J . Phys. Chem. 1993,97, 7092-7096
7092
Parameters of the Kihara Pair Potential of Anisotropic Molecules Tom66 Boublfk Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University, 128 40 Prague 2, Albertov 2030, Czechoslovakia, and Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, 165 02 Prague, Suchdol, Czechoslovakia Received: October 6. 1992
Parameters of the Kihara generalized pair potential were determined for higher n-alkanes, branched alkanes, and 1-chloroalkanes by applying the second-order perturbation theory of convex molecular fluids to systems of nonpolar molecules and its variant, the perturbation theory of polar molecular fluids, in the case of 1-chloroalkanes. Theoretical expressions were applied to sets of orthobaric data along the coexistence curves of single compounds, and parameters e / k , u, and the rod length L were determined by an optimizing procedure. Considerable regularities were found for the dependences of the single parameters on the number of C atoms in a molecule within the investigated series of compounds.
Introduction Recently, methods of statistical mechanics have found ever increasing applications in different branches of chemistry and chemical engineering. This fact has been connected with the development of the perturbation theory of molecular fluids, Le., fluids where molecules interact via a noncentral type of pair potential. Broader applications of the statistical mechanical methods are limited by availability of parameters of the pair potentials employed to characterize intermolecular forces. For nonpolar nonspherical molecules, the multicenter and generalized Kihara potentials are the most frequently used pair potentials. For both these models, perturbation expansions were proposed'" which proved to describe well the equilibrium behavior of dense gases and liquids in the broad range of temperatures and pressures. We have shown6that close relations exist between the parameters elk and u of the multicenter and Kihara potentials if the same length (end-to-end distance) is considered in both models of the studied molecule. Very recently, Wertheim7v8and Gubbins et al.9J0have shown within the theory of the associating molecules that for the fused hard sphere models (FHSM) the equilibrium behavior does not depend (within some limits) on the bond angles of sites (corresponding to single groups of atoms) of the molecule under consideration. Thus, the behavior of systems of flexible chain molecules can be determined from thermodynamic functions of the corresponding linear model. Because of the relation between the multicenter pair potential (whose special case is FHSM) and Kihara potential, an idea arose to determine thermodynamic functions of systems of flexible chain molecules by the perturbation theory of Kihara rodlike (nonpolar) molecules. This approach was followed in our recent paper," where orthobaric data along the coexistence curve of n-alkanes C&-C16H34 were employed to determine three parameters of the Kihara pair potential. When the obtained parameters were expressed as functions of the number of C atoms in a molecule, Nc, very simple relationships resulted. In this paper, the characteristic parameters of the Kihara pair potential for still higher n-alkanes are determined to investigate how far the above-mentioned-almost linear-dependences of the parameters on NCextend. Next, branched alkanes are studied by employing the same ideas and method. Finally, 1-chloro derivates of n-alkanes are considered. The recently proposed perturbation theory of polar convex molecular fluids12J3 is used to determine the Kihara parameters while the values of the permanent dipole moment are taken from the literature. This paper is organized in the following way: In the first section, a brief outline of the perturbation theory of nonpolar rodlike 0022-3654/93/2091-1092%04.00/0
molecular fluids is given, followed by the description of the perturbation expansion for systems of polar Kihara molecules. Next, the method used to evaluate the Kihara parameters from orthobaric data is discussed. The third section discusses characteristic parameters of the Kihara potential of n-alkanes Cl7H36C24H50,branched alkanes C4H148H18, and 1-chloro derivates of n-alkanes CH3Cl-C4H&1. Rules governing the dependences of the Kihara characteristic parameters on the number of C atoms are discussed in conclusion.
Theory Within this paper, it is assumed that intermolecular interactions of nonspherical molecules are described by a pair potential of the form
where wI is the vector of orientation coordinates, r the centerto-center distance, and s the surface-to-surface distance between cores ascribed to individual molecules. This distance, s, is the only variable in the Kihara pair potential
In this paper, a rod of length L is taken as a core for all the studied compounds. Interactions of the permanent dipoles, F , are characterized by the potential uD
whereqstands for theunit vector in thedirectionof themolecular axis (and of the rod) and nu for the unit vector in the direction of the center-to-center connecting line. In the case of n-alkanes and branched alkanes, it is assumed that uD equals zero. The second-order perturbation expansion of Kihara nonspherical nonpolar molecules is used to determine thermodynamic functions of alkanes. For the Helmholtz residual energy, A - A * , a t temperature T and volume V ( N is the number of molecules and k is the Boltzmann constant), it holds that
( A - A * ) / N k T = (Ao- A * ) / N k T + A y / N k T + Aq/NkT (3) In (3), an asterisk denotes the ideal gas properties and o the properties of the reference system. A! and A; are the first- and second-order perturbation terms. We employed the BarkerHenderson14 definition of the reference system; its properties were 0 1993 American Chemical Society
Kihara Pair Potential of Anisotropic Molecules
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7093
determined from the hard-body equation of state. Here, the hard convex body equation of state is used of the form I [3& - 2Y) + 5aYlY2 PV/NkT = l + 3ay (4) 1-Y (1-y)Z (1 -Y)3 where y stands for the packing fraction and a for the nonsphericity parameter. From the standard thermodynamic relationship, one obtains the following expression for the residual Helmholtz function of the reference
+
(Ao - A*)/NkT = (6a2- Sa - 1) ln(1 - y ) (15a2-9a)y (Sa-3aZ)y 2(1 -Y) The first-order perturbation term is
+
2(1 -YI2
+
AP/NkT = ( 2 n p / k g J m u ( s ) P b ( s ) [ 2 R 2 4Rs
(5)
+ s2] d s
where gbEb is the pair correlation function of hard convex bodies (spherocylinders) and R = L/4 is the mean curvature integral divided by 41rof the core. The expression for the (smaller) secondorder perturbation term differs from that of the first-order term by the different preintegration factor and the fact that uz instead of u stands in the integrand (for details see refs 4,5,and 11). For both A! and A!, analytic expressions can be written within the used approximations.6 This makes it possible to determine easily the pressure of the given one-component system. By adding the residual compressibility factor to ( A - A*)/NkT, the expression for the residual chemical potential results. Thus, the method enables quick determination of two thermodynamic functions, pressure and chemical potential, which govern the vapor-liquid equilibrium in one-component systems. In the case of polar molecule systems, the Kihara nonspherical molecular fluid itself serves as the reference in another (thirdorder) perturbation expansion. For interactionsof the permanent dipoles embedded on the nonspherical molecules, the first-order (electrostatic (ES)) perturbation term equals zero; the secondorder term can be written as
(7)
where p* = pa3 is the reduced number density and X = -h2/ kTa3. Jg(K) is an integral of the form
which depends on the axis ratio, K,of the ellipsoid of revolution in the Gaussian overlap model (GO); the shorter ellipsoid axis is ao. (The G O model is used in an approximation proposed to determine contributions of the electrostatic interactions to thermodynamic functions of nonspherical m o l e ~ u l e s . ~ ~IfJthe ~) dispersion forces are described by the Kihara pair potential, the 1 6 integral possesses the form 16(p*,T*) = JX4&J(X)
dx
353.15 373.15 393.15 413.15 423.15 433.15 453.15
6.988 6.748 6.402 6.200 6.043 5.875 5.513
1.403 2.421 3.918 6.048 7.468 8.964 12.820
-0.132 -0.088 0.053 0.043 0.090 0.147 0.278
0.001 -0.009 -0,003 0.026 -0.003 -0,021 -0.912
6.915 6.642 6.350 6.036
1.416 2.428 3.939 6.091
-0,136 -0.000 -0.059 0.000 0.031 -0.003 0.134 0.009
a T,temperature in K, PI, liquid density in mol L-I, P,pressure in 0.1 MPa.
it holds that
(6)
A r / N k T = - ~ r p * x * ~ J , ( Z6(p*,T*) ~)
TABLE I: Orthobaric Data along the Coexistence Curve: Complete (A) and Incomplete (B) Data Sets for mHexane set A set B T pi P Api AP PI P API AP
(9)
where x = r i a is the reduced distance and g L J the radial distribution function of the system of Lennard-Jones molecules. It is assumed that the corresponding LJ system has the same value of the packing fraction and reduced temperature as the GO system. Integrals of the type given in (9) were evaluated by Twu and GubbinslSJ6 and expressed in the form of a series in the reduced density and In P. The J functions were recently determined by us13 and expressed in the form of Pade approximants in the parameter K. From contributions to the thirdorder ES perturbation term, only the triangular one is nonzero;
A3,JNkT = ( 8 ~ ~ / 6 ) p * ~ x * ~ L ~ ( ~ , p * , T(10) *) Here, LS is a multidimensional integral from the product of 1112, ~ 1 3~, 2 3 and , the tripledistribution function g(3);& was determined numerically13 for a set of values of variables and expressed as a function of K and the packing fraction, y. The described perturbation theory of polar nonspherical molecule fluids was formerly tested against the simulation data of nonspherical molecules with the embedded permanent dipole or quadrupole moments, and fair agreement with the MD data was found (see refs 12 and 13 and citations given therein).
Orthobaric Data along the Coexistence Curve Parameters of the pair potentials used to characterize the behavior of dense fluids (actually of effective potentials) are usually determined from the orthobaricdata along the coexistence curveof the studied onecomponent systems. The thermodynamic conditions of the phase equilibrium are relatively severe; therefore, values of the potential parameters vary considerably less than those determined, for example, by fitting the second virial coefficients. For systems composed of relatively simple molecules, the complete orthobaric data are at our disposal.18 We adjust two or three parameters of the Kibara pair potential, Le., elk, a, and R (which for a rod equals L/4,where L is the rod length), by minimizing the objective function given as sum of squares of the relative deviations in pressure and liquid density. For systems of more complex molecules, the orthobaric data along the coexistence curve were not at our disposal; often only the saturated vapor pressures in the low-pressure range and liquid densities at room temperatures were available. In such cases, we calculated the phase points a t temperatures close to the respective boiling point employing the Antoine equation for the vapor tension and the Rackett equation for the saturated liquid densities. Ideal gas-phase behavior was then considered. In Table I, a comparison is given of orthobaric data and deviations in pressure and liquid Qensityof two equilibrium sets of data for hexane. In the former set, the data from ref 18 were employed; in the latter, the approach described above was used. In case A, values i l k = 626.0 K, Q = 0.4122nm, and R = L/4 = 0.1320nm were obtained. In case B, by fitting all three parameters we got elk = 624.9 K, a = 0.4135 nm, and R = 0.1326 nm, whereas when R = 0.1313 nm was used and only two parameters fitted, c/k = 624.3 K and a = 0.4140 nm resulted. It is apparent that the Kihara parameters determined in the latter case are in fair agreement with those obtained from the complete coexistence curve. Because for some of the systems only the low-pressure data were available, which enable one to determine only small part of the coexistence curve, only two parameters, elk and a, were determined while R was obtained from an equation found on the basis of complete orthobaric data. Table I yields also information on the accuracy of the theory via the deviations (of calculated and experimental values) in liquid
Boublfk
7094 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 15.0
8.0
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1
b 11
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4.8-tb
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8
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4) 2.0-
4
-
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----*----e---
-&----'
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density, Ap,, and pressure, AP. It appears that standarddeviations amount to0.15 mol/Land0.037 MPaforthewho1eset;excluding the point at the highest temperature, they are 0.10 mol/L and 0.0015 MPa, respectively. Relative deviations in the latter case are less than 1% and OS%, respectively. Similar accuracy was reached in the case of other systems studied here, too (for further details, see ref 12). Parameters of the Kihara Pair Potential mAhnes. Series of n-alkanes C&6 were studied in our previous paper.11 Here, we consider further members of the series, Le., C17-C24. We are not aware of complete orthobaric data along the coexistence curve for any of these compounds. Thus, vapor pressures correlated by the A n t o i n e e q u a t i ~ nand ~ ~liquid ~~~ d e n s i t i e ~ l correlated ~,~~ by the Rackett equationz1 were used to determine the liquid branch of the coexistence curve. For n-alkanes from n-heptadecane to n-tetracosane, the optimization procedure was employed in which all three parameters elk, u, and R or only the couple elk and u were determined. In the case of three fitted parameters, the dependence of the individual parameters on NC in the molecule is more scattered for NC > 20. The dependence of R on NC for NC< 20 is, however, fairly linear and agrees perfectly with the previously" determined relationship
R (nm) = 0.02244Nc - 0.0033 1
(1 1) If values of R = L/4 from (11) were used, smoothed dependences of e l k and u on NC were obtained (see Figure 1). From this figure, it is obvious that the dependence of u on NC is a very slowly increasing-almost linear-function of Nc. The formerly proposed equation Q
(nm) = 0.5082 - 1.156/(Nc
+ 6)
correctly describes also the behavior of n-alkanes studied here. In our previous paper," the following relationship was found for the dependence of c / k on NC
+
e / k (K) 2228 - 38490/(Nc 18) (Nc < 17) (13) It is obvious from Figure 1 that (1 3) correlates fairly well the c / k us NC dependence. However, for the whole series C2-C24, the relationship
+
t / k (K) = 2350 - 44947/(Nc 20) (14) has proved to yield a slightly better fit (with the standard deviation
compound ethane propane butane pentane hexane heptane octane nonane decane undecane dodecane tridecane tetradecane pentadecane hexadecane heptadecane octadecane nonadecane eicosane heneicosane docosane tricosane tetracosane
elk, K
100, nm
lOR, nm
302.6 396.2 480.9 554.6 626.0 688.6 745.4 800.5 852.6 898.8 945.0 988.5 1024.8 1061.2 1098.0 1125.95 1163.97 1196.92 1222.82 1255.31 1283.64 1311.38 1337.59
3.633 3.799 3.924 4.051 4.122 4.188 4.265 4.309 4.346 4.403 4.422 4.447 4.516 4.540 4.580 4.641 4.676 4.646 4.714 4.717 4.730 4.751 4.762
0.3945 0.6470 0.8698 1.0698 1.3197 1.5415 1.7555 1.9828 2.2342 2.4492 2.6728 2.9280 3.0948 3.3282 3.5142 3.7818 4.0060 4.2305 4.4550 4.6792 4.9038 5.1280 5.3525
ofapproximately4 K, which is withinvariationof elk). In passing, we note that an even better prediction of c / k can be obtained from a second-order polynomial expression. Branched Alkenes. Moldarconfigurations of higher branched alkanes represent a complex problem. However, it appears from the theory of associating molecular fluids that the equilibrium behavior of the corresponding fused hard sphere models (FHSM) can be described in the same manner as that of simple flexible chain molecules.22 Accordingly, the state behavior of models of branched chain molecule systems is the same as that of linear molecules with the same number of sites, diameters, and site-site distances as the given branched FHSM. As the repulsive interactions are decisive for the structure of fluids, one can expect that the equilibrium behavior of systems with both repulsive and attractive interactions would retain the same relations. Thus, we attempt to employ the method considered first for n-alkanes for branched alkanes also. In Figure 2, all three parameters elk, u, and R = L/4 of representative Kihara molecules are ploted us NC for a series of branched alkanes (2-methylpropane to 3-methylheptane,see the list of compounds in Table 11). In Figure 2, in addition to values of the parameters, curves corresponding to (1 1)-( 13) are plotted. It is apparent that they correlate fairly
Kihara Pair Potential of Anisotropic Molecules
0.0
I
4
I
I
I
5
6
7
The Journal of Physical Chemistry, Vol. 97,No.27, 1993 1095
2 8
. 4
0
5
7
1
6
NC
NC
Figure 3. Dependence of t/k X 1 t 2K (full line) and R X 10 nm (dashed line) on the number of C atoms in a molecule of branched alkanes.
Figure 4. Dependence of e l k X 1 t 2K (full line) and u X 10 nm (dashed line) on the number of C atoms in a molecule of branched alkanes.
TABLE III: Parameters elk, u, and R = L/4 of the Kihara Pair Potential for Branched Alkanes
TABLE I V Parameters elk, u, and R = L/4 of the mars Pair Potential for 1-Chloroalkanes*
compound
elk, K
loa, nm
10R, nm
compound
e / k ,K
lOu, nm
10R, nm
101*p,esu
2-methylpropane 2-methylbutane 2-methylpentane 3-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylhexane 3-methylhexane 3-ethylpentane 2-methylheptane 3-methylheptane
459.65 539.59 612.06 619.41 590.97 608.54 675.58 679.87 675.51 731.96 735.81
3.933 4.066 4.083 4.060 4.071 4.062 4.152 4.138 4.150 4.242 4.224
0.8692 1.0698 1.3195 1.3195 1.3195 1.3195 1.5415 1.5415 1.5415 1.7555 1.7555
1-chloromethane 1-chloroethane 1-chloropropane 1-chlorobutane
387.43 474.82 560.35 631.65
3.464 3.688 3.791 3.948
0.4388 0.6632 0.8875 1.1120
1.87 2.05 2.05 2.05
p,
permanent dipole moment. 7.0
6 .B
well the “experimental” values of the parameters. This is especially true for u. Therefore, we employed (12) and evaluate only parameters a/k and R. From Figure 3, it is apparent that the dependence of R on NC is well described by (.11). Finally, we employed (1 1) and adjusted a/k and u. In Table 111, values of the parameters obtained in this way are listed for 11 branched alkanes; in Figure 4, values of lO-%/k and u are plotted against Nc. It is apparent that (12) yields perfect correlation of u us NC dependence whereas (1 3) or (14) slightly overestimates values of c/k. Better correlation is obtained from
t/k (K) = 2250 - 39376/(Nc
6
6)
5.8
Y
\w 4.5
+ 18)
(15) with a standard deviation of approximately 7 K. Chloroakanes. Four 1-chloroalkanes, 1-chloromethane to 1-chlorobutane, were considered. Molecular interactions were modeled as interactions of the Kihara rodlike molecules with the embedded permanent dipole moment. Values of the dipole moments were taken from the literature;23 all other electrostatic contributions were neglected. For 1-chloromethane, all three parameters, Le., elk, u, and R, were determined by adjusting them to the complete orthobaric data along the coexistence line. The data available for the other 1-chloroalkaneswere incomplete; they did not allow reasonable determination of more then two parameters. We therefore assumed the values of R to be a sum of the value for 1-chloromethane plus a multiple (Nc - 1) of the contribution that had been found for n-alkanes: A = 0.022 44. Thus, for R = L/4 it holds that
R (nm) = 0.02244Nc
-
b
+ 0.02143
(16)
Values of e/k, u, R , and the dipole moment (esu) are given in Table IV. The dependence of u on NC as follows from the fitting
3.0
I
2
I
3
NC Figure 5. Same as in Figure 4 for 1-chloroalkanes.
of the u parameter to an expression of type (1 1) is u
(nm) = 0.5030
- 1.0914/(Nc + 6)
(17)
whereas for a/k it holds that
Elk (K) = 2187 - 34216/(Nc + 18) (18) The “experimental” data together with curves corresponding to ( 17) and ( 18) are depicted in Figure 5 . From it, a good description of both quantities determined by the last relationships is evident. In order to get some feeling about the quality of the orthobaric data prediction of the systems studied here, a comparison is given of the calculated and experimental densities and pressures along the coexistence curve for representatives of each studied group, i.e., nonadecane, 2-methylheptane, and 1-chlorobutane (see Table
7096
Boublfk
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
TABLE V Experimental Densities, Pressures, and Deviations of the Calculated Data from Experimental Data for Nonadecane, 2-Methylheptane, and 1-Chlorobutane T,K P I , mol L-I Ap,, mol L-I P , 0.1 MPa AP,0.1 MPa 570 590 610 630 650 670
2.236 2.166 2.091 2.012 1.927 1.833
300 320 340 360 380 400
6.714 6.536 6.351 6.157 5.952 5.735
320 340 360 380 400
8.979 8.753 8.516 8.267 8.004
Nonadecane -0.069 0.483 -0.049 0.762 -0.025 1.155 0.001 1.694 0.03 1 2.41 1 0.067 3.340 2-Methylhexane -0.102 0.095 -0.076 0.225 -0.046 0.475 -0.012 0.906 0.029 1.605 0.076 2.656 1-Chlorobutane -0.012 0.337 -0.012 0.689 -0.007 1.287 0.002 2.230 0.009 3.628
i
6
1 .20\
-0.007 -0.006 0.004 0.038 0.094 -0.076 0.003 0.003 -0.003 -0.013 -0.035 -0.056 0.40
0.007 0.007 -0.001 -0.021 -0,024
V). It is apparent that theaccuracy remains practically unchanged for the different systems (see also Table I).
Conclusion The principal advantage of statistical mechanical methods, proposed to describe the behavior of real fluids from the knowledge of intermolecular forces, is the fact that once the parameters of the pair potential are known all the properties of the given fluid can be determined, i.e., not only the “saturated” properties of liquids but also, for example, virial coefficients and even nonequilibrium characteristics. (This trait represents an important difference in comparison with approaches of classical thermodynamics, for example, applications of the cubic equations of state). Because of this fact, values of the pair potential parameters are of great importance. In the papers of this series, we focused on the study of molecular fluids composed of relatively large molecules. These systems, though important in practice, have not been usually considered within perturbation theories. We studied the equilibrium behavior of series of higher n-alkanes, branched alkanes, and 1-chloroalkanesby perturbation methods. All these systems are formed by flexiblechain molecules with several conformations. Different conformers can be taken into account in the case of simple molecules, e.g., n-butane. The situation is more complicated in the case of molecules with NC > 16 (studied here). In the simplest approach (for flexible chain molecules) based on the results of the associating molecule it is assumed that thermodynamic functions of a flexible molecule are equal to those of the corresponding linear model. Further, it has been shown in studies of dense fluids and the second virial coefficient that thermodynamic data of the linear multicenter LennardJones systems can be accurately calculated by methods derived for the Kihara rodlike molecules. Thus, it was assumed that dispersion interactions were characterized by the Kihara nonspherical pair potential. The perturbation expansion of convex molecular fluids was used to determine thermodynamic functions of the nonpolar fluids; contributions of the permanent dipole moment were obtained from another perturbation expansion. In the beginning, three Kihara parameters, elk, u, and R (=L/4), were determined by fitting the respective orthobaric data along thecoexistence curve. When plottingvalues of R for the members of a single series us the number of C atoms in a molecule, the linear dependence was found. This is true-within error bars of incomplete data-even for higher n-alkanes. Dependences of the other two parameters on NC are not linear. It seems that
1
5
9
13
17
21
NC Figure 6. Dependence of the ratio kT,/c on the number of C atoms in a molecule of the series of n-alkanes (full line), branched alkanes (full line), and 1-chloroalkanes (dotted line).
there exist some limiting values of e / k and u to which their values tend for large Nc. The plot (Figure 6) of the dependence of kT,/t us Nc is instructive. A smooth nonlinear dependence is found for all three series; the values for branched alkanes agree fully with those for n-alkanes. The expressions obtained for c / k , U, and R make it possible to extract improbable sets of parameters and, consequently, unreliable sets of orthobaric data (on the basis of which they were determined) to extrapolate the parameter values and evaluate on their basis equilibrium data for still higher members of individual series. It seems possible to determine the parameter dependences only on the basis of data for the first few members of the series.
Acknowledgment. This work was supported by a grant from the Academy of Science of the Czech Republic. References and Notes (1) Fischer, J. J . Chem. Phys. 1980, 72, 5371. (2) Fischer, J.; Lustig, R.; Breitensfelder-Manske, H.; Lemming, W. Mol. Phys. 1984, 52,485. (3) Bohn, M.; Fischer, J.; Kohler, F. Fluidphase Equilib. 1986,31,233. (4) Boublfk, T. Mol. Phys. 1976, 32, 1737. (5) Pavlfbk, J.; Boublfk, T. Fluid Phase Equilib. 1981, 7, 1. (6) Boublik, T. J . Chem. Phys. 1987.87, 1751. (7) Wertheim, M. S.J. Statist. Phys. 1984, 35, 19. (8) Wertheim, M. S. J. Statist. Phys. 1986, 42, 459. (9) Jackson, G.;Chapman, W. G.;Gubbins, K. E. Mol. Phys. 1988,65, 1. (10) Chapman, W. G.; Jackson,G.;Gubbins, K. E. Mol. Phys. 1988,65, 1057. (11) Boublfk, T. J . Phys. Chem. 1992, 96, 2298. (12) Boublfk, T. Mol. Phys. 1991, 73, 417. (13) Boublfk, T. Mol. Phys. 1992, 76, 327. (14) Barker, J.; Henderson, D. J . Chem. Phys. 1967, 47, 4714. (1 5) Twu, C. Ph.D. Thesis (Appendix E), University of Florida, 1976. (16) Twu, C. H.; Gubbins, K.E.; Gray, C. G. Mol. Phys. 1975,64,5186. (17) Boublfk, T. Mol. Phys. 1981,42, 209. (1 8) Vargaftik, N. B. Tableson the Thermophysical Properties ofLiquids and Gases; Wiley: New York, 1975. (19) Dreisbach, R. R. Physical Properties of Chemical Compounds; Advances m Chemistry Series 22; American Chemical Society: Washington, DC, 1959. (20) Boublfk, T.; Fried, V.; Hdla, E. The Vapour Pressures of Pure Substances; Elsevier: Amsterdam, 1973. (21) Timmermans, J. Physicochemical Constants of Pure Organic Compounds; Elsevier: Amsterdam, 1950. (22) Boublfk, T. Mol. Phys. 1989, 68, 191. (23) CRC Handbook of Chemistry and Physics, 62nd ed.; Weast, R. C., Astle, M. J., Eds.; CRC Press: Boca Raton, FL, 1981-1982.
