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# Van 't Hoff equation

The van 't Hoff equation in chemical thermodynamics relates the change in temperature (T) to the change in the equilibrium constant (K) given the enthalpy change (ΔH). The equation was first derived by Jacobus Henricus van 't Hoff.

$\frac{d \mbox{ ln K}}{dT} = \frac{\Delta H^\ominus}{RT^2}$

If the enthalpy change of reaction is assumed to be constant with temperature, the definite integral of this differential equation between temperatures T1 and T2 is given by

$\ln \left( {\frac{{K_2 }}{{K_1 }}} \right) = \frac{{ - \Delta H^\ominus }}{R}\left( {\frac{1}{{T_2 }} - \frac{1}{{T_1 }}} \right)$

In this equation K1 is the equilibrium constant at absolute temperature T1 and K2 is the equilibrium constant at absolute temperature T2. ΔHo is the enthalpy change and R is the gas constant.

Since

$\Delta G^\ominus = \Delta H^\ominus - T\Delta S^\ominus$

and

$\Delta G^\ominus = -RT \ln K$

it follows that

$\ln K = - \frac{{\Delta H^\ominus}}{RT}+ \frac{{\Delta S^\ominus }}{R}$

Therefore, a plot of the natural logarithm of the equilibrium constant versus the reciprocal temperature gives a straight line. The slope of the line is equal to minus the standard enthalpy change divided by the gas constant, ΔHo/R and the intercept is equal to the standard entropy change divided by the gas constant, ΔSo/R. Differentiation of this expression yields the van 't Hoff equation.