Before You Begin

Question #42145

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal monatomic gas is enclosed in a cylinder with a movable piston. Starting at $t=0$ , the gas expands and does work on the piston at a constant rate $W_0$ . Simultaneously, an internal electric heater supplies heat to the gas at a constant rate $P_0$ (where $P_0 > W_0$ ). The cylinder walls are poorly insulated, resulting in a time-varying heat loss to the surroundings at a rate $\frac{dQ_{\text{loss}}}{dt} = \alpha t$ , where $\alpha$ is a positive constant. If the initial temperature of the gas is $T_0$ , what is the maximum temperature reached by the gas during this process? (Here $R$ is the universal gas constant)

Options

A.
$T_0 + \frac{(P_0 - W_0)^2}{3 R \alpha}$
B.
$T_0 + \frac{(P_0 - W_0)^2}{5 R \alpha}$
C.
$T_0 + \frac{P_0^2 - W_0^2}{3 R \alpha}$
D.
$T_0 + \frac{2(P_0 - W_0)^2}{3 R \alpha}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Calorimetry with Heat Loss|Adiabatic/Isothermal Processes via KTG|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the differential application of the First Law of Thermodynamics in non-equilibrium, time-dependent processes.

A common trap for JEE Advanced students is attempting to model the expansion using standard polytropic path equations ( $PV^n = \text{constant}$ ) or assuming an average heat loss rate.

By introducing a time-varying heat sink ( $\alpha t$ ) alongside constant power and work rates, the student is forced to abandon equilibrium path assumptions, construct the rate equation $\frac{dU}{dt} = \sum P_{\text{thermal}} - P_{\text{mechanical}}$ , and integrate over time to find the extremum of the internal energy state.

Option D traps students who fail to integrate the linear time dependence correctly, while Option B traps those who carelessly use the diatomic heat capacity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Calorimetry with Heat Loss
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free