Derive the Gibbs-Duhem Equation and give its Applications?

Gibbs-Duhem Equation
The Gibbs free energy can be defined in two different ways once by subtracting off combinations of entropy S, enthalpy H and temperature T and other as a sum of chemical potentials and amounts of species. The fact that they are equal gives a new relation known as “Gibbs-Duhem Relation.” The Gibbs-Duhem relation helps us to calculate relationships between quantities as a system which remains in equilibrium. One example is the Clausius-Clapeyron equation which states that two phases at equilibrium with each other having equaled amount of a given substance must have exactly the same free energy i.e. it relates equilibrium changes in pressure to changes in temperature as a function of material parameters.

Deriving the Gibbs-Duhem equation from thermodynamics state equations is very easy. The Gibbs free energy G in equilibrium can be expressed in terms of thermodynamics as:

dG = μ₁ dn₁ + n₁ dμ₁ + μ₂ dn₂ + n₂ dμ₂……….. μ_j dn_j + n_j dμ_j
= (μ₁ dn₁ + μ₂ dn₂ + ……… μ_j dn_j) + (n₁ dμ₁ + n₂ dμ₂ +……….. n_j dμ_j)

At constant temperature and pressure, the above equation can be written as:
n₁ dμ₁ + n₂ dμ₂ +……….. n_j dμ_j = 0
∑ n_i dμ_i = 0 …………………….. (1)
Because at constant temperature and pressure, (μ₁ dn₁ + μ₂ dn₂ + ……… μ_j dn_j) = dG
The equation (1) is known as the Gibbs-Duhem equation.

Applications of Gibbs-Duhem equation:
(i) Gibbs-duhem equation is helpful in calculating partial molar quantity of a binary mixture by measuring the composition of the mixture which depends on the total molar quantity.

(ii) Gibbs-duhem equation is helpful in calculating the partial vapor pressures by calculating the total vapor pressure. All these calculations require a curve-fitting procedure. Using tabulated experimental data the accuracy of the calculated quantities was found to be comparable to the accuracy of the original experimental data.