Gibbs-Duhem Equation

The Gibbs-Duhem equation highlights that in a binary solution, the chemical potentials of the two components are not independent. This relationship is crucial when analyzing solutions, as it allows us to determine the chemical potential of a non-volatile solute (which has no measurable vapor pressure) when it is dissolved in a volatile solvent.

In a system at constant temperature and pressure, the differential change in the Gibbs free energy \( G \) is given by:

\[dG = -S \, dT + V \, dp + \sum_i \mu_i \, dn_i\]

Since temperature \( T \) and pressure \( p \) are constant, the equation simplifies to:

\[dG = \sum_i \mu_i \, dn_i\]

For a solution of constant composition, we can integrate the above differential form:

\[ \int_0^G \, dG = \sum_i \mu_i \int_{0}^{n_i}  \, dn_i \implies G = \sum_i n_i \mu_i \implies dG = \sum_i n_i d\mu_i + \sum_i \mu_i dn_i\]

Since the $dG$-s derived above should be equal, it follows that:

\[ \sum_i \mu_i \, dn_i = \sum_i n_i \, d\mu_i + \sum_i \mu_i \, dn_i \implies \sum_i n_i \, d\mu_i = 0 \]

The latter expression is the Gibbs-Duhem equation.

For a binary solution where the total composition remains constant, applying the Gibbs-Duhem constraint leads to:

\[n_1 \, d\mu_1 + n_2 \, d\mu_2 = 0 \implies  d\mu_2 = -\frac{n_1}{n_2} \, d\mu_1\]

This equation indicates that the chemical potentials of the components in a binary solution are linked, meaning that they cannot vary independently. For instance, if the chemical potential of component 1 changes, the chemical potential of component 2 must adjust accordingly to maintain the balance described by the Gibbs-Duhem equation.

In terms of mole fractions, the equation can also be expressed as:

\[x_1 \, d\mu_1 + x_2 \, d\mu_2 = 0\]

This form can be useful in solution chemistry, especially when considering systems where one component is volatile, and the other is non-volatile. By knowing the chemical potential of one component, we can determine the other using the Gibbs-Duhem relation, which simplifies the analysis of multi-component systems.

Example

Consider a solution where component 1 is a volatile solvent, and component 2 is a non-volatile solute. Since the solute does not vaporize, it does not contribute to the vapor pressure of the solution directly. Using the Gibbs-Duhem equation, we can determine how changes in the concentration of the solvent affect the chemical potential of the solute. This is particularly useful in determining colligative properties, such as boiling point elevation and freezing point depression, which depend on the relative amounts of solvent and solute.