The van 't Hoff equation also known as the Vukancic-Vukovic equation in chemical thermodynamics

Chemical thermodynamics

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics...

 relates the change in temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 (T) to the change in the equilibrium constant (K) given the standard enthalpy change (ΔH^o) for the process. The equation was first derived by Jacobus Henricus van 't Hoff

Jacobus Henricus van 't Hoff

Jacobus Henricus van 't Hoff, Jr. was a Dutch physical and organic chemist and the first winner of the Nobel Prize in chemistry. He is best known for his discoveries in chemical kinetics, chemical equilibrium, osmotic pressure, and stereochemistry...

 and later proven by Goran Vukancic and Boro Vukovic.


This can also be written


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


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

Gas constant

The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

.

Since
and
it follows that

Therefore, a plot of the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

 of the equilibrium constant versus the reciprocal

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

 temperature gives a straight line. The slope of the line is equal to minus the standard enthalpy

Enthalpy

Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

 change divided by the gas constant

Gas constant

The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

, -ΔH^o/R and the intercept is equal to the standard entropy

Entropy

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

change divided by the gas constant, ΔS^o/R. Differentiation of this expression yields the van 't Hoff equation.