On the previous page, we derived an expression which gives us a relationship between the Gibbs energy and temperature:
This relation tells us that, since everything has a positive entropy, G always decreases when the temperature is raised at constant pressure and composition. Gases have much greater entropies than liquids, which in turn have somewhat larger entropies than solids. Thus the Gibbs energies of gases are much more temperature sensitive than those of liquids, which in turn are more temperature than those of solids.
The above expression relating the Gibbs energy and temperature may be used to derive a thermodynamic result of some use. The above equation expresses the variation of G in terms of the entropy. However, we may use the definition of the Gibbs energy (G = H – TS) to replace -S by (G – H)/T. Substitution of this result into the expression gives:
We can manipulate this equation to give us an expression for the variation of G/T with T. This will prove to be a simpler expression. It is also of use as various properties (eg the equilibrium constant of a reaction) are related to G/T rather than G alone.
From the quotient rule for the derivative of one function divided by another, we may write down:
However, from inspection of the expression for the partial derivative of the Gibbs energy with respect to the temperature, we can see that
which, upon substitution into the preceding expression, gives:
which is the Gibbs-Helmholtz equation.
The equation is most useful when applied to the changes in G and H which accompany any given process. Since the Gibbs-Helmholtz equation is valid for both start and finish states, and we may write ΔG = Gf – Gi , it follows that:
By a simple change of variable in the derivative (from δT to δ(1/T), we can also derive from the Gibbs-Helmholtz equation that:
This implies that, assuming ΔH is independent of temperature over the range under consideration, a graph of ΔG/T against 1/T should give a straight line of gradient ΔH.