Use of Legendre transforms in chemical thermodynamics (IUPAC Technical Report)
Authors: Alberty Robert A. 
Published: October 2007

Abstract
The fundamental equation of thermodynamics for the internal energy U may include terms for various types of work and involves only differentials of extensive variables. The fundamental equation for U yields intensive variables as partial derivatives of the internal energy with respect to other extensive properties. In addition to the terms from the combined first and second laws for a system involving PV work, the fundamental equation for the internal energy may involve terms for chemical work, gravitational work, work of electric transport, elongation work, surface work, work of electric and magnetic polarization, and other kinds of work. Fundamental equations for other thermodynamic potentials can be obtained by use of Legendre transforms that define these other thermodynamic potentials in terms of U minus conjugate pairs of intensive and extensive variables involved in one or more work terms. The independent variables represented by differentials in a fundamental equation are referred to as natural variables. The natural variables of a thermodynamic potential are important because if a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be obtained by taking partial derivatives of the thermodynamic potential with respect to the natural variables. The natural variables are also important because they are held constant in the criterion for spontaneous change and equilibrium based on that thermodynamic potential. By use of Legendre transforms any desired set of natural variables can be obtained. The enthalpy H, Helmholtz energy A, and Gibbs energy G are defined by Legendre transforms that introduce P, T, and P and T together as natural variables, respectively. Further Legendre transforms can be used to introduce the chemical potential of any species, the gravitational potential, the electric potentials of phases, surface tension, force of elongation, electric field strength, magnetic field strength, and other intensive variables as natural variables. The large number of transformed thermodynamic potentials that can be defined raises serious nomenclature problems. Some of the transforms of the internal energy can also be regarded as transforms of H, A, or G. Since transforms of U, H, A, and G are useful, they can be referred to as the transformed internal energy U', transformed enthalpy H', transformed Helmholtz energy A', and transformed Gibbs energy G' in a context where it is clear what additional intensive natural variables have been introduced. The chemical potential μ

i
of a species is an especially important intensive property because its value is uniform throughout a multiphase system at equilibrium even though the phases may be different states of matter or be at different pressures, gravitational potentials, or electric potentials. When the chemical potential of a species is held constant, a Legendre transform can be used to define a transformed Gibbs energy, which is minimized at equilibrium at a specified chemical potential of that species. For example, transformed chemical potentials are useful in biochemistry because it is convenient to use pH as an independent variable. Recommendations are made to clarify the use of transformed thermodynamic potentials of systems and transformed chemical potentials of species.