COMMUNICATIONS
Note to the Correspondence Principle for Transport Properties of Dense Fluids
In a corresponding-states treatment dealing with transport properties of dense fluids, Tham and Gubbins showed that the coupling coefficient cy may be used as a third parameter. This communication presents a correlation for CY in terms of the known radius of gyration of the molecule.
Recently, Tham and Gubbins (1969, 1970) published two papers dealing with the correspondence principle for transport properties of dense fluids. They found that for monatomic fluids a two-parameter law of corresponding states is obeyed closely for viscosity, thermal conductivity, and self-diffusivity. If the two potential parameters E and u which are needed in the reduced equations are determined from gas-phase viscosities, the data for both gas and liquid phases are correlated well over the entire range of temperature and densities for which data are available. In the case of polyatomic fluids, the simple two-parameter principle of corresponding states only works well in the dilute gas phase (low-density limit), but does not apply accurately at high densities. These deviations from the simple principle are mainly due to the hindered rotation of the molecules a t high densities. If the rotation is hindered, a coupling between translational and rotational degrees of freedom occurs. Using a simplified harmonic oscillator cell model, Tham and Gubbins (1970) showed that a rotational coupling coefficient acts as a third parameter in a corresponding-states treatment. The rotational coupling coefficient a is defined as
a
=
(1
+ It2/m)
(1)
where I is the moment of inertia of the molecule, [ is the ratio of angular to linear amplitudes, and rn is the molecular mass. Tham and Gubbins (1970) treated a as an adjustable parameter to be evaluated from high-density experimental data. The evaluation of a is, however, tedious and then is only possible if sufficient experimental data are available, Mostly, experimental data are rare or difficult to obtain. Therefore, a reliable way must be found to predict cy for a great number of materials with sufficient accuracy. To do this, we may start from eq 1which can be rewritten as =
(1
+ R2t2)
(2) where R2 = I / m is the radius of gyration. The radius of gyration is defined as the distance from the center of mass a t which the total mass of the molecule might be concentrated without changing its moment of inertia. The radius of gyration can easily be calculated for a very great number of molecules from known bond distances and bond angles. Supposing that the atoms each are infinitely small particles; the radius of gyration of a molecule consisting of n atoms is given by cy
(3) where m , is the mass of atom i, and rL its distance from the center of mass. Generally, only the bond distances and the bond angles and not the distances of the atoms from the center of mass are known, and eq 3 cannot be used to 286
Ind. Eng.
Chem., Fundam., VoI. 13, No. 3, 1974
square radius of g y r o f m
R'
Figure 1. Plot of coupling coefficient a us. square of the radius of gyration, RZ.
calculate R. It can be shown, however, that eq 3 may be transformed to give
R 2=
1 " 2m i - 1
-
"
m,m,6,,2
(4)
,'l
where a , is the vectorial distance between atom i and j (Le., the bond distance). Further details and computed values of R2 for a great number of molecules may be found elsewhere (Bauer, 1973). The radius of gyration, therefore, may be re-garded as a known quantity. If the coupling coefficient a is plotted against R2, information about the ratio of angular to linear amplitudes [ is obtained. This was done in Figure 1. The a values used are those determined by Tham and Gubbins (1970) from high-density data. As may be seen from Figure 1, all CY values are correlated well by a single straight line. For monatomic fluids (rare gases), R2 equals zero and the three-parameter law of corresponding states correctly reduces to the simple parameter law ( a = l). (The following three materials, CH4, CD4, and 0 2 , have been omitted in Figure 1 because these values are physically meaningless according to the simple harmonic oscillator theory.) From Figure 1 one may draw the conclusion that the ratio of angular to linear amplitudes is approximately a constant for all materials investigated with a mean value of
t2 = 1.47 A-' The coupling coefficient then is given by
+
cy = 1 1.47R2 (5) From eq 5 it follows that the square radius of gyration may be used as a third parameter in a correspondingstates treatment. The same was found in other investiga-
tions dealing with the correspondence principle (Bauer,
Literature Cited
1973).
Bauer. H., VDI-Forschungsheft Nr. 556, VDI-Verlag Dusseldorf. 1973. Tham, M. J., Gubbins, K. E., Ind. Eng. Chem., Fundam., 8, 791 (1969). Tham, M . J . , Gubbins, K . E., Ind. Eng. Chem., Fundam., 9, 63 (1970).
It is clear that this correlation only holds for dense fluids, for which it is a reasonable approximation to assume that [ (the ratio of angular to linear amplitudes of oscillation) is independent of the type of molecule and also of the state condition (temperature and density). At low densities the degree of hindered rotation will depend 0, --* 0. The correlation, on the density p; thus as p therefore, is limited to dense gases and liquids for which p
-.
Institut fur Technische Therrnodynarnik und Therrnische Verfahrenstechnik Uniuersitat Stuttgart 7Stuttgart 1, West Germany
Hartmut Bauer
Received for reoiew August 23, 1973 Accepted April 5, 1974
> 2Pc.
Surface Tension of Liquids from the Coordination Number in Random Assemblage
The surface tension of liquids was derived from the Helmholtz free energy, in which the coordination number of equal spheres in a random assemblage was used. The result enables us to predict the surface tension for many sorts of liquids
Theories of surface tension of liquids can be classified into two categories: the radial distribution function method (Croxton and Ferrier, 1971; Hirschfelder, et al., 1954) and the free energy method of thermodynamics (Eyring and Jhon, 1969; Lu, et al., 1967; Hirschfelder, et al., 1954). This communication belongs to the latter and derives the surface tension of liquids from the Helmholtz free energy using the coordination number of equal spheres in a random assemblage. Surface Film In order to examine the molecular configuration in the vicinity of the vapor-liquid boundary, consider the internal energy of the fluid under consideration. The classical partition function of a system of N molecules becomes (Gotoh, 1971a,b)
where m is the molecular mass, k is the Boltzmann constant, T is the absolute temperature, h is the Planck constant, u f is the free volume of a single molecule, Z is the coordination number, and t and 0 are the Lennard-Jones (6,12) intermolecular pair potential parameters. From eq 1, the internal energy of a single molecule becomes
U
= 3kT/2
- e212
and hence the surface excess energy per unit surface area becomes (3) where 21and Z , are the coordination numbers of the bulk liquid and the liquid surface, respectively, and N s denotes the number of molecules on the unit surface area. The coordination number can be expressed by (Gotoh, 1971)
Z Z
36
= ;q5
= 20.74
- 4.35
(4 5 0.475) 0.475
d q5 5 0.7
(4b)
where 4 is the bulk mean particle volume fraction; 41, @, and & are denoted for the bulk liquid, the liquid surface, and the gas phase, respectively. We assume an equimolecular dividing surface, above and below which we consider a thin film of one molecular diameter thickness. In other
words, the surface thickness equals two molecular diameters; its bulk mean particle volume fraction is expressed by (5)
and the number of molecules on the unit surface area of the equimolecular dividing surface becomes
where the molecular diameter can be expressed by d = 2l'6 u. We know that ul = 0.64 and & = 0 a t T* = kT/c = 0.7; i . e . , the molecular configuration of spherical nonpolar fluids a t the triple point corresponds to the random close packing of equal spheres (Gotoh, 1971). From eq 3-6, we therefore obtain the following surface excess energy per unit surface area at the triple point of the spherical nonpolar fluids. T I
Experiments with Ar, Kr, Xe, Ne, NP, 0 2 , and CH4 gives 2.49 on the average (Croxton and Ferrier, 1971), which agrees fairly well with eq 7. Accordingly, we can approximately characterize the vapor-liquid boundary as a thin film of two molecular diameters thickness with the bulk mean particle volume fraction expressed by eq 5. This is reasonable judging from the calculations of Eyring and Jhon (1969). Of course if @g = 0, 4, = @ , / 2 and hence N , = @ l d / ( n d 3 / 6 ) this ; is equivalent to the surface film of one molecular diameter thickness with the particle volume fraction
Surface Tension The Helmholtz free energy per molecule can be obtained from eq 1as follows.
where d = 2amkT/h2. Hence we obtain the surface tension, i . e . , the surface excess Helmholtz free energy per unit surface area, as Ind. Eng. Chern., Fundarn., Vol. 13, No. 3,1974
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