Gibbs–Helmholtz equation

The Gibbs–Helmholtz equation is a thermodynamic equation used to calculate changes in the Gibbs free energy of a system as a function of temperature. It was originally presented in an 1882 paper entitled "Die Thermodynamik chemischer Vorgänge" by Hermann von Helmholtz. It describes how the Gibbs free energy, which was presented originally by Josiah Willard Gibbs, varies with temperature. It was derived by Helmholtz first, and Gibbs derived it only 6 years later. The attribution to Gibbs goes back to Wilhelm Ostwald, who first translated Gibbs' monograph into German and promoted it in Europe.

The equation is:

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

where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all 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.

Similar equations include


	${\textstyle U=-T^{2}\left({\frac {\partial }{\partial T}}{\frac {F}{T}}\right)_{V}}$		${\textstyle F=-S^{2}\left({\frac {\partial }{\partial S}}{\frac {U}{S}}\right)_{V}}$
$U=-V^{2}\left({\frac {\partial }{\partial V}}{\frac {H}{V}}\right)_{S}$	$U$	$\leftrightarrow U-F=TS$	$F$	${\textstyle F=-P^{2}\left({\frac {\partial }{\partial P}}{\frac {G}{P}}\right)_{T}}$
	$\updownarrow U-H=-PV$		$\updownarrow G-F=PV$
${\textstyle H=-V^{2}\left({\frac {\partial }{\partial V}}{\frac {U}{V}}\right)_{S}}$	$H$	$\leftrightarrow G-H=-TS$	$G$	${\textstyle G=-V^{2}\left({\frac {\partial }{\partial V}}{\frac {F}{V}}\right)_{T}}$
	${\textstyle H=-T^{2}\left({\frac {\partial }{\partial T}}{\frac {G}{T}}\right)_{P}}$		${\textstyle G=-S^{2}\left({\frac {\partial }{\partial S}}{\frac {H}{S}}\right)_{V}}$

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