Question

The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10. The hot reservoir of the heat pump is at a temperature of 300 K. Which of the following statements is/are correct:

• A. Work extracted from the Carnot engine in one cycle is 60 J.
• B. Temperature of the cold reservoir of the Carnot engine is 600 K.
• C. Temperature of the cold reservoir of the heat pump is 270 K.
• D. Heat supplied to the hot reservoir of the heat pump in one cycle is 540 J.

Hint:

In thermal problems involving Carnot engines and heat pumps, use efficiency and COP formulas carefully: - \( \eta = 1 - \frac{T_C}{T_H} \) for Carnot engines, - \( \text{COP}_{\text{heat pump}} = \frac{T_H}{T_H - T_C} \), and remember: Work = \( Q_H - Q_C \).

Answer 1

The Correct Option is A, B, D

Step 1: Efficiency of the Carnot engine: Given efficiency \( \eta = 0.4 \), and heat extracted \( Q_H = 150 \, \text{J} \), we find the work done per cycle:
\[ W = \eta \cdot Q_H = 0.4 \times 150 = 60 \, \text{J} \] Hence, option (A) is correct.

Step 2: Temperature of the cold reservoir of Carnot engine:
For Carnot engine: \( \eta = 1 - \frac{T_C}{T_H} \Rightarrow \frac{T_C}{T_H} = 1 - \eta = 0.6 \)
Given \( T_H = 1000 \, \text{K} \Rightarrow T_C = 0.6 \times 1000 = 600 \, \text{K} \)
Hence, option (B) is correct.

Step 3: Heat pump calculation: Work done by engine is used as input to the heat pump: \( W = 60 \, \text{J} \)
Coefficient of performance (COP) of heat pump is 10.
\[ \text{COP} = \frac{Q_H}{W} \Rightarrow Q_H = \text{COP} \times W = 10 \times 60 = 600 \, \text{J} \] Hence, option (D) is incorrect as it says 540 J instead of 600 J.

Step 4: Temperature of cold reservoir of heat pump:
COP of a heat pump: \( \text{COP} = \frac{T_H}{T_H - T_C} \Rightarrow T_C = T_H - \frac{T_H}{\text{COP}} = 300 - \frac{300}{10} = 270 \, \text{K} \)
So, option (C) is correct in value, but contradicts (D) which is incorrect.
Due to this inconsistency, we only mark A, B, and C as correct, but not (D).
Note: As per energy consistency, since \( Q_H = 600 \, \text{J} \), option (D)'s figure of 540 J is wrong.