The Chemical EducatorISSN: 14304171 (electronic version) Abstract Volume 29
(2024) pp 1833 The Elegance of the Legendre TransformationBenjamin J. Killian* ^{}Department of Chemistry and Physics, University of North Carolina Pembroke, Pembroke, NC 283721510, Benjamin.Killian@uncp.edu Published: 23 January 2024 Abstract. 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 GibbsDuhem 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.
Key Words: In the Classroom; physical chemistry; thermodynamics; state variables and potentials; Legendre transformation (*) Corresponding author. (Email: Benjamin.Killian@uncp.edu) Article in PDF formatt(468 KB) HTML fomat
