The coexistence equation in a binary system is derived from which fundamental relation?

The coexistence equation in a binary system is derived from which fundamental relation?

• A. Van der Waals equation
• B. Clapeyron equation
• C. Ideal gas law
• D. Gibbs-Duhem equation

Answer: (D)

In a binary mixture, phase coexistence requires the chemical potentials of each component to be the same in both phases. That gives two equilibrium conditions: μ1 is equal in the two phases and μ2 is equal in the two phases. The Gibbs–Duhem relation, which for a binary mixture says x1 dμ1 + x2 dμ2 = 0, links the infinitesimal changes of the chemical potentials together, reflecting that only one independent change among μ1 and μ2 exists once the composition is fixed. When you follow the coexistence line and differentiate the equilibrium conditions with respect to temperature or pressure, you can use Gibbs–Duhem to eliminate one degree of freedom and obtain the relationship that defines how T and P must change to maintain phase equilibrium for both components. That’s why the coexistence equation in a binary system fundamentally comes from the Gibbs–Duhem equation. The other options describe bulk equations of state or single-component phase boundaries and don’t capture the interdependence of μ1 and μ2 in a mixture.

In a binary mixture, phase coexistence requires the chemical potentials of each component to be the same in both phases. That gives two equilibrium conditions: μ1 is equal in the two phases and μ2 is equal in the two phases. The Gibbs–Duhem relation, which for a binary mixture says x1 dμ1 + x2 dμ2 = 0, links the infinitesimal changes of the chemical potentials together, reflecting that only one independent change among μ1 and μ2 exists once the composition is fixed. When you follow the coexistence line and differentiate the equilibrium conditions with respect to temperature or pressure, you can use Gibbs–Duhem to eliminate one degree of freedom and obtain the relationship that defines how T and P must change to maintain phase equilibrium for both components. That’s why the coexistence equation in a binary system fundamentally comes from the Gibbs–Duhem equation. The other options describe bulk equations of state or single-component phase boundaries and don’t capture the interdependence of μ1 and μ2 in a mixture.