Thermodynamic response functions on valid local charts

Fix a constitution $c$ and an open dimensionless subset $O\subseteq D_c$ in the single-phase region of Equilibrium state domains. Its physical chart is $\mathrm{\Omega }={\mathrm{\Phi }}_c(O)$, where ${\mathrm{\Phi }}_c$ is the SI coordinate map defined in Entropic temperature, pressure, and molar chemical potentials. For fixed physical amounts $n=(n_1,\dots ,n_r)$ with nonempty slice ${\mathrm{\Omega }}_n:=\{(U,V):(U,V,n)\in \mathrm{\Omega }\}$, this slice is open in ${\mathbb{R}}^2$. Use $T,p$ from Entropic temperature, pressure, and molar chemical potentials as functions of the physical coordinates on ${\mathrm{\Omega }}_n$. For a response derivative, restrict to an open neighborhood in ${\mathrm{\Omega }}_n$ where the required natural-coordinate map is a $C^1$ diffeomorphism onto an open image, so its inverse expresses the physical $(U,V)$ as $C^1$ functions of those natural coordinates ($C^k$ Euclidean maps and diffeomorphisms). Specifically, $C_V$ uses $(U,V)\mapsto (T,V)$ at fixed $n$, while $C_p$, ${\kappa }_T$, and ${\alpha }_p$ use $(U,V)\mapsto (T,p)$ at fixed $n$. The existence of each such local chart is a hypothesis for its response derivative, not a consequence of this definition. Each derivative uses the indicated natural variables and holds all other listed coordinates fixed.

The constant-volume heat capacity and constant-pressure heat capacity of the whole system are, where the derivatives exist and are finite,

$C_V:={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_{V,n},H:=U+pV,C_p:={\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }T}\right)}_{p,n}\mathrm{.}$

Here $H$ is the enthalpy in joules; both heat capacities are in $\mathrm{J}{\mathrm{K}}^{-1}$, not per mole. The isothermal compressibility and isobaric thermal expansion coefficient are

${\kappa }_T:=-\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }p}\right)}_{T,n},{\alpha }_p:=\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_{p,n}\mathrm{.}$

Since $V>0$ and the coordinates use SI units (SI quantities and unit conventions), ${\kappa }_T$ has units $\mathrm{P}{\mathrm{a}}^{-1}$ and ${\alpha }_p$ has units ${\mathrm{K}}^{-1}$. This definition asserts no sign for any response, no identity among them, and no finite value at a critical or otherwise singular chart.