Improved Free-Volume Theory of Liquids. I

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Abstract
To obtain practical results from the cell theories of liquids, it seems necessary to assume a simple symmetry for the distribution of a molecule about its lattice site. This is accomplished by replacing the interaction of the molecule with its neighbors by some ``suitably averaged'' interaction. In the Lennard-Jones and Devonshire treatment, the pair interaction is replaced by its spatial average. We find that a Boltzmann-type average is much more satisfactory. For an assumed symmetry of the distribution, our procedure provides an extension of Kirkwood's theory such that the resulting integral equation for the distribution function can be solved by numerical methods. The introduction of the Boltzmann-type averaging provides a pair wise correlation between the motions of molecules in neighboring cells. These correlative effects have been neglected in previous cell theories of the liquid phase. Numerical calculations of the equation of state and thermodynamical properties are, at present, being carried out. In evaluating the integrals which arise in this development, we have found the function L0 defined by the double surface integral ∫ ∫ f(r12,r1,r2,R)dω1dω2⧸ ∫ ∫ dω1dω2 =1R ∫ r12(min)r12(max)f(r12,r1,r2,R)L0(r1/R,r2/R,r12/R)dr12to be extremely useful. Here, r1 and r2 are the radii of two spheres, R the separation of their centers, r12 the separation of points on the two spheres, and f an arbitrary integrable function of r12/R. L0 is found to be a quadratic function of r12, different for various subranges of the interval [r12(max), r12(min)]. For a variety of physical problems such as x-ray and electron diffraction, L0 may have application.

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