Useful-work bound from the Gibbs potential

Given: A system $\sigma$ of fixed constitution $c$, an admitted record $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(\sigma )$ with equilibrium endpoints $x_i,x_f$, $T_0>0$, $p_0>0$, and the ideal reservoir/work-store model below. The core-record conditions and fixed-chart bookkeeping are those of Thermodynamic process records and transfer conventions. Write $Q=Q[P]$ and $W_{\mathrm{o}\mathrm{n}}=W_{\mathrm{o}\mathrm{n}}[P]$. Use $W_{\mathrm{o}\mathrm{u}\mathrm{t}}>0$ for energy delivered by the system, in joules.

Ideal thermal reservoir of constitution $c_R$, with fixed volume and composition: its accessed equilibrium states have entropy $S_R(E_R)=S_{R,0}+E_R\mathrm{/}{T}_0$ on the accessed energy interval, where $E_R$ is its internal energy in joules and entropy is in $\mathrm{J}{\mathrm{K}}^{-1}$. Heat $Q$ into the system changes reservoir energy by $\mathrm{\Delta }{E}_R=-Q$.

Ideal pressure and work stores: the only work on the system is $W_{\mathrm{o}\mathrm{n}}=-p_0\mathrm{\Delta }V-W_{\mathrm{o}\mathrm{u}\mathrm{t}}$, where the pressure store has energy change $p_0\mathrm{\Delta }V$ and the work store receives $W_{\mathrm{o}\mathrm{u}\mathrm{t}}$. Both stores have constant entropy; no other work coordinate is admitted.

The system, bath, and both stores form a compound $C$ with an admitted record $\widehat{P}\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(C)$ representing the same operation as $P$. Its outer boundary is isolated throughout, its constitution is fixed, and its preparations at both endpoints are equilibrium macrostates. For both records the center-of-mass kinetic and separately tracked external-field energy changes vanish. Supply store endpoint sets $Y_B,Y_W$, energy maps $E_B:Y_B\to \mathbb{R}\mathrm{J}$ and $E_W:Y_W\to \mathbb{R}\mathrm{J}$, and constant entropy maps $S_B^{\mathrm{a}\mathrm{u}\mathrm{x}}:Y_B\to \mathbb{R}$ and $S_W^{\mathrm{a}\mathrm{u}\mathrm{x}}:Y_W\to \mathbb{R}$, with values in $\mathrm{J}{\mathrm{K}}^{-1}$. The process model supplies ${\rho }_{\widehat{P}}:\{-,+\}\to \mathcal{F}\subseteq D_c\times D_{c_R}\times Y_B\times Y_W$ with ${\rho }_{\widehat{P}}(-)=(x_i,x_R^{-},y_B^{-},y_W^{-})$ and ${\rho }_{\widehat{P}}(+)=(x_f,x_R^{+},y_B^{+},y_W^{+})$. The store-energy increments are exactly those in [A2]. Here $\mathcal{F}$ is one declared feasible constituent endpoint set for the fixed totals, retained constraints and permitted exchanges of [F2]; one fixed total-energy value on $\mathcal{F}$ is $U_c(x)+U_{c_R}(x_R)+E_B(y_B)+E_W(y_W)$, including both stores from [A2]. The system and bath form an independent separated split admitted by the entropy-additivity rule required in [F2]. Every named auxiliary and work device is included in $C$; any device outside this two-constituent entropy split has explicitly constant endpoint entropy. Thus $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x,x_R,y_B,y_W)=S_c(x)+S_{c_R}(x_R)+S_B^{\mathrm{a}\mathrm{u}\mathrm{x}}+S_W^{\mathrm{a}\mathrm{u}\mathrm{x}}$ on $\mathcal{F}$ gives $\mathrm{\Delta }{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\mathrm{\Delta }S+\mathrm{\Delta }{S}_R$.

First-law energy balance for closed process records: The adopted closed-record balance is $\mathrm{\Delta }U=Q+W_{\mathrm{o}\mathrm{n}}$, with positive $Q$ into and positive $W_{\mathrm{o}\mathrm{n}}$ on the system.

Macroscopic second law for isolated compounds: For an admitted core closed record of a compound isolated throughout, with fixed constitution and equilibrium endpoint tuples associated to the same declared feasible set, the entropy change of an admitted independent separated split is nonnegative. All named devices are included, omitted entropy contributions have declared zero change, and the kinetic and separately tracked external-field endpoint energy changes vanish.

Thermodynamic process records and transfer conventions: $Q$ and $W_{\mathrm{o}\mathrm{n}}$ are process-record transfers, not state functions; an adiabatic record has $Q=0$.

SI quantities and unit conventions: $U$, $Q$, and $W_{\mathrm{o}\mathrm{n}}$ are in joules, $V$ is in cubic metres, and $\mathrm{P}\mathrm{a}{\mathrm{m}}^3=\mathrm{J}$.

Local thermodynamic Legendre potentials: On a valid local $(T,p,n)$ inverse chart, the Gibbs potential is $G=U-Ts+pV$, with $s$ the entropy coordinate.