Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 235
Binary Interaction Constants for Mixtures with a Wide Range in Component Properties David I?. Pesuit
D. R. Pesuit &
Associates, Northampton, Massachusetts 0 1060
Binary interaction constants from the literature for 48 systems are compared with calculated values. None of the equations tested is found to be accurate for the whole range of interactions considered, but it is apparent that one of two or three equations is accurate in most cases; the accuracy of these equations suggests that accurate corresponding states estimates of mixture thermodynamic properties can be obtained in many cases from the properties of pure components alone.
Introduction Binary interaction constants are of some interest in chemistry and chemical engineering. Equations of state require them as corrections to mixing rules (Chueh and Prausnitz, 1967; Zudkevich and Joffe, 1970), while calculations based upon intermolecular force functions require them as corrections to the geometric mean of pure component energy force constants (Flory, 1965; LonguettHiggins, 1951; Rowlinson, 1969); in the latter case in particular it can be shown that calculations are quite sensitive in some temperature ranges to the value of the interaction constant which is used. Present practice dictates that these constants be evaluated from mixture data, sometimes at several temperatures or pressures, before calculations or comparisons with other theoretical models are attempted. And yet, several equations do exist for obtaining a K12correction to the geometric mean c12 as a function of pure component molecular parameters (Kramer and Herschbach, 1970; Sikora, 1970; Pesuit, 1971). It is the purpose of this paper to compare such predicted values with published values for mixtures whose components offer a wide range in physical properties and thus to obtain a fairly comprehensive measure of the present state of the predictive art. Equations In calculations of mixture properties based upon intermolecular potential models, binary interaction constants are usually written as corrections to the geometric mean of pure component energy force constants.
vG2 (1- Kl2)
where the cri are distance force constants and the X i are defined for the various quantum mechanical models of molecular attraction: London (1938), X = I , where I is the first ionization potential; Slater-Kirkwood (1931), X = ( N / C U )where ~ / ~ ,cy is the mean polarizability and N is an electron number; Kirkwood-Muller (1932; 1936), X = #/cy, where # is the diamagnetic susceptibility. If, on the other hand, the Xi are eliminated between eq 2 and 3, all models of molecular attraction give
where
In both eq 4 and 5 the further assumption is made that the distance force constant for unlike interaction is the arithmetic mean of the pure component distance force constants, as would be the case for the interaction of hard spheres.
(1)
(7)
Equations based in molecular physics also obtain K12as a correction to the geometric mean energy force constant; if the London, Slater-Kirkwood, or Kirkwood-Muller potential energy of attraction is equated with the Lennard-Jones sixth-power attractive term (Kramer and Herschbach, 1970)
Since the ratio of the geometric and arithmetic means of two quantities is a measure of their difference, these equations define K12 as a function of differences in “electronic” properties and molecular size. In addition to these equations, there are other more recent equations for K12which are based upon physical models of molecular interaction at short distance (Sikora, 1970; Smith, 1972). Sikora used eq 2 and 3, but he also wrote parallel expressions for 12th power repulsion
612
=:
(3) one obtains, upon elimination of the mean polarizabilities ai
0019-7874/78/1017-0235$01.00/0
where R12 was defined by assuming the total energy of repulsion to be at a minimum and equal to the sum of the
0 1978 American Chemical Society
236 Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 Table I. A Comparison of Kong's Rule for the Distance Force Constant u with the Arithmetic Mean ;u z ~ z~~/ = F where F Is Given by Eq 18. Q, and Qo Are Ratios of Pure Component Force Constants; Q, = u I / u ,; 8, = e,/e ,
~
L
Qu
Q,
1.0
1 1.5 2 3 5 10 15 20
1.000 1.000 1.001 1.002 1.004 1.009 1.012 1.014
1.125 1.001 1.004 1.005 1.008 1.015 1.021 1.026 1.030
1.25 1.005 1.009 1.012 1.017 1.023 1.034 1.041 1.046
1.375 1.011 1.016 1.020 1.026 1.035 1.048 1.057 1.063
1.50 1.017 1.024 1.029 1.037 1.047 1.063 1.073 1.080
2
+
(tlu112)1/13 (e2u2l2)l/l3 113
Equation 16 is much like Sikora's 6-12 equation since it only lacks the electronic factor. Equation 17, however, becomes quite interesting if it is rewritten
energies of deformation of each electron cloud. Eliminating ui between eq 3 and 9 leads to where
F= while the same operation for the unlike interaction gives
Thus
)
R12
Parallel expressions can be written for an n-6 potential function, giving
The pure-component coefficients of repulsion in eq 12 and 13 are the Lennard-Jones values Ri
-
~ i ~ i "
Table I shows that eq 17 nearly always provides a combining rule for u12 which is greater than the arithmetic mean. Other equations have been suggested for the binary interaction constant which offer less in the way of fundamental justification. Kreglewski (1969) and Lin and Robinson (1971) have suggested, for example, that the harmonic mean is the correct rule for interactions not including helium.
That is, they define an effective interaction constant whose value is uninfluenced by differences in nearly all molecular parameters.
(14)
while the unlike repulsion coefficient is given by the deformation model
These equations show that equating potential energies of attraction and repulsion leads to an interaction constant whose electronic factor Xi is enhanced when compared with eq 4 and whose form is not, interestingly enough, dependent upon the physical model of molecular repulsion which Sikora used. More recently, Kong (1973) has made use of Sikora's model for the potential energy of unlike repulsion, but he has assumed as Kirkwood (1957) suggested that the potential energy of unlike molecular attraction is a geometric mean of pure component values. For the 6-12 potential he obtains
Hiza and Duncan (1970) have suggested an empirical rule
but they have indicated that this rule should not be applicable for molecules with large difference in size. In the sections below, these rules and their modifications where possible in the event that one of the molecules is much heavier and asymmetric will be compared with interaction constants for 48 mixtures offering a wide range in component properties. Tests of eq 16 are not included, since the electronic factor it lacks has been shown to be of some importance for interacting molecules that are of nearly like size (Sikora, 1970). Pure Component Data The equations above require sets of pure component properties which must be internally consistent if com-
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
Table 11. Lennard-Jones Force Constants for Argon, Determined by Nonlinear Least-Squares Fit to Data from Hecht and Donth (19688) over the Listed Temperature Ranges bo = 2 N n o '13, TR range elk, K cm3/g-mol 0.6 0.6 0.6 0.6
94.92 106.78 117.78 118.94
to 1.0 to 1.6
to 3.6 to 5.6
79.46 64.31 53.92 52.92
parisons are to be me,aningful. Force constants obtained by fitting a two-parameter potential to virial coefficients, for example, can lead to inconsistencies because the values of such constants depend upon the range of temperature over which the data are available and because several pairs of constants can give equally good fits over a given range of temperature; the former effect is clearly seen in Table I1 for argon, and it may well serve to explain why Good and Hope (1971) obtained a distance force constant for propane which is more than 30% larger than their value for butane. A more consistent set of force constants is obtained by fitting a three-parameter potential to viscosity and virial coefficient dtata (Tee et al., 1966), but such data are simply not available over a range of temperatures for many components of interest. Thus, it appears that the most consistent complete set of force constants now available is obtained from critical properties, as defined for the inert gases many years ago. u, = (I
0.828Vc,'13
(23)
= 0.0108Tc,
(Y
= 0.345(Vc.- 22.4) X
Table 111. Force Constants, Polarizabilities, and Ionization Potentials for Some Molecules of Interest 0 , '4 elk, K a , cm3 X l o z 5 I , eV 3.496 4.233 4.306 3.830 4.129 4.380 4.842 5.258 5.516 5.261 5.943 6.510 6.141 7.015 3.753 3.765 3.695 2.771 3.081 3.181 3.811 3.741 2.839 3.453 3.712 3.795 3.538 4.107 4.063
117.453 234.052 118.174 149.210 221.523 239.006 284.538 332.738 367.649 439.586 397.428 444.986 585.385 482.159 104.007 238.035 254.071 8.194 34.121 506.571 292.338 163.869 35.608 317.344 98.811 242.340 121.131 337.066 226.158
cm3
(25)
Mixture D a t a Reduction Many of the binary interaction constants in the following section were obtained by Hiza and Duncan (1970) and by Lin and Robinson (1971), whose techniques are well documented. The other values, mainly for polar-nonpolar interactions and for the interactions of molecules with unlike size, were obtained from best fits to vapor-liquid equilibrium data using the molecular corresponding states formalism developed below. Interaction constants calculated as corrections to mixing rules in equations of state (Chueh and Prausnitz, 1967) were not included since there is little reason at preslent to believe that such constants should have the same values as those defined relative to potential parameters as below. A power series in inverse volume was written for both
16.26 28.10 28.80 26.00 42.80 44.70 62.90 81.20 99.50 103.20 117.80 155.00 163.00 191.15 19.50 25.94 26.30 2.036 8.023 14.44 35.00 24.559 3.926 21.45 17.34 29.21 15.61 37.803 39.989
15.755 13.84 15.35 12.99 10.516 11.65 11.08 10.63 10.37 9.245 10.27 10.23 8.12 10.19 14.009 13.786 12.74 24.581 15.247 12.61 10.46 13.996 21.559 10.15 15.59 12.90 12.07 12.34 12.127
saturated liquid and vapor phases and truncated after the third term
(24)
Bienkowski and Chao (1975) have shown that an equation like eq 23 supplies realistic relative pure-component force constants, even for molecules as asymmetric as CIHIOor Cl,,Hz2. Equation 23 also allows for consistent modification in the event that one molecule is much larger than the other and also asymmetric as discussed in a later section. The constants in these equations cancel out in the equations for K12above. Force constants from eq 23 and 24 are listed in Table 111; pseudocritical constants are defined for He, HS, and Ne so as to approximate two-parameter correspondingstates behavior. The ionization potentials in Table I11 are taken where possible from Worley's self-consistent set (1969), while the polarizabilities are from several sources. Polarizabilities were checked against the following equation which was found to give semiquantitative results for chain molecules with critical volumes greater than 90 cm3/g mol.
237
ps =
y
(1
+ +
1;)
P
5)
E(,,,,
ps= V
where L and V are saturated volumes. Solution for p and y gives
p=
Ps
- (V
RT
+ VL + L2)
-
( V + L)
(26)
Equations 26 and 27 were combined with an expression denoting equality of fugacities for a pure component (Prausnitz, 1968) to give p,( v - L)"
RT
-
(V - L2) + 2VL In
(F)
=0
(28)
These three equations allowed p and y to be determined as functions of temperature, using vapor pressure and saturated liquid data over the whole subcritical temperature range. Figures 1 and 2 show the surprising results of several such calculations. It is apparent that these coefficients, even though they are based upon saturated P V T data, behave very much like second and third virial coefficients at temperatures near the critical. Such coefficients were spline-fitted to values from well-known correlations for second and third virial coefficients (Pitzer and Curl, 1957; Chueh and Prausnitz, 1967) to define pure component properties over the whole reduced temperature range. Mixtures were then approached using for second cross
TClZ
0
ARGON. KRYPTON. XENON
lt COiFtS
06
06
OB
01
09
10
11
REDUCED TEMPERATURE TITc
Figure 1. The reduced virial form coefficient 0 as a function of reduced temperature.
04
05
I 06
1 01
I
I
I
I
OB
09
I O
11
REDUCED TEMPERATURf 1 T c
Figure 2. The reduced virial form coefficient y as a function of reduced temperature.
coefficients a two-parameter corresponding states construction PlZ
VClZ = @II
(&)
where aIIwas determined over the whole reduced temperature range as above for argon, krypton, and xenon 6
@II
=
C Ai/TRi
i=l
(30)
where A. = -0.85473, At = 9.21321, A2 = -31.9735, A3 = 48.1530, A4 = -38.8967, A5 = 15.7747, As = -2.5319, and V,,, is the usual Lorentz mean and where
=
.\/T,T,(1- K1z)
(31)
Third cross coefficients were defined (as with virial coefficients) as geometric means. Interaction constants obtained in this manner are entirely analogous to those obtained using a two-parameter potential function (Pitzer, 1939) when used in conjunction with the above force constant equations. Interaction constants obtained in this manner were also unambiguous for most binaries, regardless of difference in physical properties, in the sense that one interaction constant independent of temperature sufficed to give a best fit over the whole range of temperature. Binaries with large differences in size were found, however, to require a temperature dependent interaction constant, which took one value at temperatures near the critical of the heavier component where most interactions are gas-phase interactions, and a smaller value at temperatures less than the geometric mean of the pure component critical temperatures where the liquid phase is well defined. The values reported in Table IX are the larger values, since it is gas-phase interactions which are of interest to us here. Needless to say, this method for developing interaction constants is unusual; an equation of state normally limited to gases at low pressures is being used to reduce highpressure vapor-liquid equilibrium data and obtain binary interaction constants for what is essentially a study of vapor-phase nonidealities. There is little doubt that the methods results are reasonable; Hiza and Duncan obtained Klz = 0.05 for CH4-HzS and Klz = 0.01 for N2-C0 compared with 0.04 and 0.003 from these calculations. But it is not clear why such a method should give reasonable values. Several papers other than this one have appeared which show similarity between vapor and liquid phases using the virial equation (Auslaender, 1968; Gopala Rao, 1960); Kammerlingh-Onnes in the original paper on the virial equation (1901) used it for liquid and gaseous phases to calculate coefficients for carbon dioxide in the gaseous phase. But a more reasonable explanation of the method’s success may lie in studies which have shown (1)that most contributions to liquid phase potential energy come from nearest-neighbor interactions, particularly in the limit of random packing (Moelwyn-Hughes, 1961; Lennard-Jones and Ingham, 1925) and (2) that the potential well may be shallower at this distance than the 6-12 potential indicates (Coulson and Davis, 1952; Guggenheim and McGlashan, 1960). It appears that an equation much like eq 4 may be written for the liquid phase binary interaction constant but with a lower power size difference factor, indicating that interaction constants could be much the same for liquid-phase and gas-phase interactions so long as the molecules were nearly like-sized. Finally, it is necessary to ask how a two-parameter equation like eq 29 is able to supply accurate interaction constants from VLE data, even though it is clear from Figure 3 and from the sensitivity analysis at the end of this paper that a third parameter becomes important to calculations of vapor phase mixture properties at reduced temperatures less than T R = 0.85. The answer is that most data available were at higher reduced temperatures; reduced temperatures from eq 31 were greater than 0.85 for 17 of the 19 systems considered, with most values falling between 0.9 IT R 11.25. Errors to be expected in Klz with this method are usually smaller than f O . O 1 and generally smaller than *0.02. Comparisons for Molecules with Nearly-Like Shapes In Tables IV-VI calculations are summarized for the
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
239
Table IV. Interaction Constants for Noble Gas Mixtures Sik612
Hizano. 1
2 3 4 5 6 7 8 9 10
binary He-Ne He-Ar He-Kr He-Xe Ne-Ar Ne-Kr Ne-Xe Ar-Kr Ar-Xe Kr-Xe
oJu1
1.02 1.26 1.35 1.47 1.23 1.32 1.43 1.07 1.16 1.09
4.34 14.40 20.00 27.67 3.31 4.60 6.37 1.40 1.92 1.39
" Hiza and Duncan (1970).
0.07" 0.2P 0.3gb 0.42b 0.141"rb 0.20" 0.338b 0.014" O.03gb 0.003b
X= HM ( N / a ) ' / , ( N / a ) l / Z eq 21
Sik612 (N/a)1/2X = I
Duncan
eJel
0.039 0.225 0.31 2 0.424 0.128 0.202 0.300 0.027 0.085 0.033
0.028 0.040 0.070 0.106 0.053 0.064 0.103 0.006 0.018 0.055
-
av abs error 0.026
0.135
-
hybrid
X=
eq4X=
0.033 0.283 0.389 0.511 0.144 0.237 0.358 0.016 0.072 0.022
-
0.082 0.248 0.345 0.451 0.160 0.21 5 0.320 0.017 0.059 0.017
-
0.055 0.040 0.074 0.109 0.073 0.073 0.144 0.008 0.019 0.006
0.030
0.020
0.127
0.220 0.507 0.574 0.633 0.156 0.234 0.315 0.013 0.051 0.013
-
0.093
Lin and Robinson (1971).
Table V. Interaction Constants for Mixtures of the Noble Gases with Nonpolar Molecules
no. 11 12 13 14 15 16 17 18 19 20 21 22
binary He-H, He-N, He-CH, He-C,H, Ne-CH, Ne-C,H, Ar-0,
Ar-CO Ar-N, Ar-CH, Ar-H, Xe-N,
o,/o
Sik612 X= I
0.064 0.063 0.100 0.137
0.116 0.326 0.416 0.548
av abs error
0.038
0.247
0.081
0.28 0.31 0.014 0.014 0.002 0.063 0.000 0.073
0.252 0.406 0.087 0.026 0.001 0.055 0.002 0.079
0.160 0.194 0.000 0.010 0.006 0.040 0.077 0.014
0.263 0.398 0.018 0.008 0.003 0.026 0.071 0.050
€,/e
I
4.16 12.0 18.2 27.0
1.11 1.34 1.38 1.58
1.35 1.45 1.01 1.06 1.07 1.10 1.14 1.09
" Hiza and Duncan (l970).
eq 4 X = (Nla)"'
0.25 0.25 0.40 0.45b
HizaDuncan 0.240 0.232 0.369 0.541
4.19 6.22 1.03 1.13 1.19 1.26 3.46 2.29
Sik612 hybrid X= X= (N/a)'/Z (N/a)'/2 0.169 0.116 0.064 0.292 0.389 0.124 0.491 0.161 0.044 0.221
- -
av abs error 0.028 0.052 0.031 (excluding Ar-H,) (0.032) (0.049) (0.025) Recalculation, using Hiza and Duncan's Table I.
HM eq 21 0.210 0.468 0.556 0.629 0.148
0.367 0.452 0.001 0.017 0.009 0.082 0.188 0.050
0.246 0.278 0.000 0.015 0,009 0.072 0.137 0.022
0.211 0.309 0.000 0.002 0.004 0.007 0.116 0.080
0.060 (0.042)
0.052 (0.021)
0.041 (0.023)
Table VI. Interaction Constants for Small Nonpolar Molecules Sik612
no. 23 24 25 26 27 28 29 30 31 32 33 34
binary H,-CH, H,-C,H, H,-C,H, N,-CO N2-02 N,-CH, N2-C2H6
CH,-CO CH,-CO, CH,-CF, C,H,-CO, C,H6-CO
u2/01
e2/el
1.24 1.34 1.42
4.37 6.49 7.00
1.01 1.05 1.03 1.18 1.02 1.02 1.12 1.10
1.17
" Hiza and Duncan (11970).
1.05 1.23 1.51 2.42 1.43 1.60 1.19 1.07 2.30
hybrid X= ( N / a ) ' / ' (N/a)"'
X=
HizaDuncan
eq 4 X = (N/a)",
Sik612 I
0.046 0.144 0.093
0.041 0.067 0.095
0.154 0.269 0.318
0.158 0.250 0.314
0.046 0.072 0.101
0.222 0.320 0.339
av abs error 0.044
0.018
0.197
0.191
0.023
0.289
0.023 0.082 0.050 0,098 0.013 0.009 0.044 0.083 0.048
0.001 0.004 0.018 0.035 0.012 0.019 0.063 0.027 0.028
0.003 0.017 0.014 0.094 0.005 0.001 0.031 0.028 0.073
0.001 0.005 0.039 0.103 0.027 0.039 0.125 0.050 0.086
0.001 0.005 0.034 0.049 0.024 0.038 0.112 0.046 0.038
0.000 0.005 0.021 0.090 0.016 0.027 0.004 0.001 0.081
av abs error 0.035
0.017
0.034
0,019
0.006
0.034
0.03" 0.07" 0.05" 0.01" 0.002" 0.03b 0.05" 0.018" 0.04b 0.110b 0.06" 0.025b
x=
-
HM eq 21
This work.
complete set of noble gas interactions, for the interactions of noble gases with some small nonpolar molecules, and for twelve interactions, between nonpolar molecules. The tables show clearly that the Hiza-Duncan rule, eq 22, gives fairly accurate predictions for all of these binaries and that the attraction rule, eq 4,with the Slater-Kirkwood substitution X = (N/(x)lI2and with N set equal to the total number of electrons per molecule is relatively poor. Hiza and Duncan reached the same conclusion when they compared their correlation with eq 4 using the London substitution, X = I. In contrast, however, the tables
indicate that the Sikora 6-12 rule, eq 13-15, with X = ( N / a ) l l Zor X = I, is nearly as accurate as the HizaDuncan rule for all interactions except those including hydrogen. The tables also include comparisons for a new hybrid rule
K 1 2= 1 -
(
24--)2 x 1 + x 2
(-------) 6
2
01
6
+
02
(32)
which combines the size difference factor found in eq 4 and
240
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
Table VII. Interaction Constants for Polar-Nonpolar Mixtures of Similar-Sized Molecules ( u 2 / u < 1-19) no.
35 36 37 38 39 40 41 42
binary C0,-H.S CH,-H;S Ar-NH, CF,-CHF, N,-NH, C0,-SO, CO,H,O C,H,-HCl
" This work.
eq 4 Sikora 6-12 hybrid HM X = I X = (N/cu)It2 X = I X = (N/cu)'/l X = I X = ( N / a ) l f 2 eq 21
Kdata
HizaDuncan
0.025" 0.050b 0.060' 0.005' 0.045' 0.020' 0.020" 0.040'
0.086 0.059 0.177 0.022 0.170 0.023 0.016 0.016
0.010
av abs error 0.045
0.019
0.006 0.024 0.002 0.026 0.007 0.022 0.020
0.020 0.015 0.053 0.033 0.045 0.021 0.017 0.035 0.007
0.017 0,009 0.052 0.002 0.019 0.018 0.018 0.050 0.016
0.019 0.012 0.047 0.003 0.048 0.009 0.021 0.021 0.012
0.016 0.005 0.045 0.002 0.022 0.006 0.024 0.036 0.013
0.005 0.054 0.111 0.009 0.149 0.015 0.067 0.000 0.034
Hiza and Duncan (1970).
Table VIII. Length-Width Ratios and Target Factors for Common Larger Molecules species
0.008 0.003 0.023 0.001 0.013 0.006 0.022 0.028 0.021
a/b 1.35 1.42 1.50 1.90 2.45 2.55 2.76 3.08 3.50 3.90 4.32
the target factor width ratio a / b
T~~
in eq 33 as a function of length-to(33)
712
0.992 0,990 0.986 0.965 0.935 0.930 0.920 0.901 0.880 0.862 0.844
5 with the squared electronic factor that was shown above to stem from equating sixth power attractive and twelfth power repulsive potential energy contributions; it is seen that this rule with the Slater-Kirkwood substitution is more accurate than any of the others for the known interactions of nonpolar molecules with each other or H2but not for the noble gas interactions or for interactions with helium. Finally, it appears that the harmonic mean rule, eq 21, is accurate for the heavier noble gases, but that for the other classes of interactions it is relatively poor. In Table VI1 comparisons are shown for the interactions of polar and nonpolar molecules with nearly like size, and it is seen that in this case the Sikora rule and the hybrid rule are both fairly accurate; calculations using the harmonic mean rule or the Hiza-Duncan rule are relatively poor. Several other comparisons have been made which are not listed in Tables IV-VII. They show, for example, that eq 5 is generally less accurate than eq 4 and that the Sikora 6-18 and 6-24 rules with either the London or SlaterKirkwood substitution are not as accurate as the Sikora 6-12 rule. Extension to Include Interactions with Large Asymmetrical Molecules Molecules such as the long-chain hydrocarbons move more slowly than small molecules in a gas phase, since mean molecular speed is an inverse function of the square root of molecular weight. Thus it is reasonable to assume that in a mixture with small molecules, such molecules are effectively stationary targets for the smaller molecules, with average collision diameters which are integrations of eq 7 over all possible angles of attack. If the long molecules are modeled as equivalent ellipsoids and if, to accommodate glancing collisions, the probability for collision with a point molecule approaching from some angle is defined as the dot product of its unit vector with the unit surface normal of the long molecule at the point of impact, computerized integrations can be performed to determine
where u2 is the volume-averaged value given by eq 23. It has been learned, however, that nearly identical values of T~~ are obtained if only center-to-center collisions are considered and if a unit collision probability is assumed for all angles of attack (Pesuit, 1977) C
T12
= - sin-1 ,.113
(t)
(34)
where (35) and
r = a/b
(36)
Table VI11 provides length-width ratios and calculated target factors for some molecules of interest. The length-width ratios were determined by comparing literature values (Prausnitz and Benson, 1954) with conformational energy calculations (Smith and Mortensen, 1961) and with fits of pure component PVT data to a modified hard-sphere equation of state (Bienkowski and Chao, 1975). The target factors in the table are less than unity, indicating a reduction in calculated interaction constants for all equations above which include distance force constants; the ratio of any two means of two positive quantities is a measure of their difference in numerical size. In Table IX binary interaction constants from the indicated equations, with and without target factor modification where applicable, are compared with values obtained using the reducing method developed above. The table clearly shows that use of the target factor modification provides substantial improvement in predictions for all rules containing a size factor, and that the new hybrid rule with modification provides the most accurate values. The table also shows that the Sikora 6-24 rule with ~~2 modification is much more accurate than the modified 6-12 rule, perhaps illustrating Prigogine's (1957) conclusion that the thermodynamic properties of globular and long-chain molecules are better reproduced by a potential with a steep repulsive well. At any rate, however, it is obvious that target factor modifications to the Sikora and hybrid rules provide much more accurate interaction constants than the Hiza-Duncan or harmonic mean rules. Sensitivity Analysis The absolute average errors above in K12are not information enough to allow estimates of engineering ac-
Ind. Eng. C h e m . Fundam., Vol. 17, No. 4, 1978
W 0,
8 0
rl
8 Lo rl
8 6
8
10
I4
12
16
18
20
REDUCED TEMPERAfURE T i T c
Figure 3. Sensitivity of the reduced second virial cross coefficient to an uncertainty of 0.01 in K I 2over a range of reduced temperatures. Based on Tsonopoulos (1974).
c.l OJ
8
curacy; differential ,analyses of the equations relating molecular parameters to thermodynamic properties are required. Consider, for example, a differential pressure calculation for an equimolar gaseous mixture at constant temperature and density using the virial equation truncated to the second term (37)
where 2, is the critical compressibility factor and VR is reduced molar volume and where BRpis the reduced second virial coefficient, written as a function of reduced temperature T R and acentric factor w (Pitzer and Curl, 1957) dBR, = [ ~ ( ( T R+)u f i ' ( T ~ ) ] d T ~
W * r n W t - d
m a m a w 9
m
*
9999"" 9
0 0 0 0 0 0
0
(38)
with (Tsonopoulis, 1974) 0 33 0.277 0.0363 + 0.004856 f 0' = L+ .~ TR2 TR~ TR~ TR9
+-
0.662
f 1' = - - -
1.269 +-+-
TR~
TR~
0.064
TR'
(39)
*
w 9 a t - 0 0 r l r i o o m m c-
r i 0 0 0 ~ 0 l 0 O O O O O O l O
(40) wamoahl m
Quick reference to ecl 31 shows that
These equations and. Figure 3 show together that the sensitivity of reducedl pressure to an error of 0.02 in the binary interaction constant is a strong function of system density and system temperature. Sensitivity is slight at low densities; a 0.02 change in KI2 produces less than a 0.1% change in reduced pressure at any reduced temperature and for any acentric factor at densities less than 1/25 of the critical. But sensitivity climbs especially at low temperature for higher densities; reduced pressure changes by as much as 2% a t TR = 0.6 with large acentric factors for the same change in interaction constant with densities about 1 / 5 of the critical; sensitivity for the same density is less than 0.4% at reduced temperatures greater than 0.8. Errors at the critical density itself are as large as 20-6070 a t low reduced tempeiratures but are reduced to less than
wr-9o(Dhl
w
0 0 0 0 0 0
0
999999 9
241
242
Ind. Eng. Chem. Fundam., Vol. 17, No.
T a b l e X. Summary T a b u l a t i o n Discussed in t h e T e x t
of A v e r a g e
s y s t e m class
1. n o b l e gases 2. H e , with s m a l l molecules o t h e r n o b l e gases, with s m a l l m o l e c u l e s 4. H,, with s m a l l molecules 5. n o n p o l a r n e a r l y l i k e sized p a i r s 6. p o l a r - n o n p o l a r n e a r l y like-sized pairs 7. b i n a r i e s with l a r g e asymmetrical molecules
3.
4, 1978
exptl points
Absolute
Errors in Binary I n t e r a c t i o n Constants, According t o Several E q u a t i o n s
attr. r u l e e q 4 X =
hybrid r u l e
31X= (NILU)"~ (N/a)",
X =Z
X = (N/cu)"Z
HizaDuncan rule e q 22
Sikora
6-12 rules
eq
harmonic mean e q 21
10 4
0.135 0.247
0.127 0.221
0.030 0.081
0.020
0.026
0.044
0.038
0.093 0.148
8
0.049
0.021
0.025
0.042
0.032
0.023
4
0.034
0.052
0.166
0.190
0.034
0.212
9
0.01 7
0.006
0.034
0.019
0.035
0.034
8
0.021
0.013
0.007
0.016
0.045
0.034
6
0.015
0.01 0
0.102
0.096
0.065
0.074
4% a t reduced temperatures greater than 1.5. Discussion With this analysis in mind, consider Table X in which absolute average errors in KI2 are summarized for all classes of interactions which have been considered. It should be apparent from the table and from the above analysis that none of these equations is able to supply accurate interaction constants for the range of interactions tested. But it should also be clear that the Hiza-Duncan rule is accurate where the Sikora and new hybrid rules are not and vice versa, indicating that one of these equations may be used in conjunction with potential functions to obtain reasonable estimates of mixture thermodynamic properties, a t least a t low and medium density, using information on the properties of pure components alone. Calculations for polar mixtures are surprisingly accurate even though it is well known that such interactions are not amenable to simple treatments; calculations for binaries with large size difference are also accurate, which may indicate that the vapor-phase thermodynamic properties of such mixtures are only weak functions of difference in electronic configuration. The poor results for the harmonic mean emphasize the importance of defining KI2in terms of several molecular parameters; such a rule also suffers from the limitation that it does not allow for systematic modification to include more complicated effects such as dimerization in the vapor phase. If indeed the Sikora and hybrid rules combined with the target model do lead to reasonable a priori corresponding states calculations of thermodynamic properties in some cases, this bodes well for further fundamental work which emphasizes the classical nature of molecular interactions at long and short distance. No attempt has been made here, for example, to include angularity dependent potential energy contributions to the interactions of polar and nonpolar polyatomic molecules with nearly like size. Nor has any attempt been made to calculate effective electron numbers (electron densities) for the Slater-Kirkwood substitution, as Wilson (1965) and as Kramer and Herschbach (1970) and Nenner et al. (1975) have done; the latter calculation would probably improve predictions for the classes of systems for which these rules now fail. But in the final analysis the success of any attempt to improve upon these models must depend upon calculated
interaction constants from the literature, which themselves depend upon the potential model which is chosen, the temperature range of interest, and the pure component force constants which are used. If investigators calculating these interaction constants were to report them using an internally consistent set of pure-component force constants, further progress might well be made.
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Received for reuiew N o v e m b e r 29, 1976 Accepted July 10, 1978
