The Carnot Cycle
Walk the four processes with the piston moving alongside, and find an efficiency that depends on nothing but two temperatures.
Skip to the animationThe Carnot cycle is the only reversible cycle possible between two reservoirs — two isothermals and two adiabatics — and its efficiency depends on nothing but the two temperatures, which makes it the ceiling every real engine is measured against.
Why its shape is forced
Reversibility forbids heat transfer across a finite temperature difference. So heat may only be exchanged while the gas is *at* the reservoir temperature — an isothermal process — and any temperature change must happen with no heat crossing at all, which is adiabatic.
Two of each is the only way to close a loop between two reservoirs. The shape is not chosen for elegance; it is the only option available.
The four processes
- 11→2 isothermal expansion at T_H. Heat
Q_His absorbed. For an ideal gasΔU = 0, so all of it leaves immediately as work. - 22→3 adiabatic expansion. Insulated, so the work done comes out of internal energy and the temperature falls to
T_C. - 33→4 isothermal compression at T_C. Heat
Q_Cis rejected — the second law's tax, made concrete. - 44→1 adiabatic compression. The temperature returns to
T_Hand every property is back where it began.
Round the loop ΔU = 0, so net heat in equals net work out — and that net work is the area enclosed on the p-V diagram.
Carnot's theorem
Efficiency is 1 − T_C/T_H, in kelvin, and no working fluid appears in it. Two corollaries follow: no engine between two reservoirs can beat a reversible one, and all reversible engines between the same two reservoirs have identical efficiency.
| T_H | T_C | η_max |
|---|---|---|
| 400 K | 300 K | 25% |
| 600 K | 300 K | 50% |
| 900 K | 300 K | 67% |
| 1500 K | 300 K | 80% |
Raising T_H helps far more than lowering T_C, which is why materials science — how hot the turbine inlet may be allowed to get — is what actually drives power plant efficiency.
Why it is unbuildable, and still the most useful cycle
Reversible means zero ΔT for heat transfer, which means zero rate: a Carnot engine takes forever to produce anything. Its value is as a ceiling.
A plant achieving 40% between 600 K and 300 K is at 80% of the Carnot limit. That ratio — the second-law efficiency — is the number that actually says whether a design is good, because it separates "limited by physics" from "limited by engineering".
The cycle also defines the thermodynamic temperature scale: since Q_H/Q_C = T_H/T_C for any reversible engine, temperature can be defined by heat ratios alone, with no reference to any substance.
Run backwards
Reversed, the cycle becomes a refrigerator or heat pump with COP_R = T_C/(T_H − T_C) and COP_HP = T_H/(T_H − T_C). Both are maxima, and both grow as the temperature difference shrinks — which is why a heat pump is efficient in mild weather and much less so in severe cold.
The numbers you will be asked for
- Carnot efficiency
η = 1 − T_C/T_H
Kelvin. No working fluid appears.
- Heat ratio
Q_H/Q_C = T_H/T_C
True for any reversible engine; it defines the absolute scale.
- Isothermal work
W = mRT·ln(V₂/V₁)
For the two isothermal legs of an ideal gas cycle.
- Second-law efficiency
η_II = η_actual / η_Carnot
The number that says whether a design is good.
- Reversed cycle
COP_R = T_C/(T_H − T_C)
Grows as the temperature lift shrinks.
Advantages and disadvantages
Advantages
- The maximum efficiency achievable between two reservoirs.
- Depends only on temperatures, so it is universal.
- Provides the yardstick for every real cycle.
- Defines a temperature scale independent of any substance.
Disadvantages
- Unbuildable: reversibility requires an infinitely slow process.
- Isothermal heat transfer cannot be arranged in a real cylinder.
- Very low work output per cycle for the swept volume used.
- The efficiency figure alone says nothing about power, which is what an engine is bought for.
Watch it work
Check yourself
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