# Legendre transform

 Legendre transformations among the four main thermodynamic potentials; the arrows indicate possible transformations of each potential; the products of conjugate variables PV and TS used in each transformation are also indicated. [5] 1820 watercolor caricature of French mathematicians Adrien-Marie Legendre eponym of the Legendre transform, by French artist Jules Boilly. [4]
In thermodynamics, Legendre transform, or "Legendre transformation", is a mathematical procedure employed to convert a specific variable of a state function equation into a more convenient form, often one in which the variable can thus be calculated readily from existing experimental data. The Legendre transformation allows one to obtain functions equivalent to U = U(S,V) dependent on more accessible laboratory coordinates or variables. [5]

Overview
The equations for a system containing heat as a variable, for example, contain entropy as a variable quantity. Entropy is an inconvenient variable, difficult to control for and hold constant as one can hold temperature, pressure, and volume constant. The Legendre transform allows a researcher to convert equations containing entropy into equations expressed only in terms of temperature, pressure, and volume. The Legendre transform can be applied correctly only under certain conditions, which must be specified.

The Legendre transform is named after French mathematician Adrien-Marie Legendre (1752-1833). [1] In mechanics, the Legendre transformation is used to go from the Lagrangian formulation to the Hamiltonian formulation. [2] The Legendre transform is an important tool in biochemical thermodynamics, in that it makes the application of thermodynamics to biochemical reaction analysis more convenient. [3]

In the 2000s, American physical chemist Robert Alberty published a number of summary papers and chapters on the overview of the use of the Legendre transform as used in chemical thermodynamics and biochemical thermodynamics. [6]

Bridgeman formulas

References
1. Adrien-Marie Legendre – Wikipedia.
2. Emanuel, George. (1987). Advanced Classical Thermodynamics (3.3: Legendre Transformation, pgs. 25-28). AIAA.
3. Alberty, Robert A. (2003). Thermodynamics of Biochemical Reactions (pg. vii; Legendre transforms for thermodynamic potentials: U, H, A, and G, pgs. 30-31). Wiley.
4. Boilly, Julien-Leopold. (1820). Album de 73 Portraits-Charge Aquarelle’s des Membres de I’Institute (watercolor portrait #29). Biliotheque de l’Institut de France.
5. Helrich, Carl S. (2009). Modern Thermodynamics with Statistical Mechanics (Legendre transformation, 24+ pgs). Springer.
6. Alberty, Robert A. (2001). “Use of Legendre Transforms in Chemical Thermodynamics” (abs), IUPAC Technical Report, Pure Appl. Chem. Vol. 73, No. 8, pp. 1349-1380.