Carnot Cycle
The most efficient heat engine possible between two temperatures. Watch the four reversible stages animate a piston while the PV and TS diagrams trace the closed loop in real time. Adjust temperatures to see why efficiency depends only on TH and TC.
η = 1 − TC / TH — maximum efficiency for any heat engine
exceed this limit
About this experiment
The Carnot cycle was conceived by French engineer Sadi Carnot in 1824 as a thought experiment: what is the most efficient heat engine that the laws of physics allow? The answer, proved by Carnot and later formalized by Clausius and Kelvin, is that no heat engine operating between a hot reservoir at temperature TH and a cold reservoir at TC can have efficiency greater than η = 1 − TC/TH. This is not a limitation of engineering but a fundamental consequence of the second law of thermodynamics.
The cycle consists of four reversible stages. In isothermal expansion, the gas absorbs heat QH from the hot reservoir while expanding at constant temperature TH — pressure falls as volume grows, tracing a hyperbola PV = nRTH on the PV diagram. In adiabatic expansion, the gas continues expanding with no heat exchange; the temperature falls from TH to TC and the curve steepens. Isothermal compression at TC rejects heat QC to the cold reservoir. Finally adiabatic compression returns the gas to its original state. The area enclosed in the PV diagram equals the net work output.
The TS (temperature-entropy) diagram reveals the symmetry most clearly. The two isothermal stages are horizontal lines (T constant, entropy changes), while the two adiabatic stages are vertical lines (entropy constant, temperature changes). The rectangle area on the TS diagram — (TH − TC) × ΔS — equals the net work, and this geometry makes the Carnot efficiency obvious: the fraction of input heat QH = TH ΔS that emerges as work W = (TH − TC) ΔS is exactly 1 − TC/TH.
Real engines fall short for several reasons. Friction converts work to heat. Heat transfer across finite temperature differences is irreversible (and fast). Combustion products are impure. The Carnot cycle requires infinitely slow (quasi-static) processes, making it infinitely slow. Engineers optimize the trade-off between efficiency and power, which is captured by the Curzon-Ahlborn efficiency ηCA = 1 − √(TC/TH). This represents maximum power (not maximum efficiency) and closely matches real power plant efficiencies.