Gibbs–Helmholtz equation

The Gibbs–Helmholtz equation is a thermodynamic equation used 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.

The equation is:[1]

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

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:[2]

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

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:

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

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,[3] that is

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

which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.[4]

Derivation

Background

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

The definition of the Gibbs function is

${\displaystyle H=G+ST\,\!}$

where H is the enthalpy defined by:

${\displaystyle H=U+pV\,\!}$

Taking differentials of each definition to find dH and dG, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes):

${\displaystyle dU=T\,dS-p\,dV\,\!}$

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:

${\displaystyle dG=-S\,dT+V\,dp\,\!}$

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.[5]

Derivation

Starting from the equation ${\displaystyle dG=-S\,dT+V\,dp\,\!}$ for the differential of G, and remembering ${\displaystyle H=G+T\,S\,\!}$ one computes the differential of the ratio G/T by applying the product rule of differentiation in the version for differentials: ${\displaystyle {\begin{aligned}d(G/T)&={\frac {T\,dG-G\,dT}{T^{2}}}={\frac {T\,(-S\,dT+V\,dp)-G\,dT}{T^{2}}}\\&={\frac {-T\,S\,dT-G\,dT+T\,V\,dp}{T^{2}}}={\frac {-(G+T\,S)\,dT+T\,V\,dp}{T^{2}}}\\&={\frac {-H\,dT+T\,V\,dp}{T^{2}}}\end{aligned}}\,\!}$ Therefore, ${\displaystyle d(G/T)=-{\frac {H}{T^{2}}}\,dT+{\frac {V}{T}}\,dp\,\!}$ A comparison with the general expression for a total differential ${\displaystyle d(G/T)=\left({\frac {\partial (G/T)}{\partial T}}\right)_{p}\,dT+\left({\frac {\partial (G/T)}{\partial p}}\right)_{T}\,dp\,\!}$ gives the change of G/T with respect to T at constant pressure (i.e. when dp = 0), the Gibbs-Helmholtz equation: ${\displaystyle \left({\frac {\partial (G/T)}{\partial T}}\right)_{p}=-{\frac {H}{T^{2}}}\,\!}$

Sources

1. Physical chemistry, P. W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7
2. Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
3. Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0-19-855148-7
4. Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
5. Physical chemistry, P. W. Atkins, Oxford University press, 1978, ISBN 0-19-855148-7

External links

• Link - Gibbs–Helmholtz equation
• Link - Gibbs–Helmholtz equation