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.
Contents

Equation 1

Chemical reactions 2

Derivation 3

Sources 4

External links 5
Equation
The equation is:^{[1]}

\left( \frac{\partial (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/T^{2}..
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 T_{2} 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, aka "master equation" (always true for reversible or irreversible processes):

dU= TdS  pdV \,\!
where S is the entropy, V is volume, (minus sign due to reversibility, in which dU = 0: work other than pressurevolume may be done and is equal to −pV) leads to the "reversed" form of the initial fundamental relation into a new master equation:

dG=  SdT + Vdp \,\!
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]}

...
Sources

^ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0198551487

^ Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0356037363

^ Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0198551487

^ Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0356037363

^ physical chemistry, p.W. Atkins, Oxford University press, 1978, ISBN 0198551487
External links

Link  Gibbs–Helmholtz equation

Link  Gibbs–Helmholtz equation
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