Van't Hoff Equation

Differential Form

Using $\Delta G^{\circ }=-RT\ln \left(K\right)$ and $\frac{\partial \Delta G^{\circ }}{\partial T}=-\Delta S^{\circ }$ with $\Delta H^{\circ }=\Delta G^{\circ }+T\Delta S^{\circ }$,

$$\boxed{\frac{\partial \ln \left(K\right)}{\partial T}=\frac{\Delta H^{\circ }(T)}{R{T}^2}}.$$

If $\Delta H^{\circ }$ is weakly $T$-dependent across the range:

$$\ln \left(\frac{K(T_2)}{K(T_1)}\right)\approx \Delta H^{\circ }\left(\frac{1}{R{T}_1}-\frac{1}{R{T}_2}\right).$$

Formation Reactions

For a formation reaction written per 1 mol of species $\mathrm{A}$, use $\Delta H_f^{\circ }(T)$ in place of $\Delta H^{\circ }$ and integrate similarly.

General Reactions

For ${\sum }_i{\nu }_i\mathrm{A}{}_{\mathrm{i}}=0$, use the reaction enthalpy: $\Delta H_{\text{rxn}}^{\circ }(T)={\sum }_i{\nu }_i{h}_i^{\circ }(T),$ then apply the same van’t Hoff relations.

Design note: $K(T)$ typically increases with $T$ for endothermic reactions ($\Delta H^{\circ }>0$) and decreases for exothermic ($\Delta H^{\circ }<0$).