\[\Delta U = q + w,\qquad dU = \delta q + \delta w\]
含义Meaning
能量守恒在热过程中的表述：体系内能的变化等于吸收的热加上环境对体系做的功（IUPAC 符号约定）。内能是状态函数，热和功不是。
Energy conservation for thermal processes: the change in internal energy equals heat absorbed plus work done on the system (IUPAC convention). U is a state function; q and w are not.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(U\) | 内能internal energy | J |
| \(q\) | 体系吸收的热heat absorbed by the system | J |
| \(w\) | 环境对体系做的功work done on the system | J |
推导Derivation
焦耳（1840 年代）用重物下落带动搅拌器加热水，证明一定量的功总产生相同的热（热功当量）。
Joule’s paddle-wheel experiments (1840s) showed a fixed equivalence of work and heat.
孤立体系能量守恒；封闭体系与环境只以热和功交换能量，所以 \(\Delta U = q + w\)。
For a closed system energy enters only as heat or work.
循环过程 \(\oint dU = 0\)：不可能制成不消耗能量而持续做功的第一类永动机。
Over a cycle ∮dU = 0: no perpetual motion of the first kind.
适用条件与局限Assumptions & limitations
注意符号约定：物理教材常用 \(\Delta U = Q - W\)（W 为体系对外做的功）。
Physics texts often write ΔU = Q − W with W done by the system.