Carnot efficiency

Carnot efficiency is the upper bound on the fraction of absorbed heat that any heat engine can convert into work while operating between a hot reservoir at absolute temperature T_h and a cold reservoir at T_c. It is given by η_Carnot = 1 − T_c/T_h, with both temperatures measured in kelvin.

Its most striking feature is universality: the bound depends only on the two temperatures, never on the working substance or the mechanical design. Helium, steam, and air engines running between the same reservoirs share exactly the same ceiling. Reaching it would require a perfectly reversible engine; all real engines, beset by friction and finite-rate heat transfer, fall short.

The formula also shows why perfect efficiency is unattainable: η = 1 would demand T_c = 0, a cold reservoir at absolute zero, which the third law places forever out of reach. Every engine pays the temperature ratio as an unavoidable tax.

Carnot derived the result in 1824 in caloric language; Clausius and Kelvin re-derived it in the 1850s from the second law and used its substance-independence to define the absolute (Kelvin) temperature scale.