Carnot Efficiency Formula + Calculator

| Added in Physics

What is Carnot Efficiency?

The Carnot efficiency represents the maximum theoretical efficiency that any heat engine can achieve when operating between two thermal reservoirs at different temperatures. Named after French physicist Sadi Carnot, this fundamental thermodynamic principle establishes an upper limit on the efficiency of all real-world heat engines.

Carnot Efficiency Formula

The Carnot efficiency is calculated using the absolute temperatures of the hot and cold reservoirs:

$$\eta = 1 - \frac{T_{C}}{T_{H}}$$

Or equivalently:

$$\eta = \frac{T_{H} - T_{C}}{T_{H}}$$

Where:

  • ฮท = Carnot efficiency (as a decimal)
  • T_H = Absolute temperature of the hot reservoir (in Kelvin)
  • T_C = Absolute temperature of the cold reservoir (in Kelvin)

To express as a percentage, multiply by 100.

How to Calculate Carnot Efficiency

Follow these steps:

  1. Identify the temperatures: Determine the absolute temperatures of both the hot reservoir (T_H) and cold reservoir (T_C) in Kelvin
  2. Apply the formula: Calculate the efficiency using ฮท = (T_H - T_C) / T_H
  3. Convert to percentage: Multiply the result by 100 to get the efficiency as a percentage

Example Calculation

Given:

  • Hot reservoir: 500 K
  • Cold reservoir: 300 K

Solution:

$$\eta = \frac{T_{H} - T_{C}}{T_{H}} = \frac{500 - 300}{500} = \frac{200}{500} = 0.4$$

Carnot Efficiency = 40%

This means the maximum theoretical efficiency of a heat engine operating between these two temperatures is 40%.

Key Points About Carnot Efficiency

  • Always use absolute temperature: Temperatures must be in Kelvin (K), not Celsius or Fahrenheit
  • Theoretical maximum: No real heat engine can exceed the Carnot efficiency
  • Temperature dependence: Higher efficiency requires a larger temperature difference between reservoirs
  • Zero at equilibrium: When T_H = T_C, the efficiency is zero (no work can be extracted)
  • Impossible to reach 100%: Perfect efficiency would require T_C = 0 K (absolute zero), which is physically impossible

Applications

The Carnot efficiency is used to:

  • Evaluate the theoretical performance limits of power plants
  • Compare real engine efficiencies to the ideal maximum
  • Design thermal systems with realistic efficiency expectations
  • Understand fundamental thermodynamic limitations