VARIATIONAL THERMODYNAMIC DERIVATION OF THE FORMULA FOR PRESSURE DIFFERENCE ACROSS A CHARGED CONDUCTING LIQUID SURFACE AND ITS RELATION TO THE THERMODYNAMICS OF ELECTRICAL CAPACITANCE

The formula for pressure difference across a charged conducting liquid surface has conventionally been derived by adding a Maxwell stress te rm to the pressure-difference formula for the field-free case. As far as can be established, no derivation applying direct energy-based meth ods to the charged-surface case has ever been clearly formulated. This paper presents a first-principles variational derivation, starting fr om the laws of thermodynamics and modelled on Gibbs's (1875) approach to the field-free case. The derivation applies to the static equilibri um situation. The method is to treat the charged liquid and its enviro nment as a heterogeneous system in thermodynamic equilibrium, and cons ider the effects of a small virtual. variation in the shape of the con ducting-liquid surface. Expressions can be obtained for virtual change s in the free energies of relevant system components and for the virtu al electrical work done on the system. By converting the space integra l of the variation in electrostatic field energy to an integral over t he surface of the liquid electrode, the usual pressure-difference form ula is retrieved. It is also shown how the problem can be formulated, in various ways, as a free-energy problem in a situation involving ele ctric stresses and capacitance. The most satisfactory approach involve s the definition of an unfamiliar form of free energy, that can be see n as the electrical analogue of the Gibbs free energy and may have use in other contexts.