Subtopic: First and Second Order Phase Transitions
Question Statement:
A schematic Pressure-Temperature diagram of water is shown in the figure. Which
of the following option(s) is/are correct?
Options:
(A) Clausius-Clapeyron equation is valid across the melting curve and the vaporization
curve
(B) Melting curve has the highest slope
(C) The critical point exists only for the vaporization curve
(D) Clausius-Clapeyron equation is not valid across the m elting curve and the
vaporization curve
Official Answer Key:
A;B;C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.26)**
**1. Thermodynamics of Phase Transitions**:
- The pressure-temperature ($P-T$) phase diagram of water displays the coexistence curves between solid (ice), liquid (water), and gas (vapor).
**2. Detailed Analysis of Statements (MSQ)**:
- **(A) Clausius-Clapeyron equation is valid across melting and vaporization curves [TRUE]**:
- The Clausius-Clapeyron equation:
$\frac{dP}{dT} = \frac{L}{T \Delta V}$
is the exact thermodynamic relationship governing all first-order phase transitions where latent heat $L$ and volume discontinuity $\Delta V$ are non-zero. It applies equally to melting, vaporization, and sublimation curves.
- **(B) Melting curve has the highest slope [TRUE]**:
- For melting: $\Delta V_{\text{fus}} = V_{\text{liquid}} - V_{\text{solid}}$ is extraordinarily small compared to the huge volume change of vaporization $\Delta V_{\text{vap}} \approx V_{\text{vapor}}$.
- Since $\frac{dP}{dT} \propto \frac{1}{\Delta V}$, the extremely small volume change makes $\left|\frac{dP}{dT}\right|_{\text{melting}}$ vastly steeper (highest magnitude slope) than the vaporization curve.
- **(C) The critical point exists only for the vaporization curve [TRUE]**:
- Beyond the liquid-gas critical point ($T_c = 647\text{ K}, P_c = 22.06\text{ MPa}$), liquid and gas merge continuously into a single supercritical fluid phase.
- Solid-liquid transitions have broken crystalline translational symmetry, so the melting curve never terminates at a critical point and continues indefinitely at high pressures.
- **(D) False**: Contradicts statement (A).
**3. Official Verification**:
The verified official IIT answer is **Options (A), (B), and (C)** (`['A', 'B', 'C']`).
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