含义Meaning
克劳修斯（1865）定义熵为可逆热温商。第二定律：孤立体系的熵永不减少，自发过程使总熵增加；“热不能自发地从低温物体传到高温物体”。
Clausius (1865) defined entropy as reversible heat over temperature. Second law: the entropy of an isolated system never decreases; spontaneous processes raise total entropy, and heat does not flow spontaneously from cold to hot.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(S\) | 熵entropy | J·K⁻¹ |
| \(\delta q_\text{rev}\) | 可逆过程的热reversible heat | J |
推导Derivation
卡诺循环表明 \(\oint\delta q_\text{rev}/T = 0\)，所以 \(\delta q_\text{rev}/T\) 是某个状态函数的全微分。
Carnot cycles give ∮δq_rev/T = 0, so it is an exact differential.
不可逆循环效率低于卡诺，推得 \(\oint\delta q/T \le 0\)（克劳修斯不等式），即 \(dS \ge \delta q/T\)。
Irreversible cycles give the Clausius inequality.
孤立体系 δq = 0，所以 dS ≥ 0：熵增原理。
For an isolated system δq = 0, so dS ≥ 0.
适用条件与局限Assumptions & limitations
熵增针对孤立体系（体系 + 环境）；体系自身熵可以减少（如水结冰），只要环境熵增更多。
Applies to system + surroundings; a system’s entropy can fall if the surroundings gain more.