The perfect heat engine that no real machine can beat. A gas is taken around four reversible steps between a hot reservoir and a cold one — expanding while it draws heat in, being squeezed while it dumps heat out — and the loop it traces on a PV diagram encloses the net work. Because it is perfectly reversible, its efficiency hits the absolute ceiling, η = 1 − T꜀/Tʜ, set only by the two temperatures. Slide the hot and cold temperatures apart and watch the efficiency climb, the loop swell and the work grow. It all runs on your device.
Widen the gap between the hot and cold temperatures and watch efficiency climb
Efficiency η = 1 − T꜀/Tʜ62.5%
Work per cycle—
Heat in Qʜ—
Heat out Q꜀—
Only the two temperatures decide the ceiling. Run a gas around a perfectly reversible loop between a hot reservoir at Tʜ and a cold one at T꜀, and the fraction of the absorbed heat you can turn into work is fixed at η = 1 − T꜀/Tʜ — nothing about the gas, the size or the speed changes it. The work per cycle is the area of the loop, the difference Qʜ − Q꜀ between the heat drawn from the hot side and the heat dumped to the cold side. Push the reservoirs further apart and the loop, the work and the efficiency all grow.
How It Works
Four reversible steps, a loop whose area is the work, and an efficiency set only by temperature.
1
Two expansions, two compressions
Starting hot, the gas expands at constant temperature Tʜ while touching the hot reservoir, drawing in heat Qʜ. Insulated, it expands further and cools to T꜀. Touching the cold reservoir it is compressed at constant T꜀, dumping heat Q꜀. Insulated again, it is compressed back to the start and reheats to Tʜ. The red and blue curves on the diagram are the two isotherms; the grey ones are the insulated adiabatic steps.
2
The loop's area is the work
Each trip around the closed loop returns the gas to exactly where it began, so all the energy that went in as heat comes out as work or waste heat. The net work per cycle is the area enclosed by the loop on the PV diagram — shaded here — equal to Qʜ − Q꜀. A wider loop means more work.
3
The efficiency ceiling
Divide the work by the heat you paid for and, for a reversible Carnot engine, it comes out as exactly η = 1 − T꜀/Tʜ — the highest any engine can reach between those two temperatures. It hits 100 percent only if the cold reservoir is at absolute zero. Real engines have friction and finite-speed heat flow, so they always sit below this line; Carnot tells you where the line is.
What is the Carnot cycle?
The Carnot cycle is an idealised, perfectly reversible sequence of four steps that a heat engine can run between a hot reservoir and a cold one: isothermal expansion at the hot temperature, adiabatic expansion, isothermal compression at the cold temperature, and adiabatic compression back to the start. On a pressure-volume diagram it forms a closed loop whose enclosed area is the net work done per cycle. It is a theoretical benchmark, not a real machine, and no engine working between the same two temperatures can be more efficient.
What is the Carnot efficiency?
The Carnot efficiency is η = 1 − T꜀/Tʜ, where Tʜ and T꜀ are the absolute (kelvin) temperatures of the hot and cold reservoirs. It depends only on those two temperatures, not on the gas or the details of the engine. A bigger temperature gap gives a higher efficiency, but it can never reach 100 percent unless the cold reservoir is at absolute zero. Change the temperatures in this simulator and watch the efficiency track 1 − T꜀/Tʜ exactly.
Why can't a real engine reach Carnot efficiency?
Because the Carnot cycle assumes every step is perfectly reversible: no friction, no turbulence, and heat flowing only across an infinitesimal temperature difference, which would take infinitely long. Real engines have friction, finite-speed heat transfer, and other irreversible losses that generate entropy and waste energy, so they always fall short of the Carnot limit. The Carnot efficiency is the unreachable ceiling that tells engineers the best they could ever hope for between two temperatures.
What are the four steps of the Carnot cycle?
First, isothermal expansion: the gas touches the hot reservoir, absorbs heat Qʜ and expands at constant hot temperature. Second, adiabatic expansion: insulated from everything, it keeps expanding and cools from Tʜ to T꜀. Third, isothermal compression: it touches the cold reservoir and is compressed, dumping heat Q꜀ at constant cold temperature. Fourth, adiabatic compression: insulated again, it is compressed back to the start and warms from T꜀ to Tʜ. The net work equals Qʜ − Q꜀.