**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 = μ_{1} dn_{1} + n_{1} dμ_{1} + μ_{2} dn_{2} + n_{2} dμ_{2}……….. μ_{j} dn_{j} + n_{j} dμ_{j}

= (μ_{1} dn_{1} + μ_{2} dn_{2} + ……… μ_{j} dn_{j}) + (n_{1} dμ_{1} + n_{2} dμ_{2} +……….. n_{j} dμ_{j})

At constant temperature and pressure, the above equation can be written as:

**n _{1} dμ_{1} + n_{2} dμ_{2} +……….. n_{j} dμ_{j} = 0** …………………….. (1)

∑ n_{i} dμ_{i} = 0

Because at constant temperature and pressure, (μ

_{1}dn

_{1}+ μ

_{2}dn

_{2}+ ……… 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.

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