![Thermodynamic square](https://upload.wikimedia.org/wikipedia/commons/f/fb/Thermodynamic_square.svg)
The thermodynamic square (also known as the thermodynamic wheel, Guggenheim scheme or Born square) is a
mnemonic
A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding.
Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
diagram attributed to
Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
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
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation— ...
while the sides represent
thermodynamic potentials
A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermod ...
. 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
file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...
(in
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
) 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
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal
In thermodynamics, an isotherma ...
. 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
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
. 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
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
. The following procedure should be considered:
# Place oneself in the thermodynamic potential of interest, namely (
,
,
,
). In our example, that would be
.
# The two opposite corners of the potential of interest represent the coefficients of the overall result. If the coefficient lies on the left hand side of the square, a negative sign should be added. In our example, an intermediate result would be