English: Thermodynamic Octaeder

Inspired by: L. T. Klauder, American Journal of Physics, 1968, 36(6), 556-557 [https://dx.doi.org/10.1119/1.1974977.

This is intended for memorizing the Maxwell relations. It is a generalization of the Thermodynamic square under changes in particle number (extension from two to three degrees of freedom).

Opposing corners represent conjugate state functions: S entropy, T temperature, p pressure , V volume, N particle number, μ chemical potential. Faces are thermodynamic potentials, H enthalpy, U internal energy, F Helmholtz free energy, G Gibbs free energy. Each potential is thus bounded by the three variables on which it depends. The black triangles containing the letter G are just for representational purposes, showing that the front-facing faces are labelled G, G'.

The figure thus represents eight equation, e.g. the equation represented by the face "U" is:

dU = T·dS -p·dV + μ·dN

The 1968 paper uses m and H for N and μ, respectively [I suppose m is mass as a stand-in for particle number, so H must be chemical potential?], and the enthalpy is labeled E.

The convention of "primed" potentials U' etc. is taken from the 1968 paper, these are those potentials that depend on changes in chemical potential with fixed particle number. [? This doesn't correspond to a physical process unless chemical reactions are taking place (in which case the situation becomes more complicated anyway because the reaction itself will release or consume energy), but you can still plot how things depend on chemical potential; I don't think these "prime potentials" have actual names, they aren't named (even with a symbol) in the en:Thermodynamic potential article, but they are briefy alluded to, "using the other potentials we can get equations such as...".].