Statistical-Mechanical Derivation of the Partial Molecular Stress Tensors in Isothermal Multicomponent Systems

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Abstract
Commencing with the general equations of molecular dynamics as used by Bearman and Kirkwood, the hydrodynamic equations of motion are derived for each component in an isothermal multicomponent system consisting of sensibly spherical nonreacting molecules. It was the primary purpose of the investigation of which this is a report to attempt to obtain a microscopic statistical basis for the macroscopic phenomenologic development of Snell and Spangler, especially in regard to their new formulation for the partial molecular stress tensors. An expansion of the number densities in pair space of perturbed nonequilibrium states alternate to that used by Bearman and Kirkwood is employed. Our expansion esplicitly recognizes the position vector arguments of both the number densities in singlet space and the local mean velocities of the individual components, as well as for the equilibrium pair correlation function. The quantities are then expanded about the central position vector in a Taylor's series. The results are not only corroborative of the previous phenomenologic development when the flows of the individual components are solenoidal and attention is confined to systems in which the kinetic contribution to the stress tensor is negligible in comparison to the contribution arising from intermolecular interaction, but they also provide a new microscopic theory of hydrodynamical flow in multicomponent systems.