Carnot Efficiency

Divide the cold by the hot absolute temperature and subtract the result from one: η = 1 − T_cold/T_hot. Both temperatures must be inserted in kelvin. Example: with T_hot = 500 K and T_cold = 300 K you get η = 1 − 300/500 = 0.40, that is 40 percent. The Carnot efficiency is the theoretically highest possible efficiency of a heat engine operating between these two reservoirs. Real machines always fall below it because of friction, heat losses and irreversible processes. The larger the temperature difference between hot and cold, the higher the maximum achievable efficiency.

Because the Carnot efficiency contains a ratio of two temperatures and this only makes physical sense on an absolute scale. The Kelvin scale starts at absolute zero, so ratios such as T_cold/T_hot are unambiguous. On the Celsius scale the ratio would be meaningless, because 0 °C is not a true zero but merely the freezing point of water. If you accidentally insert Celsius values, you get completely wrong and even negative efficiencies. Therefore always convert first: T in kelvin equals T in Celsius plus 273.15. Only then form the ratio and subtract it from one.

An efficiency of 100 percent would require η = 1, that is T_cold/T_hot = 0. This would only work if the cold temperature were exactly at absolute zero or the hot one infinitely large, both practically impossible. Physically the second law of thermodynamics forbids heat from being fully converted into work; part must always be released as waste heat to the cold reservoir. Even the ideal, reversible Carnot engine is subject to this limit. Real machines lie well below it because of additional losses; a modern power plant reaches about 45 percent against a Carnot limit of roughly 65 percent.

First isolate the fraction: T_cold/T_hot = 1 − η. Then multiply by the hot temperature and get T_cold = T_hot·(1 − η). The cold temperature is therefore the product of the hot temperature and the factor one minus efficiency. Example: with T_hot = 800 K and η = 0.5 it follows that T_cold = 800·0.5 = 400 K. If you want the hot temperature instead, rearrange to T_hot = T_cold/(1 − η). In all cases you work in kelvin and convert the final result to Celsius only afterwards if needed.

The Carnot efficiency is a theoretical upper bound that only a reversible, lossless cycle between two temperatures could reach. The real efficiency is the actually measured fraction of supplied heat converted into usable work, and always lies below it. Causes of the deviation are friction, heat conduction, incomplete combustion and irreversible processes that create additional entropy. The Carnot value serves as a benchmark: it shows how much would be possible at best. Problems often ask for both values, so you can measure the quality of a real machine against its theoretical maximum.