The Elegance of the Legendre Transformation

Thermodynamics provides the relationship between the extensive state variables that define a system and the internal energy of that system. The Legendre transformation is the mathematical tool allowing one to convert to a subset of conjugate intensive variables, thus defining new thermodynamic potentials. A mathematical description of the Legendre transformation is provided that assumes only proficiency at the level of the calculus. The transformation is applied to a classical single component system with expansion work to derive the four common, and several uncommon, thermodynamic potentials, as well as the Gibbs-Duhem relation. The method is extended to electrical work, and the Nernst equation is derived using the transformation. Lastly, the transformation is employed in the definition of eight statistical ensembles and to relate the macroscopic random variables to the partition functions for the ensembles.