An Introduction to Thermal Physics · Problem 5.85

Free Energy and Chemical Thermodynamics — Daniel Schroeder

Physics — Statistical Mechanics · vant-hoff-equation

Problem

Derive the van't Hoff equation, dln⁡KdT=ΔH∘RT2,\frac{d\ln K}{dT} = \frac{\Delta H^\circ}{RT^2}, which gives the dependence of the equilibrium constant on temperature. Here ΔH∘\Delta H^\circ is the enthalpy change of the reaction, for pure substances in their standard states (1 bar pressure for gases). Notice that if ΔH∘\Delta H^\circ is positive (loosely speaking, if the reaction requires the absorption of heat), then higher temperature makes the reaction tend more to the right, as you might expect. Often you can neglect the temperature dependence of ΔH∘\Delta H^\circ; solve the equation in this case to obtain ln⁡K(T2)−ln⁡K(T1)=ΔH∘R(1T1−1T2).\ln K(T_2) - \ln K(T_1) = \frac{\Delta H^\circ}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right).

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