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Question #42140

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The Question

A block of ice of mass $m$ and density $\rho_i$ is released from rest from a height $H$ above the free surface of a liquid of density $\rho_L$ ( $\rho_L < \rho_i$ ). The liquid has mass $M$ and is contained in a perfectly insulated, rigid, closed cylindrical calorimeter. The space above the liquid is a vacuum. The initial depth of the liquid is $d$ . The ice block falls, sinks to the bottom of the calorimeter, and then melts completely. The initial temperature of the liquid is $T_0 > 0^\circ\mathrm{C}$ and the ice is at $0^\circ\mathrm{C}$ . The specific heat capacity of the liquid is $s_L$ , the specific heat capacity of water is $s_w$ , and the latent heat of fusion of ice is $L$ . The density of water is $\rho_w$ . Assume the liquid and water are immiscible, and the cross-sectional area of the calorimeter is very large such that the depth of the melted water layer is negligible. Which of the following equations correctly represents the energy balance to find the final equilibrium temperature $T_f$ of the mixture?

Options

A.
$mg\left(H + d - d\frac{\rho_L}{\rho_w}\right) + M s_L (T_0 - T_f) = m L + m s_w T_f$
B.
$mg\left(H + d - d\frac{\rho_L}{\rho_i}\right) + M s_L (T_0 - T_f) = m L + m s_w T_f$
C.
$mg(H + d) + M s_L (T_0 - T_f) = m L + m s_w T_f$
D.
$mgH + M s_L (T_0 - T_f) = m L + m s_w T_f$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Specific Heat Capacity of Phase Mixtures|Mechanical Energy Conservation|Work, Energy, and Power

Why This Question Is Tricky

Common Mistakes

This question tests the 'Energy Balance' misconception in phase changes under submerged conditions.

Students typically calculate the work done by buoyancy using the initial volume of the ice (leading to Option B) or ignore buoyancy entirely (Option C).

They fail to account for the work done by the surrounding liquid pressure as the ice melts and its volume decreases.

Equivalently, from a potential energy perspective, they forget that as the ice shrinks into denser water, the entire column of liquid above it must drop to fill the void, releasing additional gravitational potential energy that contributes to the heat of the mixture.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Specific Heat Capacity of Phase Mixtures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free