Thermodynamics & Heat
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Carnot Efficiency

Maximum efficiency of a heat engine.

Formula

η=1−TCTH\eta = 1 - \frac{T_C}{T_H}

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Variables

T_CCold reservoir (K)
T_HHot reservoir (K)

Example

T_C=300,T_H=600 ⇒ 50%

How it works

Carnot efficiency gives the upper limit on how much heat can be converted into useful work by any engine operating between two thermal reservoirs. It depends only on the absolute temperatures of the hot and cold reservoirs, because a reversible Carnot engine sets the best possible case allowed by the second law of thermodynamics: some energy must always be rejected to the cold reservoir. As the cold reservoir approaches the hot reservoir, the efficiency falls toward zero; as the cold reservoir gets much colder relative to the hot reservoir, the maximum possible efficiency rises.

When to use it

Use this formula when you want the theoretical maximum efficiency of a heat engine, not the efficiency of a real machine. It is especially useful in thermodynamics problems involving power plants, engines, refrigerators viewed in reverse, and comparisons between operating temperatures.

Worked example, step by step

Given: T_H = 900 K, T_C = 450 K Step 1: Write the Carnot efficiency formula: η = 1 - T_C / T_H Step 2: Substitute the temperatures: η = 1 - 450 / 900 Step 3: Evaluate the fraction: 450 / 900 = 0.5 Step 4: Compute the efficiency: η = 1 - 0.5 = 0.5 Step 5: Convert to a percentage: η = 50% Final answer: The maximum possible efficiency is 50%.

Common mistakes to avoid

  • Using degrees Celsius instead of kelvin for T_H and T_C
  • Forgetting that this is a maximum theoretical efficiency, not the actual efficiency of a real engine
  • Reversing the temperatures or using T_H / T_C, which gives an impossible result

Did You Know?

Carnot proved no heat engine can beat his efficiency limit, a ceiling set purely by the two temperatures involved.

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Reviewed & maintained by the FormulaWon editorial teamLast reviewed September 2026