# 吉布斯－亥姆霍茲方程

${\displaystyle \left({\frac {\partial ({\frac {G}{T}})}{\partial T}}\right)_{p\,}=-{\frac {H}{T^{2}}}}$

${\displaystyle H\,}$
${\displaystyle T\,}$ 絕對溫度
${\displaystyle G\,}$ 吉布斯能

${\displaystyle \left({\frac {\partial ({\frac {\Delta G}{T}})}{\partial T}}\right)_{p\,}=-{\frac {\Delta H}{T^{2}}}}$

${\displaystyle {\frac {\Delta G,T_{2}}{T_{2}}}-{\frac {\Delta G^{\circ },T_{1}}{T_{1}}}=\Delta H^{\circ }(P)*({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}})}$

${\displaystyle \ \Delta G^{\circ }=-RT\ln K}$

## 證明

${\displaystyle dG=-SdT+VdP\,}$

${\displaystyle dG_{p\,}=-SdT\,}$

${\displaystyle \left({\frac {\partial G}{\partial T}}\right)_{p\,}=-S\,}$

${\displaystyle \left({\frac {\partial ({\frac {G}{T}})}{\partial T}}\right)_{p\,}={\frac {\left({\frac {\partial G}{\partial T}}\right)_{p\,}}{T}}-{\frac {G}{T^{2}}}={\frac {T\left({\frac {\partial G}{\partial T}}\right)_{p\,}-G}{T^{2}}}={\frac {-ST-G}{T^{2}}}={\frac {-H}{T^{2}}}}$

${\displaystyle \left({\frac {\partial ({\frac {G}{T}})}{\partial \left({\frac {1}{T}}\right)}}\right)_{p\,}=H}$