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:

To determine the correct statements, we analyze the Carnot engine and the heat pump step-by-step.

Step 1: Analyze the Carnot Engine

The Carnot engine operates with a hot reservoir at temperature $T_1=1000\text{K}$ and has an efficiency $\eta =0.4$. It extracts heat $Q_1=150\text{J}$ per cycle from this reservoir.

The work extracted from the engine per cycle ($W$) is given by:
$W=\eta \times Q_1$
$W=0.4\times 150\text{J}=60\text{J}$
Thus, the work extracted from the Carnot engine in one cycle is indeed 60 J.

The efficiency of a Carnot engine is also related to the temperatures of the hot ($T_1$) and cold ($T_2$) reservoirs by the formula:
$\eta =1-\frac{T_2}{T_1}$
Substituting the given values:
$0.4=1-\frac{T_2}{1000}$
$\frac{T_2}{1000}=0.6$
$T_2=600\text{K}$
Thus, the temperature of the cold reservoir of the Carnot engine is 600 K.

Step 2: Analyze the Heat Pump

The work input to the heat pump is the work output of the Carnot engine, so $W=60\text{J}$. The coefficient of performance ($\mathrm{COP}$) of the heat pump is 10, and its hot reservoir is at temperature $T_H=300\text{K}$.

For a Carnot heat pump, the coefficient of performance is defined as:

where $T_C$ is the temperature of the cold reservoir of the heat pump. Substituting the known values:
$10=\frac{300}{300-T_C}$
$300-T_C=\frac{300}{10}$
$300-T_C=30$
$T_C=270\text{K}$
Thus, the temperature of the cold reservoir of the heat pump is 270 K.

The heat supplied to the hot reservoir of the heat pump ($Q_H$) is given by:
$Q_H=\mathrm{COP}\times W$
$Q_H=10\times 60\text{J}=600\text{J}$
Therefore, the statement claiming the heat supplied to the hot reservoir of the heat pump is 540 J is incorrect.

Conclusion

The correct statements are:
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.