The Carnot cycle, explained

The four processes

A Carnot engine runs a working fluid between a hot reservoir at T_H and a cold one at T_C, through four reversible steps. First, isothermal expansion at T_H: the fluid takes in heat Q_H while its temperature stays fixed. Second, adiabatic expansion: no heat crosses the boundary, the fluid keeps doing work, and its temperature falls from T_H to T_C. Third, isothermal compression at T_C: the fluid rejects heat Q_C. Fourth, adiabatic compression back up to T_H, which closes the loop.

On a temperature–entropy diagram the cycle is a rectangle. The two isotherms are horizontal lines, the two reversible adiabats are vertical, because a reversible adiabatic process is isentropic. The area inside the rectangle is the net work per cycle, and the area under the top line is the heat taken in.

Why it is the limit

Carnot’s theorem has two parts. No engine working between two reservoirs can be more efficient than a reversible one between the same reservoirs, and every reversible engine between them has the same efficiency. Both follow from the second law: if a better engine existed, you could pair it with a reversed Carnot engine and build a machine that turns heat from a single reservoir entirely into work, which the Kelvin–Planck statement forbids.

The efficiency depends only on the two temperatures, and they must be absolute temperatures in kelvin. It rises when the hot side gets hotter or the cold side gets colder. That single fact explains why turbine inlet temperatures keep climbing and why power plants sit next to cold water.

Why nobody builds one

Reversible heat transfer needs an infinitesimal temperature difference, which means infinite time or infinite area. An engine that takes forever per cycle produces no power. Gas-based Carnot cycles also have a tiny net work compared with the work moved back and forth, so friction would eat the output.

Real plants run cycles that are practical to build: Rankine for steam, Brayton for gas turbines, Otto and Diesel for piston engines. Carnot is the yardstick they are measured against. Stirling and Ericsson cycles with ideal regeneration reach the Carnot efficiency on paper, which is why they keep coming back.

Where the name comes from

Sadi Carnot published the argument in 1824, in Réflexions sur la puissance motrice du feu, decades before the first law was settled. CARNOT, the Mechanical Engineering student club at Tel Aviv University, is named after the same idea: the limit on what any machine can do.

A worked number

Steam enters a turbine at about 820 K and the condenser rejects heat at about 300 K. The Carnot limit is 1 − 300/820 ≈ 0.63, so 63%. Real steam plants at those temperatures reach roughly 40%. The gap is irreversibility: heat transferred across finite temperature differences, friction, and throttling.

For a refrigerator holding 275 K in a 300 K kitchen, the Carnot COP is 275/25 = 11. Real domestic fridges manage around a quarter to a third of that.

Common mistakes

• Putting temperatures in Celsius. 1 − 27/327 gives 92% instead of the correct 50%.
• Treating the Carnot efficiency as the efficiency of a real cycle. It is an upper bound, not a prediction.
• Mixing up the refrigerator and heat pump COP. They differ by exactly 1, because the heat pump counts the work as delivered heat too.