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. [2025]

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: [2025]

Smart Achievers · Accepted Answer

(1, 2, 3)

$Efficiency of engine η = \frac{W}{Q_{1}} and coefficient of performance (COP) of heat pump = \frac{Q_{3}}{W}$

$η = 0.4$

${(COP)}_{HP} = 10$

$W = η \times Q_{1} = 0.4 \times 150 = 60 J$

$So, option (1) is correct.$

$10 = \frac{Q_{3}}{W} = \frac{Q_{3}}{60}$

$∴$ $Q_{3} = 600 J$

$So option (2) is correct.$

$Also η = 0.4 = 1 - \frac{T_{2}}{T_{1}}$

$T_{2} = 600 K$

$COP = \frac{T_{3}}{T_{3} - T_{4}} = 10 = \frac{300}{300 - T_{4}} = 10$

$∴$ $T_{4} = 270 K$

$So option (3) is correct.$

Solution

1 Work extracted from the Carnot engine in one cycle is 60 J.

2 Temperature of the cold reservoir of the Carnot engine is 600 K.

3 Temperature of the cold reservoir of the heat pump is 270 K.

4 Heat supplied to the hot reservoir of the heat pump in one cycle is 540 J.

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