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Question #42169

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The Question

A cylindrical vessel of constant cross-sectional area $A$ contains a liquid of mass $M$ . Initially, the liquid is at a uniform temperature $T_0$ and has height $H$ . The density of the liquid varies with temperature as $\rho(T) = \rho_0[1 - \gamma(T - T_0)]$ , where $\gamma$ is the volumetric expansion coefficient (assume $\gamma \Delta T \ll 1$ ). The liquid is subjected to two different heating processes:

Process I: The liquid is heated uniformly to a final temperature $T_0 + \Delta T$ . Process II: The liquid is heated such that its temperature varies linearly with height $z$ from the bottom, with the bottom at $T_0$ and the top surface at $T_0 + 2\Delta T$ .

Let $U_I$ and $U_{II}$ be the final gravitational potential energies of the liquid in Process I and Process II, respectively. Assuming the vessel does not expand, find the difference $U_I - U_{II}$ .

Options

A.
$\frac{1}{6} M g H \gamma \Delta T$
B.
$\frac{1}{3} M g H \gamma \Delta T$
C.
$\frac{1}{2} M g H \gamma \Delta T$
D.
0

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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

State Function vs Path Function|Gravitational Potential Energy (System)|Variation of Density with Temperature

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that gravitational potential energy is a pure state function of macroscopic variables (like total mass and average temperature) or depends only on the final geometric boundaries of the system.

Students often calculate the final height of the fluid, find it to be identical in both processes ( $H_f \approx H(1 + \gamma \Delta T)$ ), and incorrectly conclude that the potential energies must be equal (Option D).

The trap exposes the deep coupling between thermodynamics and mechanics: the spatial distribution of density (the thermal path) shifts the Center of Mass independently of the geometric boundaries.

In the gradient process, the lower half is cooler and denser, concentrating more mass at a lower height and resulting in a lower overall potential energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	State Function vs Path Function
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free