The Gibbs-Duhem Equation
We have stated on the previous page that the total Gibbs energy of a binary mixture is given by:
Since the chemical potentials depend upon the composition of the solution, we might expect that an infinitesimal change in the composition would bring about a change in G that obeyed:
(The first two terms on the right express the effects of the changes in the amounts of each substance. The second two terms express the effects of the changes in chemical potential of the two substances.)
However, we have previously used the result that at constant temperature and pressure:
which implies that at constant temperature and pressure:
This is a special case of the Gibbs-Duhem Equation:
This equation simply means that the changes in the chemical potential of each substance, when multiplied by the amount of that substance that is present and summed together, equal zero. i.e. the chemical potential of one component of a mixture cannot change independently of the chemical potentials of the other components.
In a binary mixture, an increase in one partial molar quantity of substance A must be balanced by a decrease in that partial molar quantity for substance B:
The same reasoning can be applied to other partial molar quantities, and in practice, the Gibbs-Duhem Equation is most often used to determine the partial molar volume of one component of a binary mixture from measurements of the partial molar volume of the other component.