A tentative statistical theory of Macleod’s equation for surface tension, and the parachor

Abstract

Macleod’s (1923) empirical formula for the surface tension σ of a liquid in equilibrium with its own vapour, namely σ = const. ( ρ liq . ─ ρ vap ) 4 , (1) where the ρ ’s are densities in g./c. c., must impress everyone with the feeling that such a simple and successful relation must have some equally simple foundation in thermodynamic or statistical theory. This feeling is only intensified when one realizes the wide use to which this formula has been put by Sugden and his followers. Sugden (1930) writes M *σ ¼ / ρ liq . ─ ρ vap . = P , (2) where M * is the (chemical) molecular weight of the substance, and calls P the parachor . The comparative study of the parachors for various liquids seems to have proved fruitful. This study is described by Sugden as the comparative study of molecular volumes at equal surface tensions, and therefore, to the best of our opportunities, at equal “internal pressures”. We need not here enquire into the precise significance (if any) of the phrase ‘‘internal pressure”. It is sufficient to recognize that the quantity P defined by (2) is in fact a constant (very nearly) for any given substance, independent of the temperature over a wide range, from the critical temperature T c downwards—and having recognized this important empirical fact, to attempt to derive it as a theorem in statistical mechanics applied to a reasonable model, and to give a formula for P in terms of molecular diameters and (or) intermolecular forces.