The Gibbs-Duhem equation can be expressed as which of the following?

The Gibbs-Duhem equation is a fundamental relationship in thermodynamics that describes how the chemical potential of components in a thermodynamic system varies with changes in their composition at constant temperature and pressure. This equation can be represented by the expression Σn_Jdμ_J = 0, where n_J represents the amount of each component in the system and dμ_J represents the differential change in the chemical potential of that component.

This formulation indicates that for a closed system with multiple components, the changes in the chemical potentials, weighted by the amounts of the components, sum to zero. This fundamental relationship ensures that as the composition of a mixture changes, the chemical potentials of the individual constituents will adjust in a way that maintains the system's equilibrium.

The other expressions provided, while they are also relevant to thermodynamic relationships, do not represent the Gibbs-Duhem equation. The second option describes the differential changes in Gibbs free energy, linking pressure and temperature. The third option relates to the correlation between vapor pressure and temperature changes during phase transitions, while the fourth option relates to changes in pressure and temperature along a phase transition curve. These expressions serve different purposes in thermodynamics and do not capture the essence of the Gibbs-Duhem relationship.