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Question #42538

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly insulated cylinder of cross-sectional area $S$ contains $n$ moles of an ideal monoatomic gas at initial pressure $P_0$ and volume $V_0$ . A wire of unstretched length $L$ , cross-sectional area $A$ , Young's modulus $Y$ , and breaking stress $\sigma_b$ connects the piston to the rigid bottom of the cylinder. The wire is completely enclosed within the gas. The piston is slowly pulled outward by an external force until the wire breaks. Immediately after breaking, the piston is locked in place. Assuming all the elastic potential energy of the wire is dissipated as heat into the gas, which of the following expressions gives the final pressure $P_f$ of the gas in equilibrium? (Let $V_1 = V_0 + \frac{S \sigma_b L}{Y}$ be the volume of the gas just before the wire breaks. Neglect the volume and heat capacity of the wire.)

Options

A.
$P_0 \left( \frac{V_0}{V_1} \right)^{5/3} + \frac{\sigma_b^2 A L}{3 Y V_1}$
B.
$P_0 \left( \frac{V_0}{V_1} \right)^{5/3} + \frac{\sigma_b^2 A L}{2 Y V_1}$
C.
$P_0 \left( \frac{V_0}{V_1} \right)^{5/3} + \frac{2 \sigma_b^2 A L}{3 Y V_1}$
D.
$P_0 \left( \frac{V_0}{V_1} \right)^{5/3}$

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

Internal Energy Dependency|Breaking Stress Concept|Mechanical Energy Conservation

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of solid mechanics and thermodynamics.

The primary trap is the assumption that the entire process remains a standard reversible adiabatic expansion (Option D).

Students often fail to recognize that material failure introduces a discontinuous, irreversible injection of thermal energy.

When the wire snaps, the stored elastic strain energy cannot simply vanish; it dissipates as heat into the closed system.

By locking the piston, the process becomes an isochoric heat addition.

This forces students to apply the First Law of Thermodynamics across a brittle fracture event, bridging mechanical energy conservation and internal gas energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Internal Energy Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free