# Carnot heat engine
A Carnot heat engine is an idealized engine that operates on the [Carnot cycle](https://www.edgepedia.org/carnot-cycle), a closed sequence of four reversible processes, and it defines the maximum thermal efficiency any heat engine can achieve while working between two heat reservoirs at fixed temperatures.[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) The cycle is named for Sadi Carnot (1796–1832), the French engineer who established this theoretical limit in 1824 while investigating the efficiency of steam engines.[2](https://eprints.ukh.ac.id/id/eprint/267/1/2014_Book_ThermodynamicsAndEnergyConvers.pdf)
| Key fact | Detail |
|---|---|
| Cycle stages | Reversible isothermal heat addition at T_H, adiabatic expansion, reversible isothermal heat removal at T_L, adiabatic compression[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) |
| Efficiency | \( \eta_{\mathrm{Carnot}} = 1 - T_L/T_H \), with absolute (kelvin) temperatures[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) |
| Carnot's theorem | All reversible engines between the same two reservoirs have the same efficiency; irreversible engines are less efficient[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) |
| Work per cycle | \( W = Q_h - Q_c \); the net work equals the area enclosed by the cycle in a pressure–volume diagram[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) • [4](https://farside.ph.utexas.edu/teaching/sm1/lectures/node57.html) |
| Reverse operation | Run backward, the cycle is a Carnot refrigerator or heat pump with the highest coefficient of performance between the same reservoirs[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) |
| Power limit | A single Carnot cycle requires infinite time, so the ideal engine has zero power; finite-time engines trade power against efficiency[5](https://arxiv.org/abs/2206.10153) |
| Practical benchmark | A Carnot cycle between 250 °C steam and an 18 °C lake gives \( \eta = 0.443 \); a real 500-MW Rankine plant on the same reservoirs was projected at about 35%[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) |
## The four processes
The engine consists of a working fluid, typically a gas in a cylinder with a piston, brought into contact with a hot reservoir at temperature \( T_H \) and a cold reservoir at \( T_L \). The cycle runs as follows[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/):
1. **Isothermal expansion at \( T_H \)**. The gas expands while in contact with the hot reservoir, absorbing heat \( Q_H \). The expansion is slow enough that the gas temperature stays essentially at \( T_H \); heat flows only across an infinitesimal temperature difference, which is what makes the transfer reversible.[6](https://www.feynmanlectures.caltech.edu/I_44.html)
2. **Adiabatic expansion**. The gas is thermally isolated and continues to expand, doing work at the expense of its internal energy, so its temperature falls from \( T_H \) to \( T_L \).[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
3. **Isothermal compression at \( T_L \)**. In contact with the cold reservoir, the gas is compressed while rejecting heat \( Q_L \).[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
4. **Adiabatic compression**. Isolated again, the gas is compressed and its temperature rises from \( T_L \) back to \( T_H \), returning it to the initial state.[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
Because the working fluid returns to its starting state, the internal energy change over a full cycle is zero, and the first law of thermodynamics gives the net work as \( W = Q_h - Q_c \).[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) In a pressure–volume diagram the net work is the area enclosed by the closed curve.[4](https://farside.ph.utexas.edu/teaching/sm1/lectures/node57.html)
## Efficiency and Carnot's theorem
The thermal efficiency is the ratio of work output to heat absorbed, \( \eta = W/Q_h = 1 - Q_c/Q_h \).[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) For an ideal gas Carnot cycle, the isothermal heat exchanges are \( Q_h = N k_B T_h \log(V_2/V_1) \) and \( Q_c = N k_B T_c \log(V_3/V_4) \), and the adiabatic relations force the logarithmic factors to be equal, so the volume ratios cancel and[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html)
\[ \eta_{\mathrm{Carnot}} = 1 - \frac{T_c}{T_h}. \]
Two conclusions follow. First, since \( T_c \) must be positive, the efficiency is always less than unity; some heat must be rejected.[6](https://www.feynmanlectures.caltech.edu/I_44.html) Second, the result is universal. [Carnot's theorem](https://www.edgepedia.org/carnots-theorem-thermodynamics) states that the efficiency of a reversible engine depends only on the temperatures of the two reservoirs, not on the design or the working substance. The proof couples two reversible engines with different assumed efficiencies: running the less efficient one backward off the work of the more efficient one would produce a net spontaneous flow of heat from cold to hot, violating the second law of thermodynamics, so their efficiencies must be identical.[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) The same argument shows that any irreversible engine working between the same reservoirs must be less efficient, since otherwise it could be paired with a reversible engine to break the theorem.[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) The Carnot efficiency is therefore the maximum efficiency that can occur in an engine working between two given temperatures.[7](https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node44.html)
## The absolute temperature scale
The efficiency formula is what makes the thermodynamic temperature scale meaningful. The scale is defined so that for a reversible cycle \( (Q_L/Q_H)_{\mathrm{rev}} = T_L/T_H \), which fixes temperature ratios without reference to any particular substance.[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) The kelvin, the SI unit of thermodynamic temperature, was anchored for decades to the triple point of water, assigned exactly 273.16 kelvins by the 10th CGPM in 1954.[8](https://www.nist.gov/pml/special-publication-330/sp-330-appendix-1) In the 2018 revision of the SI the kelvin was instead defined by fixing the [Boltzmann constant](https://www.edgepedia.org/boltzmann-constant) at exactly \( 1.380649 \times 10^{-23} \) J/K, connecting thermodynamic temperature to energy.[9](https://www.nist.gov/si-redefinition/kelvin-introduction)
## Entropy from the Carnot cycle
Comparing the Carnot efficiency with the general definition of efficiency shows that \( Q_c/Q_h = T_c/T_h \), or \( Q_h/T_h = Q_c/T_c \): the heat absorbed per unit temperature equals the heat rejected per unit temperature.[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) This quantity is the entropy. For a reversible process the entropy change is \( \mathrm{d}S = \delta Q/T \), and around a complete Carnot cycle the net change is zero.[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html) Because any closed path in the state space can be approximated by many small Carnot cycles, entropy is a state function, depending only on the thermodynamic state and not on the path taken.[3](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html)
## Running the cycle in reverse
Because every step of the Carnot cycle is reversible, the engine can be operated backward: work is supplied, heat \( Q_L \) is taken from the cold reservoir, and heat \( Q_H \) is delivered to the hot reservoir. That device is a refrigerator.[10](https://oyc.yale.edu/physics/phys-200/lecture-23) Among all refrigerators or heat pumps operating between the same reservoirs, the Carnot versions consume the least power and achieve the highest coefficient of performance:[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
\[ \mathrm{COP}_{R,\mathrm{rev}} = \frac{T_L}{T_H - T_L}, \qquad \mathrm{COP}_{HP,\mathrm{rev}} = \frac{T_H}{T_H - T_L}. \]
The heat-pump coefficient of performance falls as the outdoor temperature \( T_L \) drops, which is why heat pumps perform better in mild winter conditions.[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
## Real engines and finite-time operation
No real engine reaches the Carnot limit. [Heat transfer](https://www.edgepedia.org/heat-transfer) across finite temperature differences, friction, unrestrained expansion, and other irreversibilities all lower the efficiency below \( 1 - T_L/T_H \).[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) The ideal cycle also has a structural limitation: reversible heat transfer requires infinitesimal temperature differences, so a single Carnot cycle takes infinite time and the ideal engine delivers zero power.[5](https://arxiv.org/abs/2206.10153) [Experiments](https://www.edgepedia.org/experiments) with a finite-time Carnot cycle using sealed dry air as the working substance confirm a tradeoff relation between power and efficiency.[5](https://arxiv.org/abs/2206.10153)
The limit still sets the design direction. Since efficiency improves when \( T_H \) rises or \( T_L \) falls, and the sink temperature is usually fixed by the ambient environment, engineers push the heat-source temperature as high as material limits allow; inlet steam in large power plants runs at 300 to 600 °C (573 to 873 K).[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/) In a worked comparison, a Carnot cycle between 250 °C steam and an 18 °C lake yields \( \eta = 0.443 \), rising to 0.492 if the steam temperature increases to 300 °C, while the projected efficiency of the actual [Rankine cycle](https://www.edgepedia.org/rankine-cycle) on the same reservoirs was about 35%.[1](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
## References
1. [6.4 Carnot cycles, Introduction to Engineering Thermodynamics (BCcampus)](https://pressbooks.bccampus.ca/thermo1/chapter/6-4-carnot-cycles/)
2. [Thermodynamics and Energy Conversion (chapter naming Sadi Carnot, 1796–1832)](https://eprints.ukh.ac.id/id/eprint/267/1/2014_Book_ThermodynamicsAndEnergyConvers.pdf)
3. [14. Heat engines, entropy, and free energy, TU Delft, Introduction to particle and continuum physics](https://interactivetextbooks.tudelft.nl/nb1140/content/entropy.html)
4. [Heat engines, University of Texas, Statistical Mechanics lecture notes](https://farside.ph.utexas.edu/teaching/sm1/lectures/node57.html)
5. [Experimental implementation of finite-time Carnot cycle, arXiv:2206.10153](https://arxiv.org/abs/2206.10153)
6. [The Feynman Lectures on Physics, Vol. I Ch. 44: The Laws of Thermodynamics](https://www.feynmanlectures.caltech.edu/I_44.html)
7. [6.1 Limitations on the Work that Can be Supplied by a Heat Engine, MIT 16 Unified Thermodynamics notes](https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node44.html)
8. [SP 330 Appendix 1: 10th CGPM, 1954, NIST](https://www.nist.gov/pml/special-publication-330/sp-330-appendix-1)
9. [Kelvin: Introduction, NIST SI Redefinition](https://www.nist.gov/si-redefinition/kelvin-introduction)
10. [PHYS 200 Lecture 23: The Second Law of Thermodynamics and Carnot's Engine, Open Yale Courses](https://oyc.yale.edu/physics/phys-200/lecture-23)
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*Topic: Encyclopedia › Technology and the built world › Engineering and manufacturing › Mechanical engineering › Heating, cooling, refrigeration, and heat pumps*
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