# 26.7: The van 't Hoff Equation

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We can use Gibbs-Helmholtz to get the temperature dependence of $$K$$

$\left( \dfrac{∂[ΔG^o/T]}{∂T} \right)_P = \dfrac{-ΔH^o}{T^2} \nonumber$

At equilibrium, we can equate $$ΔG^o$$ to $$-RT\ln K$$ so we get:

$\left( \dfrac{∂[\ln K]}{∂T} \right)_P = \dfrac{ΔH^o}{RT^2} \nonumber$

We see that whether $$K$$ increases or decreases with temperature is linked to whether the reaction enthalpy is positive or negative. If temperature is changed little enough that $$ΔH^o$$ can be considered constant, we can translate a $$K$$ value at one temperature into another by integrating the above expression, we get a similar derivation as with melting point depression:

$\ln \dfrac{K(T_2)}{K(T_1)} = \dfrac{-ΔH^o}{R} \left( \dfrac{1}{T_2} - \dfrac{1}{T_1} \right) \nonumber$

If more precision is required we could correct for the temperature changes of $$ΔH^o$$ by using heat capacity data.

How $$K$$ increases or decreases with temperature is linked to whether the reaction enthalpy is positive or negative.

The expression for $$K$$ is a rather sensitive function of temperature given its exponential dependence on the difference of stoichiometric coefficients One way to see the sensitive temperature dependence of equilibrium constants is to recall that

$K=e^{−\Delta_r{G^o}/RT}\label{18}$

However, since under constant pressure and temperature

$\Delta{G^o}= \Delta{H^o}−T\Delta{S^o} \nonumber$

Equation $$\ref{18}$$ becomes

$K=e^{-\Delta{H^o}/RT} e^{\Delta {S^o}/R}\label{19}$

Taking the natural log of both sides, we obtain a linear relation between $$\ln K$$and the standard enthalpies and entropies:

$\ln K = - \dfrac{\Delta {H^o}}{R} \dfrac{1}{T} + \dfrac{\Delta{S^o}}{R}\label{20}$

which is known as the van ’t Hoff equation. It shows that a plot of $$\ln K$$ vs. $$1/T$$ should be a line with slope $$-\Delta_r{H^o}/R$$ and intercept $$\Delta_r{S^o}/R$$. Figure 26.7.1 : Endothermic Reaction (left) and Exothermic Reaction van 't Hoff Plots (right). Figures used with permission of Wikipedia

Hence, these quantities can be determined from the $$\ln K$$ vs. $$1/T$$ data without doing calorimetry. Of course, the main assumption here is that $$\Delta_r{H^o}$$ and $$\Delta_r{S^o}$$ are only very weakly dependent on $$T$$, which is usually valid.

This page titled 26.7: The van 't Hoff Equation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mark E. Tuckerman.