Alloy Phase Equilibrium Condition

For two phases, denoted as $\alpha$ and $\beta$, within an alloy to be in thermodynamic equilibrium, specific conditions must be met:

• Temperature must be uniform across both phases ($T^{\alpha} = T^{\beta}$).
• Pressure must be uniform across both phases ($P^{\alpha} = P^{\beta}$).
• The chemical potential of each component must be the same in both phases.

The chemical potential of component $i$ in a phase is represented by its partial molar Gibbs free energy, $\overline{G_i}$. Therefore, the condition for equilibrium between phases $\alpha$ and $\beta$ is:

$ \overline{G_i^{\alpha}} = \overline{G_i^{\beta}} $ for every component $i$ present in the alloy.

Analysis of Options

Let's examine why the other options are incorrect:

• Option 1: $c^{\alpha}_p = c^{\beta}_p$ - This suggests the concentration of component $p$ is the same in both phases. However, different phases in equilibrium can have distinct compositions.
• Option 2: $V_m^{\alpha} = V_m^{\beta}$ - This implies the molar volumes are equal, which is not a requirement for phase equilibrium. Molar volumes often differ between phases.
• Option 3: $G_m^{\alpha} = G_m^{\beta}$ - This equates the total molar Gibbs free energy of each phase. While the total Gibbs energy of the system is minimized at equilibrium, this specific equality is not the fundamental condition for equilibrium between *components* across phases.
• Option 4: $\overline{G_i^{\alpha}} = \overline{G_i^{\beta}}$ - This correctly states that the partial molar Gibbs free energy (chemical potential) of each component $i$ must be identical in both phases $\alpha$ and $\beta$. This ensures no net transfer of material occurs between the phases, leading to stable equilibrium.

Conclusion: The condition for thermodynamic equilibrium between two phases in an alloy is the equality of the chemical potential for each component across the phases.