The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of $1000\,\text{K}$ is $0.4$. It extracts $150\,\text{J}$ 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 $300\,\text{K}$. Which of the following statements is/are correct:

Step 1: Analyze the Carnot engine.

The efficiency $\eta$ of a Carnot engine is related to the temperatures of its hot reservoir $T_H$ and cold reservoir $T_C$.

$$\eta = 1 - \frac{T_C}{T_H}$$

Substitute the given values $T_H = 1000\text{ K}$ and $\eta = 0.4$ to find $T_C$.

$$0.4 = 1 - \frac{T_C}{1000}$$ $$T_C = 600\text{ K}$$

The work output $W$ per cycle is calculated from the heat extracted from the hot reservoir $Q_H = 150\text{ J}$.

$$W = \eta Q_H$$ $$W = 0.4 \times 150 = 60\text{ J}$$

Step 2: Analyze the heat pump's energy transfer.

The work output $W = 60\text{ J}$ runs the heat pump. Use the coefficient of performance ($\text{COP} = 10$) to find the heat delivered to the hot reservoir $Q_H'$.

$$\text{COP} = \frac{Q_H'}{W}$$ $$10 = \frac{Q_H'}{60}$$ $$Q_H' = 600\text{ J}$$

The heat extracted from the cold reservoir of the heat pump $Q_C'$ is given by conservation of energy.

$$Q_C' = Q_H' - W$$ $$Q_C' = 600 - 60 = 540\text{ J}$$

Thus, $540\text{ J}$ of heat is taken from the cold reservoir, while $600\text{ J}$ is delivered to the hot reservoir.

Step 3: Find the cold reservoir temperature of the heat pump.

For a Carnot heat pump, the COP is related to the reservoir temperatures $T_H'$ and $T_C'$.

$$\text{COP} = \frac{T_H'}{T_H' - T_C'}$$

Substitute $T_H' = 300\text{ K}$ and $\text{COP} = 10$.

$$10 = \frac{300}{300 - T_C'}$$ $$300 - T_C' = 30$$ $$T_C' = 270\text{ K}$$

Step 4: Evaluate the statements.

Comparing the calculated values with the statements:

• The work extracted from the engine is $60\text{ J}$ (Correct)
• The temperature of the cold reservoir of the engine is $600\text{ K}$ (Correct)
• The temperature of the cold reservoir of the heat pump is $270\text{ K}$ (Correct)
• The heat supplied to the hot reservoir of the heat pump is $540\text{ J}$ (Incorrect, it is $600\text{ J}$)

Answer: The correct statements are:

1. The work extracted from the engine is $60\text{ J}$.
2. The temperature of the cold reservoir of the engine is $600\text{ K}$.
3. The temperature of the cold reservoir of the heat pump is $270\text{ K}$.