Local thermodynamic Legendre potentials

Fix a material constitution $c$ with finite species list $(1,\dots ,r)$ and a local energy representation $U_c=U_c(s,V,n_1,\dots ,n_r)$ from Local energy representation and thermodynamic conjugates. Here $s$ is entropy in $\mathrm{J}{\mathrm{K}}^{-1}$, $V>0$ is volume in ${\mathrm{m}}^3$, each $n_j>0$ is in $\mathrm{m}\mathrm{o}\mathrm{l}$, $U_c$ is in joules, and the conjugates are $T=U_{c,s}$ in kelvins, $p=-U_{c,V}$ in pascals, and ${\mu }_j=U_{c,n_j}$ in $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$. The derivatives hold every other listed natural coordinate fixed; the physical-coordinate and rest-frame conventions are those of Equilibrium state domains and SI quantities and unit conventions.

For an open chart, a local Legendre transformation means the following partial transformation of $U_c$: select one or more of its extensive coordinates $q$, take their conjugate coordinates $x=D_q{U}_c$, and use only a neighborhood on which $(q,z)\mapsto (x,z)$, with $z$ the untransformed coordinates, has a $C^1$ inverse. Such a neighborhood is a valid local transform chart in the sense of Thermodynamic response functions on valid local charts. No inverse or potential as a function of the proposed natural coordinates is asserted outside that chart.

On their respective valid charts define the enthalpy, Helmholtz potential, Gibbs potential, and grand potential by

$H_c:=U_c+pV,F_c:=U_c-Ts,G_c:=U_c-Ts+pV,{\mathrm{\Omega }}_c:=U_c-Ts-{\sum }_{j=1}^r{\mu }_j{n}_j\mathrm{.}$

Their natural-coordinate charts are respectively $(s,p,n)$, $(T,V,n)$, $(T,p,n)$, and $(T,V,{\mu }_1,\dots ,{\mu }_r)$. In the volume transform the conjugate coordinate is $-p=U_{c,V}$, which accounts for the plus sign in $H_c$ and $G_c$; in the chemical transform the conjugates are ${\mu }_j=U_{c,n_j}$, which accounts for the minus signs in ${\mathrm{\Omega }}_c$. All four potentials have units of joules. A displayed differential or global convex-duality characterization is not part of this definition; critical, coexistence, and other non-C2 boundaries are excluded.