\[w = -\int_{V_1}^{V_2}p_\text{ex}\,dV,\qquad w = -p_\text{ex}\Delta V\ (\text{恒外压})\]\[w = -\int_{V_1}^{V_2}p_\text{ex}\,dV,\qquad w = -p_\text{ex}\Delta V\ (\text{const. external p})\]
含义Meaning
体系膨胀反抗外压做功（w < 0），被压缩时环境对体系做功（w > 0）。功的大小与路径有关：可逆过程（外压始终与内压相差无穷小）做功最多。
Expansion against external pressure does work on the surroundings (w < 0); compression has w > 0. Work depends on path, and a reversible path gives the maximum.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(p_\text{ex}\) | 外压external pressure | Pa |
| \(V\) | 体积volume | m³ |
推导Derivation
活塞面积 A，外力 \(F = p_\text{ex}A\)，移动 dx：环境做功 \(-F\,dx = -p_\text{ex}\,dV\)。
A piston of area A moving dx against \(p_\text{ex}A\).
积分得总功；恒外压时直接相乘。在 p–V 图上即曲线下面积。
Integrate; on a p–V diagram it is the area under the curve.
向真空自由膨胀 \(p_\text{ex} = 0\)，w = 0。
Free expansion into vacuum does no work.
适用条件与局限Assumptions & limitations
只计体积功；电功、表面功等非体积功另计。
Expansion work only.
例题Worked example
1 mol 理想气体 298 K 下由 1 bar 恒外压膨胀到 10 L→ 24.8 L：w = −1×10⁵×0.0148 = −1.48 kJ；可逆等温膨胀相同体积则做功 −2.25 kJ。
Irreversible vs reversible expansion between the same states: −1.48 vs −2.25 kJ.