Carnot Cycle: The Absolute Limit on Engine Efficiency

A ceiling nobody can raise

In 1824, French engineer Sadi Carnot proved something that still governs every power plant and combustion engine built today: no heat engine operating between two fixed temperatures can be more efficient than a fully reversible engine running between the same two temperatures — and he derived exactly what that maximum efficiency is. A heat engine takes in heat Q_h from a hot reservoir, converts part of it into useful work W, and dumps the rest, Q_c, into a cold reservoir. Energy conservation (the first law) gives W = Q_h − Q_c, and thermal efficiency is η = W/Q_h. The second law, in its Kelvin-Planck form, forbids Q_c from ever reaching zero in a cyclic process — some waste heat is unavoidable, no matter how clever the design.

Four reversible steps

The Carnot cycle threads an idealised gas through four steps, alternating between letting heat flow and forbidding it entirely. Step one is an isothermal expansion at the hot temperature T_h: the gas expands slowly enough to stay at T_h throughout, absorbing heat Q_h as it does work. Step two is an adiabatic expansion: the gas keeps expanding but now with no heat exchange at all, so its temperature falls all the way from T_h to T_c purely from doing work. Step three is an isothermal compression at T_c, rejecting heat Q_c to the cold reservoir. Step four is an adiabatic compression back to the starting state, with temperature climbing from T_c back to T_h with no heat exchange. The two adiabatic legs contribute equal and opposite work, so they cancel exactly — the net work of the whole cycle comes entirely from the difference between the two isothermal legs.

Where the clean formula comes from

Plotted on a pressure-volume diagram, the area enclosed by the four curves is the net work extracted per cycle — isotherms trace hyperbolas (PV = nRT), adiabats trace steeper curves (PV^γ = const). Working through the heat absorbed and rejected on the two isothermal legs, the logarithmic volume-ratio terms cancel exactly because the adiabatic legs constrain the volume ratios to match, leaving a strikingly simple result that depends on temperature alone:

eta_Carnot = 1 - Qc/Qh = 1 - Tc/Th        (T in kelvin, absolute scale)

Example - steam turbine, Th = 600 K, Tc = 300 K:
  eta_max = 1 - 300/600 = 0.50 = 50%
  best real steam plants reach only ~42% (irreversibilities cost the rest)

Notice what is not in that formula: no working fluid, no engine geometry, no fuel type. Carnot's theorem guarantees that every reversible engine running between the same two reservoirs shares exactly this efficiency — if one design beat another, you could couple the winner forward to the loser running in reverse and pull net work out of a system with no net heat flow, which the second law forbids outright. That is the proof, and it is why raising T_h or lowering T_c is the only lever that ever improves a real engine's ceiling — a perfect engine running between body temperature and room temperature (310 K and 293 K) tops out at a mere 5.5%, no matter how well it is built.

The same cycle, seen through entropy

Entropy, defined through reversible heat exchange as dS = δQ_rev/T, turns the Carnot cycle into a perfect rectangle on a temperature-entropy diagram: entropy rises by Q_h/T_h during the hot isotherm, stays flat through both adiabats (no heat, no entropy change), then falls by Q_c/T_c during the cold isotherm. Because the cycle is reversible, those two changes must exactly cancel — Q_h/T_h = Q_c/T_c — which is precisely the algebraic step that produces the clean efficiency formula above. Any irreversible engine instead generates net entropy over a cycle, satisfying the Clausius inequality ∮δQ/T ≤ 0 rather than equality, and that extra generated entropy is exactly the efficiency it sacrifices relative to Carnot.

How real engines fall short — on purpose

No practical engine runs the Carnot cycle, because reversibility demands infinitely slow, infinitesimally small temperature differences at every step — an engine that slow delivers essentially zero power. Real cycles trade some efficiency for finite power output. The Otto cycle (petrol engines) swaps the isothermal legs for constant-volume ones and reaches roughly 25–35% in practice. The Diesel cycle uses a higher compression ratio for a modest edge, typically 35–45%. The Rankine cycle, the workhorse of steam power plants, chains a pump, boiler, turbine and condenser and reaches 40–42% with reheat and regeneration. The Stirling cycle is the interesting exception: built from two isotherms and two isochores with a regenerator that recycles heat internally, it can in principle match Carnot efficiency exactly — real regenerators are imperfect, so it falls short in practice, but the theoretical result is why Stirling engines appear in solar-dish concentrators and quiet submarine propulsion.

Run the entire cycle in reverse — consuming work instead of producing it — and you get a refrigerator or heat pump, with the Carnot bound now capping a coefficient of performance rather than an efficiency: COP_fridge = T_c/(T_h−T_c), COP_{heat pump} = T_h/(T_h−T_c). Engineers optimising for maximum power rather than maximum efficiency use the Curzon-Ahlborn formula η_CA = 1 − √(T_c/T_h), which more realistically predicts what finite-time, finite-rate real engines actually achieve.

How the simulation here uses it

The Carnot Cycle simulation on this site draws the four legs live on a PV diagram as you adjust T_h, T_c, and the compression ratio, recomputing Q_h, Q_c, net work and η = 1 − T_c/T_h after every change. Watching the enclosed loop area — literally the net work per cycle — grow and shrink as you widen or narrow the temperature gap is the fastest way to feel why the formula depends on nothing but two numbers: however you reshape the isotherms and adiabats, the area they enclose is bounded by exactly that temperature ratio.

Frequently asked questions

Try it live

Everything above runs in your browser — open Carnot Cycle and drag the hot and cold reservoir temperatures while it runs to watch the PV loop and the efficiency readout respond together. Nothing is installed, nothing is uploaded, the whole model lives in one tab.