First-law energy balance for closed process records

For every admitted $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ with equilibrium endpoints $x_P^{-},x_P^{+}\in D_c$, fixed material constitution, zero change in center-of-mass kinetic energy, and zero change in separately tracked external-field potential energy in the fixed chart selected by the process model, the adopted model balance is $U_c(x_P^{+})-U_c(x_P^{-})=Q[P]+W_{\mathrm{o}\mathrm{n}}[P],$ where $U_c$ is the dimensioned energy-coordinate map of Equilibrium state domains.

Here the internal-energy coordinate is evaluated in the center-of-mass rest frame as in Galilean inertial charts and the center-of-mass rest frame, while $Q[P]$ and $W_{\mathrm{o}\mathrm{n}}[P]$ are the protocol transfers defined by Thermodynamic process records and transfer conventions, in joules as in SI quantities and unit conventions. The sign convention is positive $Q$ into the system and positive $W_{\mathrm{o}\mathrm{n}}$ for work done on it. If $P$ is adiabatic, no thermal energy crosses its selected outer boundary throughout the record, so $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$ and hence $Q[P]=0$; the same adopted balance gives $W_{\mathrm{o}\mathrm{n}}[P]=U_c(x_P^{+})-U_c(x_P^{-})$.

This equation is an adopted physical postulate of the stated framework. It is not deduced here from a mechanical model or from experimental observations. Records with matter exchange or nonzero changes in omitted kinetic or external-field energy require a separately specified balance and are outside this item.