indicating a relatively low surfactant adsorption capacity (equilibrium) of the Cr(II1) hydroxide particulates. The fraction of the feed retained as the effluent was fit by a second-order model in XI, L , Hi, and G. As the feed Cr concentration was increased at constant XI 121and decreased and 1 - Z&/ operating variables, 1 zIL pulled away from 1 - zh ’zi,indicating greater losses of the feed as the collapsed foam stream. The chemical and aeration costs for the precipitate flotation of a 50-mg per liter Cr suspension would be about 75$ per lb Cr. The effluent stream would contain 5 mg per liter Cr and the collapsed foam would contain 905 mg per liter Cr, leading to ready Cr recovery. One hundred-mg per liter Cr suspensions could be precipitate floated to 5 mg per liter, using two columns in series, with the only operating expense of the second column being the low cost of aeration.
zh/zi
Nomenclature
B = flow rate of effluent stream, l./min G = air rate, l./min H , = height above foam-suspension interface t o which foam rises in column before being taken off, cm Hi = height of liquid column (foam-suspension interface) above base of column, cm L = feed rate to column, l./min Xh = surfactant (SDS) concentration in effluent stream, mgi 1. XI = surfactant (SDS) concentration in feed stream, mg/l. = chromium(II1) hydroxide (Cr) concentration in effluent stream, mg/l. 2, = chromium(II1) hydroxide (Cr) concentration in feed stream, mg/l. 8, = surfactant-precipitate mixing time, min
z*
Literature Cited
Baarson, R. E., Ray, C. L., in “Unit Processes in Hydrometallurgy,” Vol. 24, M. E. Wadsworth and F. T. Davis, Eds., Gordon and Breach, New York, N.Y., 1964, p 656. Bhattacharyya, D., Carlton, J. A., Grieves, R. B., AIChE J . , 17, 419 (1971). Cross, J. T., Analyst, 90, 315 (1965). Grieves, R. B., J . Water Pollut. Contr. Fed., 42 (part 2), R336 (1970). Grieves, R . B., Bhattacharyya, D., Separ. Sci., 4, 301 (1969). Grieves, R. B., Schwartz, S.M., AIChE J . , 12, 746 (1966). Grieves, R. B., Malone, D . P., Conger, W. L., J . Amer. Water Works Ass., 62, 310 (1970a). Grieves, R. B., Ogbu, I. U., Conger, W. L., Separ. Sci., 5 , 583 (1970b). Grieves, R. B., Ettelt, G. A., Schrodt, J. T., Bhattacharyya, D., J . Sanit. Eng. Diu., Amer. SOC.Civil Eng., 95, 515 (1969). Kalman, K. S., PhD Thesis, McGill University, Montreal, Quebec, Canada, 1970. Mahne, E. J., Pinfold, T. A,, J . Appl. Chem., 18, 52 (1968). Mahne, E. J., Pinfold, T. A , , ibid., 19, 57 (1969). Rubin, A. J., J . Amer. Water Works Ass., 60, 832 (1968). Skrylev, L. D., Mokrushin, S. G., Zh. Prihl. Khim. (Leningrad), 34, 2403 (1961). Wood, R. K., Tran, T., Can. J . Chem. Eng., 44, 322 (1966). RECEIVED for review July 3, 1970 ACCEPTED February 3, 1971 Support for this work was from the Chicago Bridge and Iron Co., through a research grant.
Macroscopic Combining Rules for Second Virial Coefficient of Nonpolar Mixtures Larry I. Eisenman and Leonard 1. Stiel’ Department of Chemical Engineering and Metallurgy, Syracuse University, Syracuse, N . Y . 13210 F o r the calculation of the thermodynamic properties of pure fluids and mixtures, an equation of state is quite useful, especially for the dilute gas region. The virial equation of state is of particular importance, since theoretical relationships for the temperature dependence of its parameters can be derived by the use of a realistic intermolecular potential function for a group of fluids. I n addition, theoretical relationships for the parameters of the virial equation for mixtures are available in terms of the parameters for the pure components. In the virial equation of state, the compressibility factor is expressed as a series expansion in reciprocal volume. The coefficient of the I Present address, Department of Chemical Engineering, University of Missouri, Columbia, Mo. 65201. To whom correspondence should be addressed.
second term is called the “second virial coefficient” and is obtained from experimental PVT data. For nonpolar fluids, the Kihara core potential enables the calculation of the second virial coefficient to within the experimental accuracy of the data. Kihara (1953) proposed combining rules for the pure component parameters which permit the theoretical relationships for various types of core to be extended to nonpolar mixtures. Good agreement with experimental data is generally obtained by the use of the Kihara combining rules. Several studies (Dantzler et al., 196813; Huff and Reed, 1963; Prausnitz and Gunn, 1958) have been concerned with the development of macroscopic combining rules for the calculation of the second virial coefficients of nonpolar mixtures. The conclusion usually has been reached that second virial coefficients for mixtures containing dissimilar Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971 395
Procedures have been developed for the calculation of interaction virial coefficients for nonpolar mixtures, based on theoreticai relationships for the Kihara core potential. Correlations previously presented for the parameters of the Kihara spherical core potential in terms of the critical constants and acentric factor permit the accurate calculation of second virial coefficients for pure normal fluids. Improved relationships for nonpolar fluids of large size can be obtained by the inclusion of an additional shape parameter, as provided by the planar core model. Combining rules for the molecular parameters permit the calculation of interaction virial coefficients from the spherical core relationships for pure fluids to within the experimental error for nonpolar mixtures, including mixtures containing components of dissimilar size. Relationships are also developed for the interaction parameters T,,, and P,,, of the mixture. The only input data needed to apply the relationships of this study are the critical constants and acentric factors of the pure com'ponents, without the requirement of experimental data for the specific mixture.
components cannot be calculated to within the accuracy of the data without the use of binary interaction terms established from the experimental data. In the present study, it is shown that macroscopic relationships consistent with a three-parameter theorem of corresponding states permit the calculation of second virial coefficients from the Kihara core expressions to within the accuracy of the experimental data for a wide range of nonpolar mixtures, including mixtures containing dissimilar components. The development of macroscopic relationships for the parameters of nonpolar mixtures is important not only for the second virial coefficient, but also for the calculation of other properties such as the viscosity and diffusivity of gaseous mixtures. I n addition, these relationships have considerable implications for the calculation of the thermodynamic properties of dense gaseous mixtures (Pitzer and Hultgren, 1958; Prausnitz and Gunn, 1958) and liquid solutions (Rowlinson, 1969). The Second Virial Coefficient of Nonpolar Fluids
The intermolecular potential model of Kihara (1953, 1955, 1967) allows a convex core of any shape to be assigned to a molecule, and the potential energy is expressed in terms of p , the shortest distance between core surfaces. The following relationship results for the second virial coefficient for the Kihara potential:
B/Nop:= (2 x/3)Fs(Z) + (MO/PO)FZ(Z)+ [So/P02+ Yi x ( M O / P O ) L ] F I+( Z )
[ Yi ( M o l P o ) (SO/P,2) + ( v o / P : )
I
(1)
where M , is the mean curvature of the molecular core integrated over the core surface, So is the surface area of the core, V, is the core volume, and p o is the shortest distance between molecular centers when the potential energy is zero. Also 2 = t/KT where c is the minimum potential energy, and m
F,(t/KT) = (-s/12) J
-0
( 2 j / j ! ) ( t / ~ T ) ( i / ~ + ~ /j ~- ~ s ))/ r1[2(] 6 (2) For a spherical core of radius a, M , = 4 T U , So = 4 TU', V , = % TU^, and the Kihara potential can be expressed in terms of r , the distance between molecular centers, The relationship for the second virial coefficient for the spherical core potential is
B/a3= (2 x N , / 3 ) [ ~ +* ~3(2)1'6U*2Fi(Z) + 3(2)1'3u*F2(Z)+ 21'2F3(Z)](1 + 396
(3)
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
where u = 2 - 1 6 p , + 2 a and a* = 2 U / ( U - 2 a ) . Tee et al. (1966) developed the following generalized relationships for the parameters of the Kihara spherical core potential in terms of macroscopic parameters: ~ / K=Tf ~~( w )= 1.0043 3.045 w (4)
+
a(Pc/Tc)l= f i ( w ) = 2.2639 U*
where
w
0.8586 = f3(w) = 0.1501 + 2.3724 w -
w
(5) (6)
is the acentric factor (Pitzer, 1955) w =
-log PR,T R= 0.7 - 1.00
(7)
For 14 nonpolar substances, Equations 3 to 7 enabled virial coefficient data to be reproduced to within approximately 10 cc per mole. Inherent in the acentric factor approach is the assumption that a single dimensionless parameter is sufficient to represent deviations of the property from simple fluid behavior. An analysis of the general Kihara relationship (Equation 1) indicates that this assumption is strictly correct for the second virial coefficient for any shape core if the following linear approximation is sufficient (Pitzer, 1955) :
B/pi = (2 ~ / 3 ) N , [ F 3 ( 2 + ) xFi(Z)]
(8)
where x = 3 M,/2 ~ p , . For w > 0.1, second- and higherorder terms become significant and a realistically shaped core should be employed for a particular fluid t o obtain best accuracy for the second virial coefficient. Therefore, to test the accuracy of the spherical core relationships for molecules of appreciable size, calculations were also performed with the planar core model used by Connally and Kandalic (1960). For a planar core, M , = x ( l l + 1 2 ) , So = 2 1 1 1 2 , and V , = 0 , where l I is the plane width and l2 the plane length. Planar core parameters for n-alkanes (propane to n-octane) were presented by Connally (1961) for constant values of l1 = 0.89 and of po = 2.70. This value of P o represents an overall average which enables reproduction of the data to within experimental error. The available planar core parameters were utilized to obtain the following relationships ~ / K=T1.650 ~ 0.978 w (9)
+
po(Pc/Tc)'= 1.678 - 2.830 w + 2.425 w 2 (10) x = 3 [(Li + /2)/2 p o l = -0.437 + 14.983 w (11) These equations are applicable only for w 2 0.1 and for a ratio of l l / p o = (0.89/2.70), for which condition the planar core model contains one shape parameter. Equations 9 to 11 with Equation 1 yielded results compara-
ble to those obtained from the spherical core relationships for n-alkanes, alkenes, and isomeric hexanes (Eisenman, 1968). For these fluids, further improvement can possibly be obtained for the planar core model with variable values of the additional shape parameter, l l / p , , . Both the spherical core relationships (Equations 4 to 6) and the planar core relationships (Equations 9 to 11) with l l / p o = (0.89/2.70), can be combined with the theoretical expressions for the second virial coefficient into the following functional form:
B* = BPc/RT,=g(T.q,w) (12) The macroscopic relationship of Pitzer and Curl (1957) is also of the form of Equation 12. Values of B" calculated from the spherical core, planar core, and Pitzer-Curl relationships were compared a t equal values of w and reduced temperature. The results for the three methods converge for high values of T R and fall approximately on a single curve throughout the entire range, as shown in Figure 1 for o = 0.2. This behavior suggests that the relationships for the three methods are approaching the equation which produces the optimum accuracy for the second virial coefficient of nonpolar fluids by the use of a single shape parameter. Therefore, because of their applicability for most nonpolar fluids, the spherical core relationships were chosen in this study for the development of macroscopic combining rules for nonpolar mixtures, based on a threeparameter theorem of corresponding states.
2.00
1.75
1.50
-
I. 25
P
y
1.00
0" m I
1
I
*m
0.75
0.50
0.25
0.75
1.00 TR
1.25
Figure 1 . Relationship between B*
A
Spherical care
Planar core
1.50
1.75
2.00
and TR for w = 0.20 X Pitzer a n d Curl
(1957)
The Second Virial Coefficient of Nonpolar Mixtures
The second virial coefficient of a binary mixture can be expressed as
B , = $Bii
+ 2 yiyzBi2 + Y%zz
(13)
where B I 2is the interaction virial coefficient of the mixture. Various combining rules for the molecular parameters have been suggested for the calculation of Blz (Fender and Halsey, 1962; Hudson and McCoubrey, 1960; Magasanik and Ellington, 1963). Guggenheim and McGlashan (1951) developed relationships for the calculation of interaction virial coefficients of simple fluids from the critical constants of the pure components. More complicated relationships in terms of the ionization potentials of the components were developed by Cruickshank et al. (1966) and Huff and Reed (1963). In the latter study, the Pitzer-Curl 2 ( w , + w 2 ) / 2 , and an relationship was utilized with ~ 1 = average error of 10% was claimed for the mixtures examined. Prausnitz and Gunn (1958) presented the following combining rules for the calculation of interaction virial coefficients from the Pitzer-Curl relationship:
TC,)= ( T c l T c 2 )+1ATc,> '2 / 2 + A VcI2 Vc12= ( Vc: + Vc2) w12
=
(w1
+ w2)/2
(14) (15) (16)
The critical pressure, P,,?, is calculated from Tcli and VC,> [with Z,,? = ( 2 , + 2 , > ) / 2 ]The . correction terms, and AVC,>,were determined from experimental data for the binary system. Although good results are obtained by this procedure, experimental second virial coefficient data for all the binary systems of a multicomponent mixture are required. For nonpolar mixtures, Kihara derived the following relationship for the interaction virial coefficient, analogous to Equation 1:
+ where the mixture parameters are taken as 2 1 2
= tlz/~T = (t1t2)'"/KT
(18)
and
(19) Po,, = ( P O ] + P o , ) / 2 Several investigators tested Equations 17 to 19 for a number of mixtures. Brewer (1962) obtained excellent results for mixtures of argon with nitrogen and hydrogen. Prausnitz and Myers (1963) obtained very good agreement for carbon dioxide mixtures with hydrogen and nitrogen a t low temperatures. Connolly (1961) obtained excellent agreement for hydrogen-benzene mixtures and fair results for the system hydrogen-n-octane. As a test of Equations 17 to 19, calculations of B I 2 were made for a number of normal mixtures for which virial coefficient data were reported in the literature. The spherical core parameters determined by Tee et al. (1966) were used for argon, krypton, xenon, methane, nitrogen, ethane, ethylene, carbon dioxide, and neopentane, while planar core parameters (Connolly, 1961) were employed for the other normal alkanes through octane. I n general, good results were obtained for the mixtures considered (Eisenman, 1968). For a spherical core, Equation 17 can be expressed as follows Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
397
Blz/u;2
= (2 7rNo/3)[alx2+ 3(21b)alY’2K(zn)+ 3(2)1’3alY’2Fz(zi2) + 2”F3(Z12)](1 + a&)-’
(20)
where aB = 2 a l ? / ( u l r- 2 a d , al? = (al + a2)/2, and u12 = 2-l pol. + 2 all = (ul + a 2 ) / 2 from Equation 19. The parameters clz, uI2, and a12 of Equation 20 can be calculated from the combining rules and Equations 4 to 6 for the parameters of the pure components as
’
€n/K
= [TC,fi(wi)Tclfi(~2)I1
rn
1,3
rn
(21)
1,3
relationships for the pure component parameters yielded results which are within the experimental error of the second virial coefficient data for most cases (Eisenman, 1968). Good results are also obtained by this procedure for mixtures containing dissimilar components, as shown in Table I (Method 11) for normal alkane and argonhydrocarbon mixtures. The virial coefficients calculated by this method are usually slightly lower than those calculated from the spherical core relationships with the combining rules suggested by Kihara. Relationships for the macroscopic parameters T,,,, P,,, , and wI2 can be derived by the extension of Equations 4 to 6 as follows:
P
4
a12
ala: 1+a;
+ *-)1uza; +a2
=-
=
where g(w) = f 2 ( w ) f 3 ( w ) . For the mixtures studied, this procedure yielded results as good as those obtained by the use of actual parameters with the Kihara relationship for B12. In addition, values of B I 2 can be obtained for substances for which no core data are available, and the comparisons were extended to mixtures containing neon, carbon tetrafluoride, 2-methylpentane, 2,2-dimethylbutane, benzene, cyclohexane, and chloroform (Eisenman, 1968). The data of Dantzler and co-workers for normal alkane mixtures (Dantzler et al., 196813) and argon-hydrocarbon mixtures (Dantzler et al., 1968a) were particularly valuable for these comparisons. These investigators determined B12 from an experimental excess quantity, [,
5 = B12 - (B11+ B22)/ 2
(24)
where Bll and B22 for the pure components were calculated from the relationship of McGlashan and Potter (1962). I n the present study, the values of B12were revised from Equation 24 by the calculation of Bll and B D from the spherical core relationships (Equations 3 to 6). For most of the measurements of these investigators, the error in is estimated to be less than 3 cm3 per mole, so that the errors in the revised values of B12 should be approximately the same as those resulting from the use of the spherical core relationships for the pure components (about 10 cm3 per mole). For normal alkane and argonhydrocarbon mixtures, the standard errors between values of B l zcalculated with Equations 20 to 23 and the spherical core relationships of Tee and co-workers (Method I ) are shown in Table I. Significantly, this method enables second virial coefficient data to be reproduced to within the experimental error for mixtures containing molecules of dissimilar size. A simplified procedure also has been developed for the application of the spherical core relationships. I n place of the combining rules for pol? and alzsuggested by Kihara, arithmetic mean rules are assumed for uI2 and a$. Equations 2 1 and 22 are then still applicable for t l z and u12, but in place of Equation 23 the following relationship results for a;: aE = [ f 3 ( W 1 ) + f 3 ( W 2 ) ] / 2 (25) Equations 20 to 22, Equation 25, and the spherical core 398
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
7
c
3
If Equation 25 is used for a t , an arithmetic mean rule results from Equation 28 for w12 (Equation 16). This relationship was used in previous three-parameter correlations
Table 1. Comparison of Calculated Values of 6 1 2 with Experimental Data
Methods use Equation 20, Equations 4-6 for spherical core parameters, and Method I uses Equations 21-23; Method 11, Equations 21, 22, and 25; and Method 111, Equations 16,31, and 32 Standard error of estimatea, ccfmole Normal alkane mixtures (Dantzler et al., 1968b)
Method
No. of points
I
II
111
4 4 4 4 4 4 4 4 4 4 4 4 4 3 3
2.7 3.2 3.1 6.9 4.8 2.2 1.8 2.9 4.7 2.3 1.7 6.0 2.6 4.2 6.2
3.2 3.9 3.1 7.6 6.6 2.1 1.8 2.7 3.1 2.4 2.5 5.3 3.0 5.6 6.5
1.9 2.1 9.5 16.0 24.7 1.2 4.9 13.6 26.7 1.1 5.7 14.9 1.0 4.9 4.8
1 1 1 2
0.1 7.8 3.3 9.5
1.6 7.5 2.5 10.0
21.2 32.8 21.3 26.2
Methane-ethane Methane-propane Methane-n-butane Methane-n-pentane Methane-n-hexane Ethane-propane Ethane-n-butane Ethane-n-pentane Ethane-n-hexane Propane-n-butane Propane-n-pentane Propane-n-hexane n-Butane-n-pentane n-Butane-n-hexane n-Pentane-n-hexane Argon-hydrocarbon mixtures (Dantzler et al., 1968a)
Argon-2,2-dimethylbutane Argon-2-methylpentane Argon-n-pentane Argon-n-hexane “Standard error of estimate =
for the second virial coefficient (Huff and Reed, 1963; Prausnitz and Gunn, 1958). Equations 29 and 30 suggest a simplified set of combining rules for the critical constants when w 1 = w 2 = 0
Rules of this type are correct only for a two-parameter intermolecular potential function, such as the LennardJones potential, but are commonly employed with threeparameter relationships. The use of these rules with Equations 26 to 28 and Equation 16 for w12 does not enable the prediction of B12 from Equation 20 to within the experimental accuracy of the data for mixtures containing dissimilar components, as shown in Table I (Method 111). For example, for methane-n-hexane, a standard error of 24.7 cc per mole results from the use of these rules. Several investigators (Dantzler et al., 1968b; Prausnitz and Gunn, 1958) have indicated that Equation 31 provides for a value of T,,? which is too high. T o confirm this effect, the ratio of T,,. from Equations 29 and 31 is
Tc,?/T:,2 =[
Zlfi(~~z)
f i ( ~ ) f i ( d ] ~
0.96
0'92
I I
1
0.I
Discussion
A comparison between experimental values of B I 2for the data of Dantzler et al. (196813) for normal alkane mixtures a t 100°C and those calculated by Method I1 is presented in Table 11. Dantzler et al. (196813) also calculated second virial coefficients for this condition with the relationship of McGlashan and Potter (1962) by the use of the geometric mean rule for T,,2 (Equation 31), which are included in Table 11. I n Table 11, both the values of BI2determined by Dantzler et al. (1968b) and those recalculated by the authors by the use of the spherical core relationships are presented. Dantzler and co-workers also tested a more complicated version of the geometric mean rule suggested by Fender and Halsey (1962), and the equation for Tc12proposed by Hudson and McCoubrey (1960) involving ionization potentials and critical volumes. For mixtures containing dissimilar components, the deviations for the methods based on the geometric mean rule and the relationships of Fender and Halsey (1962) and Hudson and McCoubrey (1960) are beyond the experimental uncertainties involved. The method of Huff and Reed (1963) utilizes the relationship of Hudson and McCoubrey for T,,?. For w1 = 0 2 , no correction is provided by either Method I or Method I1 over the simple combining rules for the critical constants (Method 111). Therefore, for mixtures of benzene and chloroform (Saxena et al., 1967) which have the same value of W , poor results are obtained by all three methods. The value of the acentric factor for chloroform includes polarity effects, while the value of
0.4
*P
Figure 2 . Relationship between ( T C 2 / T : , ? )and w2 for values of w1
Table II. Comparison between Experimental and Calculated Values of -812 (Cc/M) for Data of Dantzler et al. (1968b) at 100°C
(33)
The right-hand side of Equation 33 is always less than one, and the greatest discrepancy in T",,,- T,? is when the absolute value of ( w I - w2) is the largest. This behavior is plotted is illustrated in Figure 2 in which Tc!:/T:> against w e for varying values of wl. The values of P,,? calculated by the two approaches are quite close for most mixtures.
I
0.3
0.2
System
Exptla
New exptlb
Geometric mean'
Method II
CH,-C,Hs CH,-C3Ha CH,-CJIio CHd-CjHir CH*-C~HII C>Hs-C3Ha C2Hs-C8io C2H6-CrHi, C?Hs-CsHia C3Hs-CdHia C3HrCsH12 C3Ha-csH1, C8io-CaHiz CdHio-CsHi, CoHi2-CsHi4
56 75 100 122 145 165 215 272 319 316 399 478 540 650 838
54.8 76.6 97.1 117.0 142.2 165.4 212.9 265.9 315.0 317.6 397.6 471.7 535.1 647.0 831.0
53 80 109 137 167 167 223 279 337 321 404 490 542 660 841
51.8 76.6 100.8 123.7 145.9 163.8 215.9 267.9 319.7 315.3 394.1 473.9 531.2 644.6 829.5
"Values presented by Dantzler et al. (1968b). bData revised from original source by the use of Equations 3 t o 6.
w required to characterize the interaction of chloroform with a nonpolar substance is that which accounts only for the shape of the molecule. Erratic results were also obtained by the methods of this study for mixtures containing carbon dioxide which has strong quadrupole effects (Eisenman, 1968). These comparisons indicate that the properties of a mixture are more sensitive to the exact type of molecular interaction than the properties of a pure fluid, and the relationships of this study are not recommended without modification for mixtures containing slightly polar components which generally follow the acentric factor approach. Improvement beyond the approach of this study for mixtures containing polar components can be obtained by the inclusion of additional parameters, such as a fourth parameter characterizing dipole-dipole interactions. I n addition, improved results for nonpolar mixtures containing large or complex molecules may ultimately be obtained by the use of an additional shape parameter, as provided by the planar core model. The relationships for the second virial coefficient permit the calculation of the compressibility factor and derived
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
399
thermodynamic properties of normal fluids and mixtures in the dilute gaseous region. The combining rules developed in this study should also be applicable for the calculation of the transport properties of gas mixtures. As pointed out by Hudson and McCoubrey (1960), the effect of the correction in t I 2 on the calculated transport properties is considerably smaller than the effect on the virial coefficient, because of the small variation of the collision integrals for the transport properties with t . The relationships of this study for T , , P , , and w12 should also be useful for the calculation of thermodynamic properties of mixtures in the dense gaseous and liquid regions. Joffe and Zudkevitch (1966) determined a value of Tc, = 274 for methane-n-butane by the application of the Redlich-Kwong equation of state to mixtures, as compared to the value of 278 from Equation 32, 271 from the study of Prausnitz and Gunn (1958), and the geometric mean value of 285. In this study, the combining rules for u 1 2 , ( t / ~ ) ~ 2 and , a& and relationships for the Kihara spherical core potential are shown to be sufficient for the calculation of second virial coefficients for nonpolar mixtures. The significance is that the properties of a wide range of mixtures in the dilute gaseous region can be calculated from the use of only the critical constants and acentric factors of the pure components, without the requirement of experimental data for the specific mixture. Acknowledgment
The authors thank the Syracuse University Computing Center for their cooperation during the course of this investigation, and Michelle Millot for help in manuscript preparation. Nomenclature -. - -
a = radius of a spherical core a* = dimensionless shape parameter for Kihara spherical core potential, 2 a / ( -~ 2 a ) B = second virial coefficient, cm3/g-mol B” = reduced second virial coefficient, BP,/ R T , fl(W)
+ 3.0454 w 0.1501 + 2.3724 w
= 1.0043
f d w ) = 2.2639 - 0.8586 w
= = rectangular dimensions of core M , = mean curvature of molecular core integrated over
f3(W)
11, 1 2
the core surface
No P, PR r R
= Avogadro’s number = critical pressure, atm
= reduced vapor pressure = distance between molecules = gas constant, 1.987 cal/g-mol K so= surface area af core T = temperature, K T , = critical temperature, K TR = reduced temperature, T / T , v, = critical volume, cm3/g-mol V” = volume of core x = shape parameter, 3 M,/ 2 r p 0 Y = mole fraction C/KT 2, = critical compressibility factor, P,V,/ R T ,
z=
400
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
Greek Letters = maximum energy of attraction, ergs = Boltzmann constant, 1.3805 x erg/” K w = acentric factor, defined in Equation 7 t
K
p po
= shortest distance between molecular cores = shortest distance between molecular cores when
u
= distance between molecular centers when poten-
the potential energy is zero [ =
tial energy is zero excess quantity, defined in Equation 24
Subscripts and Superscripts
1, 2 = components of a mixture 12 = interaction term for mixture s = simplified combining rules literature Cited
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