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Question #81632

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly insulated rigid vessel of volume $V_0$ contains an ideal gas with adiabatic exponent $\gamma$ at an initial pressure $P_0$ and density $\rho_0$ . A small orifice of cross-sectional area $A$ is opened, allowing the gas to escape into a surrounding vacuum. The thermal conductivity of the vessel walls is $k = 0.05 \text{ W/(m}\cdot\text{K)}$ and the acceleration due to gravity is $g = 9.8 \text{ m/s}^2$ . Let $P$ , $\rho$ , $m$ , and $U$ be the instantaneous pressure, density, mass, and total internal energy of the gas remaining inside the vessel at time $t$ . Which of the following mathematical relations regarding the instantaneous state of the system is INCORRECT?

Options

A.
$U \propto m^{\gamma}$
B.
$\frac{dU}{dt} = \frac{V_0}{\gamma - 1} \frac{dP}{dt}$
C.
$\frac{dm}{dt} = -A \sqrt{2P\rho}$
D.
$\frac{1}{P}\frac{dP}{dt} = \frac{\gamma}{\rho}\frac{d\rho}{dt}$

Check Your Answer

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

Internal Energy Dependency|Torricelli’s Efflux Velocity|Integration

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to distinguish between incompressible fluid dynamics and compressible thermodynamics.

A common trap is applying Torricelli's law ( $v = \sqrt{2\Delta P/\rho}$ ) to gases, which erroneously assumes the density remains constant as the gas accelerates through the orifice.

In reality, the gas undergoes adiabatic expansion, and flow into a vacuum becomes choked at the local speed of sound.

Furthermore, the question requires students to recognize that the gas remaining in the vessel undergoes a reversible adiabatic expansion, allowing them to derive the correct time-dependent relations for internal energy, pressure, and mass without being distracted by extraneous thermal conductivity or gravitational data.

The incorrect option is dimensionally consistent, making it a lethal trap for students who rely solely on unit checking.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Internal Energy Dependency
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free