A variational technique which is based on two different inequalities for the Helmholtz free energies is used to calculate the equilibrium thermodynamic properties of simple fluids. A system with hard-sphere potential function is used as the reference system. Helmholtz free energy of the original system is calculated by variation around the Helmholtz free energy of the reference system, and the other thermodynamic properties are calculated from free energy. By choosing a hard-sphere reference system, it is possible to calculate the equilibrium thermodynamic properties of fluids from very low densities to densities close to solid, and from high temperatures in the gas phase to low temperatures in the liquid phase, in the ranges where experimental and machine-calculated data are available. It is shown that the present variational technique is a better approach to the prediction of the equilibrium thermodynamic properties of liquids and vapor-liquid phase transition than any other approach so far developed. While the variational calculation based on a hard-sphere reference system does not predict the liquid-solid phase transition, it is argued that this might be due to the neglect of the orientation, or ordering in the formulation of the working inequality for fluids.

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A variational approach to the equilibrium thermodynamic properties of an original system based upon an inequality for the Helmholtz free energy of that system is introduced. A system with molecules obeying the cell model of Lennard-Jones and Devonshire, and having a harmonic-oscillator-type potential function inside their cells, is used for a reference system to produce the inequality for the Helmholtz free energy of the original system. Optimization upon this inequality indicates that a variational calculation based on a reference system with highly ordered structure, as the cell model, predicts the properties of the solid phase better than the liquid phase. Also, it shows that by an ordered-structure reference model, it is possible to predict the liquid–solid phase transition. Equilibrium thermodynamic properties of the solid phase and liquid–solid phase equilibria are calculated and are compared with the machine-calculated and the experimental data.

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