含义Meaning
等温大气中压强（和分子数密度）随高度指数下降，是玻尔兹曼分布在重力场中的体现。
In an isothermal atmosphere pressure falls exponentially with height — the Boltzmann distribution in a gravitational field.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(h\) | 高度height | m |
| \(g\) | 重力加速度gravitational acceleration | m·s⁻² |
| \(M\) | 摩尔质量molar mass | kg·mol⁻¹ |
| \(H = RT/Mg\) | 标高（压强降为 1/e 的高度）scale height | m |
推导Derivation
厚度 dh 的气层受力平衡：\(dp = -\rho g\, dh\)。
Hydrostatic balance of a layer dh: \(dp = -\rho g\, dh\).
由 \(\rho = pM/RT\) 得 \(dp/p = -(Mg/RT)\,dh\)。
With \(\rho = pM/RT\): \(dp/p = -(Mg/RT)\,dh\).
T 恒定时积分即得指数式；等价于 \(N(h) \propto e^{-mgh/k_\mathrm{B}T}\)。
Integrating at constant T gives the exponential, equivalent to \(N(h) \propto e^{-mgh/k_\mathrm{B}T}\).
适用条件与局限Assumptions & limitations
假设等温、g 不变、气体组成均匀；真实对流层温度随高度降低（约 6.5 K/km）。
Assumes isothermal air, constant g and uniform composition; the real troposphere cools by ~6.5 K/km.
每种气体有自己的标高；100 km 以上大气按组分分层。
Each gas has its own scale height; above ~100 km the atmosphere stratifies by species.
例题Worked example
T = 288 K、M = 0.029 kg/mol：标高 H ≈ 8.4 km，珠峰顶（8.85 km）气压约 p₀/e ≈ 35 kPa（实测约 33 kPa）。
T = 288 K, M = 0.029: H ≈ 8.4 km; Everest’s summit is at about p₀/e ≈ 35 kPa (measured ≈ 33 kPa).