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]

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

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

The typical applications are to chemical reactions. The equation reads:[2]

\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,[3] 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.[4]

Derivation

Background

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.[5]

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}}}\,\!

Sources

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

External links

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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