Translation of J. D. van der Waals' ``The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density''

Van der Waals justifies the choice of minimization of the (Helmholtz) free energy as the criterion of equilibrium in a liquid-gas system (Sections 1-4). If density ρ is a function of height h then the local free energy density differs from that of a homogeneous fluid by a term proportional to ( d ² ρ/ dh ²); the extra term arises from the energy not from the entropy (Section 5). He uses this result to show how ρ varies with h (Section 6), how this variation leads to a stable minimum free energy (Section 7), and to calculate the capillary energy or surface tension σ (Section 9). Near the critical point σ varies as ( τ _k - τ)^3/2, where τ _k is the critical temperature (Section 11). The paper closes with short discussions of the thickness of the surface layer (Section 12), of the difficulty of assuming that ρ varies discontinuously with height (Section 14), and of the possible effect of derivatives of higher order than ( d ² ρ/ dh ²) on the free energy and surface tension (Section 15).