\[\left(\frac{\partial(G/T)}{\partial T}\right)_p = -\frac{H}{T^2},\qquad \left(\frac{\partial(\Delta G/T)}{\partial T}\right)_p = -\frac{\Delta H}{T^2}\]
含义Meaning
由焓求吉布斯能随温度的变化，是推导范特霍夫方程、沸点升高、凝固点降低等的关键。
Gives the temperature dependence of G from H — the key step in deriving the van ’t Hoff equation and colligative properties.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(G/T\) | 吉布斯能除以温度Gibbs energy over temperature | J·K⁻¹ |
推导Derivation
\((\partial G/\partial T)_p = -S\)（由 dG = V dp − S dT）。
From dG = V dp − S dT, (∂G/∂T)_p = −S.
\(\partial(G/T)/\partial T = \frac1T\frac{\partial G}{\partial T} - \frac{G}{T^2} = -\frac{S}{T} - \frac{G}{T^2} = -\frac{G + TS}{T^2} = -\frac{H}{T^2}\)。
Apply the quotient rule and G + TS = H.
适用条件与局限Assumptions & limitations
恒压下成立；积分时需知 ΔH(T)。
At constant pressure; integration needs ΔH(T).