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Question #42171

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal, thermally insulated cylinder of cross-sectional area $A$ contains an ideal gas confined by a frictionless piston of mass $M$ . The external pressure acting on the piston from the outside varies with the piston's position $x$ as $P_{ext} = \alpha x^2$ , where $\alpha$ is a positive constant. The closed end of the cylinder is at $x = 0$ . Initially, the piston is held at rest at $x = L$ . The piston is then released and moves outward, momentarily coming to rest at $x = 2L$ . What is the change in the internal energy of the gas during this process?

Options

A.
$-\frac{5}{2} \alpha A L^3$
B.
$+\frac{7}{3} \alpha A L^3$
C.
$-\frac{7}{3} \alpha A L^3$
D.
$-4 \alpha A L^3$

Check Your Answer

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

Sign Convention Trap|Equipartition Theorem|Work-Energy Theorem Validity

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Work-Energy Theorem with the First Law of Thermodynamics in a non-quasi-static process.

The primary trap is the 'Sign Convention Trap' and the false assumption of reversibility.

Students often attempt to find the final state variables using $PV^\gamma = \text{constant}$ , which is invalid for an irreversible expansion where the piston accelerates.

Furthermore, students frequently confuse the work done by the gas with the work done against the external force, or use an average pressure formula like $W = P_{avg}\Delta V$ , leading to incorrect magnitudes or sign errors in $\Delta U$ .

By applying the Work-Energy Theorem to the piston ( $\Delta K = 0$ ), one can equate the expansion work of the gas directly to the integral of the variable external force, bypassing the need for intermediate thermodynamic states.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Sign Convention Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free