Liquids: Surface tension, compressibility, and invariants

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Abstract
A new equation has been dervied which relates the surface tension (σ) to a liquid’s isothermal compressibility (κ) and mass density (ρ). The derivation is based on a generalized square‐gradient approximation for the free energy density of a nonuniform fluid. The equation is σ(κ/ρ)1/2=A1/20=constant in the normal liquid range. Except for water, A0 is temperature independent for a variety of inorganic, organic, and polymer liquids. Among 50 nonpolar and polar organic liquids, including hydrogen bonding liquids, A1/20 appears to be an invariant with a value of 2.78±0.13×10−4 (erg cm2/g)1/2. Among the diatomic elements (except hydrogen), A1/20 is an invariant with a value of 1.8×10−4. Among the heavy noble elements, A1/20 is an invariant with a value of 1.36×10−4. For the quantum noble elements helium and neon, A1/20=1.0×10−4. The constant A0 is shown to be proportional to a second moment of a direct correlation function. A semiempirical formula has been derived for A0 relating it to the parameters ε0 and σ0 that characterize the pair interaction potential. For a Lennard‐Jones 6‐12 potential, it is shown that A1/20=0.26(ε0σ20/M)1/2, where M is molecular weight. This result combined with the experimental evaluations of A0 implies that the parameter combination (ε0σ20/M) is an invariant for certain classes of molecules. It appears that this surprising observation has never been made before; its physical implications remain unclear.

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