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# Gibbs-Helmholtz equation

The Gibbs-Helmholtz equation is a thermodynamic equation useful for calculating changes in the Gibbs energy of a system as a function of temperature. It is named after Josiah Willard Gibbs and Hermann von Helmholtz:

$\left( \frac{\partial ( \frac{G} {T} ) } {\partial T} \right)_{p\,} = - \frac {H} {T^2}$

With:

$H\,$ the enthalpy
$T\,$ the absolute temperature
$G\,$ the Gibbs free energy

at constant pressure $P\,$. The equation states that the change in the G/T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor (H/T2).

For a chemical reaction the equation reads:

$\left( \frac{\partial ( \frac{\Delta G} {T} ) } {\partial T} \right)_{p\,} = - \frac {\Delta H} {T^2}$

with $\Delta G\,$ as the change in Gibbs energy and $\Delta H\,$ as the enthalpy change (which is considered independent of temperature).

which can rearrange to:

$\frac{\Delta G,T_2}{T_2} - \frac{\Delta G^\circ,T_1}{T_1} = \Delta H^\circ(P)*(\frac{1}{T_2} - \frac{1}{T_1})$

This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T2 with knowledge of just the Standard Gibbs free energy change of formation and the Standard enthalpy change of formation for the individual components at 25°C and 1 bar.

Through:

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

which relates Gibbs energy to an equilibrium constant, the van 't Hoff equation is derived.

## Proof

The Gibbs free energy for a closed system

$dG = - SdT + VdP \,$

at constant pressure $P\,$ (dP = 0) reduces to

$dG_{p\,} = - SdT \,$

or

$\left(\frac{\partial G}{\partial T}\right)_{p\,} = - S \,$

The dependence of the G/T ratio on T is found with the aid of the quotient rule:

$\left( \frac{\partial ( \frac{G} {T} ) } {\partial T} \right)_{p\,}= \frac{\left( \frac{\partial G}{\partial T} \right)_{p\,}}{T} - \frac{G}{T^2} = \frac{T\left ( \frac{\partial G}{\partial T} \right)_{p\,}- G}{T^2} = \frac{-ST - G}{T^2} = \frac{-H}{T^2}$

From times it can be found like this:

$\left( \frac{\partial ( \frac{G} {T} ) } {\partial\left(\frac{1}{T}\right)}\right)_{p\,}= H$