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Question #42540

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal, thermally insulated cylinder of cross-sectional area $A$ contains an ideal gas with adiabatic exponent $\gamma = 1.5$ . A massless piston encloses the gas and is attached to the closed end of the cylinder by a linear elastic wire. Initially, the gas has volume $V_0 = 1 \text{ m}^3$ and pressure $P_0 = 8 \times 10^5 \text{ Pa}$ , and the wire is unstretched. The external pressure $P_{\text{ext}}$ on the piston is slowly reduced from $P_0$ to zero, causing the gas to expand reversibly and adiabatically. The wire breaks exactly when $P_{\text{ext}}$ reaches zero. Given that the breaking force of the wire is $F_b = 10^5 A$ , what is the total work done by the gas strictly against the external pressure $P_{\text{ext}}$ during this process?

Options

A.
$650 \text{ kJ}$
B.
$800 \text{ kJ}$
C.
$150 \text{ kJ}$
D.
$1200 \text{ kJ}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Molar Heat Capacity (Cv vs Cp)|Breaking Stress Concept|Calculus/Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the First Law of Thermodynamics coupled with structural mechanics (elastic potential energy).

The primary trap is the 'energy partitioning' concept during expansion.

Students often calculate the total thermodynamic work done by the gas ( $W = \int P dV$ ) using the standard adiabatic work formula and incorrectly assume this entire amount is the work done against the external environment.

However, the gas must also do work to deform the containment structure (the elastic wire).

The total work done by the gas is the sum of the work done against the external pressure and the elastic potential energy stored in the wire up to its breaking stress.

Failing to subtract the strain energy leads to the classic distractor of 800 kJ.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Molar Heat Capacity (Cv vs Cp)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free