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Question #42152

Difficulty: 8/10 | 🟢 Free

The Question

A thin spherical shell of radius $r$ and thickness $t$ ( $t \ll r$ ) is completely filled with a liquid of mass $M$ and specific volume $v_0$ at its freezing temperature $T_0$ . The shell material has Young's modulus $Y$ , Poisson's ratio $\nu$ , coefficient of linear thermal expansion $\alpha$ , and tensile yield stress $\sigma_y$ . The system is slowly cooled. Upon freezing, the substance transforms into a solid with a larger specific volume $v_s$ (let $\Delta v = v_s - v_0 > 0$ ), releasing a latent heat of fusion $L$ . The expansion during freezing increases the internal pressure, which in turn lowers the freezing point via the Clausius-Clapeyron relation. Assuming the specific volumes of the liquid and solid phases are independent of pressure and temperature, what fraction $f$ of the liquid has frozen exactly when the shell begins to yield?

Options

A.
$\frac{3 v_0 \sigma_y}{\Delta v} \left[ \frac{1-\nu}{Y} - \frac{2 \alpha t T_0 \Delta v}{r L} \right]$
B.
$\frac{3 v_0 \sigma_y (1-\nu)}{Y \Delta v}$
C.
$\frac{3 v_0 \sigma_y}{\Delta v} \left[ \frac{1}{Y} - \frac{2 \alpha t T_0 \Delta v}{r L} \right]$
D.
$\frac{3 v_0 \sigma_y}{\Delta v} \left[ \frac{1-\nu}{Y} - \frac{\alpha t L}{r T_0 \Delta v} \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Latent Heat and Volume Change|Thermal Expansion of Area and Volume|Poisson's Ratio Limits

Why This Question Is Tricky

Common Mistakes

This question tests the coupled thermodynamic and mechanical equilibrium of a constrained phase transition.

The core trap lies in assuming the container is either perfectly rigid or only expands elastically, neglecting the thermal contraction of the shell as the temperature drops to maintain the phase boundary.

Furthermore, students often misapply the 3D Hooke's law for a spherical shell, forgetting the Poisson's ratio factor $(1-\nu)$ in the volume strain $\frac{\Delta V}{V} = \frac{3Pr}{2tY}(1-\nu)$ .

The problem forces the integration of the Clausius-Clapeyron equation with the yield criterion of a thin pressure vessel, demanding strict dimensional and physical consistency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Latent Heat and Volume Change
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free