Potentials And Responses — Examples

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.

Fix a material constitution and amount $n$ and stipulate, on $U>0$ and $0<V<1\mathrm{/}b$, the entropy chart

$S(U,V)=C\ln (U\mathrm{/}{U}_{\ast })+aV-\frac{ab}{2}{V}^2+S_0,$

where $C>0$ is in $\mathrm{J}{\mathrm{K}}^{-1}$, $U_{\ast }>0$ is in joules, $a>0$ is in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, $b>0$ is in ${\mathrm{m}}^{-3}$, and $S_0$ is in $\mathrm{J}{\mathrm{K}}^{-1}$. Let $s$ be a closed system of this constitution and amount. Stipulate that the process model admits a record $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ as in Thermodynamic process records and transfer conventions, using one fixed Galilean inertial chart and satisfying $\mathrm{\Delta }{K}_{\mathrm{c}\mathrm{m}}=\mathrm{\Delta }{U}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$. Its equilibrium state path is piecewise-C1, starting at $(T_1,V_1)$ and following, in order, a constant-temperature leg $T=T_1$ from $V_1$ to $V_2$, a constant-volume leg $V=V_2$ to $T_2$, a constant-temperature leg $T=T_2$ from $V_2$ to $V_1$, and a constant-volume leg $V=V_1$ back to $T_1$, where $T_1>T_2>0$ and $0<V_1<V_2<1\mathrm{/}b$. The moving boundary is a planar piston of constant area. Throughout the volume-changing legs it remains in frictionless mechanical equilibrium with a load tracking the equilibrium pressure, as required by Mechanical pressure-volume work coupling; controlled heat exchange is permitted, and no other work coordinate acts. In this model, the finite work transfer $W_{\mathrm{o}\mathrm{n}}[P]$ is defined as the line integral of the incremental piston work along this piecewise-C1 path. This is a stipulated idealized model record, not a reported observation. Its pressure-work integral is not an endpoint increment.

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.