含义Meaning
把压强解释为大量分子撞击器壁的动量传递；温度是分子平均平动动能的量度。
Pressure is the momentum delivered by molecular impacts on the walls; temperature measures the mean translational kinetic energy.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(N/V\) | 分子数密度number density | m⁻³ |
| \(m\) | 一个分子的质量mass of one molecule | kg |
| \(\overline{u^2}\) | 速率平方的平均值mean square speed | m²·s⁻² |
推导Derivation
边长 L 的立方容器中，一个分子以 \(u_x\) 撞击右壁，动量变化 \(2mu_x\)，往返一次用时 \(2L/u_x\)。
In a cube of side L, a molecule hitting the wall changes momentum by \(2mu_x\) every round trip \(2L/u_x\).
它对该壁的平均作用力 \(F = 2mu_x/(2L/u_x) = mu_x^2/L\)。
Its average force on that wall is \(F = mu_x^2/L\).
N 个分子求和，压强 \(p = F/L^2 = Nm\overline{u_x^2}/V\)。
Summing over N molecules: \(p = Nm\overline{u_x^2}/V\).
各向同性：\(\overline{u_x^2} = \overline{u^2}/3\)，得 \(pV = \tfrac13 Nm\overline{u^2}\)。
Isotropy: \(\overline{u_x^2} = \overline{u^2}/3\), so \(pV = \tfrac13 Nm\overline{u^2}\).
与 \(pV = Nk_\mathrm{B}T\) 比较，得平均平动动能 \(\tfrac12 m\overline{u^2} = \tfrac32 k_\mathrm{B}T\)。
Comparing with \(pV = Nk_\mathrm{B}T\): \(\tfrac12 m\overline{u^2} = \tfrac32 k_\mathrm{B}T\).
适用条件与局限Assumptions & limitations
假设分子是无体积质点、只有弹性碰撞、无相互作用；分子间碰撞不影响结果（只是重新分配速度）。
Point particles, elastic collisions, no interactions; intermolecular collisions only redistribute speeds.
只计平动；转动、振动能量要用能量均分定理另算（影响热容，不影响压强）。
Translation only; rotation and vibration affect heat capacity, not pressure.