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Question #42507

Difficulty: 8/10 | 🟢 Free

The Question

A Carnot refrigerator operates to maintain a cold reservoir at a constant temperature $T_C$ . It extracts heat from the cold reservoir at a constant rate $P_C$ . The refrigerator rejects heat into an ambient sealed room containing an ideal gas of mass $M$ and specific heat capacity at constant pressure $C_p$ . The room is designed to expand isobarically to maintain a constant pressure $P_0$ . Initially, the ambient room air has a temperature $T_0$ and density $\rho_0$ . Assuming the room is perfectly thermally insulated from the outside environment, what is the time $t$ required for the density of the ambient room air to decrease to $\frac{\rho_0}{e}$ ?

Options

A.
$\frac{M C_p T_C}{P_C}$
B.
$\frac{M C_p T_C}{P_C} (e - 1)$
C.
$\frac{M C_p T_0}{P_C} (e - 1)$
D.
$\frac{M C_p T_0}{P_C}$

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|Refrigerator Coefficient of Performance|Variation of Density with Temperature|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize that the Coefficient of Performance (COP) of a refrigerator is a dynamic variable when the hot reservoir is finite and its state changes.

A common lethal mistake is treating the COP as a constant determined by the initial ambient temperature $T_0$ , leading to a constant heat rejection rate and a linear temperature increase (Option B).

By coupling the ambient temperature to the air density via the ideal gas law for an isobaric process, the student is forced to set up a differential equation for entropy generation.

The problem elegantly demonstrates that the time constant of the system depends purely on the cold reservoir temperature $T_C$ , completely independent of the initial ambient temperature $T_0$ .

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