Carnot Efficiency Calculator
Carnot heat engine limit with fridge and heat pump COPs.
What it does
No heat engine beats Carnot between two temperatures: 1 minus cold over hot in kelvin caps every steam plant and car motor, with fridge and heat pump COPs riding the same pair. Real machines fall short.
The tool above takes both reservoir temps. It returns efficiency with both reversed COPs.
The default 500 K over 300 K caps at 40% with COPs 1.5 and 2.5. A 600 over 300 K pair caps at 50%.
Limits follow OpenStax thermodynamics with HyperPhysics cycles behind them. Both demand kelvin and state the same forms used here.
How to use
- Pick Celsius or Fahrenheit.
- Enter hot reservoir temp.
- Enter cold reservoir temp.
- Read efficiency first.
- Read fridge COP beside it.
- Read heat pump COP.
Formula
- Th
- Hot reservoir kelvin.
- Tc
- Cold reservoir kelvin.
- eta
- Fractional ideal efficiency.
Fahrenheit converts first ((F-32)x5/9+273.15). Equal reservoirs reject; negative kelvin rejects. Efficiency prints 1 decimal percent with both COPs beside it.
Worked example
Example: 500 K over 300 K
Metric defaults: 226.85 C hot, 26.85 C cold.
- Hot 226.85 C = 500 K, cold 26.85 C = 300 K.
- Ratio: 300/500 = 0.6.
- Efficiency: 1 - 0.6 = 40%.
- Fridge COP: 300/(500-300) = 1.5.
- Heat COP: 500/200 = 2.5.
- Per 1,000 J heat: 400 J work, 600 J out.
- Tool shows 40%, 1.5 fridge COP, 2.5 heat COP.
Reference table
| Reservoirs (K) | Efficiency | Fridge COP |
|---|---|---|
| 500 / 300 | 40% | 1.5 |
| 600 / 300 | 50% | 1.0 |
| 400 / 300 | 25% | 3.0 |
Limits
- Reversible ideal only.
- Equal temps give 0%.
- Absolute zero excluded.
- Steady reservoirs assumed.
Common mistakes
- Feeding Celsius as kelvin.
- Expecting real engines to match.
- Forgetting reversed COP forms.
- Dividing by zero spans.
- Chasing 100% efficiency.
Try it yourself
Quick checks. Answers sit tucked under each question.
600 K over 300 K?
1 - 300/600 = 50%, COPs 1.0 and 2.0.
Equal reservoirs?
Th must exceed Tc: equal spans reject.
Imperial defaults?
440.33 F = 500 K mirrors the default 40%.
Questions
What is Carnot efficiency?
1 minus cold over hot in kelvin. 500 over 300 gives 40%.
Can real engines beat it?
No engine between the same reservoirs beats a reversible one; real engines run worse.
What are the reversed COPs?
Fridge Tc/(Th-Tc), heat pump Th/(Th-Tc). Here 1.5 and 2.5.
Can I enter Celsius?
Yes: convert first (K = C + 273.15). Kelvin-only math follows.
What if reservoirs match?
Efficiency hits 0% and both COPs divide by zero. Hot must exceed cold.
Is 100% possible?
No: absolute zero never arrives, so 100% stays theoretical.