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Question #42504

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal monoatomic gas is confined in an adiabatic cylinder by a massless, frictionless piston subject to a constant external pressure $P_0$ . The initial temperature of the gas is $T_0$ . A heating element inside the gas is connected to a constant DC voltage source $V$ . The electrical resistance of the heating element varies with the gas temperature $T$ as $r(T) = r_0 \left(\frac{T}{T_0}\right)$ . Assuming the heat capacity of the heater is negligible, what is the time $t$ required for the volume of the gas to double? (Let $R$ be the universal gas constant).

Options

A.
$\frac{15 R r_0 T_0}{4 V^2}$
B.
$\frac{5 R r_0 T_0}{2 V^2}$
C.
$\frac{9 R r_0 T_0}{4 V^2}$
D.
$\frac{3 R r_0 T_0}{2 V^2}$

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

Entropy and Reversibility|Internal Resistance Variation|Ideal Gas Assumptions|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of the First Law of Thermodynamics with a temperature-dependent power source.

The primary 'Lethal Collision' occurs when students incorrectly assume a constant heating rate $P = V^2/r_0$ , failing to account for the feedback loop where rising gas temperature increases resistance, thereby slowing the rate of heat addition over time (leading to Option B).

A secondary trap involves neglecting the work done by the gas during expansion, incorrectly using $C_v$ instead of $C_p$ for the isobaric process (leading to Options C or D).

The correct approach requires setting up and integrating a non-linear differential equation.

Quick Facts About This Question

Question Information

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