\[\frac{d\ln p}{dT} = \frac{\Delta_\text{vap}H}{RT^2},\qquad \ln\frac{p_2}{p_1} = -\frac{\Delta_\text{vap}H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\]
安托万经验方程Antoine equation
\[\lg\frac{p}{\mathrm{mmHg}} = A - \frac{B}{C + t/{}^\circ\mathrm C}\]含义Meaning
液（固）–气平衡时饱和蒸气压随温度的变化。ln p 对 1/T 为直线，斜率 −ΔvapH/R；可用来求汽化焓、估算不同压强下的沸点。
How vapour pressure varies with temperature: ln p versus 1/T is linear with slope −ΔvapH/R, giving vaporisation enthalpies and boiling points at other pressures.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(p\) | 饱和蒸气压vapour pressure | Pa |
| \(\Delta_\text{vap}H\) | 摩尔汽化焓molar enthalpy of vaporisation | J·mol⁻¹ |
推导Derivation
从克拉珀龙方程出发：液–气相变 \(\Delta V \approx V_m(\mathrm g) = RT/p\)（忽略液体体积、视蒸气为理想气体）。
Start from Clapeyron with ΔV ≈ RT/p.
\(dp/dT = p\Delta H/(RT^2)\)，即 \(d\ln p/dT = \Delta H/RT^2\)。
Rearrange to d ln p/dT.
ΔH 视为常数积分得两点式。
Integrate with constant ΔH.
适用条件与局限Assumptions & limitations
远离临界点；温度范围大时 ΔvapH 随温度减小（临界点处为零），需用安托万方程等经验式。
Away from the critical point; ΔvapH decreases with T, so use Antoine over wide ranges.
例题Worked example
水在 100 °C 蒸气压 101.3 kPa、ΔvapH = 40.7 kJ/mol：在珠峰顶（约 33.7 kPa）沸点约 71 °C。
Water boils at ~71 °C at Everest’s summit (33.7 kPa).