含义Meaning
熵是体系微观状态数的对数（玻尔兹曼，1877；普朗克写成此式）。熵增即体系走向微观状态数更多、更“混乱”的宏观态。此式刻在玻尔兹曼的墓碑上。
Entropy is the logarithm of the number of microstates (Boltzmann 1877, written so by Planck). Entropy increase means moving to macrostates with more microstates. The formula is engraved on Boltzmann’s tombstone.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(W\) | 微观状态数number of microstates | 1 |
| \(k_\mathrm B\) | 玻尔兹曼常数Boltzmann constant | J·K⁻¹ |
推导Derivation
熵可加、W 可乘：两独立体系 \(W = W_1W_2\)、\(S = S_1 + S_2\)，所以 S ∝ ln W。
S adds while W multiplies, so S ∝ ln W.
比例系数通过与理想气体熵（如 \(\Delta S = nR\ln(V_2/V_1)\)）比较确定为 \(k_\mathrm B = R/N_A\)：N 个分子体积加倍，\(W_2/W_1 = 2^N\)。
Matching ideal-gas expansion (W₂/W₁ = 2^N) fixes k_B = R/N_A.
适用条件与局限Assumptions & limitations
适用于孤立体系、等概率微观态；一般情况用吉布斯熵 \(-k\sum p\ln p\)。
For equal-probability microstates; generally use Gibbs’s form.
例题Worked example
1 mol 理想气体向真空膨胀到两倍体积：ΔS = k ln 2^{N_A} = R ln 2 = 5.76 J/K。
Free expansion of 1 mol to double volume: ΔS = R ln 2.