Van 't Hoff equation

Equation relating the change in equilibrium constant with temperature

The Van 't Hoff equation is an important equation in chemical thermodynamics that relates the change in the equilibrium constant of a chemical reaction to the change in temperature. It is named after the Dutch physical chemist Jacobus Henricus van 't Hoff, who was awarded the first Nobel Prize in Chemistry in 1901.

Equation

The Van 't Hoff equation is expressed as:

$\frac{d\ln K}{dT}=\frac{\mathrm{\Delta }{H}^{\circ }}{R{T}^2}$

where:

• $K$ is the equilibrium constant of the reaction,
• $T$ is the temperature in Kelvins,
• $\mathrm{\Delta }{H}^{\circ }$ is the standard enthalpy change of the reaction,
• $R$ is the universal gas constant.

This differential form of the Van 't Hoff equation can be integrated to give:

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

where $K_1$ and $K_2$ are the equilibrium constants at temperatures $T_1$ and $T_2$, respectively.

Applications

The Van 't Hoff equation is used to estimate the effect of temperature on the position of equilibrium in chemical reactions. It is particularly useful in predicting whether a reaction will be more favorable at higher or lower temperatures.

Endothermic Reactions

For an endothermic reaction, where $\mathrm{\Delta }{H}^{\circ }>0$, the equilibrium constant $K$ increases with an increase in temperature. This is illustrated in the following plot:

File:Endothermic Reaction van't Hoff Plot.png

Endothermic reaction Van 't Hoff plot

Exothermic Reactions

For an exothermic reaction, where $\mathrm{\Delta }{H}^{\circ }<0$, the equilibrium constant $K$ decreases with an increase in temperature. This behavior is shown in the plot below:

File:Exothermic Reaction van't Hoff Plot.png

Exothermic reaction Van 't Hoff plot

Van 't Hoff Plot

A Van 't Hoff plot is a graph of $\ln K$ versus $1/T$. The slope of the line is equal to $-\mathrm{\Delta }{H}^{\circ }/R$, and the intercept can be used to determine the standard entropy change $\mathrm{\Delta }{S}^{\circ }$.

File:Van't Hoff analysis.png

Van 't Hoff analysis

Mechanism Studies

Van 't Hoff plots are also used in reaction mechanism studies to determine the enthalpy and entropy changes associated with different steps in a reaction mechanism.

File:Van't Hoff plot in Mechanism study.png

Van 't Hoff plot in mechanism study

Temperature Dependence

The Van 't Hoff equation provides insight into the temperature dependence of reaction equilibria, which is crucial for understanding and optimizing chemical processes.

File:Temperature dependence van't Hoff plot.png

Temperature dependence Van 't Hoff plot

Related pages

• Le Chatelier's principle
• Gibbs free energy
• Chemical equilibrium

References

• Atkins, P., & de Paula, J. (2006). Physical Chemistry. Oxford University Press.
• Laidler, K. J. (1987). Chemical Kinetics. Harper & Row.