Gibbs–Helmholtz equation

The Gibbs–Helmholtz equation is a thermodynamic equation used for calculating 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 Vorgange" by Hermann von Helmholtz. It describes how the Gibbs free energy, which was presented originally by Josiah Willard Gibbs, varies with temperature.

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.

Chemical reactions

Main article: Thermochemistry

The typical applications are to chemical reactions. The equation reads:

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

with ΔG as the change in Gibbs energy and ΔH as the enthalpy change (considered independent of temperature). The o denotes standard pressure (1 bar).

Integrating with respect to T (again p is constant) it becomes:

${\frac {\Delta G^{\ominus }(T_{2})}{T_{2}}}-{\frac {\Delta G^{\ominus }(T_{1})}{T_{1}}}=\Delta H^{\ominus }(p)\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)$

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.

Also, using the reaction isotherm equation, that is

${\frac {\Delta G^{\ominus }}{T}}=-R\ln K$

which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.

Derivation

Background

Main articles: Defining equation (physical chemistry), Enthalpy, and Thermodynamic potential

The definition of the Gibbs function is $$H=G+ST$$ where H is the enthalpy defined by: $$H=U+pV$$

Taking differentials of each definition to find dH and dG, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes): $$dU=T\,dS-p\,dV$$ where S is the entropy, V is volume, (minus sign due to reversibility, in which dU = 0: work other than pressure-volume may be done and is equal to −pV) leads to the "reversed" form of the initial fundamental relation into a new master equation: $$dG=-S\,dT+V\,dp$$

This is the Gibbs free energy for a closed system. The Gibbs–Helmholtz equation can be derived by this second master equation, and the chain rule for partial derivatives.

Derivation

Starting from the equation $$dG=-S\,dT+V\,dp$$ for the differential of G, and remembering $$H=G+T\,S,$$ one computes the differential of the ratio G/T by applying the product rule of differentiation in the version for differentials: $${\begin{aligned}d\left({\frac {G}{T}}\right)&={\frac {T\,dG-G\,dT}{T^{2}}}={\frac {T\,(-S\,dT+V\,dp)-G\,dT}{T^{2}}}\\&={\frac {-T\,S\,dT-G\,dT+T\,V\,dp}{T^{2}}}={\frac {-(G+T\,S)\,dT+T\,V\,dp}{T^{2}}}\\&={\frac {-H\,dT+T\,V\,dp}{T^{2}}}\end{aligned}}\,\!$$

Therefore, $$d\left({\frac {G}{T}}\right)=-{\frac {H}{T^{2}}}\,dT+{\frac {V}{T}}\,dp\,\!$$

A comparison with the general expression for a total differential $$d\left({\frac {G}{T}}\right)=\left({\frac {\partial (G/T)}{\partial T}}\right)_{p}\,dT+\left({\frac {\partial (G/T)}{\partial p}}\right)_{T}\,dp$$ gives the change of G/T with respect to T at constant pressure (i.e. when dp = 0), the Gibbs-Helmholtz equation: $$\left({\frac {\partial (G/T)}{\partial T}}\right)_{p}=-{\frac {H}{T^{2}}}$$

External links

• Link - Gibbs–Helmholtz equation, by W. R. Salzman (2004).

[1] Gibbs-Helmholtz Equation, by P. Mander (accessed 17 March 2022)