振动配分函数 Vibrational partition function — 公式推导、符号与局限

\[q_\text{vib} = \frac{1}{1 - e^{-\Theta_\text{vib}/T}}\ (\text{从 } v = 0 \text{ 算起}),\qquad \Theta_\text{vib} = \frac{hc\tilde\nu}{k_\mathrm{B}}\]\[q_\text{vib} = \frac{1}{1 - e^{-\Theta_\text{vib}/T}}\ (\text{from} v = 0 \text{}),\qquad \Theta_\text{vib} = \frac{hc\tilde\nu}{k_\mathrm{B}}\]

振动平均能与热容（每摩尔每个模式）mean vibrational energy and heat capacity per mode

\[\bar\varepsilon_\text{vib} = \frac{hc\tilde\nu}{e^{\Theta_\text{vib}/T} - 1},\qquad C_{V,\text{vib}} = R\left(\frac{\Theta_\text{vib}}{T}\right)^2\frac{e^{\Theta_\text{vib}/T}}{(e^{\Theta_\text{vib}/T} - 1)^2}\]

含义Meaning

谐振子能级等间距，配分函数是几何级数。大多数分子 Θ_vib 达数千 K，室温下振动几乎“冻结”，对热容贡献很小。

Equally spaced oscillator levels give a geometric series. Most molecules have Θ_vib of thousands of kelvin, so vibrations are nearly frozen at room temperature.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(\Theta_\text{vib}\)	振动特征温度vibrational temperature	K
\(\tilde\nu\)	振动波数vibrational wavenumber	cm⁻¹

推导Derivation

以零点能为能量零点：\(\varepsilon_v = vhc\tilde\nu\)，\(q = \sum_{v=0}^\infty x^v\)，\(x = e^{-\Theta_\text{vib}/T}\)。
Measured from the zero-point level, \(q = \sum x^v\) with \(x = e^{-\Theta_\text{vib}/T}\).
几何级数求和 \(q = 1/(1-x)\)。
Sum the geometric series.
\(\bar\varepsilon = k_\mathrm{B}T^2\,\partial\ln q/\partial T\) 得平均能；再对 T 求导得热容：高温趋于 R（能量均分），低温指数趋零。
Differentiate for the mean energy and heat capacity: R at high T, exponentially small at low T.

适用条件与局限Assumptions & limitations

谐振近似；高温下非谐性和解离使其偏离。多原子分子对每个简正模式分别相乘。
Harmonic; anharmonicity matters at high T. Multiply over all normal modes.

例题Worked example

N₂（2359 cm⁻¹）：Θ_vib = 3394 K，298 K 时 q_vib = 1.00001；I₂（214 cm⁻¹）：Θ_vib = 308 K，q_vib = 1.55。

N₂: Θ_vib = 3394 K, q ≈ 1.00001 at 298 K; I₂: Θ_vib = 308 K, q = 1.55.