The Gibbs–Helmholtz equation
[∂(G/T)/ ∂T]P = −H/T2
The Gibbs–Helmholtz equation expresses the temperature dependence of the ratio of G/T at constant pressure, which is a composite function of T as G itself also depends on the temperature. This fundamental equation is very important, since it is the starting point to the van’t Hoff equation, expressing the temperature dependence of the equilibrium constant that interprets quantitatively the shift of chemical equilibrium upon temperature change, predicted in the Le Chatelier–Brown principle.
Starting with the definition of the Gibbs free energy for one mole of a substance,
G = H − TS
dividing by T and taking the derivative of both sides with respect to T at constant pressure, we obtain
[∂(G/T)/ ∂T]P = 1/T (∂H/∂T)P − H /T2 − (∂S/∂T)P
Since CP = (∂H/∂T)P and CPT(∂S/∂T)P, we obtain the Gibbs–Helmholtz equation.
For an isobaric change of state of a closed system of fixed composition, it gives the relation of the change in G to the change in H as
[∂(ΔG/T)/ ∂T]P = −ΔH/T2
This equation is of particular use in experimental thermodynamics, as it allows ΔG to be obtained from a measurement of ΔH. As well as ΔH, the heat of a reaction, to be obtained from a measurement of the variation of ΔG, the free energy change for the reaction, with temperature. The usefulness of this equation will be developed and applied later in the text to the calculation of partial molar heats of solution as well as to the calculation of the temperature variation of the equilibrium constants of reactions in systems.