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Question #43947

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal, perfectly insulated cylinder contains an ideal monatomic gas at an initial pressure $P_0$ and volume $V_0$ . The cylinder is fitted with a massless, perfectly insulated piston of cross-sectional area $A$ . The piston expands into a vacuum ( $P_{ext} = 0$ ) but experiences a velocity-dependent frictional force $f = \alpha v$ from the cylinder walls, where $v$ is the instantaneous velocity of the piston and $\alpha$ is a positive constant. Assume that all the heat generated by this friction is continuously and entirely absorbed by the gas. What is the time taken for the volume of the gas to double from its initial value?

Options

A.
$\frac{3 \alpha V_0}{2 P_0 A^2}$
B.
$\frac{3 \alpha V_0}{8 P_0 A^2} \left( 2^{8/3} - 1 \right)$
C.
$\frac{\alpha V_0}{P_0 A^2}$
D.
$\frac{3 \alpha V_0}{10 P_0 A^2} \left( 2^{10/3} - 1 \right)$

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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

Adiabatic Processes|Newton’s Second Law|Integration

Why This Question Is Tricky

Common Mistakes

Students often conflate an 'adiabatic container' with an 'isentropic (reversible adiabatic) process'.

The trap relies on the assumption that $dQ = 0$ for the gas from the outside environment, leading students to incorrectly apply the relation $PV^\gamma = \text{constant}$ (Option B).

However, the friction makes the expansion irreversible.

The mechanical work done by the gas to push the piston is dissipated by friction into heat, which is then re-absorbed by the gas.

Consequently, the net energy leaving the gas is zero ( $dU = 0$ ), making the process isothermal despite the insulated walls.

This tests the generalized application of the First Law of Thermodynamics over rote memorization of adiabatic formulas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Adiabatic/Isothermal Processes via KTG
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free