Variational Thermodynamic Derivation of the Formula for Pressure Difference across a Charged Conducting Liquid Surface and Its Relation to the Thermodynamics of Electrical Capacitance
Abstract
The formula for pressure difference across a charged conducting liquid surface has conventionally been derived by adding a Maxwell stress term to the pressuredifference formula for the fieldfree case. As far as can be established, no derivation applying direct energybased methods to the chargedsurface case has ever been clearly formulated. This paper presents a firstprinciples variational derivation, starting from the laws of thermodynamics and modelled on Gibbs's (1875) approach to the fieldfree case. The derivation applies to the static equilibrium situation. The method is to treat the charged liquid and its environment as a heterogeneous system in thermodynamic equilibrium, and consider the effects of a small virtual variation in the shape of the conductingliquid surface. Expressions can be obtained for virtual changes in the free energies of relevant system components and for the virtual electrical work done on the system. By converting the space integral of the variation in electrostatic field energy to an integral over the surface of the liquid electrode, the usual pressuredifference formula is retrieved. It is also shown how the problem can be formulated, in various ways, as a freeenergy problem in a situation involving electric stresses and capacitance. The most satisfactory approach involves the definition of an unfamiliar form of free energy, that can be seen as the electrical analogue of the Gibbs free energy and may have use in other contexts.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 July 1995
 DOI:
 10.1098/rspa.1995.0078
 Bibcode:
 1995RSPSA.450..177L