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Abstract:
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In this work, a fundamental integral equation in statistical thermodynamics, the Ornstein-Zernike (OZ) equation, is solved analytically for the first time for arbitrary intermolecular potentials with a hard core. The proposed solution which combines the perturbation method and the Hilbert transform is applied for both pure fluids and mixtures. Subsequently, the radial distribution function (RDF), which is a crucial function to determine the structure and thermodynamics of fluids, is obtained for a number of typical fluids including the two well-known square-well (SW) and Lennard-Jones (LJ) fluids. RDFs for the SW and LJ fluids are successfully compared with computer simulation data, and the thermodynamic properties derived from these RDFs are found to be better than or comparable to other liquid theories. Furthermore, a two-Yukawa function is suggested in this work to map the LJ potential. The mapping successfully brings the calculations of the structure and thermodynamics of the LJ fluid in a straightforward and analytical manner. The thermodynamics of LJ mixtures are developed analogously. The developed expressions are utilized to describe various types of behavior of LJ mixtures, including vapor-liquid and liquid-liquid equilibria. The description surpasses considerably other liquid theories. The applicability of the present equation of state (EOS) for simple real fluids is also investigated and the superiority of the present EOS over other empirical equations is evidenced by yielding better thermodynamic consistency. Accompanying the above development, a new strategy for the inverse Laplace transform is proposed to obtain RDF in a completely explicit and analytical manner. A new version of RDF for both pure hard spheres and hard-sphere mixtures is developed based on the first-order OZ solution obtained. A new approximation is suggested to improve the prediction of RDF of the LJ fluid. A RDF for a nonspherical molecule, hard-sphere chain, is also analytically obtained. |
