Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 183-187 Graef, M. 0.;Allan, G. G.; Krieger, 8. B. Am. Chem. SOC.Div. Pet. Chem. frepr. 1979, 24(2), 432. Harkin, J. M. fortschr. Chem. Forsch. 1988, 6 , 47. Hayatsu, R.; Winans, V. E.; Beth, M.; Scott, R. L.; Moore, L. P.: Studier, M. H. Am. Chem. SOC.Div. fuel Chem. frepr. 1979, 24(1), 110. IatrMis, B.; Gavalas, G. R. I n d . Eflg. Chem. Prod. Res. Dev. 1979, 18, 127. Jegers, H. E. M.Ch.E. Thesis, University of Delaware, Newark, DE, 1982. Klrshbaum. I. 2.: Dombura. G. E.: Seraeeva. V. N. Khim. Drev. 1976. No. 4 . 96. Kistiakowsky, G. B.; Smith, W. R. J. Am. Chem. SOC. 1934, 56, 636. Klein, M. T. Sc.D. Thesis, M.I.T., Cambridge, MA, 1981. Klein, M. T.; Virk, P. S. M.I.T. Energy Lab Report MIT-EL-61-005, 1981
183
Marton, J. I n "Lignins: Occurrence, Formation, Structure and Reactions", Sarkanen, K. V.: Ludwig, C. H., Ed.: Wiley-Interscience: New York, 1971. Panvelker, S. V.; Shah, Y. T.; Cronauer, D. C. Ifld. Eflg. Chem. fundam. 1982, 21, 236. Petrocelli, F. P.; Klein, M. T. Macromolecules 1984; 17, 161-169. Raj, S . Am. Chem. SOC.Dlv. fuel Chem. frepr. 1979, 24(3), 251. Sweetlng, J. W.: Wilshire, J. F. K. Aust. J. Chem. 1982, 15, 89.
Received for review October 4,1983 Revised manuscript received April 4,1984 Accepted April 19,1984
Interaction Second Virial Coefficients of Polar Mixtures from Parameters of the Components Nlkos J. P. Marls and Leonard I. Stlel' Polytechnic Institote of New Yo&, Brooklyn, New York 11201
Procedures are presented for the calculation of interaction second virial coefficients B 12 of polar mixtures from the parameters of the components. Relationships for the second virial coefficients of polar fluids for the Kihara spherical core potentlal with temperatwedependent molecular constants are utilized in conjunction with combining rules for the mixture parameters. The input data required are the critical temperatures, critical pressures, acentric factors, and polarity factors of the components. For 111 experimental points this approach resulted in an average deviation of 25 cm /mol in comparison with available experimental values of 8 12. Comparable results are obtained with a simplified procedure based on the use of the Pitzer-Curl equations for the second virial coefficient with temperaturedependent critical constants. A procedure for the separation of induction contributions for nonpolar-polar mixtures is also described which results in improved values of 8 12 for most of the systems considered.
Interaction second virial coefficients of polar gas mixtures are required for many applications, including the calculation of vapor-liquid equilibrium values at moderate pressures. Most of the previous methods for the determination of this property require experimental data to obtain reliable values. It is desirable to have an improved procedure for polar mixtures to enable the calculation of values for systems for which data are not available and to permit suitable combining rules for the mixture parameters to be established. In this study a method has been developed for the determination of interaction second virial coefficients of polar mixtures from parameters of the components by the use of a four-parameter intermolecular potential function for polar fluids.
Previous Studies for Polar Mixtures OConnell and Prausnitz (1967) presented a relationship for the second virial coefficients of polar fluids of the following form
where B*(O)and B*(') are the reduced functions of Pitzer and Curl (1957) for nonpolar fluids, Oh is the acentric fador of a nonpolar substance having a similar size, p~ = loSp2Pc/T~ is a reduced dipole group, and q is a parameter representing the effects of association. Analytical expressions were obtained for B,* and B,*. For polar mixtures, OConnell and Prausnitz presented relationships for the calculation of T,,,, Pc,,, wh,,, PR~,, and qlz from the parameters of the components. Average errors of up to 150 cm3/mol were obtained for various types of polar 0196-4305/85/1124-0183$01.50/0
mixtures. For several polar-polar mixtures an interaction constant q12 obtained from the experimental data was required to obtain reasonable results. Halm and Stiel(1971) utilized experimental second virial coefficient data for 13 polar fluids to determine the coefficients B*(2),...,B*(5)of the following relationship as functions of reduced temperature B* = B*(O)+ &*(I) + XB*W + X2B*(3)+ u6B*(4)+ ,XB*(5) ( 2 ) where B*(O)and B*(l)are the values for the Pitzer-Curl equation and x is the polarity factor of the fluid. For polar substances, Halm and Stiel developed relationships between available molecular parameters for a four-parameter intermolecular potential function and critical constants, acentric factors, and polarity factors. These relationships were utilized with combining rules for the molecular parameters to obtain equations for the mixture parameters T,,,, Pc, , w12, and xlz from the parameters of the components. $hese mixture parameters resulted in an average error of 7.0% in mixture second virial coefficients for nine polar systems. Tsonopolous (1974) proposed a relationship for the second virial coefficient of polar fluids of the form
where Po)and f ( l ) are modified functions similar to B*(O) and B*(l),and a and b are constants determined from data for each substance. For non-hydrogen bonding substances b = 0, and separate relationships between a and the dipole group p R were developed for several groups of substances 0 1984 American Chemical Society
184
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
including ketones, ethers, and alkyl halides. For mixtures, Tsonopolous proposed the following combining rules for use with eq 3
TC,, = (TclTc2)1'2(1 -
012
= 0.5(Wl
+
W2)
(4)
the experimental error (less than 10 cm3/mol) for nonpolar mixtures, including systems containing components of dissimilar size such as the methane-n-hexane system. Eisenman and Stiel also derived the following relationand oI2of ships for the interaction parameters Tc,,,Pc12, nonpolar mixtures by the use of eq 9-14
(6)
where k12is a constant determined from experimental data. For nonpolar-polar mixtures a12and blz = 0, and for polar-polar mixtures these parameters are arithmetic averages of the constants of the components. For nonpolarpolar mixtures this procedure resulted in low deviations from the reported values of B12,but experimental data are required for the desired temperature range. Somewhat higher deviations resulted for less accurate data for mixture virial coefficients BM,including values for several polarpolar systems. Hayden and O'Connell (1975) presented a method for the calculation of second virial coefficients of nonpolar and polar fluids and mixtures from the parameters of the components. Input data are the criticaI temperature, critical pressure, radius of gyration, dipole moment, and for some fluids or mixtures a group association parameter. For a number of nonpolar or polar mixtures the average error in B12was 26.6 cm3/g-mol, with largest errors for polar-polar systems. Second Virial Coefficients of Nonpolar Mixtures The approach utilized in this study is an extension of the procedure of Eisenman and Stiel (1971) for the interaction virial coefficients of nonpolar mixtures. For the Kihara spherical core potential, the second virial coefficient can be calculated as B - = 2irN0[a*3+ 3(21/6)a*2F1(z)+ 3(21/3)a*Fz(Z)+ pO3 3 (21/2)J'3(z)I (7) where Z = t/KT, a* = 2a/po, and
w12 = [(2al2/pOl2) - 0.1501]/2.3724
where fl and f2 are the functions of eq 12 and 13. Interaction second virial coefficients can be calculated for nonpolar mixtures to within about 20 cm3/moI by the use of eq 15-17 and the Pitzer-Curl relationships (or similar versions) expressed in the form
Interaction Second Virial Coefficients for Polar Mixtures In this study, the relationships of Lin and Stiel (1977) for the second virial coefficients of polar fluids have been utilized. Lin and Stiel considered the following four-parameter potential function consisting of the Kihara spherical core potential combined with a pre-averaged dipole-dipole interaction term .rp)-4€[(;)"-(;)"]-(--)
.Tp6
(19)
For this model, the solution for the second virial coefficient is given by eq 7 and 8, expressed in terms of temperature-dependent parameters E' and p( which are related to t and po of eq 19 by the relationships €' = €(I Y€/KT)' (20)
+
Po' = Po(1
Interaction virial coefficients are obtained for nonpolar mixtures from eq 7 by the use of the following combining rules for the molecular energy, distance, and size parameters
(17)
+ YC/KT)-lis
(21)
where y = l/6(P4/t2p:). Lin and Stiel (1977) utilized experimental second virial coefficient data for polar fluids to develop the following relationships for the molecular groups of this model C / K T=~1.0042 + 3.0454~- 58.883~+ 8 7 . 1 9 7 9 ~+~ 6 . 8 8 1 3 ~- ~4 . 2 4 2 2 ~(22) ~ p0(Pc/Tc)1/3 = L1.0554 - 0.4003~+ 10.8594~1 8 . 7 8 5 9 ~+~23.5418~'- 0.0871~']/(1 + a*) (23)
+
2.37240 - 25.3457~+ 35.79750~- 64.3636~'- 2 . 0 5 4 0 ~(24) ~
a* = 2a/po = 0.1501
The molecular parameters of the components required in eq 9-11 are calculated from the relationships of Tee et al. (1966) €/.Tc = 1.0042 + 3.0454~= f l ( w ) (12) 1.0554 - 0.4003~ = fi(W) (13) p0(Pc'Tc)'/3 1.1501 + 2.3724~ 2a/po = 0.1501 2.3724~= f 3 ( ~ ) (14)
+
Eisenman and Stiel (1971) found that eq 7-14 enable the calculation of interaction second virial coefficientsto within
+
y = 17.4290~ - 2 4 . 0 5 1 6 ~-~1 1 . 2 3 0 1 ~ ~0.7319~' (25)
where the polarity factor x is defined as x = log PR,CjITR=0,6 + 1 . 7 0 ~+ 1.552
(26)
Lin and Stiel treated the group y as an adjustable parameter, so that p is an effective dipole moment for this procedure. For nonpolar fluids, eq 22-24 reduce to eq 12-14 with the u2terms taken as zero. Because of the use of a preaveraged dipole term, these relationships for the second virial coefficient of polar fluids are most accurate for
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
185
Table I. ComDarison of Interaction Second Virial Coefficients for Polar Mixtures mixture ethanol-argon ethanol-nitrogen acetone-argon 2-propanol-argon ethanol-argon ethanol-methane methanol-argon methanol-methane methanol-nitrogen methanol-nitrogen water-argon water-nitrogen water-methane methyl chloride-argon methyl chloride-argon acetone-methyl chloride acetone-n-butane acetone-methane acetone-nitrogen ammonia-argon ammonia-krypton CHBF-NZ CH3F-Ar CHF3-Ar HCl-Ar HCl-Kr HCl-CSH, CHC1F2-Ar CHClFZ-Nz CClzF2-CZH4Fz CC13F-CHClF2 CC12Fz-CHClF2 CClF3-CHClF2 CClF3-CHF3 CHClFZ-CHF, CC13F-CHF3
reference Vigdergauz and Semkin (1971) Vigdergauz and Semkin (1971) Vigdergauz and Semkin (1971) Vigdergauz and Semkin (1971) Gupta et al. (1973) Gupta et al. (1973) Hemmaplardh and King (1972) Hemmaplardh and King (1972) Hemmaplardh and King (1972) Neogi and Kudchadker (1977) Rigby and Prausnitz (1968) Rigby and Prausnitz (1968) Rigby and Prausnitz (1968) Bottomley and Spurling (1967) Lichtenthaler and Schafer (1969) Bottomley and Spurling (1967) Kappallo et al. (1963) Hicks and Prausnitz (1981) Hicks and Prausnitz (1981) Michels (1958) Michels (1958) Michels (1958) Copeland and Cole (1976) Copeland and Cole (1976) Dymond and Smith (1980) Glockler et al. (1933b) Glockler et al. (1933a) Brewer (1967) Brewer (1967) Sinka and Murphy (1967) Bougard and Jadot (1976) Bougard and Jadot (1976) Bougard and Jadot (1976) Bougard and Jadot (1976) Bougard and Jadot (1976) Bougard and Jadot (1976)
moderate and high reduced temperatures. For polar mixtures, interaction virial coefficients are calculated from eq 7 and 8 expressed in terms of the pap’012and 2a12/p’011, where rameters d12, t’12
= C12(1 + 3 ’ 1 2 6 1 ~ / K T ) ~
(27)
The mixture parameters t12,pOl2,and a12are obtained from the combining rules eq 9-11, and y12from the additional relationship
-)
y12= (y1y2)1/2( p
012
(29)
For nonpolar-polar mixtures y12= 0, and therefore d12 = c12 and p’o12 = pol. The procedure for the calculation of B12from eq 7-11 for these systems is similar to that for nonpolar mixtures, with the molecular parameters calculated from eq 12-14 for the nonpolar component and eq 22-24 for the polar component. In Table I, the results of comparisons between experimental interaction virial coefficients and values calculated from these relationships and the constants of the components are presented for a number of nonpolar-polar mixtures containing water, ammonia, acetone, alcohols, and halocarbons with argon, krypton, nitrogen, and hydrocarbons. The values of T,, P,, o,and x utilized for the polar components are listed in Table 11. It can be seen that in general good results are obtained and that the deviations are close to the probable experimental errors of BI2. Comparable results are obtained with the actual molecular parameters determined by Lin and Stiel(l977)
temp range, K 323.15-353.15 353.15 353.15 323.15-353.15 298.15-348.15 298.15-348.15 288.15-333.15 288.15-333.15 288.15-333.15 303.2-348.2 298.15-373.15 298.15-373.15 298.15-373.15 293.82-430.25 288.2-313.2 323.15-427.78 282.3-321.0 223.15-323.15 223.15-308.15 298.15 298.15 298.15 323.15 323.15 190.0-480.0 328.7-368.0 368.7-389.5 273.15-298.15 273.15-298.15 298.15-413.15 302.26 300.56 300.71 300.31 300.31 302.26
no. of points 2 1
av abs dev, cm3/mol 41.0 21.0
1 2 3 3 5 5 5 4 4 4 4 6 4 6 4 4 5 1 1 1 1 1 13 2 2 2 2 7 1 1 1 1 1 1
11.0
7.0 16.0 32.0 31.0 30.0 31.0 25.0 26.0 18.0 31.0 50.0 23.0 37.0 48.0 39.0 19.0 5.0 25.0 1.0 8.0 14.0 8.0 33.0 39.0 8.0 5.0 9.0 7.0 23.0 35.0 3.0 11.0 80.0
Table 11. Values of T,,P,, w, and x for Polar Fluids Studied substance To,K P,,MPa w X water 647.3 22.11 0.344 0.023 11.38 ammonia 406.0 0.254 0.016 6.13 ethanol 514.0 0.641 Q.000 4.76 2-Propanol 508.4 0.633 -0.053 7.95 methanol 513.2 0.556 0.037 4.72 0.312 acetone 508.7 0.014 8.26 HCl 324.6 0.126 0.008 6.68 416.3 0.149 CHSC1 0.002 5.88 CH3F 317.8 0.191 0.012 4.98 CHClFz 369.2 0.222 0.004 4.12 385.0 0.176 CClzFz 0.0003 0.0 3.86 CClF3 301.9 0.17 4.41 471.2 0.185 0.0 CCl3F 0.004 4.83 CHF3 299.7 0.251 0.007 4.50 386.6 0.265 C2H4F2
for the potential function of eq 19. For example, for mixtures of water with argon, methane, and nitrogen (Rigby and Prausnitz, 1968) the average absolute deviation is 21 cm3/mol with the actual molecular parameters for water and 24 cm3/mol with the calculated parameters. For the argon-methyl chloride system, closer agreement is obtained for the data of Lichtenthaler and Schafer (1969) than for the values of Bottomley and Spurling (1967) for which larger experimental errors are indicated. The data of Lazalde et al. (1980) for mixtures of methanol with methane and nitrogen are considerably more negative than the calculated values, particularly at low temperatures. Similar deviations are obtained for these data with values calculated by the method of Tsonopolous (1974) with parameters determined from data at higher temperatures for these systems. These larger deviations are
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
possibly due to the influence of the third virial coefficient on the experimental values at low temperatures. Larger deviations result for systems with components which have appreciable quadrupole moments such as carbon dioxide and ethylene. For polar-quadrupolar systems modifications in the procedure are required such as those presented by Ramaiah and Stiel (1973) for nonpolar-quadrupolar mixtures. For polar-polar mixtures, the additional combining rule eq 29 is utilized, and interaction virial coefficients are calculated from eq 7 and 8 with the values of P'~~,, and 2a12/p'012.The parameters of the components are calculated from eq 22-25. Equation 25 for y was developed from data primarily for substances with large polarity, and it has been found that the following alternate relationship should be utilized for substances with small polarity factors y = 14.206~ (o50.35; x 50.005) (30) Experimental data for B12are scarce to test the procedure for polar-polar systems. For the acetonemethyl chloride and CHC1F2-CHF, (R-22-R-23) systems, the deviations between calculated and experimental values which are included in Table I are similar to those for nonpolar-polar systems. For 111 experimental points the comparisons presented in Table I result in an average deviation of 25 cm3/mol for the interaction second virial coefficient. Values of second virial coefficients of polar fluids and mixtures can also be calculated with the Pitzer-Curl relationships with temperature-dependent macroscopic parameters. For pure polar fluids, the functional form is
The parameters P:, T:, and w' are related to the molecular parameters e', p d , and 2alpd as follows
(1)-polar (2) mixture, eq 19 becomes
Equation 35 can be represented alternatively as
where c12 and p!!," result from dispersion effects and a1 is the pofarizability of the nonpolar component. The parameters of the two forms of the potential are related through the following relationships €12
= €12"(1 + f 1 2 ) 2
(37)
where 512 = ~ 1 ~ 2 ~ / 4 ~ 1. 2 , ~ ~ 0 The parameters e12, and8npoIy,can be obtained from the following combining rules €12" =
(€1€2,)1'2
(39)
- Po, + Po,,
-
(40) 2 The parameters t2, and po, for the polar component are calculated from the following relationships which result from the separation of dispersion and induction contributions in eq 19 €2, = E Z ( 1 - E d 2 (41) PO," = PO,(l - E2)-1/6 (42) where tZ= a+22/2e2po,6 Relationships for the parameters t12and pol, modified for induction effects result from eq 37-42 €12 = (€1€2)1'2(1- 52)(1 + E12I2 (43) c
(33) 2a 7= f3(w') Po
(34)
where fi-f3 are the functions of eq 12-14. For polar mixtures, values of B12are calculated from eq 31 with interaction constants TICl2,PIC,,, and d12 determined from eq 32-34. The corresponding values of d12,pf0 and 2al~/p'0,~ are calculated in the same manner as utifzed with eq 7 and 8. The values of the molecular parameters are obtained from eq 12-14 for nonpolar components and eq 22-25 for polar components. For nonpolar-polar mixtures the interaction constants are independent of temperature, and eq 32-34 are equivalent to eq 15-17. The values of B or B12calculated from eq 31-34 agree closely with the corresponding values determined from eq 7 and 8 at high values of T/Tl,with a deviation of about 15 cms/mol at TIT,' = 0.75. The procedure based on eq 31 for pure polar fluids or polar mixtures represents a considerable simplification for hand calculations. Separation of Induction Contributions For the potential function of eq 11 the effects of dispersion and dipole-induced dipole interactions are combined in the attractive portion of the spherical core model. Therefore, the parameters el2 and p0,, should be modified to account for the different induction contributions for the pure polar components and the mixture. For a nonpolar
The parameters E and po are calculated from eq 12 and 13 for the nonpolar component and eq 22 and 23 for the polar component. The value of p2 required in eq 43 and 44 is calculated from the corresponding value of y2through the equation ~ 2 '= (6~2)"~€2~b, (45) Similar expressions as eq 43 and 44 can be derived for modified parameters for polar-polar mixtures. The modified parameters can be used to determine Blz from eq 7 and 8 or from the Pitzer-Curl relationships, eq 31-34. For most of the nonpolar mixtures considered, the values of B12calculated with the parameters el2 and polz determined from eq 9 and 10 are more negative than the corresponding experimental values. The modified parameters calculated by eq 43 and 44 with the polarizability of the nonpolar component result in values of B12which are more positive and in general closer to the experimental values. As shown in Table I11 the average absolute deviations between experimental values of B12and values calculated from eq 7 and 8 with the modified parameters for a number of mixtures are lower than the corresponding errors presented in Table I. For the argon-HC1 system the modified procedure results in an average absolute error of 2.3 cm3/mol, indicating that highly accurate results are possible by this procedure. The use of the modified parameters with eq 31-34 resulted in almost identical errors for these systems.
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
Table 111. Average Deviations for Nonpolar-Polar Mixtures with Parameters Modified for Induction Effects no. of a v abs dev, mixture reference points cm3/mol CH3C1-Ar Lichtenthaler and Schafer (1969) 4 14.0 H20-Ar Rigby and Prausnitz (1968) 4 16.0 HzO-Nz Rigby and Prausnitz (1968) 4 8.0 H20-CH4 Rigby and Prausnitz (1968) 4 17.0 NH,-Ar Michels (1958) 1 3.0 HClGlockler et al. (1933a) 2 28.0 C3Hs HCl-Ar Dymond and Smith (1980) 13 2.3
For mixtures of methanol with nonpolar components, the values of Blz calculated with the unmodified parameters are less negative than the experimental values, and the modified parameters result in larger errors. Similarly, for acetone mixtures the values of Blzcalculated with the unmodified parameters are more negative than the experimental values at high temperatures but less negative at low temperatures. Therefore, for mixtures containing these components the values of el, and pOl2calculated with eq 9 and 10 should be used. For methanol and acetone there is considerable scatter in the available second virial coefficient data. The anomalous results with modified parameters for these substances may be due to the choice of data utilized by Lin and Stiel(1977) in the development of the correlations for the molecular parameters, eq 22-25. Discussion For nonpolar-polar mixtures, the method of Tsonopolous (1974) with parameters calculated from eq 4-6 is similar to the procedure of this study utilizing the Pitzer-Curl relationship, eq 31, and eq 32-34 for T,,,, PCl2, and w12. The method of Tsonopolous can only be used for systems for which experimental data are available for B12 to establish the constant itlz of eq 4. In addition, wl2 is calculated in eq 6 as an arithmetic average of the acentric factors of the components. However, the value of w12 determined from eq 34 by the procedures of this study can be substantially smaller than the average of the component acentric factors for a nonpolar-polar system, since the effect of the dipole moment of the polar component is excluded in the calculation of the mixture parameters. For example, for the argon-water system, wI2 = 0.17 from eq 6 and 0.042 from eq 34. For simple polar systems the approach of Hayden and O'Connell (1975) is similar to that of the present study. This approach requires the mean radius of gyration of the molecule which is not easily accessible. Specific association parameters may also be required for this procedure for mixtures containing highly polar substances. The results of this study indicate that interaction second virial coefficients can be calculated with good accuracy from the parameters of the components for most polar systems. Because of the complexity of the molecular interactions for polar mixtures, it is more difficult than for nonpolar systems to calculate values of Blz with comparable accuracy as the best experimental data. For some polar systems specific interaction effects may also be present which cannot be characterized solely by the properties of the components. The procedure based on the use of the Kihara spherical core relationships, eq 7 and 8, with temperature-dependent molecular parameters enables reliable results to be obtained for the mixtures considered. Comparable results are obtained with the simplified procedure utilizing the Pitzer-Curl equations. For most nonpolar-polar systems, more accurate values of E,, result from the separation of induction contributions by the use of the polarizability of the nonpolar component.
187
Improvement beyond the results of this study, particularly for low reduced temperatures, can be obtained with a more accurate four-parameter model for polar fluids. Improved correlations for the molecular parameters can also be developed by the simultaneous treatment of pure component and interaction second virial coefficient data. Nomenclature a, b = constants of eq 3 a = radius of spherical core, A a* = 2a/po B = second virial coefficient, cm3/mol B* = reduced second virial coefficient, BPJRT, klz = interaction constant of eq 4 No = Avogadro's number P, = critical pressure, MPa P w = reduced vapor pressure R = gas constant T = temperature, K T, = critical temperature, K T R = reduced temperature, T I T , V , = critical volume, cm3/mol Y =
1/6(~4/&06)
z = €/KT
Greek Letters a = polarizability e = energy parameter, J 1) = association parameter in eq 1 K = Boltzmann constant, 1.3806 X J/K /I = actual or effective dipole moment p~ = reduced dipole group, wzP,/T: x lo5 F = induction group, apz/tpo6 p = distance between the surfaces of cores of molecules, A po = distance parameter x = polarity factor, defined in eq 26 w = acentric factor wh = third parameter characteristic of shape effects Subscripts and Superscripts ' = temperature-dependent parameter 1, 2 = parameters of components of mixture 12 = interaction parameter for mixture m = mixture property n = parameter characterizing dispersion effects Literature Cited Bottomley, G. A.; Spurling, T. H. Aust. J. Chem. 1067, 2 0 , 1789. Bougard, J.; Jadot, R. J. Chem. Phys. 1078, 73,415. Brewer, J. "Determination of Mixed Vlrial Coefficients", Final Report, Contract No. AF 49 (838pl620,Alr Force Office of Scientiflc Research, ArIlngton, VA, Dec 1967. Copeland, T. G.; Cole, R. H. J. Chem. Phys. 1076, 6 4 , 1747. Dymond, J. H.; Smith, E. 8. "The Vlrial Coefflcients of Pure Gases and Mixtures"; Oxford University Press: New York, 1980;p 289. Eisenman, L. I.; Stiei, L. I. Ind. Eng. Chem. Process Des. Dev. 1071, 10,
395. Giockler, G.; Fuller, D. L.; Roe, C. P. J . Chem. Phys. 1033a, 7 , 709. Glockler, G.; Roe, C. P.; Fuller, D. L. J. Chem. Phys. 1033b, 7 , 703. Gupta, S.K.; Lesslle, R. D.; King, A. D., Jr. J. Phys. Chem. 1072, 76,2170. Halm, R. L.; Stiel, L. 1. AIChE J. 1071, 17,259. Hayden, J. G.; O'Conneli, J. P. Ind. Eng. Chem. Process Des. Dev. 1075, 14, 209. Hemmaplardh, G.; King, A. D., Jr. J. Phys. Chem. 1072, 76,2170. Hicks, P. J., Jr.; Prausnitz, J. M. J . Chem. Eng. Data 1081. 26, 74. Kappallo, W.; Lund, N.; Schafer, K. 2. Phys. Chem. (Frankfurt am Main) 1063, 37, 196. LazaMe, H.; Breedveld, G.; Prausnitz, J. M. AIChE J. 1080, 26, 462. Lichtenthaler, R. N.; Schafer, K. Ber. Bunsenges . Phys . Chem. 1080, 73,
43. Lin, H.-H.; Stiel, L. 1. Can. J. Chem. Eng. 1077, 55, 597. Michels, A. Nuovo Cimento Suppl. 1058, 9 ,358. Neogl, P.; Kudchadker, A. P. J. Chem. Soc., Farraday Trans. 7 1977, 73,
385. O'Connell, J. P.; Prausnitz, J. M. I n d . Eng. Chem. Process Des. Dev. 1967, 6,245. Pitzer, K. S.;Curl, R. F., Jr. J. Am. Chem. SOC. 1057. 79,2369, Ramaiah, V.; Stiel, L. I. I n d . Eng. Chem. Process Des. Dev. 1073, 12,
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Rigby. M.; Prausnitz, J. M. J. Phys. Chem. 1066. 72, 330. Sinka, J. V.; Murphy, K. P. J. Chem. Eng. Data 1067, 72, 315. Tee, L. S.; Gotoh, S.;Stewart, W. E. I n d . Eng. Chem. Fundam. 1086, 5 ,
363. Tsonopoulos, C. AIChE J. 1074, 2 0 , 263. Vigdergauz, M.;Semkln, V. J. Chromatog. 1071, 5 8 , 95.
Received for review June 17, 1983 Accepted February 21, 1984
