An Elementary Deduction of Gibbs' Adsorption Theorem

The Gibbsian treatment of the thermodynamics of surfaces is very long and On the other hand the equations designated as ``Gibbs' adsorption theorem'' in all texts that have come to the authors' attention are only special cases, incompletely or inexactly derived, of the Gibbsian equation for constant temperature, $$\mathrm{d\sigma =-}{\displaystyle \sum _{\text{i=1}}^K}{\Gamma }_i{\mathrm{d\mu }}_i.$$ An exact elementary deduction of this equation is therefore desirable. Such a deduction is given by means of a thermodynamic engine. Various closely related equations, all for constant temperature, are also deduced. The relation of the equations obtained to the most general Gibbsian equations, which include variations of temperature, is indicated. The assumptions underlying the Gibbsian treatment of the thermodynamics of surfaces are discussed briefly.