# Derive the Gibbs-Duhem equation and give its application.

The Gibbs Duhem equation was introduced by Josiah Willard Gibbs and Pierre Duhem. This equation describes the relationship between the changes in chemical potential of components in a thermodynamic system at constant temperature and pressure.

Let us consider a system that comprises n types of constituents with n_{1}, n_{2}, n_{3} ... ni moles.
Being an extensive property, the partial molar free energy depends not only on the temperature and pressure but also on the number of moles of different components. Mathematically, we can write as -

G = ƒ(T,P, n_{1}, n_{2}, n_{3}, ... ni) --- Eq:1

Where n_{1}, n_{2}, n_{3}, ... ni are the number of moles of various constituents.

Differentiating Eq:1, we get-

We know that-

dG = − SdT + VdP --- Eq:4

Comparing Eq:3 and Eq:4 we get-

Putting these values in Eq:2, we get-

dG = − SdT + VdP + μ_{1}dn_{1} + ... + μ_{i}dn_{i} --- Eq:5

At constant temperature and pressure Eq:5 reduces to-

(dG)_{T,P} = μ_{1}dn_{1} + ... + μ_{i}dn_{i} --- Eq:6

Integrating Eq:6 we get the following for a system of definite composition:

(dG)_{T,P} = μ_{1}n_{1} + ... + μ_{i}n_{i} --- Eq:7

Differentiating Eq:7, we get-

(dG)_{T,P} = μ_{1}dn_{1} + n_{1}dμ_{1} + ... μ_{i}dn_{i} + n_{i}dμ_{i} --- Eq:8

Comparing Eq:6 and Eq:8, we get-

n_{1}dμ_{1} + ... + n_{i}dμ_{i} = 0 --- Eq:9

Σn_{i}dμ_{i} = 0 --- Eq:10

Eq:10 is called the Gibbs Duhem equation. It is applicable to a system at constant temperature and pressure.

The physical significance of the Gibbs-Duhem equation can be understood by taking the example of binary solutions i.e., a system of two components only. The Gibbs-Duhem equation for such systems is-

n_{1}dμ_{1} + n_{2}dμ_{2} = 0

n_{1}dμ_{1} = − n_{2}dμ_{2} = 0

dμ_{1} = − (n_{2}/n_{1})dμ_{2} = 0

Hence, the chemical potential of one constituent is not independent of another component in binary solutions. In other words, the chemical potentials or partial molar free energies of two components of the binary system are mutually dependent, and if one increases the other decreases.

## Applications of the Gibbs-Duhem equation

Gibbs-Duhem equation has several important applications. Some of which are given below-

1. Gibbs-Duhem equation helps to understand and analyze the phase equilibria in multi-component systems. It gives relationships between the chemical potentials of the components, which is essential for determining the conditions under which different phases coexist in equilibrium.

2. In non-ideal solutions, the Gibbs-Duhem equation is used to calculate activity coefficients, which measure the deviation of a real solution from ideal behaviour.

3. Gibbs-Duhem equation helps to calculate the fugacity coefficients in gases, which are important for describing real gas behaviour.

4. It provides a way to relate the chemical potentials of the components in a mixture. This is especially useful to understand how the chemical potential of one component affects the others in a multi-component system.

5. This equation is used to calculate the excess functions such as excess Gibbs energy, excess enthalpy, and excess volume. These excess functions are important for characterizing the thermodynamic properties of non-ideal mixtures.

6. The Gibbs-Duhem equation helps in the study of the mixing properties of solutions, such as the heat of mixing, volume change upon mixing, and entropy of mixing. These properties are important for designing separation processes and understanding solution behaviour.

7. In the study of electrolyte solutions, the Gibbs-Duhem equation aids in understanding the behavior of ions in solution and their interactions. It helps in deriving expressions for activities and activity coefficients of ions in electrolyte solutions.

8. For binary and multicomponent systems, the Gibbs-Duhem equation provides constraints on the composition and chemical potentials, which are necessary for the development of phase diagrams and understanding the behaviour of complex mixtures.

9. This equation is also applied in surface and interface thermodynamics to relate the changes in surface tension or interfacial tension to the changes in the chemical potentials of the components at the interface.

10. Gibbs-Duhem equation is used to calculate partial molar properties such as partial molar volume and partial molar enthalpy from the overall properties of the mixture.

11. Gibbs-Duhem equation is helpful in calculating the partial vapor pressures by calculating the total vapor pressure.