双语化学辞典Chem Lexicon
本科Undergraduate 统计热力学 Statistical thermodynamics

分子配分函数Molecular partition function

\[q = \sum_i g_i\,e^{-\varepsilon_i/k_\mathrm{B}T},\qquad q = q_\text{trans}\,q_\text{rot}\,q_\text{vib}\,q_\text{el}\]
正则配分函数canonical partition function
\[Q = \frac{q^N}{N!}\ (\text{理想气体，全同不可区分分子}),\qquad Q = q^N\ (\text{可区分，如晶格中的原子})\]\[Q = \frac{q^N}{N!}\ (\text{ideal gas , indistinguishable molecules}),\qquad Q = q^N\ (\text{distinguishable , e.g. atoms in a lattice})\]

含义Meaning

配分函数是按玻尔兹曼因子加权的有效可及态数，包含体系全部热力学信息。分子能量可分解时，配分函数可写成各运动形式配分函数之积。

The partition function counts thermally accessible states weighted by Boltzmann factors and contains all thermodynamic information. When the molecular energy separates, q factorises into contributions from each motion.

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(q\)分子配分函数molecular partition function1
\(Q\)体系（正则）配分函数canonical partition function1
\(N\)分子数number of molecules1

推导Derivation

  1. 玻尔兹曼分布的归一化因子即 \(q\)；T → 0 时 \(q \to g_0\)，T → ∞ 时 q 趋于可及态总数。

    q normalises the Boltzmann distribution; it tends to \(g_0\) as T → 0 and grows with T.

  2. 若 \(\varepsilon = \varepsilon_t + \varepsilon_r + \varepsilon_v + \varepsilon_e\) 且各自独立，指数和可拆成乘积。

    If ε = εt + εr + εv + εe independently, the sum factorises.

  3. N 个不可区分分子的体系：交换分子不产生新状态，除以 N! 避免重复计数（吉布斯修正）。

    For N indistinguishable molecules divide by N! (Gibbs correction).

适用条件与局限Assumptions & limitations

  • 能量可分离是近似（振–转耦合、非谐性）；\(Q = q^N/N!\) 只在经典极限下成立。

    Separability is approximate; \(Q = q^N/N!\) holds in the classical limit.