Van 't Hoff equation

Remarks

Since $\ln K = -\Delta_r G^\circ/(RT)$, the claim is $\frac{d}{dT}\left(\frac{\Delta_r G^\circ}{T}\right) = -\frac{\Delta_r H^\circ}{T^2}$, which is the Gibbs–Helmholtz equation applied to each species and summed with the stoichiometric coefficients. If $\Delta_r H^\circ$ is constant over a range of temperatures, integration gives $\ln\frac{K(T_2)}{K(T_1)} = -\frac{\Delta_r H^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$. For an exothermic reaction, $\Delta_r H^\circ < 0$, so $K$ decreases as $T$ increases; the synthesis of ammonia is an example, and this is why its yield falls at high temperature. A plot of $\ln K$ against $1/T$, the van 't Hoff plot, is then nearly a straight line of slope $-\Delta_r H^\circ/R$.

Going deeper

For a student

It is the quantitative form of Le Chatelier's principle for temperature changes. It has the same shape as the Clausius–Clapeyron relation, which is the special case of an evaporation "reaction" from liquid to vapour with $K$ proportional to the vapour pressure. In practice it is used in both directions: to predict $K$ at a new temperature, or to measure $\Delta_r H^\circ$ from equilibrium data at several temperatures.