Carnot Cycle
The Carnot cycle, described by Sadi Carnot in 1824, is an idealised reversible cycle consisting of an isothermal expansion at the hot temperature Th, an adiabatic expansion, an isothermal compression at the cold temperature Tc, and an adiabatic compression back to the start. Its efficiency, η = 1 − Tc/Th, is the maximum possible for any heat engine operating between those two temperatures (Carnot's theorem).
Curious
In 1824 a young French engineer, Sadi Carnot, asked a simple question: how much work can you get from heat? He imagined an ideal engine with no friction and no heat leaking across temperature differences. Its efficiency, he found, depends only on the temperatures of the hot source and the cold sink, not on the working substance, whether steam, air or anything else.
The result is a limit (see Second Law): a power station, a car engine or a jet engine always rejects some heat to its surroundings. The hotter the source and the colder the sink, the better the best possible engine can do.
Exploring
| Step | Process | Heat | Temperature |
|---|---|---|---|
| 1 → 2 | Isothermal expansion | Qh absorbed | Th |
| 2 → 3 | Adiabatic expansion | none | falls Th → Tc |
| 3 → 4 | Isothermal compression | Qc rejected | Tc |
| 4 → 1 | Adiabatic compression | none | rises Tc → Th |
Deep Dive
Carnot's theorem. If any engine were more efficient than a reversible one between the same reservoirs, coupling it to a reversed Carnot engine would move heat from cold to hot with no net work, violating the Clausius statement of the second law. All reversible engines between the same two temperatures therefore have the same efficiency, a function of the temperatures alone, which Kelvin used to define absolute temperature: Qc/Qh = Tc/Th. In entropy terms, the reversible cycle exchanges entropy Qh/Th = Qc/Tc, so the total entropy change is zero (see Entropy).
Historical note. Carnot reasoned within the caloric theory, treating heat as a conserved fluid falling from hot to cold like water through a mill wheel. His efficiency argument nevertheless survived the switch to the mechanical theory of heat, once Clausius and Kelvin reformulated it around 1850 (see Rudolf Clausius). The cycle was drawn on a pressure–volume diagram by Émile Clapeyron in 1834.
Finite-time engines. Real engines must deliver power, which requires finite temperature differences for heat flow. The Curzon–Ahlborn (1975) analysis, anticipated by Novikov and Chambadal in 1957, gives efficiency at maximum power as 1 − √(Tc/Th), which is often closer to observed power-plant efficiencies; its generality is debated, but it illustrates the trade-off between efficiency and power.
Concept Relationship Map
Requires: second law, ideal gas law
Enables: heat engines, absolute temperature scale, refrigerators
Commonly confused with: a practical engine design
Sources
- Carnot, S. (1824). Réflexions sur la puissance motrice du feu. Paris: Bachelier.
- Curzon, F.L., Ahlborn, B. (1975). “Efficiency of a Carnot engine at maximum power output.” American Journal of Physics, 43, 22–24.
- Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). Wiley.