by a simple iteration procedure: Initially the reduced density in the righthand side is set equal to 1 in order to have the first calculated value. This value is substituted in the righthand side of the equation for a new estimation of the reduced density. This is repeated a few times until two consecutive estimates differ by less than a preset small value. Once the reduced density is known, one obtains the density of the system at the given temperature and pressure from eq 3, namely, F ) F*F˜ . Group Contribution Method The group contribution method that will be used for the evaluation of the LF scaling constants of the polymers is the model of Constantinou and Gani (1994), which has been proven to be very successful in estimating various thermophysical properties of simple fluids. Only the essentials of the method will be presented here. A detailed presentation may be found in the original references.
According to this method, the property estimation is done at two levels. The basic level has contributions from first-order functional groups such as those currently applied for the estimation of mixture properties (Derr and Deal, 1969; Fredenslund et al., 1977). The next level has a set of second-order groups which have the first-order groups as building blocks. The definition and identification of second-order groups is based on the concept of conjugation operators. According to the method, the molecular structure of a compound is viewed as a hybrid of a number of conjugate forms (alternative formal arrangements of the valence electrons) and the property of a compound is a linear combination of these conjugate form contributions. Each conjugate form is an idealized structure with integer-order localized bonds and integer charges on the atoms. The purely covalent conjugate form is the dominant conjugate form, and the ionic forms are the recessive conjugates, which can be obtained from the dominant form by a rearrangement of electron pairs. A conjugation operator defines a particular pattern of electron rearrangement and, when applied to a dominant conjugate, yields an entire class of recessive conjugates. The property of a compound is estimated by determining and combining the properties of its conjugate forms through conjugation operators. The group identification follows precise principles and focuses on the operators which correspond to the important conjugate forms, that is, the operators with significantly higher contributions than the others. The structure of a second-order group should incorporate the distinct subchain of at least one important conjugation operator; for example, the CH3COCH2 second-order group incorporates the OdCsC, the OdCsCsH, and the CsCsCsH operators. The structure of a secondorder group should have adjacent first-order groups as building blocks, and it should be as small and simple as possible. Of course, the performance of second-order groups (as with first-order groups) is independent of the molecule in which the groups occur. On the basis of this method, Constantinou and Gani (1994) have presented extensive lists of first-order and second-order groups. One molecule has definitely firstorder groups but may or may not have second-order groups. The first- and second-order groups, which have been used in this work, are presented in Tables 1 and 2, respectively. The selection of the groups was dictated by the type of polymers whose scaling constants are known. It should be stressed that this list is a partial list which can, and will, be extended when additional information on new polymers becomes available. The proposed method for the estimation of a property X of a compound is as follows. Let Ci be the contribution of the first-order group of type i, which occurs Ni times, and Dj be the contribution of the second-order group of type j, which occurs Mj times in the compound. By selecting a simple function f(X) of the property X, we may write
f(X) )
∑i NiCi + W∑j MjDj
(7)
The constant W is equal to 1 when there are secondorder groups contributing to the property and equal to 0 when only first-order groups are contibuting. The selection of the function f(X) is based on some principles as well. It has to achieve additivity in the contributions Ci and Dj and to demonstrate the best possible fit of the experimental data. In addition, it should be able to
3970 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 1. First-Order Groups and Their Contributions for the LF Equation-of-State Parameters for Polymer Liquids group
T*1i, K
P*1i, MPa
F*1i, kg/m3
CH3 CH2 CH C CHdCH ACH ACCH3 ACCH CH3COO COO CH3O CH2O ACO CH2Cl CHCl Cl(CdC) CF3 CF2 CdO CHCN SiO AC CH2COO
-18.08 -7.65 70.40 97.41 -129.64 -114.89 -130.12 674.27 -139.69 -76.38 -72.69 -8.14 311.48 -76.21 61.71 169.00 -54.57 -24.47 213.58 193.71 -154.78 250.60 71.67
-105.41 -29.67 -36.29 82.23 -6.11 -17.51 -29.45 1.24 85.50 168.96 -70.50 7.89 131.12 -2.06 -46.78 -31.00 -134.86 -59.23 93.72 76.22 23.37 -9.65 17.24
-100.69 -46.65 16.94 136.60 63.84 -18.53 -41.47 223.28 293.24 363.43 110.24 182.84 339.05 257.75 491.19 294.00 67.12 614.27 71.07 257.19 285.91 -77.53 294.14
Table 2. Second-Order Groups and Their Contributions for the LF Equation-of-State Parameters for Polymer Liquids group
T*2j, K
P*2j, MPa
F*2j, kg/m3
(CH3)2CH CH2CH2 ACOC CH2CHCH3 CHCOO CH2CHCH2
-62.13 -0.61 2.64 59.39 -41.40 41.45
1.94 -0.69 5.11 -37.08 6.30 8.72
16.67 -4.37 22.16 -37.07 10.82 -11.71
provide sufficient extrapolating behavior and, therefore, a wide range of applicability. The determination of the (adjustable) parameters Ci and Dj of the model is divided into a two-step regression analysis. In the first step, W is set equal to 0 and the regression is carried out in order to determine the contributions Ci of the first-order groups. Having determined the Ci’s, in the second step, W is set equal to 1 and the contributions Dj of the second-order groups are estimated through regression. In this way, the contribution of the first-order groups is independent of that of the second-order groups and the contribution of the second-order groups serves as a correction to the first-order approximation. Application We will apply in this section the above group contribution method for the estimation of the LF scaling constants of the polymers. Contrary to the critical constants, the LF scaling constants cannot be determined directly from experiment. They are determined indirectly by fitting usually the LF equation of state to experimental PVT data. In a recent review (Sanchez and Panayiotou, 1994), an updated compilation of these constants for ca. 30 polymers has been presented. This is the information which has been used in this work in order to select and estimate the group contributions to each of the three LF scaling constants. The selected function f(X) of eq 7 has the form
f(X) ) X - X0
(X ) T*, P*, or F˜ *)
(8)
Table 3. Values of the Additional Adjustable “Universal” Parameters parameter
value
T*0 P*0 F*0
666.95 K 489.46 MPa 1019.47 kg/m3
The constants X0 are “universal” constants and have the same value for all polymers. In this work, the selection of the groups has been based on the repeating unit of the polymer. As is traditional, the repeating unit of polyethylene was considered to be CH2CH2 rather than CH2. The first- and second-order groups which are needed in order to describe all 30 polymers (for which information on the LF constants is available) are reported in Tables 1 and 2, respectively. In the same tables are reported the contributions of each group to each LF constant. These contributions were obtained by a leastsquares fit of eq 7 to the available LF constants by using a modified Levenberg algorithm. Besides the group contributions, this regression has also given the universal constants X0 which are reported in Table 3. This is all information which is needed in order to estimate the LF constants of any polymer whose repeating unit consists of groups appearing in Tables 1 and 2. The way this is done is presented in the Appendix, where the method is applied to two typical cases: (1) The LF constants of the polymer have been used in the regression and contain only first-order groups. (2) The LF constants of the polymer are not available, and its repeating unit contains both first-order and secondorder groups. As shown in the Appendix, the application of the method is very simple. However, a more meaningful test of the method would be the comparison of the estimated volumetric properties of the polymers with the experimental ones. Since, in most cases, the LF equation of state can correlate the experimental data to within experimental error, an equivalent test of our method would be the comparison of the estimated volumetric properties with the calculated ones by eq 4 when using the original LF constants (from the compilation of Sanchez and Panayiotou (1994)). This has been done by estimating the density for all these polymers and over the range of temperatures and pressures for which experimental information is available. The results are reported in Table 4 along with the mean absolute deviation, the maximum deviation, and the P and T conditions where the maximum deviation appears. As observed in Table 4, the mean deviation is 0% for some polymers. This is due to the fact that in the repeating unit of these particular polymers, there appear groups which do not appear in any other polymer of the table and, consequently, the group contributions were adjusted to fit the experimental data (through the predetermined LF constants). The observed average deviation in the rest of the polymers is of the order of 1%. Relatively higher deviations of the order of 5% are observed for three polymers, polyisobutylene, poly(1butene), and poly(4-methyl-1-pentene). The structures of these polymers are interrelated, but with the available data, we could not identify a second order-group that would reduce the deviations. A better picture of the above deviations is obtained by means of Figures 1, 2, and 3, where are compared the estimated values by the present method PVT data with the “experimental” data, that is, the calculated values by eq 4 when using the LF constants of the
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3971 Table 4. Performance of the Method in the Estimation of the Volumetric Properties of the Polymers polymer
T, K
max P, MPa
mean error, %
max error, %
polyethylene polypropylene polyisobutylene poly(1-butene) poly(1,4-cis-butadiene) poly(4-methyl-1-pentene) polystyrene poly(o-methylstyrene) poly(vinyl acetate) poly(methyl methacrylate) poly(ethyl methacrylate) poly(n-butyl methacrylate) poly(cyclohexyl methacrylate) poly(ethylene oxide) poly(2,6-dimethylphenylene oxide) poly(vinyl methyl ether) poly(tetrahydrofuran) polycarbonate bisphenol A tetramethyl bisphenol A hexafluoro bisphenol A poly(ethylene terephthalate) poly(methyl acrylate) poly(ethyl acrylate) poly(-caprolactone) poly(acrylonitrile) poly(vinyl chloride) poly(chloroprene) poly(epichlorohydrin) poly(dimethylsiloxane) poly(tetrafluoroethylene)
426-473 473-553 326-383 423-503 277-318 513-583 388-468 412-471 308-373 397-432 319-377 307-473 396-472 361-497 493-533 400 335-440
100 20 100 20 100 20 200 160 80 200 50 200 200 60 0 0 80
1.968 0 4.69 5.19 0 4.8 0.394 0.392 0 1.66 2.085 0.164 0 2.395 0 0 1.29
2.028 (0.1 MPa, 473 K)
493-553 493-543 493-553 543-615 320-490 320-490 373 423-473 373-423 373 333-413 298-343 603-743
50 50 50 0 200 200 0 50 100 0 100 100 20
0.172 0.271 0 0 0.327 0.343 0.786 0 0 0 0 0 0
0.265 (0.1 MPa, 553 K) 0.366 (0.1 MPa, 543 K)
Figure 1. Comparison of the estimated densities of polystyrene by the present method (lines) with the calculated ones by the original LF model (symbols) at various temperatures and pressures.
original compilation (Sanchez and Panayiotou, 1994). The three figures correspond to three representative cases of polymers with typical deviations. In order to further test the reliability of the method, we have estimated the LF constants for poly(isobutyl methacrylate). The LF constants for this polymer have been determined from experimental PVT data, which became recently available and were not used in the regression. The calculations by the present method are shown as example 3 in the Appendix. In Figure 4 are compared the predictions of the present method with the experimental PVT data for this test polymer. The agreement between the two sets of values is rather satisfactory and is indicative of the potential of the method. Overall and in view of the limited information that has been used in the regression, the deviations in Table
4.84 (95 MPa, 326 K) 5.322 (0.1 MPa, 426 K) 4.93 (0.1 MPa, 513 K) 0.878 (0.1 MPa, 468 K) 0.827 (0.1 MPa, 471 K) 2.493 (0.1 MPa, 432 K) 2.23 (46.5 MPa, 319 K) 0.34 (195 MPa, 473 K) 2.72 (0.1 MPa, 497 K)
1.57 (0.1 MPa, 440 K)
0.896 (0.1 MPa, 490 K) 0.98 (0.1 MPa, 490 K)
Figure 2. Comparison of the estimated densities of poly(tetrahydrofuran) by the present method (lines) with the calculated ones by the original LF model (symbols) at various temperatures and pressures.
4 and in Figures 1-4 are rather small. In general, the maximum deviations occur, as expected, at the highest temperatures and the lowest pressures. It must be noticed that the most extensively tested predictive method of polymer densities, so far, is the method of Bicerano (1993). Although his method calculates the density at one P and T point only, the average deviation from experiment is much larger than the corresponding deviation of the present method. Conclusions The combination of the group contribution method of conjugation operators and the LF model, proposed in this work, is a promising method for estimating the thermodynamic behavior of polymers. The method is very simple in its application, and the agreement between estimated and experimental properties is rather
3972 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 6 group
occurrence
contribution to T*
first order CH3 CH2 CH C COO
3 1 1 1 1
-18.09 × 3 -7.65 × 1 70.4 × 1 97.41 × 1 -76.38 × 1 ∑NiT*1i ) 29.52
second order (CH3)2CH
1
-62.13 × 1 ∑MjT*2j ) -62.13
occurrence
contribution to F*
Table 7 group Figure 3. Comparison of the estimated densities of tetramethyl bisphenol-A polycarbonate by the present method (lines) with the calculated ones by the original LF model (symbols) at various temperatures and pressures.
first order CH3 CH2 CH C COO
3 2 1 1 1
-100.69 × 3 -46.65 × 2 16.94 × 1 136.6 × 1 363.43 × 1 ∑NiF*1i ) 121.6
second-order (CH3)2CH
1
16.67 × 1 ∑MjF*2j ) 16.67
Appendix In this Appendix, we will present three typical examples of the application of the proposed method for the estimation of the LF constants. Example 1: Poly(o-methylstyrene). The LF constants for this polymer were used in the regression. Let us evaluate its characteristic density F*. Its repeating unit is CH2
CH CH3
Figure 4. Comparison of the estimated densities of poly(isobutyl methacrylate) by the present method (lines) with the calculated ones by the original LF model (symbols) (T* ) 621 K, P* ) 394 MPa, F* ) 1164 kg/m3) at various temperatures and pressures. Table 5 first-order group
occurrence
contribution to F*
CH2 ACH ACCH ACCH3
1 4 1 1
-46.65 × 1 -18.53 × 4 223.28 × 1 -41.47 × 1 ∑NiF*1i ) 61.06
and has first-order groups only, as shown in Table 5. Thus, by using Tables 1 and 3, we obtain
F* )
satisfactory. Unfortunately, there is available information on the LF constants for only a limited number of polymers. Information on additional polymers would lead to a better estimation of the group contributions reported in this work. In particular, experimental information on polymers with new functional groups would expand the range of applicability of the method. It is hoped that this work will inspire experimentalists to measure thermodynamic properties and, in particular, volumetric properties of judiciously chosen homopolymers and copolymers with sufficient structural information that would provide further tests of the proposed method and expand its applicability. Such an expansion in the estimation of the glass transition temperatures of polymers is underway in our laboratory. Acknowledgment We are gratefull to Professor I. C. Sanchez for providing us with the experimental PVT data of the test polymer.
∑NiF*1i + F*0 ) 61.06 + 1019.47 kg/m3 )
1080.53 kg/m3
The corresponding value in the original compilation (Sanchez and Panayiotou, 1994) is 1079 kg/m3. Example 2: Poly(isopropyl methacrylate). The LF constants of this polymer are not available. Let us evaluate its characteristic temperature T*. Its repeating unit is CH3 CH2
C C
O
O CH3
CH
CH3
and it has both first-order and second-order groups, as shown in Table 6. Thus, by using Tables 1-3, we obtain
T* )
∑NiT*1i + W∑MjT*2j + T*0 )
29.52 - 62.13 + 666.95 ) 634.34 K
This value will be a further test of the method when experimental PVT information are available for this polymer.
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3973
Example 3: Poly(isobutyl methacrylate). This is the test polymer whose scaling constants have not been used in the regression. Let us evaluate its characteristic density. Its repeating unit is CH3 CH2
C C
O
O CH2 CH H3C
CH3
and it has both first-order and second-order groups, as shown in Table 7. Thus, by using Tables 1-3, we obtain
F* )
∑NiF*1i + ∑MjF*2j + F*0 )
121.6 + 16.67 + 1019.46 ) 1158 kg/m3
The “true” value of F* for this polymer is 1164 kg/m3. Literature Cited Bicerano, J. Prediction of Polymer Properties; Marcel Dekker: New York, 1993. Constantinou, L.; Gani, R. New Group Contribution Method for Estimating Properties of Pure Compounds. AIChE J. 1994, 40 (10), 1697. Constantinou, L.; Prickett, S. E.; Mavrovouniotis, M. L. Estimation of Thermodynamic and Physical Properties of Acyclic Hydrocarbons Using the ABC Approach and Conjugation Operators. Ind. Eng. Chem. Res. 1993, 32 (8), 1734. Derr, E. L.; Deal, C. H., Jr. Analytical Solutions of Groups. Inst. Chem. Eng., Symp. Ser. (London) 1969, 3, 40. Flory, P. J. Thermodynamics of Polymer Solutions. Faraday Soc. Discuss. 1970, 49, 7. Fredenslund, Aa.; Gmehling, J.; Rasmussen, P. Vapor Liquid Equilibria using UNIFAC; Elsevier Scientific: Amsterdam, 1977.
Horvath, A. L. Molecular Design; Elsevier: Amsterdam, 1992. Joback, K. G.; Reid, R. C. Estimation of Pure-Component Properties from Group Contributions. Chem. Eng. Commun. 1983, 57, 233. Kehiaian, H. V. Group Contribution Methods for Liquid Mixtures: A Critical Review. Fluid Phase Equilib. 1984, 13, 243. Kurata, M.; Isida, S. Theory of Normal Paraffin Liquids. J. Chem. Phys. 1955, 23, 1126. Lyman, W. J.; Reehl, W. F.; Rosenblatt, D. H. Handbook of Chemical Property Estimation Methods; American Chemical Society: Washington, DC, 1990. Mavrovouniotis, M. L. Estimation of Properties from Conjugate Forms of Molecular Structures: The ABC Approach. Ind. Eng. Chem. Res. 1990, 32, 1734. Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of r-mer Fluids and their Mixtures. Polymer J. 1982, 14, 681. Patterson, D.; Delmas, G. Corresponding States Theories and Liquid Models. Discuss. Faraday Soc. 1970, 49, 98. Prigogine, I. (with the collaboration of A. Bellemans and V. Mathot) The Molecular Theory of Solutions; North Holland: Amsterdam, 1967. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. Sanchez, I. C.; Lacombe, R. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145. Sanchez, I. C.; Panayiotou, C. Equations of State Thermodynamics of Polymer and Related Solutions. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S. I., Ed.; Dekker: New York, 1994; pp 187-285. Smirnova, N. A.; Victorov, I. Thermodynamic Properties of Pure Fluids and Solutions from the Hole Group-Contribution Model. Fluid Phase Equilibr. 1987, 34, 235. Tsonopoulos, C.; Tan, Z. The Critical Constants of Normal Alkanes from Methane to Polyethylene. II. Application of the Flory Theory. Fluid Phase Equilib. 1993, 83, 127. Wu, S. E.; Sandler, S. I. Use of ab Initio Quantum Mechanics Calculations in Group Contribution Methods: 1. Theory and the Basis for Group Identifications. Ind. Eng. Chem. Res. 1991, 30, 881.
Received for review March 20, 1997 Revised manuscript received May 27, 1997 Accepted May 27, 1997X IE970242G X Abstract published in Advance ACS Abstracts, August 1, 1997.
