What this calculator does
The Carnot efficiency is a hard ceiling. No heat engine operating between two given temperatures can beat it, regardless of how it is built or what working fluid it uses. That makes it the benchmark every real engine is measured against.
The result depends only on the two absolute temperatures, and it says something uncomfortable: unless the cold side is at absolute zero, some heat must always be rejected. A heat engine that converted all its input heat into work is not merely difficult to build, it is impossible.
The formula
Convert both temperatures to kelvin, then subtract the ratio of cold to hot from one. Multiplying that efficiency by the heat input gives the maximum work available, and the remainder is the heat that must be dumped to the cold reservoir.
| Term | Meaning |
|---|---|
| Carnot efficiency | The theoretical maximum fraction of heat input that can become work, set only by the two temperatures. |
| Hot reservoir | The source the engine takes heat from, such as a combustion chamber or a boiler. |
| Cold reservoir | The sink the engine rejects heat to, usually the surrounding air or cooling water. |
The inputs explained
| Field | What to enter |
|---|---|
| Hot reservoir temperature (°C) | The hot reservoir temperature in degrees celsius, converted to kelvin internally. |
| Cold reservoir temperature (°C) | The cold reservoir temperature in degrees celsius. It must be colder than the hot reservoir for the engine to work at all. |
| Heat input per cycle (J) | The heat supplied per cycle, in joules, used to calculate the maximum work available. |
When to use it
Judging a real engine's performance
Comparing an actual efficiency against the Carnot figure for the same temperatures shows how much of the theoretical maximum is being achieved.
Understanding why power stations run hot
Efficiency rises with the hot-side temperature, which is why plants push materials to their thermal limits.
Assessing an efficiency claim
Any claimed efficiency above the Carnot figure for its operating temperatures is impossible, which makes this a quick credibility test.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does efficiency rise with the hot-side temperature?
A fixed cold reservoir with the hot side varied.
| Hot reservoir | Carnot efficiency | Maximum work output |
|---|---|---|
| 200°C | 35.9% | 359.3 J |
| 300°C | 47.1% | 471.1 J |
| 500°C | 60.8% | 607.9 J |
| 800°C | 71.8% | 717.5 J |
Questions
Why can no engine beat this?
It follows from the second law of thermodynamics. An engine exceeding Carnot efficiency could be combined with a reversed one to move heat from cold to hot with no work input, which is exactly what the second law forbids.
Why must temperatures be in kelvin?
Because the formula uses a ratio of temperatures, and ratios only mean something on an absolute scale. Using celsius would give a different answer depending on an arbitrary zero point, which is a common mistake.
Do real engines get close to it?
Not especially. Large combined-cycle power stations reach perhaps 60 per cent of their Carnot ceiling; car engines do considerably worse. Friction, incomplete combustion and heat loss all take their share.
How do heat pumps fit in?
They run the cycle backwards, using work to move heat rather than extracting work from it. Their performance can exceed 100 per cent in apparent terms, because they move existing heat instead of creating it, so efficiency is not the right measure for them.
For the heat required to change a material's temperature, see the specific heat calculator. For the gas behaviour behind many engine cycles, see the ideal gas law calculator.