Question

Consider the phase diagram of a one-component system given below. \( V_{\alpha} \), \( V_{\beta} \), and \( V_{{Liquid}} \) are the molar volumes of \( \alpha \), \( \beta \), and liquid phases, respectively. Which one of the following statements is TRUE? Given: The change in molar enthalpies, \( \Delta H_{\alpha \to \beta} \) and \( \Delta H_{\beta \to {Liquid}} \), are positive.

• A. \( V_{\alpha}<V_{\beta} \) and \( V_{\beta}<V_{{Liquid}} \)
• B. \( V_{\alpha}>V_{\beta} \) and \( V_{\beta}<V_{{Liquid}} \)
• C. \( V_{\alpha}<V_{\beta} \) and \( V_{\beta}>V_{{Liquid}} \)
• D. \( V_{\alpha}>V_{\beta} \) and \( V_{\beta}>V_{{Liquid}} \)

Hint:

For phase transitions where the enthalpy change is positive, the volume typically increases from the denser phase (e.g., solid) to the less dense phase (e.g., liquid). This leads to a sequence of molar volumes: \( V_{\alpha}<V_{\beta}<V_{{Liquid}} \).

Answer 1

The Correct Option is B

In the phase diagram, the key information provided is that the enthalpy changes for the transitions \( \alpha \to \beta \) and \( \beta \to {Liquid} \) are positive. This suggests that both transitions require heat absorption, which typically corresponds to a change from a denser phase to a less dense phase. Let's analyze the phases and their associated molar volumes: 
- The transition from \( \alpha \) to \( \beta \) involves an increase in volume, so \( V_{\beta}>V_{\alpha} \). 
- The transition from \( \beta \) to liquid also involves an increase in volume, so \( V_{{Liquid}}>V_{\beta} \). 
Thus, the correct relationship between the molar volumes is: \[ V_{\alpha}<V_{\beta}<V_{{Liquid}} \] This matches Option B. 
Step 1: Analysis of each option - Option (A): \( V_{\alpha}<V_{\beta} \) and \( V_{\beta}<V_{{Liquid}} \) - Incorrect: While \( V_{\alpha}<V_{\beta} \) is correct, the condition \( V_{\beta}<V_{{Liquid}} \) does not align with the enthalpy changes. The liquid phase has a larger volume than the \( \beta \)-phase, so this option is incorrect. 
- Option (B): \( V_{\alpha}>V_{\beta} \) and \( V_{\beta}<V_{{Liquid}} \) - Correct: This matches the expected sequence of molar volumes based on the phase transitions and the positive enthalpy changes. The volume of the liquid phase is greater than that of the \( \beta \)-phase, which is larger than that of the \( \alpha \)-phase. 
- Option (C): \( V_{\alpha}<V_{\beta} \) and \( V_{\beta}>V_{{Liquid}} \) - Incorrect: This contradicts the typical behavior of phase transitions. The liquid phase should have a larger volume than the \( \beta \)-phase, making this option incorrect. 
- Option (D): \( V_{\alpha}>V_{\beta} \) and \( V_{\beta}>V_{{Liquid}} \) - Incorrect: This option suggests that the \( \beta \)-phase has a larger volume than the liquid phase, which is not true. Hence, this option is also incorrect. 
Step 2: Conclusion The correct option is Option B, where \( V_{\alpha}>V_{\beta} \) and \( V_{\beta}<V_{{Liquid}} \). This corresponds to the expected volume changes based on the enthalpy and phase transitions.