Carnot Efficiency Calculator
This Carnot efficiency calculator gives the highest possible efficiency of a heat engine working between a hot and a cold reservoir: e = 1 − Tc/Th, with temperatures in kelvin. Between 750 K and 300 K it is 60 %. It also shows the coefficients of performance of a Carnot refrigerator and heat pump and the heat flows per cycle.
Example: Known values: Both temperatures · Temperature unit: Kelvin (K) · Hot reservoir Th 750 · Cold reservoir Tc 300 · Energy per cycle: Work W done by the engine · Amount of energy 300 J → Carnot efficiency: Efficiency 60.0 % · fridge COP 0.67 · heat pump COP 1.67. Source: OpenStax University Physics Volume 2, 4.5 The Carnot Cycle – e = 1 − Tc/Th (Eq. 4.5), K_R = Tc/(Th − Tc) (Eq. 4.6), K_P = Th/(Th − Tc) (Eq. 4.7), W = Qh − Qc, Carnot’s principle; Example 4.2 (e = 0.60, Tc = 300 K → Th = 750 K, W = 300 J → Qh = 500 J, Qc = 200 J), Example 4.3 (heat pump 0 °C/20 °C, 30.0 kJ → about 2 kJ) (read 10.10.2026). Updated: .
How it is calculated
The Carnot efficiency is e = 1 − Tc/Th, where Th and Tc are the absolute temperatures (in kelvin) of the hot and the cold reservoir. The textbook OpenStax University Physics derives this formula for the Carnot cycle – two isothermal and two adiabatic steps – and states Carnot’s principle: no engine working between two reservoirs at constant temperatures can be more efficient than a reversible engine. The Carnot efficiency is therefore an upper limit, not the efficiency of a real engine.
Carnot efficiency for typical temperatures
| Hot (°C) | Cold (°C) | Efficiency e | Fridge K_R | Heat pump K_P |
|---|---|---|---|---|
| 1000 | 100 | 70.7 % | 0.41 | 1.41 |
| 600 | 30 | 65.3 % | 0.53 | 1.53 |
| 400 | 30 | 55.0 % | 0.82 | 1.82 |
| 100 | 20 | 21.4 % | 3.66 | 4.66 |
| 35 | 5 | 9.7 % | 9.27 | 10.27 |
| 25 | 0 | 8.4 % | 10.93 | 11.93 |
| 20 | 0 | 6.8 % | 13.66 | 14.66 |
The temperatures must be converted to kelvin first (T = t + 273.15): between 400 °C and 30 °C, e = 1 − 303.15/673.15 = 0.550. OpenStax Example 4.2 works backwards: an efficiency of 0.60 with a cold reservoir at 300 K needs Th = 300 K ÷ (1 − 0.60) = 750 K. If that engine does 300 J of work per cycle, it takes Qh = W/e = 500 J from the hot reservoir and gives off Qc = 500 J − 300 J = 200 J.
Refrigerator and heat pump
Run in reverse, the Carnot cycle is a refrigerator or a heat pump. Their coefficients of performance are K_R = Tc/(Th − Tc) for the refrigerator and K_P = Th/(Th − Tc) for the heat pump; both grow when the two temperatures are close together. In Example 4.3 a Carnot heat pump delivers 30.0 kJ into a house at 20 °C from outside air at 0 °C. It needs W = 30 kJ × (293 K − 273 K)/293 K, which the book rounds to 2 kJ; with 273.15 the calculator shows 2.05 kJ. The colder it is outside, the smaller K_P and the more work is needed.
- Always use kelvin: with degrees Celsius the formula gives wrong results.
- Th must be higher than Tc; at Th = Tc no work can be gained.
- The values are ideal limits – real engines, fridges and heat pumps stay below them.
The energy needed to heat water is found with the energy to heat water calculator; work and kinetic energy are covered by the kinetic energy calculator.
Frequently asked questions
What is the formula for Carnot efficiency?
e = 1 − Tc/Th, with Th and Tc the temperatures of the hot and cold reservoir in kelvin. Multiply by 100 for percent.
What is the Carnot efficiency between 400 °C and 30 °C?
In kelvin: 673.15 K and 303.15 K. e = 1 − 303.15/673.15 = 0.550, so 55.0 %.
Why do I have to use kelvin?
The formula uses the ratio of absolute temperatures. With degrees Celsius the ratio is wrong – and a cold side at 0 °C would give 100 %.
Can a real engine reach the Carnot efficiency?
No. Carnot’s principle says that no engine between two reservoirs is more efficient than a reversible one; real engines lose energy and stay below the limit.
What is the COP of a Carnot refrigerator?
K_R = Tc/(Th − Tc). For a fridge between 0 °C and 25 °C: 273.15/25 = 10.93. A heat pump between the same temperatures reaches K_P = Th/(Th − Tc) = 11.93.
Sources and legal basis
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