Van't Hoff Equation

The Van't Hoff equation gives the precise, quantitative relationship between temperature and the equilibrium constant K. It follows directly from two relations already familiar from thermodynamics: the standard free energy change of a reaction is related to its equilibrium constant by

and $\Delta G^0$ is also related to the standard enthalpy and entropy changes by

Substituting (1) into (2) gives $-RT\ln K = \Delta H^0 - T\Delta S^0$, and dividing through by $-RT$ and rearranging gives

Differentiating equation (3) with respect to temperature (treating $\Delta H^0$ and $\Delta S^0$ as constant over a small temperature range) gives the differential form of the Van't Hoff equation:

Integrating equation (4) between two temperatures $T_1$ and $T_2$, with their respective equilibrium constants $K_1$ and $K_2$, gives the integrated form of the Van't Hoff equation:

Worked Problem. For an equilibrium reaction, $K_P = 0.0260$ at $25^\circ$C, with $\Delta H = 32.4$ kJ mol$^{-1}$. Calculate $K_P$ at $37^\circ$C. Here $T_1 = 25+273 = 298$ K, $T_2 = 37+273 = 310$ K, $\Delta H = 32400$ J mol$^{-1}$, $R = 8.314$ J K$^{-1}$mol$^{-1}$. Then

so $\dfrac{K_{P2}}{K_{P1}} = \text{antilog}(0.2198) = 1.6588$, and $K_{P2} = 0.026\times1.6588 = 0.0431$. …