15 Thermodynamics of Molecular Fluids and Their Mixtures 1
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Y. SINGH and K. P. SHUKLA Downloaded by KTH ROYAL INST OF TECHNOLOGY on September 15, 2015 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch015
Banaras Hindu University, Department of Physics, Varanasi-221005, India
The problem of calculating the equilibrium properties of fluids having nonspherical rigid molecules of arbitrary symmetry and their mixtures is studied. A perturbation expansion in which all tensor interactions (anisotropic pair and three-body nonadditive interactions) are taken as a perturbation of the central pair potential is discussed. Theoretical expressions are given and calculations made for the virial coefficients of the equation of state, Helmholtz free energy, configurational energy, entropy, and pressure. These are compared with experimental data for nitrogen, oxygen, carbon monoxide, carbon dioxide, and methane. The theoretical predictions for binary mixtures are compared with experimental results for argon-nitrogen, argon-oxygen, argon-carbon monoxide, nitrogen-oxygen, and nitrogen-carbon monoxide at the zero pressure isobar and temperature equal to 83.82 K. Agreement with experiment is very satisfactory for all of these systems.
T
H E POTENTIAL ENERGY of IV interacting molecules has nonadditive
interaction terms in addition to the sum of pair potentials. A substantial improvement in the quantitative understanding of the behavior of real, dense fluids can be made if the nonadditive interactions are included in theoretical calculations. For atomic fluids, it has been shown that the long-range triple-dipole three-body dispersion (Axilrod-Teller) interaction (1) contributes substantially to the thermodynamic properties of fluids (2, 3) and to low-order cluster integrals appearing in the density expansion of the radial distribution function (4-6). Barker, Henderson, and Smith (3) have found that the calculated pressure, internal energies, and critical constants of dense gaseous argon are in reasonable agreement 1
Current address: University of Illinois, School of Chemical Sciences, Urbana, IL 61801. Current address: University of Duisburg, Fachgebiet Thermodynamik, Duisberg, Federal Republic of Germany. 2
0065-2393/83/0204-0365$08.25/0 © 1983 American Chemical Society In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
Downloaded by KTH ROYAL INST OF TECHNOLOGY on September 15, 2015 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch015
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MOLECULAR-BASED STUDY OF FLUIDS
with experiment provided that the three-body contributions are included. Goldman (7) has found that the effect of the triple-dipole threebody dispersion interaction on the Henry's law constant for krypton in argon is so large that it cannot possibly be treated accurately by any effective pair potential. The effect of the three-body nonadditive dispersion interactions on the structure factor of liquid rare gases has, however, been found negligibly small (8, 9). The extent to which the three-body nonadditive interaction and the anisotropy in pair interaction contribute to the equilibrium properties of dense polyatomic fluids has been a subject of considerable interest in recent years (10-12). Singh and Singh (13, 14) have found that both the dielectric and equation-of-state virial coefficients and the dilute-gas viscosity of polyatomic fluids can be explained satisfactorily with one set of force parameters, provided that the molecular asymmetry and the nonadditivity of the (three-body) interactions are explicitly taken into account in calculating the virial coefficients. The calculation of thermodynamic properties and correlation functions of molecular fluids in the presence of three-body forces is relatively difficult. The solutions of integral equations such as the hypernettedchain (HNC) equation, the Percus-Yevick (PY) equation, the meanspherical approximation (MSA), or the optimized random phase approximation (ORPA) are difficult to obtain even in the absence of three-body forces. This is because the solution of these equations involves, even for axially symmetric rigid molecules, repetitive sixfold, numerical integrations (more in the presence of three-body forces) and requires the calculation of the full anisotropic pair correlation function, a procedure that is numerically very complicated but that can be accomplished by a spherical harmonic expansion (see, e.g., 15). Another method that can be applied to molecular fluids with relative ease is a use of a perturbation scheme (11, 12, 16-26) in which quantities of interest are obtained by applying a perturbation correction to the corresponding quantities of some reference system. In this chapter we describe a method for computing the thermodynamic properties of fluids and their mixtures, assuming rigid nonspherical molecules of arbitrary symmetry. The procedure is based upon a perturbation expansion (11,12, 25, 26) in which all tensor interactions (angledependent pair and triplet potentials) are taken as a perturbation of the central pair potential. The procedure applies most directly to molecules with electric multipoles embedded in a core that deviates only marginally from spherical symmetry, so that its nonsphericity can be treated as a perturbation of the spherical core. In this type of perturbation expansion, in which a spherically symmetric reference potential (SSRP) is used, the series is summed up by using the Pade approximant of Stell et al. (24).
In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
Downloaded by KTH ROYAL INST OF TECHNOLOGY on September 15, 2015 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch015
15.
SINGH AND SHUKLA
Thermodynamics of Molecular Fluids
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The Pade approximant has been found to yield results for systems with dipolar and/or quadrupolar forces that are in good agreement with machine simulation results even when the electric multipole moments are very large (27, 28). Even for anisotropic overlap forces, the Pade approximant has been found to work well for not too elongated molecules (29). The details of the pair and triplet (three-body nonadditive) interactions used in the calculation are given in the first section below. In the following section we review a thermodynamic perturbation theory of a multi-component mixture (12). Molecular asymmetry of very general types arising from permanent electric multipole moments, induced dipole moments, and anisotropic dispersion forces are considered. Permanent moment interactions involving dipoles and quadrupoles are treated through the third-order of the perturbation, while all other anisotropic interactions including three-body nonadditive interactions are treated to the second-order term only. The contribution of the higher-order terms in the expansion, arising because of the first few permanent moment interactions, are approximated by means of a simple [1,0] Pade extrapolation procedure. An application of the theory to the description of the equilibrium properties of some nonpolar fluids is then presented. In the last section we apply the theory to predict the properties of binary mixtures. Molecular
Interactions
We consider a fluid mixture consisting of t components contained in a volume V at temperature T. The system is described by a potential function that depends on the orientation and the relative center of mass coordinates of molecules, but that is independent of rotational momentums and internal vibrational states. We approximate the total potential energy of interaction of the system as a sum of the interaction energies of isolated pairs and triplets. The pair potential energy,
