The figure shows a heat engine (HE) working between two reservoirs. The amount of heat (Q2) rejected by the heat engine is drawn by a heat pump (HP). The heat pump receives the entire work output (W) of the heat engine. If temperature, T₁ > T₃ > T₂, then the relation between the efficiency (η) of the heat engine and the coefficient of performance (COP) of the heat pump is

Correct Answer:
The correct relation is: COP = η^-1

Step-by-Step Explanation:

1. Understanding the Heat Engine (HE)
The heat engine operates between reservoirs to produce work output. Let the heat engine absorb heat $Q_1$ from the high-temperature reservoir at $T_1$, produce work output $W$, and reject heat $Q_2$ to the reservoir at $T_2$.
By the first law of thermodynamics:
$$Q_1=W+Q_2$$
The thermal efficiency ($\eta$) of the heat engine is defined as the ratio of net work output to the heat input:
$$\eta =\frac{W}{Q_1}=\frac{W}{W+Q_2}$$
Taking the reciprocal of the efficiency expression:
$${\eta }^{-1}=\frac{W+Q_2}{W}$$

2. Understanding the Heat Pump (HP)
The heat pump is driven by the entire work output ($W$) from the heat engine. It extracts the rejected heat $Q_2$ from the reservoir at $T_2$ and delivers heat $Q_3$ to the reservoir at $T_3$.
Applying the first law of thermodynamics to the heat pump, the heat delivered is:
$$Q_3=W+Q_2$$
The coefficient of performance (COP) of the heat pump is defined as the desired heating effect divided by the work input:
$$\text{COP}=\frac{Q_3}{W}=\frac{W+Q_2}{W}$$

3. Relating Efficiency and COP
By comparing the two derived equations, we can observe that:
$$\text{COP}={\eta }^{-1}$$
This confirms that the coefficient of performance of the heat pump is the reciprocal of the efficiency of the heat engine.