平均自由程与碰撞频率 Mean free path and collision frequency — 公式推导、符号与局限

\[\lambda = \frac{k_\mathrm{B}T}{\sqrt2\,\sigma\, p}, \qquad z = \frac{\sqrt2\,\sigma\,\bar v\, p}{k_\mathrm{B}T}, \qquad \sigma = \pi d^2\]

含义Meaning

分子两次碰撞间平均走过的距离 λ，以及每个分子每秒的碰撞次数 z。

λ is the mean distance travelled between collisions; z is the collision rate of one molecule.

符号Symbol	含义 Meaning	单位 Unit
\(\lambda\)	平均自由程mean free path	m
\(z\)	碰撞频率collision frequency	s⁻¹
\(\sigma\)	碰撞截面collision cross-section	m²
\(d\)	分子碰撞直径collision diameter	m

一个分子运动时扫过截面 \(\sigma = \pi d^2\) 的圆柱，单位时间扫过体积 \(\sigma \bar v_{\text{rel}}\)。
A moving molecule sweeps a cylinder of cross-section \(\sigma = \pi d^2\), volume \(\sigma \bar v_{\text{rel}}\) per second.
其中所含分子数即碰撞次数 \(z = \sigma \bar v_{\text{rel}}\, N/V\)；相对速率 \(\bar v_{\text{rel}} = \sqrt2\,\bar v\)。
Molecules in it give \(z = \sigma \bar v_{\text{rel}} N/V\) with \(\bar v_{\text{rel}} = \sqrt2\,\bar v\).
\(\lambda = \bar v / z = 1/(\sqrt2\,\sigma N/V)\)，再用 \(N/V = p/k_\mathrm{B}T\)。
\(\lambda = \bar v/z = 1/(\sqrt2\,\sigma N/V)\) with \(N/V = p/k_\mathrm{B}T\).

硬球模型；真实分子的有效截面随温度略变。
Hard-sphere model; effective cross-sections vary somewhat with T.
λ 远大于容器尺寸时（高真空）进入分子流区，扩散、黏度规律改变。
When λ exceeds the vessel size (high vacuum) the gas enters the molecular-flow regime.

N₂（σ ≈ 0.43 nm²）在 25 °C、1 atm：λ ≈ 68 nm，z ≈ 7×10⁹ s⁻¹。

N₂ (σ ≈ 0.43 nm²) at 25 °C, 1 atm: λ ≈ 68 nm, z ≈ 7×10⁹ s⁻¹.