[[File:Thermodynamic square.svg|thumb|The thermodynamic square with potentials highlighted in red. Key: G = Gibbs free energy p = Pressure H = Enthalpy S = Entropy U = Internal energy V = Volume F = Helmholtz free energy T = Temperature]] The thermodynamic square (also known as the thermodynamic wheel, Guggenheim scheme or Born square) is a mnemonic diagram attributed to Max Born and used to help determine thermodynamic relations. Born presented the thermodynamic square in a 1929 lecture. The symmetry of thermodynamics appears in a paper by F.O. Koenig. The corners represent common conjugate variables while the sides represent thermodynamic potentials. The placement and relation among the variables serves as a key to recall the relations they constitute. A mnemonic used by students to remember the Maxwell relations (in thermodynamics) is "Good Physicists Have Studied Under Very Fine Teachers", which helps them remember the order of the variables in the square, in clockwise direction. Another mnemonic used here is "Valid Facts and Theoretical Understanding Generate Solutions to Hard Problems", which gives the letter in the normal left-to-right writing direction. Both times A has to be identified with F, another common symbol for Helmholtz free energy. To prevent the need for this switch the following mnemonic is also widely used:"Good Physicists Have Studied Under Very Ambitious Teachers"; another one is Good Physicists Have SUVAT, in reference to the equations of motion. One other useful variation of the mnemonic when the symbol E is used for internal energy instead of U is the following: "Some Hard Problems Go To Finish Very Easy".

Use

Derivatives of thermodynamic potentials

The thermodynamic square is mostly used to compute the derivative of any thermodynamic potential of interest. Suppose for example one desires to compute the derivative of the internal energy U. The following procedure should be considered: The Gibbs–Duhem equation can be derived by using this technique. Notice though that the final addition of the differential of the chemical potential has to be generalized.

Maxwell relations

The thermodynamic square can also be used to find the first-order derivatives in the common Maxwell relations. The following procedure should be considered: By rotating the \sqcup shape (randomly, for example by 90 degrees counterclockwise into a \sqsupset shape) other relations such as: can be found.

Natural variables of thermodynamic potentials

Finally, the potential at the center of each side is a natural function of the variables at the corner of that side. So, G is a natural function of p and T, and U is a natural function of S and V.