In 1824 the young French engineer Sadi Carnot published a small treatise entitled Réflexions sur la puissance motrice du feu — Reflections on the Motive Power of Fire. The question was practical: what is the maximum work you can get out of a steam engine? The answer became one of the deepest insights of thermodynamics: the efficiency depends only on the temperatures of the hot and the cold reservoir — not on the working fluid, not on the design of the engine.
The cycle consists of four reversible processes: two isothermal (at constant temperature) and two adiabatic (with no heat exchange). Press play below and follow the gas round the loop.
Adjust the cycle's parameters
Slide Th higher or Tc lower — you will see the efficiency rise. That is the essence of the second law of thermodynamics.
The four processes — formulae
For an ideal gas with n moles, gas constant R and adiabatic index γ:
① Isothermal expansion at Th
Qh = nRTh · ln(V₂/V₁) ΔU = 0 ⇒ W₁ = QhThe gas is in contact with the hot reservoir. It expands slowly, takes in heat Qh and does an equal amount of work.
② Adiabatic expansion
T·Vγ−1 = constant W₂ = nCv · (Th − Tc)The cylinder is insulated. The gas expands further, but now at the expense of its own internal energy — the temperature falls from Th to Tc.
③ Isothermal compression at Tc
Qc = nRTc · ln(V₃/V₄) ΔU = 0 ⇒ W₃ = −QcThe gas is in contact with the cold reservoir. It is compressed, gives out heat Qc, and work is done on the gas.
④ Adiabatic compression
T·Vγ−1 = constant W₄ = −nCv · (Th − Tc)Insulated again. The continued compression raises the gas temperature from Tc back to Th. The cycle is closed.
The beautiful result
The adiabats ensure that V₂/V₁ = V₃/V₄. So the net work is:
The efficiency depends solely on the two temperatures — not on the gas, not on the volumes.
Worked example
A steam power station · Th = 773 K (500 °C), Tc = 298 K (25 °C)
A modern coal-fired power station has a boiler at 500 °C and a condenser at 25 °C (ambient temperature). What is the theoretical maximum efficiency?
Question: How much work do we get per kg of coal burned (heat of combustion ≈ 30 MJ/kg)?
In a real power station (η = 40%): W = 0.40 × 30 MJ = 12 MJ per kg of coal
Mole-by-mole calculation · 1 mol helium, Th = 600 K, Tc = 300 K, V₂/V₁ = 2
Comparison with real engines
The Carnot efficiency is the ceiling no real engine can reach. Here is a selection of engines, their typical operating temperatures and how close they come:
| Engine | Th | Tc | ηCarnot | η (real) |
|---|---|---|---|---|
| Steam locomotive (classic) | 473 K (200 °C) | 373 K (100 °C) | 21.1 % | ~6–10 % |
| Car engine (petrol) | 2300 K | 400 K | 82.6 % | ~30 % |
| Diesel engine (ship) | 2100 K | 500 K | 76.2 % | ~50 % |
| Coal-fired power station | 773 K (500 °C) | 298 K (25 °C) | 61.4 % | ~40 % |
| Combined-cycle gas power station | 1700 K | 300 K | 82.4 % | ~60 % |
| Geothermal power station | 473 K (200 °C) | 290 K (17 °C) | 38.7 % | ~12 % |
Notice the interesting point: the car engine has the highest Carnot ceiling (over 80%) but reaches only about 30%. It is limited not mainly by thermodynamics but by irreversibilities — fast combustion, friction, heat loss through the cylinder, incomplete exhaust. The coal-fired power station, by contrast, comes considerably closer to its own Carnot ceiling: it is a large, steady-state plant in which heat is recovered at several stages (feed-water heating, reheating) and losses can be kept small.
Refrigerator and heat pump — Carnot in reverse
If the Carnot cycle is run backwards (anticlockwise in the P-V diagram), it becomes a refrigerator: work is put in, and heat is pumped from the cold to the hot reservoir. The measure of performance is not η but the COP (coefficient of performance):
A heat pump that lifts heat from 0 °C outdoors (273 K) to 35 °C in underfloor heating (308 K) has a Carnot COP = 308/35 ≈ 8.8 — that is, 8.8 kWh of heat delivered per kWh of electricity. Real heat pumps reach 3–5. That is still a factor of 3–5 better than direct electric heating, and explains why heat pumps are thermodynamics' best friend in the green transition.
What Carnot taught us
Carnot died of cholera in 1832, only 36 years old. His treatise attracted little attention in his own time, but when Rudolf Clausius and William Thomson took it up in the late 1840s and 1850s, it became the cornerstone of the whole of thermodynamics. Three deep consequences:
- The second law of thermodynamics: heat flows spontaneously from hot to cold — not the other way — and every conversion of heat into work has a fundamental ceiling.
- The absolute temperature scale: Kelvin defined T so that ηCarnot = 1 − Tc/Th is exact. Temperature was defined through the laws of thermodynamics, not through a particular liquid.
- Entropy: Clausius discovered that Qh/Th = Qc/Tc for the Carnot cycle. This became the definition of a new state function — the entropy S — and gave birth to the concept of irreversibility.