Q.A Carnot engine takes in heat from a source at $600\ \text{K}$ and rejects heat to a sink at $300\ \text{K}$. Calculate the efficiency of the engine.

The Carnot Engine: Why Nature Puts a Ceiling on Efficiency

Imagine you have a steam engine. You burn coal, heat water, make steam, push a piston. You get work out. But you also always waste heat — the exhaust steam is still hot. Could you ever build an engine that converts all the heat into work? The Carnot engine is the thought experiment that answers that question with a firm no.

The intuition is simple: heat flows spontaneously only from hot to cold. To make it do work, you must let some of it "fall" to a lower temperature, just as a water wheel can only extract work from water falling downhill — you never get all the potential energy back, because the water ends up at the bottom. The Carnot engine is the perfect water wheel: it wastes the absolute minimum heat required by the second law of thermodynamics.

The Precise Definition

A Carnot engine is an idealised, reversible heat engine that operates in a cycle between two thermal reservoirs at temperatures $T_H$ (hot) and $T_C$ (cold), where $T_H > T_C$. The cycle consists of exactly four reversible processes:

1. Isothermal expansion at $T_H$ — the working substance (usually an ideal gas) absorbs heat $Q_H$ from the hot reservoir while expanding slowly, doing work on the surroundings.
2. Adiabatic expansion — the gas is thermally insulated and continues expanding, doing work. Its temperature drops from $T_H$ to $T_C$ with no heat exchange.
3. Isothermal compression at $T_C$ — the gas is in contact with the cold reservoir. It is compressed slowly, rejecting heat $Q_C$ to the cold reservoir.
4. Adiabatic compression — the gas is insulated again and compressed further, raising its temperature back to $T_H$ with no heat exchange, returning to the starting state.

The Efficiency — The One Result You Must Know

The efficiency $\eta$ of any heat engine is defined as:

From energy conservation over one cycle, $W = Q_H - Q_C$, so:

For a Carnot engine, the ratio of heat exchanged is exactly equal to the ratio of absolute temperatures:

This is the maximum possible efficiency for any engine operating between $T_H$ and $T_C$. No real engine can beat it — not because of engineering limitations, but because of the second law of thermodynamics.

Why This Is the Maximum …