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Question #42549

Difficulty: 8/10 | 🟢 Free

The Question

$\nu$ moles of an ideal monoatomic gas are confined in a cylinder with a movable piston. One end of the cylinder has area $A$ and acts as a perfect blackbody (emissivity $\epsilon = 1$ ), radiating heat into deep space at $0\text{ K}$ . The rest of the cylinder is perfectly insulated. The piston is pushed inward such that the temperature of the gas increases at a constant rate $\frac{dT}{dt} = \alpha$ . At the instant the gas temperature is $T$ , the process can be modeled as an instantaneous polytropic process $PV^n = \text{constant}$ . Which of the following expressions gives the instantaneous polytropic index $n$ ? (where $\sigma$ is the Stefan-Boltzmann constant and $R$ is the universal gas constant)

Options

A.
$\frac{2\sigma A T^4 + 5\nu R\alpha}{2\sigma A T^4 + 3\nu R\alpha}$
B.
$\frac{2\sigma A T^4 - 5\nu R\alpha}{2\sigma A T^4 - 3\nu R\alpha}$
C.
$\frac{\sigma A T^4 + 5\nu R\alpha}{\sigma A T^4 + 3\nu R\alpha}$
D.
$\frac{2\sigma A T^4 + 3\nu R\alpha}{2\sigma A T^4 + 5\nu R\alpha}$

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

Polytropic Process|Stefan-Boltzmann Power Law|Internal Energy Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle equilibrium dynamics where the polytropic index 'n' is not a fixed constant but a dynamic variable driven by radiative heat loss.

The primary trap lies in the sign convention of the radiated heat and the assumption that 'n' must be derived from purely mechanical constraints.

Students often incorrectly assume dQ/dt is positive (leading to Option B) or attempt to use adiabatic relations (n=5/3) because the cylinder is 'insulated' everywhere except one face.

By forcing the student to equate the radiative power loss to the instantaneous definition of molar heat capacity (C = dQ/dT), the problem exposes the misconception that polytropic processes are strictly closed-form paths rather than instantaneous state differentials.

Quick Facts About This Question

Question Information

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