A smooth chart with no isothermal pressure inverse

Fix the amount of substance and stipulate the local entropy representative

$S(U,V)=C\ln (U\mathrm{/}{U}_{\ast })+aV+S_0$

on the open domain $U>0$, $V>0$, where $C>0$ has units $\mathrm{J}{\mathrm{K}}^{-1}$, $U_{\ast }>0$ has units joules, $a>0$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, and $S_0$ has units $\mathrm{J}{\mathrm{K}}^{-1}$. This chart is finite, C2, concave, and has $S_U>0$. It illustrates that a smooth positive-temperature chart need not admit pressure as a local coordinate at fixed temperature.

The displayed entropy is $S^{\mathrm{S}\mathrm{I}}$ in physical SI coordinates. For this constructed local model, choose the positive-amount open neighborhood stated here (or any positive neighborhood of the fixed amount, extending this entropy independently of amount) and let $O_c$ be the numerical preimage of that product patch under ${\mathrm{\Phi }}_c(\widehat{U},\widehat{V},\widehat{n})=(\widehat{U}\mathrm{J},\widehat{V}{\mathrm{m}}^3,\widehat{n}\mathrm{m}\mathrm{o}\mathrm{l})$. Declare $O_c$ as the model's local state-domain patch and set $S_c=S^{\mathrm{S}\mathrm{I}}\circ {\mathrm{\Phi }}_c$. The derivatives below are those of $S^{\mathrm{S}\mathrm{I}}=S_c\circ {\mathrm{\Phi }}_c^{-1}$ on ${\mathrm{\Phi }}_c(O_c)$, exactly as in Entropic temperature, pressure, and molar chemical potentials. This supplies the numerical-to-SI chart identification for the formal witness, with no assertion of a measured material or a global constitutive extension.