Mechanical pressure-volume work coupling

For a simple-compressible system whose only admitted mechanical equilibrium coordinate is volume, model the piston as a planar boundary of fixed area $A>0$ in ${\mathrm{m}}^2$ that translates only along its outward unit normal without changing shape. Let $\mathrm{\ell }$ in metres be its outward displacement coordinate, so the enclosed volume is $V_{\ast }+A\mathrm{\ell }$ for a fixed reference value $V_{\ast }$. Let $\mathrm{\Omega }$ be an open positive-temperature single-phase chart of equilibrium states, with equilibrium states as in Equilibrium state domains.

A quasistatic piston leg in this postulate is a map $\theta \mapsto (U(\theta ),V(\theta ),n,\mathrm{\ell }(\theta ))$ on a compact interval $J=[\alpha ,\beta ]\subseteq \mathbb{R}$, with $\alpha <\beta$, of a dimensionless parameter, fixed composition $n$, $V(\theta )=V_{\ast }+A\mathrm{\ell }(\theta )$, and the system in an equilibrium state at every $\theta \in J$. Each coordinate function is continuous on $J$, differentiable on $(\alpha ,\beta )$, and has a derivative extending continuously to $J$; equivalently the leg is $C^1$ in its Euclidean coordinates. A piecewise-$C^1$ quasistatic piston path is a finite concatenation of these legs with matching endpoint state and piston position. The path is frictionless when the tangential contact force vanishes on every leg. It is in mechanical equilibrium when the net normal force on the piston is zero at every path point; thus the external load balances the system's outward normal force. Write $F_{\mathrm{s}\mathrm{y}\mathrm{s}}(\theta )$ for the signed outward normal force component, measured in newtons, exerted by the system on the piston, and define the oriented mechanical pressure by $p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}(\theta ):=F_{\mathrm{s}\mathrm{y}\mathrm{s}}(\theta )\mathrm{/}A$, in pascals.

For every such path, adopt the identification

$p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=p_c,$

where $p_c$ is the derivative-defined pressure of Entropic temperature, pressure, and molar chemical potentials, evaluated at $(U(\theta ),V(\theta ),n)$. With $W_{\mathrm{o}\mathrm{n}}$ positive for work done on the system, the system's outward work on the piston is $F_{\mathrm{s}\mathrm{y}\mathrm{s}}d\mathrm{\ell }$, so the coupling assigns the work increment along the path

$\delta W_{\mathrm{o}\mathrm{n}}=-p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}Ad\mathrm{\ell }=-p_{c}dV\mathrm{.}$

Here $\delta W_{\mathrm{o}\mathrm{n}}$ denotes an increment along the stated path; no path-independence is asserted. This is an adopted, regime-limited physical assumption, not a deduction from the entropy derivative or from the first-law balance First-law energy balance for closed process records. It makes no claim for friction, paths outside the quasistatic regime, changing piston area, shear, surface forces, fields, composition transfer, or other generalized coordinates.