The first law for closed systems: work is area

The accountant's law

• A closed system 闭口系统 has no mass crossing its boundary, and the first law is its ledger: $\Delta U = Q - W$, with work by the system positive. Energy changes only by heat and work crossing the boundary - nothing else.
• Boundary work 边界功 is $W = \int_1^2 p\,dV$, the area under the path on a p-V diagram. Because the integral runs along the path $p(V)$, work is a path dependence 路径依赖 quantity: same endpoints, different paths, different answers.

Three paths, one pair of states

• Air starts at $(200\ \mathrm{kPa}, 0.3\ \mathrm{m^3})$ and triples its volume. An isobaric process gives $W = p\Delta V = 200 \times 0.6 = \mathbf{120}\ \mathrm{kJ}$. An isothermal process 等温过程 through the same ends gives $W = p_1 V_1 \ln 3 = \mathbf{65.9}\ \mathrm{kJ}$ - the same endpoints, $54.1\ \mathrm{kJ}$ apart, and the difference is exactly the area between the two curves.
• Now an adiabatic process 绝热过程 ($Q = 0$): compress from the same start to $0.15\ \mathrm{m^3}$: $pV^{1.4} = \mathrm{const}$ gives $p_2 = \mathbf{527.8}\ \mathrm{kPa}$ and $W = (p_2 V_2 - p_1 V_1)/(1 - \gamma) = \mathbf{-48}\ \mathrm{kJ}$ - work ON the gas, and $\Delta U = -W = +48\ \mathrm{kJ}$ closes the ledger.

Sign and relation slips

Recap