A. DABGUPTAAND T. S. STORVICK
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Transport Properties of Polar Gases. Collision Integrals for the Kihara Spherical Core Plus Dipole-Dipole Potential by A. Das Gupta and T. S . StorVicky Chemical Engineering Department, University of Missouri-Columbia, (Received August 18, 1971)
Columbia, Missouri 65801
Publication costs assisted by the Engineering Experiment Station, University of Missouri-Columbia
The transport collision integrals for the Kihara spherical core plus dipole-dipole terms are reported. The calculations were performed at fixed orientation during an encounter, and the average collision integral for all orientations was calculated assuming all orientations are equally probable. The potential function parameters are evaluated using the viscosity data for ten polar gases. The experimental data are reproduced within experimental error for all of these gases. The value of the spherical core diameter was used as a fitting parameter. The size of this core was found to be smaller than expected from physical argument.
Introduction The low-density properties of gases provide a primary source of information about the intermolecular potential functions of simple molecules. The transport collision integrals that come from the Chapman-Enskog solution of the Boltzmann equation have been used extensively to evaluate parameters in the empirical potential functions that are used in these studies. Tabulated values of these collision integrals are available for the 12-6,’ 9-6129a2&7,3 e ~ p - 6 Morsels ,~ Kihara,6 and the sperical-shell potentials.’ The more elaborate models have been used to describe the properties of quasispherical, polyatomic molecules with some encouraging success. All real polyatomic molecules have dipole or higher multipole moments, and it is important to compute the effect that these forces have on the transport properties. Monchiclr and Mason8used the Stockmayer 12-6-3 potential model to compute the collision integrals for dipolar gases. The properties of these gases are significantly altered by including the dipole forces. The agreement with the experimental data for polar gases was as good as the previous work for nonpolar gases. Smith, Munn, and Mason9 have used the 12-6-5 potential model t o represent the transport properties of quadrupolar gases. The effect of the quadrupole moment is much less than the dipole moment, but these corrections should not be neglected when accurate representation is necessary. The Kihara potential has been extcnsively used to model the propcrties of polyatomic gases. It is the purpose of this paper to report the transport collision integrals for the Kihara spherical-core potential with point dipoles located at the center of the core. The collision integrals are computed assuming a fixed orientati0n~8~ during an encounter, and the average collision integral is obtained by averaging over all orientations The Journal of Physical Chemistry, Vol. 76, No. 10,1972
assuming each configuration is equally probable. Potential function parameters obtained from the viscosity data for ten polar gases are reported.
Calculations The Kihara sperical-core potential has been described in detail elsewhere.6~10The extension of this model to include a point dipole at the center of the core has been used by O’Connell and Prausnitz6with others‘l to compute the second virial coefficients of polar gases. The form of this potential with the dipole-dipole term added is
u
= 4e[(---)’2 u - 2a
r - 2a
(4
- u - 2a r - 2a
-
where E is the minimum potential energy and u is the separation when U = 0 that would be obtained when (1) J. 0. Hirschfelder, C . F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N. Y . , 1954, pp 1126-1127. (2) G. M. Roe, L. F. Epstein, and M. D. Powers, J . Chem. Phys., 20, 1665 (1952); L. F. Epstein, ibid., 20, 1670 (1952). (3) F. J. Smith, E. A. Mason, and R. J, Munn, ibid., 42, 1334 (1965). (4) E. A. Mason, ibid., 23, 49 (1955). (5) F. J. Smith and R. J. Munn, ibid., 41, 3560 (1964). (6) J. P. O’Connell and J. M . Prausnitz, “Advances in Thermophysical Properties at Extreme Temperatures and Pressures,” ASME, United Engineering Center, New York, N. Y., 1965, pp 19-31. Tabulated collision integrals available as Document No. 8432, AD1 Auxiliary Publications Project. (7) A. G. DeRocco, T. 8. Storvick, and T. H. Spurling, J . Chem. Phys., 48, 997 (1968). (8) L. Monchick and E. A. Mason, ibid., 35, 1676 (1961). (9) F. J. Smith, R. J. Munn, and E. A. Mason, ibid., 46, 317 (1967). (10) A. E. Sherwood and J. M. Prausnitz, ibid., 41, 429 (1964). (11) C . S. Lee, J. P. O’Connell, C. D. Myrat, and J. M. Prausnitz, Can. J . Chem., 48, 2993 (1970).
TRANSPORT PROPERTIES OF POLAR GASES
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the dipole moments, 1.11 and pz, are zero; 2a is the Kihara sperical core diameter; r is the separation of the core centers; and G(&, B2, 4) is the dipole orientation function12
G(&, 02, 4 ) = 2 cos el cos e2 - sin el sin e2 cos 4 ( 2 )
. .
I
I n eq 2, 01 and 02 are the angles of inclination of the dipole vectors with the line connecting the centers of the Kihara cores and 4 is the azimuthal angle between the two dipole vectors. The value of G is in the range -2.0 6 G 2.0 for all orientations. Since we are considering only like molecule interactions, eq 1 can be rewritten as
