Animated Solution for Physics - Thermodynamics: 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:

Imagine a perfectly synchronized factory where the exhaust of one machine powers the next. This problem presents us with a beautiful thermodynamic duo: a Carnot engine and a heat pump working in tandem. The Carnot engine acts as the powerhouse, drawing thermal energy from a blazing $1000\text{ K}$ reservoir and converting a portion of it into useful work. But this work isn't wasted; it is immediately fed into a heat pump, which uses it to force heat into a $300\text{ K}$ room.

Our mission is to dissect this system, step by step, and verify the claims made in the options. Let's break it down!

We start with the powerhouse: the Carnot engine. We are given two crucial pieces of information: its efficiency $\eta =0.4$ and the heat it extracts from the hot reservoir $Q_1=150\text{ J}$.

The efficiency of any heat engine is defined as the ratio of the useful work output $W$ to the total heat input $Q_1$.

By rearranging this, we can easily find the work produced: This confirms that Option (A) is absolutely correct. The engine generates $60\text{ J}$ of work per cycle.

But what about the cold reservoir where the engine dumps its waste heat? For a reversible Carnot engine, the efficiency is intrinsically linked to the absolute temperatures of its reservoirs:

Substituting our known values: This perfectly matches Option (B). The engine's cold sink is sitting at $600\text{ K}$.

Now, let's follow the energy. The $60\text{ J}$ of work produced by the engine is channeled directly into the heat pump. We are told this heat pump has a Coefficient of Performance (COP) of $10$ and delivers heat to a hot reservoir at $T_3=300\text{ K}$.

For a heat pump, the COP is the ratio of the desired heating effect (heat delivered to the hot reservoir, $Q_4$) to the work required to achieve it ($W$). In terms of temperatures for a reversible cycle, it is:

Let's find the temperature of the heat pump's cold reservoir ($T_4$): This confirms that Option (C) is correct. The heat pump is extracting heat from a chilly $270\text{ K}$ source.

Finally, let's evaluate the actual heat energy transferred by the heat pump. How much heat is it dumping into the $300\text{ K}$ reservoir? We return to the fundamental definition of COP:

The heat pump supplies $600\text{ J}$ to its hot reservoir. Option (D) claims this value is $540\text{ J}$. Where did $540\text{ J}$ come from? If we calculate the heat extracted from the cold reservoir ($Q_3$), we get $Q_3=Q_4-W=600-60=540\text{ J}$. Option (D) is a classic trap, confusing the heat supplied with the heat extracted! Therefore, Option (D) is incorrect.

Our final correct statements are (A), (B), and (C).