# 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 + n11 + μ2 dn2 + n22……….. μj dnj + njj
= (μ1 dn1 + μ2 dn2 + ……… μj dnj) + (n11 + n22 +……….. njj)

At constant temperature and pressure, the above equation can be written as:
n11 + n22 +……….. njj = 0
∑ nii = 0
…………………….. (1)
Because at constant temperature and pressure, (μ1 dn1 + μ2 dn2 + ……… μj dnj) = 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.

Category: Second Law of Thermodynamics

### 3 Responses to " Derive the Gibbs-Duhem Equation and give its Applications? "

1. Noon says:

Thank you. 🙂

2. Rajesh A. Chauhan says:

Gibbs Duhem equation solution is very good. My concept is cleared from this theory.

3. anil says:

very good