Heat Engines & Carnot Efficiency
Heat Engines & the Carnot Limit
A heat engine takes in heat from a hot source, turns part of it into useful work, and dumps the rest as waste heat into a cooler place. No engine — no matter how well built — can convert all of that heat into work; some always has to be thrown away. The best possible efficiency an engine could ever reach depends only on how hot and how cold its two reservoirs are.
The formula
η = 1 − T_c / T_h
- η — efficiency (—): the fraction of the heat energy the engine converts into useful work
- T_c — cold reservoir temperature (K): the temperature of the cold place where waste heat is dumped
- T_h — hot reservoir temperature (K): the temperature of the hot source supplying heat to the engine
Worked example
A power plant's hot reservoir, steam, is at 400 kelvin, and its cold reservoir, river water, is at 300 kelvin. What is the best possible efficiency for an engine working between them?
- T_h = 400 K
- T_c = 300 K
- η = 1 − T_c / T_h
- η = 1 − 300 K / 400 K
η = 0.25 = 25%
Test yourself
A heat engine runs between a hot reservoir at 800 kelvin and a cold reservoir at 200 kelvin. What is its maximum possible efficiency?
- 25%
- Correct answer: 75%
- 50%
Right! η = 1 − 200 K / 800 K = 1 − 0.25 = 75%.
Which change would raise a heat engine's maximum possible efficiency?
- Raise the cold reservoir's temperature
- Lower the hot reservoir's temperature
- Correct answer: Lower the cold reservoir's temperature
Yes! A cooler cold reservoir makes T_c / T_h smaller, so one minus that ratio — the efficiency — gets bigger.
Where you see this
Real power plants are built around this limit: steam from a boiler at hundreds of kelvin drives turbines, and the leftover heat must be dumped into a river, a lake, or those iconic cooling towers — the cold reservoir is as much a part of the plant as the furnace. Between steam at 400 K and river water at 300 K, no engine ever built could beat 25%.
Common mistakes
The persistent myth is that a clever enough engineer could convert heat entirely into work — the Carnot limit depends only on the two reservoir temperatures, not on craftsmanship. The algebra trap is stopping at the ratio: η = 1 − T_c / T_h means 800 K hot and 200 K cold gives 1 − 0.25 = 75%, not 25%. And the temperatures are kelvin, as always in thermodynamics.
How it connects
This is where mechanics meets thermodynamics: the engine's ledger is energy conservation — heat in equals work out plus heat dumped — which is why this lesson's prerequisite is energy conservation itself. The unreachability of 100% is the second law announcing itself early: the waste heat is entropy's tax, made precise in the next lesson.