# Carnot cycle
The **Carnot cycle** is an ideal, fully reversible thermodynamic cycle consisting of two isothermal processes and two adiabatic processes, proposed by Sadi Carnot in 1824 as the model of a perfect heat engine operating between a hot reservoir at temperature \( T_h \) and a cold reservoir at temperature \( T_c \)[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle) • [2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH). Its efficiency, \( e = 1 - T_c/T_h \), depends only on the two reservoir temperatures and sets the maximum possible efficiency for any heat engine working between them[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH) • [3](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/03%3A_The_Second_Law_of_Thermodynamics/3.03%3A_Consequences_of_the_Second_Law).
| Key fact | Detail |
|---|---|
| Four steps | Isothermal expansion at \( T_h \), adiabatic expansion (\( T_h \to T_c \)), isothermal compression at \( T_c \), adiabatic compression (\( T_c \to T_h \))[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle) |
| Heat ratio | \( Q_c/Q_h = T_c/T_h \), the ratio of heat dumped to heat absorbed equals the ratio of absolute reservoir temperatures[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH) |
| Efficiency | \( e = 1 - T_c/T_h \), with temperatures in kelvins[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle) |
| Carnot's theorem | No engine between two given reservoirs exceeds Carnot efficiency; all reversible engines between two given temperatures have the same efficiency, independent of the working substance[3](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/03%3A_The_Second_Law_of_Thermodynamics/3.03%3A_Consequences_of_the_Second_Law) |
| Work output | Net work \( W = Q_h - Q_c \), equal to the area enclosed by the loop on the pV diagram[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle) |
| Power | Zero, because each step must run infinitesimally slowly to remain reversible; real engines reach about 0.7 of the Carnot efficiency at best[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH) |
| Reversed operation | Run backward it is a Carnot refrigerator with \( K_R = T_c/(T_h - T_c) \), or a Carnot heat pump with \( K_P = T_h/(T_h - T_c) \)[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle) |
## The four processes
The cycle is conventionally traced on a pressure–volume diagram through four states[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle):
1. **Isothermal expansion.** The gas, in thermal contact with the hot reservoir at \( T_h \), absorbs heat \( Q_h \) and expands, doing work. For an ideal gas the internal energy depends only on temperature, so \( \Delta E_{\text{int}} = 0 \) and, by the first law, the heat absorbed equals the work done[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle).
2. **Adiabatic expansion.** The gas is thermally isolated and expands further, doing work while its temperature falls from \( T_h \) to \( T_c \). The process obeys \( T V^{\gamma - 1} = \text{constant} \), so \( T_h V_N^{\gamma-1} = T_c V_O^{\gamma-1} \)[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle).
3. **Isothermal compression.** In contact with the cold reservoir at \( T_c \), the gas is compressed and gives up heat \( Q_c \) to the cold reservoir[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle).
4. **Adiabatic compression.** The gas is isolated again and compressed, its temperature rising from \( T_c \) back to \( T_h \), returning it to its initial state[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle).
Because the system returns to its initial state, the net change of internal energy over the cycle is zero, and the first law gives the net work as \( W = Q_h - Q_c \). This work equals the area enclosed by the loop on the pV diagram[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle).
## Efficiency
For an ideal gas, the isothermal heats are \( Q_h = n R T_h \ln(V_N/V_M) \) and \( Q_c = n R T_c \ln(V_O/V_P) \). Dividing the adiabatic relations of steps 2 and 4 shows \( V_O/V_P = V_N/V_M \), so the logarithmic factors cancel and \( Q_c/Q_h = T_c/T_h \). The efficiency is then[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle)
\[ e = 1 - \frac{T_c}{T_h}. \]
**Why this is the maximum.** [Carnot's theorem](https://www.edgepedia.org/carnots-theorem-thermodynamics), proved from the second law of thermodynamics, states that no engine operating between two specified heat reservoirs can be more efficient than a Carnot engine. The proof runs a supposed more efficient engine \( E_2 \) against a Carnot engine driven in reverse as a refrigerator; if \( E_2 \) were more efficient, the combined pair would convert heat from a single reservoir entirely into work, violating the Kelvin statement of the second law[3](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/03%3A_The_Second_Law_of_Thermodynamics/3.03%3A_Consequences_of_the_Second_Law). Two corollaries follow: all perfectly reversible engines operating between two given temperatures have the same efficiency, and that efficiency is independent of the working material[3](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/03%3A_The_Second_Law_of_Thermodynamics/3.03%3A_Consequences_of_the_Second_Law). Irreversible engines are strictly less efficient, since the reverse inequality cannot be established for them[3](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/03%3A_The_Second_Law_of_Thermodynamics/3.03%3A_Consequences_of_the_Second_Law).
The efficiency formula carries two practical messages. Efficiency rises as the ratio \( T_c/T_h \) shrinks, so the hottest possible hot reservoir and coldest possible cold reservoir give the greatest efficiency[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH). And 100% efficiency would require \( T_c = 0 \) K, a cold reservoir at absolute zero, which is a practical and theoretical impossibility[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH).
## Why the ideal engine has zero power
Reversibility requires each step to proceed through equilibrium states, that is, infinitesimally slowly, so a single Carnot cycle must be performed in infinite time and the ideal Carnot engine delivers zero power[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH). This makes it a benchmark rather than a design: no real heat engine can match the Carnot efficiency, and an actual efficiency of about 0.7 of this maximum is usually the best that can be accomplished[2](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH). The ratio of an actual engine's efficiency to the ideal efficiency at the same reservoir temperatures is used as a measure of how closely the engine approaches the ideal[4](https://www.sciencedirect.com/topics/mathematics/carnot-cycle).
## Reversed cycle: refrigerator and heat pump
Run in reverse, the Carnot cycle becomes an ideal refrigerator or heat pump. The refrigerator extracts heat \( Q_c \) from the cold reservoir during isothermal expansion, compresses adiabatically to \( T_h \), discards \( Q_h \) during isothermal compression, and returns to its initial state by adiabatic expansion. The work input per cycle is again \( W = Q_h - Q_c \), and the analysis gives \( Q_c/T_c = Q_h/T_h \), leading to a coefficient of performance[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle)
\[ K_R = \frac{T_c}{T_h - T_c} \]
for the Carnot refrigerator and
\[ K_P = \frac{T_h}{T_h - T_c} \]
for the Carnot heat pump[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle). These formulas, like the engine efficiency, hold for any working substance even though the textbook derivation assumes an ideal gas[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle).
## Historical role
Carnot summarized his study of the cycle in what is now called Carnot's principle: no engine working between two reservoirs at constant temperatures can have a greater efficiency than a reversible engine. This principle is equivalent to the Kelvin and Clausius statements of the second law, and the cycle provided the basis on which the second law was later formulated[1](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle) • [5](https://uomus.edu.iq/img/lectures21/MUCLecture_2022_8179701.pdf). A practical contrast illustrates the cycle's special position: other reversible engine designs, such as one with heating at constant volume like the [Otto cycle](https://www.edgepedia.org/otto-cycle), would need a whole series of heat reservoirs at intermediate temperatures to run reversibly, whereas the Carnot cycle achieves reversibility with only two[6](https://www.theory.physics.manchester.ac.uk/~judith/stat_therm/node20.html).
## References
1. [4.5 The Carnot Cycle, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle)
2. [15.4 Carnot's Perfect Heat Engine: The Second Law of Thermodynamics Restated, College Physics, OpenStax](https://openstax.org/books/college-physics/pages/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated?modal=MH)
3. [3.3: Consequences of the Second Law, Physics LibreTexts (Nair)](https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_%28Nair%29/03%3A_The_Second_Law_of_Thermodynamics/3.03%3A_Consequences_of_the_Second_Law)
4. [Carnot Cycle, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/carnot-cycle)
5. [Thermodynamics lecture notes, University of Mosul](https://uomus.edu.iq/img/lectures21/MUCLecture_2022_8179701.pdf)
6. [2.3 Carnot cycles, Statistical Thermodynamics, University of Manchester](https://www.theory.physics.manchester.ac.uk/~judith/stat_therm/node20.html)
---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Processes and cycles*
*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license