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Coordination numbers for rigid spheres of different size—estimating the number of next-neighbour interactions in a mixture
Fluid Phase Equilibria, 8 (1982) 123-129 Elsevier Scientific Publishing Company, Amsterdam-Printed
123 in The Netherlands
COORDINATION NUMBERS FOR RIGID SPHERES OF DIFFERENT SIZE-ESTIMATING THE NUMBER OF NEXT-NEIGHBOUR INTERACTIONS IN A MIXTURE
ULRICH
K. DEITERS
Lehrstuhl fiir Physikalische (Received
Chemie II,
Ruhr-Universitijt, Bochum (F.R. G.)
August 7th, 1981; accepted in revised form October 23rd. 1981)
ABSTRACT Deiters, U.K., 1982. Coordination numbers for rigid spheres of different size-estimating the number of next-neighbour interactions in a mixture. Fiuid Phase Equilibria, 8: 123-129. The maximum number of rigid spheres which can be attached to a central sphere of different diameter is calculated numerically. It is shown that this coordination number is not proportional to the surface ratio of the spheres, but roughly to power 1.2 of the diameter ratio. This has consequences for the methods by which averages of molecular properties are calculated; for several binary liquid mixtures it is shown that replacing molar fractions by contact fractions makes plots of excess properties versusconcentrationappearmore symmetrical, thereby reducing the number of terms required in Redlich-Kister correlations.
INTRODUCTION
The statistical treatment of mixtures containing molecules of different size is of great importance for a large number of applications (calculation of densities, phase equilibria, etc.) and therefore has been the subject of many publications. The common problem is always the calculation of the number and energy contributions of molecular interactions. With short-range intermolecular potential functions only, next-neighbour interactions are important. Indeed, most theories, especially lattice theories, ignore other interactions (see Guggenheim, 1952). Thus it is extremely important to know the average size and composition of the first coordination shell. The maximal coordination number for molecules of equal size is zii = 12 (The actual coordination number in fluids is somewhat lower (8-10) due to random packing (Kohler, 1972; Bernal, 1960)). For molecules of different size, however, the maximal coordination number depends on the molecular size ratio in a non-trivial way. 0378-3812/82/0000-0000/$02.75
6 1982 Elsevier Scientific Publishing Company
124 The main object of most statistical approaches is the calculation of the contribution of each molecular species to the total configurational energy. The simplest method to estimate these contributions uses molar fractions: u=
z,,x:
+ zc,,x,x,
+ E2*X:
with 1,2 denoting molecular species, eii interaction energies, and x, molar fractions. This may be called the zeroth approximation, since size differences are ignored. This approach is useful for mixtures of similar molecules, or for low densities. Indeed, a mixing rule of this type is generally used for virial coefficients (Hirschfelder et al., 1967). A major improvement is the first approximation, which uses contact fractions instead of molar fractions; each species is weighted according to the number of next-neighbour interactions (contacts) it can undergo:
with 4i
=
xisi/(xIsI
The si denote rewritten
q2 =
+
x*s*)
a contact
number
per molecule.
The definition
of qi can be
&x2/(x, + S2,x2) where%, = Q/S,
The characteristic parameter of the binary mixture is the contact ratio S,, (the argument is easily generalized to multicomponent mixtures). It is our aim to establish a relationship between this contact ratio and the molecular diameter ratio, which is defined as Rij= ai/oj, with ai and aj denoting molecular diameters. Since R ,, = R,, = 1 and R,, = R;,‘, only R,, needs to be considered. Molecules are thought to interact when their ‘surfaces’ are close or touch each other. Therefore it is often tacitly assumed that averaging interactionenergy/surface ratios (e.g. Kleintjens and Koningsveld, 1980) or assigning statistical weights according to surface fractions (e.g. Beret and Prausnitz, 1975; Donohue and Prausnitz, 1978; both publications primarily aimed at chain molecules) yields the best results. For spherical molecules this implies a quadratic relationship between S,, and R,, (S,, = R:,). A preliminary investigation of this problem (Deiters, 1979), however, indicated that the matter is more complicated. The number of contacts of a molecule is its coordination number; therefore we can equate the contact ratio to the coordination number ratio
125 TABLE
I
Diameter
ratios of several symmetric
Triangle Tetrahedron Octahedron Cube
configurations
I
R
3 4 6 8
0.1547 0.2247 0.4142 0.7321
Icosahedron
L
R
12 14 24
l.OOcQ 1.1221 1.7611
For a few symmetric arrangements of neighbour molecules, it is possible to calculate the diameter ratio R,, and the corresponding coordination number ratio .S,, analytically. The results are summarized in Table 1. The first entry of this table corresponds to a plane triangular configuration, the next four entries to platonic polyhedra (these data have long been used to explain stability limits of crystal structures by Dekker (1957)). The remaining entries correspond to less symmetric but nevertheless dense structures. We have calculated maximal coordination numbers for intermediate diameter ratios numerically, using a desk-top computer, hp 9845T. The calculation procedure was as follows. A molecule of species I was placed as the central molecule. From the locations of two other molecules of species2 touching the central molecule and each other, the computer calculates the coordinates of a third sphere of species2 on the surface of the central molecule touching the other spheres. After checking for overlap with other spheres, the computer then uses this third sphere, in combination with one of the other spheres, to calculate the location of a fourth sphere, etc. The computation continues until no further spheres can be inserted into the coordination shell. The results are displayed in Fig. 1. As a by-product we found that the coordination numbers 5 and 11 are not possible for a ‘dense’ first coordination shell. If the coordination numbers were proportional to the molecular surface ratio (= Rz,), the coordination range patterns would follow the dotted line. However, the patterns deviate markedly from the quadratic approximation. The assumption that the maximal coordination number is proportional to the diameter ratio (linear approximation, dashed line in Fig. 1) is better, but still not satisfactory. If we are determined to approximate the maximal coordination number, with respect to the contact ratio, by a function of R,, which is antisymmetric with respect to commutation of species 1 and 2: s,, =f(R,,) S,, = S,;’ =f( R,‘)
Fig. 1. Maximal coordination numbers of rigid spheres (coordination with molecules of different size). X, Calculated analytically; -, calculated numerically; ----- -, linear . . . . . . , quadratic approximation. approximation; The bars represent the range of R,, over which the coordination number indicated is the largest possible.
which leads to functions of the type f( R2,) = R’;;, we can obtain m from a plot of the ratio In S,,/ln R,,. From Fig. 2 a value m = 1.2 may be derived, i.e. the maximal coordination number varies with the power 1.2 of R,, for 0.4 -= R,, c 2.5.
I
0
io I2
Fig. 2. Ratio of the logarithms around the value 1.2.
20 of contact
30
I
ratio and diameter
ratio.
The bars are grouped
127 A better representation of the numerical results is accomplished by 22, -2=
(z,, - 2)R\f3
but this formula lacks the desired symmetry. For R,,-)cm(corresponding to spheres on a plane) a quadratic dependence is to be expected. However, for spherical molecules, molecular diameter ratios do not cover a very large range; even the extreme pair helium/xenon has only R,,= 2.24, R,,= 0.44. Therefore our results indicate that for molecules of spherical shape, best averaging of interaction parameters will be obtained by using contact ratios R\f instead of molecular surface ratios. As an illustration we apply the concept of contact fractions to excess properties in binary liquid mixtures. The dependence of an excess property XE (volume, Gibbs energy, enthalpy) on concentration is often described by a simple quadratic approximation (Porter’s function: XE = Ax,x,). However, the approximation is often insufficient even for mixtures of simple molecules, because plots of XE versus x1 usually exhibit a slight skewness. Therefore, Redlich-Kister functions are generally preferred for correlating
TABLE 2 The concentration for which substance mentioned first) System
R2,
excess properties
x , Calculated
m=l.O
m=l.2
m=2.0
attain
extreme
values
(subscript
Property
Ref.
Paas 1977 Simon and Knobler 1971 Lewis et al. 197s Lewis et al. 197s Lewis et al. 197s Calado et al. 1978 Calado et al. 1978 Trappeniers and Schouten 1974
1.138
0.532
0.538
0.564
0.529 0.518
GE at 93 K GEatPOK
h/CH4
1.123
0.529
0.535
0.558
OS36
Ha at 92 K
0, /Ns
1.033
0.508
0.510
0.516
OSOS
HE at 80 K
0, /Ar
1.030
0.508
0.509
0.515
0.508
He at 87 K
Kr/C,H,
1.172
OS39
OS47
0,579
OS31
HE at 118 K
0.529
GEat
0.620
GE at 163 K
1.320
0.569
0.583
0.635
to
x, Exp.
CH,/CF,
Ne/Kr
1 refers
116K
128 excess properties. That skewness, however, is often removed, when molar fractions are replaced by contact fractions. Table2 lists for several simple mixtures of approximately spherical molecules the molecular diameter ratio, the molar fraction at which an extremum of GE or HE was observed, and the predictions of our model ( XE = Aq,q,) for different values of the exponent m. The a, required to calculate the R,, were either taken from literature (Hirschfelder et al., 1967) or fitted to critical point data by means of an equation of state (Deiters, 1979, 1981); usually both ways led to the same values of R *,. Literature does not always agree on excess-property data, and the limited precision of the experiments makes it difficult to discriminate between m = 1.0 and m = 1.2, but it is evident from Table 2 that the use of surface fractions (m = 2) for spherical molecules tends to overcompensate the skewness. Some care is necessary when applying this concept to GE or SE, because SE does not solely depend on contact statistics. The above considerations apply to tightly packed first coordination shells only. This does not mean that they are only applicable to dense liquids; molecular dynamics simulations for systems of Lennard-Jones molecules show that the number of next neighbours (integral over the first peak of the radial distribution function) varies slowly with density, when the density is larger than the critical density (MD data taken from Verlet (1968)). Therefore, we expect that the results of this work will be useful also for moderately dense fluid mixtures. ACKNOWLEDGEMENTS
The author thanks work, and Prof. Dr. cial support by the Nordrhein-Westfalen
Prof. Dr. G.M. Schneider for friendly support of this F. Kohler for critically reading the manuscript. FinanMinister fiir Wissenschaft und Forschung des Landes is gratefully acknowledged.
LIST OF SYMBOLS
4i ‘i
Sii xi ‘ij
cij Rij ai
contact fraction contact number per molecule i contact ratio of species i versus species j molar fraction of species i coordination number of a molecule i surrounded species j next-neighbour interaction energy diameter ratio of species i versus species j molecular diameter of species i
by molecules
of
129 REFERENCES Beret, S. and Prausnitz, J.M., 1975. Perturbed hard-chain theory-an equation of state for fluids containing small or large molecules. AIChE J., 2 I: 1123- 1132. Bemal, J.D., 1960. Co-ordination of randomly packed spheres. Nature, 188: 910-911. Calado, J.C.G., Nunes da Ponte, M., Soares, V.A.M. and Staveley, L.A.K., 1978. The thermodynamic excess functions of krypton+ethene liquid mixtures. J. Chem. Thermodyn., 10: 35-44. Deiters, U., 1979. Entwickhmg einer semiempirischen Zustandsgleichung fiir fluide Stoffe und Berechnung von Fluid-Phasengleichgewichten in bit&en Mischungen bei hohen Drticken. Ph.D. thesis, Ruhr-Universitat, Bochum. Deiters, U., 1981. A new semiempirical equation of state for fluids. I. Derivation. Chem. Eng. Sci., 36: 1139- 1146; II. Application to pure substances. Chem. Eng. Sci., 36: 1147- 1151. Dekker, A.J., 1957. Solid State Physics, Chapters 5, 6, Prentice Hall, Englewood Cliffs. Donahue, M.D. and Prausnitz, J.M., 1978. Perturbed hard chain theory for fluid mixturesThermodynamic properties for mixtures in natural gas and petroleum technology. AIChE J., 24: 8499860. Guggenheim, E.A., 1952. Mixtures, Clarendon Press, Oxford. Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., 1967. Molecular Theory of Gases and Liquids, Wiley, New York, p. 153. Kleintjens, L.A. and Koningsveld, R., 1980. Equation of state from phase equilibria. Proceedings of the 2nd International Conference on Phase Equilibria and Fluid Properties in the Chemical Industry. Part I: 24, DECHEMA, Frankfurt. Kohler, F., 1972. The Liquid State, Monographs in Modem Chemistry, Vol. I, Verlag Chemie, Weinheim. Lewis, K.L., Saville, G. and Staveley, L.A.K., 1975. Excess enthalpies of liquid oxygen+argon, oxygen+nitrogen, and argon+ methane. J. Chem. Thermodyn., 7: 389-400. Paas, R., 1977. Phasenverhalten und Thermodynamik fluider Stoffe bei tiefen Temperaturen und hohen Drtlcken. Ph.D. thesis, Ruhr-Universitit, Bochum. Simon, M. and Knobler, C.M., 1971. The excess Gibbs energy of liquid CH, +CF., at 98K. J. Chem. Thermodyn., 3: 657-662. Trappeniers, N.J. and Schouten, J.A., 1974. Vapour-liquid and gas-gas equilibria in simple systems. III. The system neon-krypton. Physica (Utrecht), 73: 546-555. Verlet, L., 1968. Computer “experiments” on classical fluids. 11. Equilibrium correlation functions. Phys. Rev., 165: 201-214.
123 in The Netherlands
COORDINATION NUMBERS FOR RIGID SPHERES OF DIFFERENT SIZE-ESTIMATING THE NUMBER OF NEXT-NEIGHBOUR INTERACTIONS IN A MIXTURE
ULRICH
K. DEITERS
Lehrstuhl fiir Physikalische (Received
Chemie II,
Ruhr-Universitijt, Bochum (F.R. G.)
August 7th, 1981; accepted in revised form October 23rd. 1981)
ABSTRACT Deiters, U.K., 1982. Coordination numbers for rigid spheres of different size-estimating the number of next-neighbour interactions in a mixture. Fiuid Phase Equilibria, 8: 123-129. The maximum number of rigid spheres which can be attached to a central sphere of different diameter is calculated numerically. It is shown that this coordination number is not proportional to the surface ratio of the spheres, but roughly to power 1.2 of the diameter ratio. This has consequences for the methods by which averages of molecular properties are calculated; for several binary liquid mixtures it is shown that replacing molar fractions by contact fractions makes plots of excess properties versusconcentrationappearmore symmetrical, thereby reducing the number of terms required in Redlich-Kister correlations.
INTRODUCTION
The statistical treatment of mixtures containing molecules of different size is of great importance for a large number of applications (calculation of densities, phase equilibria, etc.) and therefore has been the subject of many publications. The common problem is always the calculation of the number and energy contributions of molecular interactions. With short-range intermolecular potential functions only, next-neighbour interactions are important. Indeed, most theories, especially lattice theories, ignore other interactions (see Guggenheim, 1952). Thus it is extremely important to know the average size and composition of the first coordination shell. The maximal coordination number for molecules of equal size is zii = 12 (The actual coordination number in fluids is somewhat lower (8-10) due to random packing (Kohler, 1972; Bernal, 1960)). For molecules of different size, however, the maximal coordination number depends on the molecular size ratio in a non-trivial way. 0378-3812/82/0000-0000/$02.75
6 1982 Elsevier Scientific Publishing Company
124 The main object of most statistical approaches is the calculation of the contribution of each molecular species to the total configurational energy. The simplest method to estimate these contributions uses molar fractions: u=
z,,x:
+ zc,,x,x,
+ E2*X:
with 1,2 denoting molecular species, eii interaction energies, and x, molar fractions. This may be called the zeroth approximation, since size differences are ignored. This approach is useful for mixtures of similar molecules, or for low densities. Indeed, a mixing rule of this type is generally used for virial coefficients (Hirschfelder et al., 1967). A major improvement is the first approximation, which uses contact fractions instead of molar fractions; each species is weighted according to the number of next-neighbour interactions (contacts) it can undergo:
with 4i
=
xisi/(xIsI
The si denote rewritten
q2 =
+
x*s*)
a contact
number
per molecule.
The definition
of qi can be
&x2/(x, + S2,x2) where%, = Q/S,
The characteristic parameter of the binary mixture is the contact ratio S,, (the argument is easily generalized to multicomponent mixtures). It is our aim to establish a relationship between this contact ratio and the molecular diameter ratio, which is defined as Rij= ai/oj, with ai and aj denoting molecular diameters. Since R ,, = R,, = 1 and R,, = R;,‘, only R,, needs to be considered. Molecules are thought to interact when their ‘surfaces’ are close or touch each other. Therefore it is often tacitly assumed that averaging interactionenergy/surface ratios (e.g. Kleintjens and Koningsveld, 1980) or assigning statistical weights according to surface fractions (e.g. Beret and Prausnitz, 1975; Donohue and Prausnitz, 1978; both publications primarily aimed at chain molecules) yields the best results. For spherical molecules this implies a quadratic relationship between S,, and R,, (S,, = R:,). A preliminary investigation of this problem (Deiters, 1979), however, indicated that the matter is more complicated. The number of contacts of a molecule is its coordination number; therefore we can equate the contact ratio to the coordination number ratio
125 TABLE
I
Diameter
ratios of several symmetric
Triangle Tetrahedron Octahedron Cube
configurations
I
R
3 4 6 8
0.1547 0.2247 0.4142 0.7321
Icosahedron
L
R
12 14 24
l.OOcQ 1.1221 1.7611
For a few symmetric arrangements of neighbour molecules, it is possible to calculate the diameter ratio R,, and the corresponding coordination number ratio .S,, analytically. The results are summarized in Table 1. The first entry of this table corresponds to a plane triangular configuration, the next four entries to platonic polyhedra (these data have long been used to explain stability limits of crystal structures by Dekker (1957)). The remaining entries correspond to less symmetric but nevertheless dense structures. We have calculated maximal coordination numbers for intermediate diameter ratios numerically, using a desk-top computer, hp 9845T. The calculation procedure was as follows. A molecule of species I was placed as the central molecule. From the locations of two other molecules of species2 touching the central molecule and each other, the computer calculates the coordinates of a third sphere of species2 on the surface of the central molecule touching the other spheres. After checking for overlap with other spheres, the computer then uses this third sphere, in combination with one of the other spheres, to calculate the location of a fourth sphere, etc. The computation continues until no further spheres can be inserted into the coordination shell. The results are displayed in Fig. 1. As a by-product we found that the coordination numbers 5 and 11 are not possible for a ‘dense’ first coordination shell. If the coordination numbers were proportional to the molecular surface ratio (= Rz,), the coordination range patterns would follow the dotted line. However, the patterns deviate markedly from the quadratic approximation. The assumption that the maximal coordination number is proportional to the diameter ratio (linear approximation, dashed line in Fig. 1) is better, but still not satisfactory. If we are determined to approximate the maximal coordination number, with respect to the contact ratio, by a function of R,, which is antisymmetric with respect to commutation of species 1 and 2: s,, =f(R,,) S,, = S,;’ =f( R,‘)
Fig. 1. Maximal coordination numbers of rigid spheres (coordination with molecules of different size). X, Calculated analytically; -, calculated numerically; ----- -, linear . . . . . . , quadratic approximation. approximation; The bars represent the range of R,, over which the coordination number indicated is the largest possible.
which leads to functions of the type f( R2,) = R’;;, we can obtain m from a plot of the ratio In S,,/ln R,,. From Fig. 2 a value m = 1.2 may be derived, i.e. the maximal coordination number varies with the power 1.2 of R,, for 0.4 -= R,, c 2.5.
I
0
io I2
Fig. 2. Ratio of the logarithms around the value 1.2.
20 of contact
30
I
ratio and diameter
ratio.
The bars are grouped
127 A better representation of the numerical results is accomplished by 22, -2=
(z,, - 2)R\f3
but this formula lacks the desired symmetry. For R,,-)cm(corresponding to spheres on a plane) a quadratic dependence is to be expected. However, for spherical molecules, molecular diameter ratios do not cover a very large range; even the extreme pair helium/xenon has only R,,= 2.24, R,,= 0.44. Therefore our results indicate that for molecules of spherical shape, best averaging of interaction parameters will be obtained by using contact ratios R\f instead of molecular surface ratios. As an illustration we apply the concept of contact fractions to excess properties in binary liquid mixtures. The dependence of an excess property XE (volume, Gibbs energy, enthalpy) on concentration is often described by a simple quadratic approximation (Porter’s function: XE = Ax,x,). However, the approximation is often insufficient even for mixtures of simple molecules, because plots of XE versus x1 usually exhibit a slight skewness. Therefore, Redlich-Kister functions are generally preferred for correlating
TABLE 2 The concentration for which substance mentioned first) System
R2,
excess properties
x , Calculated
m=l.O
m=l.2
m=2.0
attain
extreme
values
(subscript
Property
Ref.
Paas 1977 Simon and Knobler 1971 Lewis et al. 197s Lewis et al. 197s Lewis et al. 197s Calado et al. 1978 Calado et al. 1978 Trappeniers and Schouten 1974
1.138
0.532
0.538
0.564
0.529 0.518
GE at 93 K GEatPOK
h/CH4
1.123
0.529
0.535
0.558
OS36
Ha at 92 K
0, /Ns
1.033
0.508
0.510
0.516
OSOS
HE at 80 K
0, /Ar
1.030
0.508
0.509
0.515
0.508
He at 87 K
Kr/C,H,
1.172
OS39
OS47
0,579
OS31
HE at 118 K
0.529
GEat
0.620
GE at 163 K
1.320
0.569
0.583
0.635
to
x, Exp.
CH,/CF,
Ne/Kr
1 refers
116K
128 excess properties. That skewness, however, is often removed, when molar fractions are replaced by contact fractions. Table2 lists for several simple mixtures of approximately spherical molecules the molecular diameter ratio, the molar fraction at which an extremum of GE or HE was observed, and the predictions of our model ( XE = Aq,q,) for different values of the exponent m. The a, required to calculate the R,, were either taken from literature (Hirschfelder et al., 1967) or fitted to critical point data by means of an equation of state (Deiters, 1979, 1981); usually both ways led to the same values of R *,. Literature does not always agree on excess-property data, and the limited precision of the experiments makes it difficult to discriminate between m = 1.0 and m = 1.2, but it is evident from Table 2 that the use of surface fractions (m = 2) for spherical molecules tends to overcompensate the skewness. Some care is necessary when applying this concept to GE or SE, because SE does not solely depend on contact statistics. The above considerations apply to tightly packed first coordination shells only. This does not mean that they are only applicable to dense liquids; molecular dynamics simulations for systems of Lennard-Jones molecules show that the number of next neighbours (integral over the first peak of the radial distribution function) varies slowly with density, when the density is larger than the critical density (MD data taken from Verlet (1968)). Therefore, we expect that the results of this work will be useful also for moderately dense fluid mixtures. ACKNOWLEDGEMENTS
The author thanks work, and Prof. Dr. cial support by the Nordrhein-Westfalen
Prof. Dr. G.M. Schneider for friendly support of this F. Kohler for critically reading the manuscript. FinanMinister fiir Wissenschaft und Forschung des Landes is gratefully acknowledged.
LIST OF SYMBOLS
4i ‘i
Sii xi ‘ij
cij Rij ai
contact fraction contact number per molecule i contact ratio of species i versus species j molar fraction of species i coordination number of a molecule i surrounded species j next-neighbour interaction energy diameter ratio of species i versus species j molecular diameter of species i
by molecules
of
129 REFERENCES Beret, S. and Prausnitz, J.M., 1975. Perturbed hard-chain theory-an equation of state for fluids containing small or large molecules. AIChE J., 2 I: 1123- 1132. Bemal, J.D., 1960. Co-ordination of randomly packed spheres. Nature, 188: 910-911. Calado, J.C.G., Nunes da Ponte, M., Soares, V.A.M. and Staveley, L.A.K., 1978. The thermodynamic excess functions of krypton+ethene liquid mixtures. J. Chem. Thermodyn., 10: 35-44. Deiters, U., 1979. Entwickhmg einer semiempirischen Zustandsgleichung fiir fluide Stoffe und Berechnung von Fluid-Phasengleichgewichten in bit&en Mischungen bei hohen Drticken. Ph.D. thesis, Ruhr-Universitat, Bochum. Deiters, U., 1981. A new semiempirical equation of state for fluids. I. Derivation. Chem. Eng. Sci., 36: 1139- 1146; II. Application to pure substances. Chem. Eng. Sci., 36: 1147- 1151. Dekker, A.J., 1957. Solid State Physics, Chapters 5, 6, Prentice Hall, Englewood Cliffs. Donahue, M.D. and Prausnitz, J.M., 1978. Perturbed hard chain theory for fluid mixturesThermodynamic properties for mixtures in natural gas and petroleum technology. AIChE J., 24: 8499860. Guggenheim, E.A., 1952. Mixtures, Clarendon Press, Oxford. Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., 1967. Molecular Theory of Gases and Liquids, Wiley, New York, p. 153. Kleintjens, L.A. and Koningsveld, R., 1980. Equation of state from phase equilibria. Proceedings of the 2nd International Conference on Phase Equilibria and Fluid Properties in the Chemical Industry. Part I: 24, DECHEMA, Frankfurt. Kohler, F., 1972. The Liquid State, Monographs in Modem Chemistry, Vol. I, Verlag Chemie, Weinheim. Lewis, K.L., Saville, G. and Staveley, L.A.K., 1975. Excess enthalpies of liquid oxygen+argon, oxygen+nitrogen, and argon+ methane. J. Chem. Thermodyn., 7: 389-400. Paas, R., 1977. Phasenverhalten und Thermodynamik fluider Stoffe bei tiefen Temperaturen und hohen Drtlcken. Ph.D. thesis, Ruhr-Universitit, Bochum. Simon, M. and Knobler, C.M., 1971. The excess Gibbs energy of liquid CH, +CF., at 98K. J. Chem. Thermodyn., 3: 657-662. Trappeniers, N.J. and Schouten, J.A., 1974. Vapour-liquid and gas-gas equilibria in simple systems. III. The system neon-krypton. Physica (Utrecht), 73: 546-555. Verlet, L., 1968. Computer “experiments” on classical fluids. 11. Equilibrium correlation functions. Phys. Rev., 165: 201-214.