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:

The Carnot engine (CE) and the heat pump (HP) work in tandem, the work output of the CE being the work input of the HP.

Carnot engine

Efficiency of a Carnot engine is defined as $$\eta = 1 - \frac{T_C}{T_H}$$, where $$T_H$$ and $$T_C$$ are the absolute temperatures of the hot and cold reservoirs respectively.

Given $$\eta = 0.4$$ and $$T_H = 1000\,\text{K}$$, we have

$$0.4 = 1 - \frac{T_C}{1000}$$

$$\frac{T_C}{1000} = 1 - 0.4 = 0.6$$

$$T_C = 0.6 \times 1000 = 600\,\text{K}$$

Thus the cold-reservoir temperature of the Carnot engine is $$600\,\text{K}$$  →  Option B is correct.

The work obtained from the Carnot engine in one cycle is

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

Hence the work delivered per cycle is $$60\,\text{J}$$  →  Option A is correct.

Heat pump

The entire $$60\,\text{J}$$ of work produced by the engine drives the heat pump. For a heat pump, the coefficient of performance (COP) is defined as

$$\text{COP} = \frac{Q_H^{(\text{HP})}}{W}$$

Given $$\text{COP} = 10$$ and $$W = 60\,\text{J}$$, the heat delivered to the hot reservoir of the pump is

$$Q_H^{(\text{HP})} = \text{COP}\times W = 10 \times 60\,\text{J} = 600\,\text{J}$$

The cold-reservoir heat extracted by the pump is

$$Q_C^{(\text{HP})} = Q_H^{(\text{HP})} - W = 600\,\text{J} - 60\,\text{J} = 540\,\text{J}$$

The problem states that the heat pump operates between $$T_H^{(\text{HP})}=300\,\text{K}$$ and an (as yet) unknown cold-reservoir temperature $$T_C^{(\text{HP})}$$. For an ideal (Carnot) heat pump,

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

Substituting $$\text{COP}=10$$ and $$T_H = 300\,\text{K}$$,

$$10 = \frac{300}{300 - T_C}$$

$$300 - T_C = \frac{300}{10} = 30$$

$$T_C = 300 - 30 = 270\,\text{K}$$

Thus the cold-reservoir temperature of the heat pump is $$270\,\text{K}$$  →  Option C is correct.

Checking Option D

Option D claims that the heat supplied to the hot reservoir of the pump is $$540\,\text{J}$$ per cycle. The calculated value is $$Q_H^{(\text{HP})}=600\,\text{J}$$, so Option D is incorrect.

Correct statements: Option A, Option B, Option C.