含义Meaning
最概然速率、平均速率和方均根速率，比值约为 1 : 1.128 : 1.225。
Most probable, mean and root-mean-square speeds, in the ratio ≈ 1 : 1.128 : 1.225.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(v_p\) | 最概然速率most probable speed | m·s⁻¹ |
| \(\bar v\) | 平均速率mean speed | m·s⁻¹ |
| \(u_\mathrm{rms}\) | 方均根速率root-mean-square speed | m·s⁻¹ |
| \(M\) | 摩尔质量（kg/mol！）molar mass (kg/mol!) | kg·mol⁻¹ |
推导Derivation
由 \(\tfrac12 m\overline{u^2} = \tfrac32k_\mathrm{B}T\) 直接得 \(u_\mathrm{rms} = \sqrt{3k_\mathrm{B}T/m} = \sqrt{3RT/M}\)。
From \(\tfrac12 m\overline{u^2} = \tfrac32k_\mathrm{B}T\): \(u_\mathrm{rms} = \sqrt{3RT/M}\).
对麦克斯韦分布 \(f(v)\) 求极大（\(df/dv = 0\)）得 \(v_p\)；求 \(\int_0^\infty v f(v)\,dv\) 得 \(\bar v\)。
Maximise the Maxwell distribution for \(v_p\); integrate \(\int_0^\infty v f(v)\,dv\) for \(\bar v\).
适用条件与局限Assumptions & limitations
基于麦克斯韦–玻尔兹曼分布（经典、理想、平衡态）。
Based on the Maxwell–Boltzmann distribution (classical, ideal, equilibrium).
M 必须以 kg/mol 代入。
Insert M in kg/mol.
例题Worked example
25 °C 时 N₂：u_rms = √(3×8.314×298/0.028) = 515 m/s，接近声速的 1.5 倍。
N₂ at 25 °C: u_rms = 515 m/s, about 1.5 × the speed of sound.