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A new model is proposed for the study of the liquid-vapor phase transition. The potential energy of a given configuration of N molecules is defined by U(rJ, ••• , rN) = [W(rJ, ••• , rN) -NvoJ./vo:$O, where TV is the volume covered by N interpenetrating spheres each of volume Vo and each centered on one molecule, and where, is an arbitrary positive energy. This continuum model proves to have a line of symmetry comparable with those found hitherto only in lattice models. The line is that of the diameter, or mean orthobaric density P = !Pl+!Pe, below the critical point, and continues through the one-phase region to infinite temperature. The existence of this line allows some of the properties to be obtained explicitly, the most unusual of which is that the diameter has a singularity comparable with that in Cv ; hence the law of the rectilinear diameter is not obeyed. An exact solution of the model is obtained in one dimension, in which there is no phase transition, and a mean-field solution in three dimensions. The latter preserves the symmetry. The model is shown to be thermodynamically equivalent to a two-component system in which the pair potential between molecules of like species is zero, while that between molecules of unlike species implies a mutually excluded volume of Vo. In this transcription the symmetry of the model becomes obvious. I. INTRODUCTIONone dimension where, as expected, it shows no phase transition) . Of the models hitherto proposed for studying liquid-In the next section we review the definition and symvapor equilibrium and the attendant critical phenom-metry properties of the lattice gas, casting them in a enon the most successful by far has been the lattice gas.! form appropriate for comparison with their analogs in The discovery by Onsager2 of what amounts to! the the new model. In Sec. III the new model is defined and partition function of the two-dimensional lattice gas discussed, and the equation of state of its one-dimenat a fixed density equal to the critical density, has sional version is obtained. The underlying symmetry of inspired a vast program of analytical and numerical the general model is derived in Sec. IV, where it is studies of these hypothetical fluids. The three-dimen-shown that the properties of the original one-component sional lattice gas is now known to yield a realistic fluid may be transcribed from those of an equivalent critical point with thermodynamic singularities in close two-component system in which there is a manifest accord with those found experimentally.3 symmetry between the components. Some of the conOne of the attractive features of the model is an sequences of the basic symmetry that bear directly on obvious hole-particle symmetry that has important experiment are derived in Sec. V. Among these is the analytical consequences, and much of what is known conclusion that the rate of change of density with temexactly about the lattice gas is deducible from this perature along the diameter of the coexistence curve has symmetry.! An unattractive feature of...

Measurements are reported of the vapour pressures, the heats of mixing, the densities and the phase relationships of solutions of polyethylene glycol and polypropylene glycol in water. The free energies, heats and entropies of dilution, the volume changes on mixing and the excess partial volumes have been derived. The results show significant differences from those for other polar polymer solutions. A comparison is made with the results for aqueous solutions of dioxane which is the cyclic dimer of the repeating unit in polyethylene glycol. Some new measurements of the vapour pressures of dioxane + water solutions at temperatures between 100" C and 156" C and of the heat of mixing at 25" C , are reported in an appendix.

The gas-liquid surface of a system of Lennard-Jones (12, 6 ) molecules has been simulated by Monte Carlo and by Molecular Dynamic methods at temperatures which span most of the liquid range. For systems of 255 molecules the two methods lead to similar results and this agreement confirms that the density profile, as a function of height, falls monotonically from the density of the bulk liquid to that of the gas. The thickness of the surface layer is sensitive to the surface area, and appears to approach its thermodynamic limit for surface areas of 4000~ for a system of 4080 molecules. The density profile can be represented by a hyperbolic tangent of an appropriately scaled height. The thickness of the surface is of the order of two molecular diameters at temperatures near the triple point and increases rapidly as the critical point is approached. The computed surfacetens ions agree well with those calculated by statistical perturbation theory.

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