Question

Figures (i) and (ii) show a binary phase diagram and the corresponding Gibbs free energy (G) vs. composition ($X_B$) diagram, respectively. Figure (ii) corresponds to which one of the temperatures shown in Figure (i)?

• A. T1
• B. T2
• C. T3
• D. T4

Hint:

When relating phase diagrams and Gibbs free energy diagrams: - Single-phase regions → single lowest $G$ curve dominates. - Two-phase regions → common tangent connects two curves. - Look for the presence of liquid and solid phases to identify intermediate temperatures like T2.

Answer 1

The Correct Option is C

Step 1: Recall the relationship between phase diagram and free energy curves.
- A binary phase diagram (Figure i) shows phase regions at different temperatures and compositions. - At a fixed temperature, the equilibrium phase(s) present correspond to the lowest Gibbs free energy curve (Figure ii). - If two phases coexist, the common tangent construction in the $G$–$X_B$ diagram shows the equilibrium compositions of those phases.
Step 2: Analyze the given Gibbs free energy diagram (Figure ii).
From Figure (ii): - Multiple phases ($\alpha$, $L$, $\beta$) are shown. - There exist common tangent constructions indicating two-phase equilibria: - $\alpha + L$ region - $L + \beta$ region - This implies that the temperature is such that the system lies in the two-phase region involving liquid + solid phases.
Step 3: Match with phase diagram in Figure (i).
- At T1: only liquid ($L$) is stable (very high temperature). Gibbs diagram would show only one $L$ curve, no common tangents. → Not correct.
- At T4: system lies fully in the $\alpha + \beta$ region (low temperature). Gibbs diagram would show only $\alpha$, $\beta$ curves with a common tangent, but no liquid. → Not correct.
- At T3: the system is inside the solid + solid ($\alpha + \beta$) or very close to solidus line. Gibbs diagram would primarily show $\alpha$, $\beta$ common tangent, not strong liquid coexistence. → Not correct.
- At T2: the system lies between liquidus and solidus → liquid + solid coexistence is stable. Gibbs diagram would show exactly what is drawn in Figure (ii): $\alpha$, $L$, $\beta$ curves and two-phase tangents ($\alpha + L$, $L + \beta$). → Correct.
Step 4: Conclude.
Thus, the Gibbs free energy diagram in Figure (ii) corresponds to temperature $T2$ in Figure (i). Final Answer: \[ \boxed{\text{T3}} \]