
The thermodynamic square (Born diagram) drawn where each potential (U, F, G, H), is between its its natural independent variable (SV, VT, TP, PS); whereby, derivatives are obtained by following the diagonal of the square, and the sign is determined by the direction of the arrow. [6]

The thermodynamic square (Born diagram) drawn where each potential (U, F, G, H), is between its its natural independent variable (SV, VT, TP, PS); whereby, derivatives are obtained by following the diagonal of the square, and the sign is determined by the direction of the arrow. [6]

In thermodynamics, thermodynamic square, or Born diagram or Koenig-Born thermodynamic square, is a type of mnemonic memory trick flow diagram or schematic, depicted adjacent, for recalling the exact differentials of the thermodynamic potentials U, F, G, and H, the associated Maxwell relations, and conditions for equilibrium under various sets of imposed conditions. [1]

Etymology
The “thermodynamics square” concept was developed German physicist Max Born in his late 1920s University of Gotttingen lectures on thermodynamics, on the subject of on Maxwell’s relations. One of his 1929 lectures was attended by Hungarian-born American physicist Laszlo Tisza, of the MIT school of thermodynamics, who later incorporated the diagram into his 1966 Generalized Thermodynamics. [6]

The first literature publication of the thermodynamic square method appeared in the 1935 article “Families of Thermodynamic Equations: The Methods of Transformation by the Characteristic Group” by American chemist Frederick Koenig, wherein he attempts to regroup the important equations of thermodynamics into so-called "families", via of a precise definition. [3]

Koenig followed this with a part two 1972 article subtitled “Families of Thermodynamic Equations: The Case of Eight Characteristic Function” (see: characteristic function). [4]

The thermodynamic square diagram next seems to have appeared in Tisza’s student American physicist Herbert Callen’s famous 1985 second edition Thermodynamics an Introduction to Thermostatics textbook. [4]

Variants
If more potentials and varialbes are used, other more advanced variants of the square can be made such as a “thermodynamic cuboctahedron”. [7] Likewise, a the so-called “thermodynamic cube” can be made using a sheet of paper cut and folded a certain way. [8]


A thermodynamic cube, a more encompassing advanced variation of the thermodynamic square. [8]

A thermodynamic cube, a more encompassing advanced variation of the thermodynamic square. [8]

References
1. Ganguly, Jimbamitra. (2008). Thermodynamics in Earth and Planetary Sciences (3.5: thermodynamics square, pgs. 59-). Springer.
2. Born, Max. (1929). “Lecture on Maxwell’s Relations”, Gottingen Lectures on Thermodynamics.
3. Koenig, Frederick O. (1935). “Families of Thermodynamic Equations. I. The Methods of Transformation by the Characteristic Group” (abs), J. Chem. Phys. 3:29-35; and 56:4556, 1972.
4. Koenig, Frederick O. (1972). “Families of Thermodynamic Equations. II. The Case of Eight Characteristic Function” (abs), J. Chem. Phys. 56:4556.
5. Callen, Herbert. (1985). Thermodynamics an Introduction to Thermostatics (7.2: A Thermodynamic Mnemonic Diagram, pgs. 183-86). Wiley.
6. Tisza, Laszlo. (1966). Generalized Thermodynamics (Born diagram, pg. 64). MIT Press.
7. Fox, Ronald F. (date). “The Thermodynamic Cuboctahedron”, Publication.
8. Pate, Stephen F. (1999). “The Thermodynamic Cube: A Mnemonic and Learning Device for Students and Users of Classic Thermodynamics”, American Journal of Physics, 67(12):1111-13.

Further reading
● Klauder, L.T. (1968). “Article”, Am. Journ. Phys. 36:556.
● Cheng, Chien-Chung. (1970). “General Born Diagram and Legendre Transformation” (abs), American Journal of Physics, 33(8):956
● Rock, Peter. (2003). Chemical Thermodynamics (5.6: The Thermodynamic Square, pg. 126-). University Science Books.

External links
● Thermodynamic square – Wikipedia.
● A question about the Thermodynamic Square (2010) – ChemicalForums.com.