Physics/Option Modules/Engineering Physics/Carnot Cycle
Physics · Engineering Physics · A2 Level

Carnot Cycle

If energy conservation were the only rule that mattered, nothing would stop an engine from converting heat completely into work with perfect efficiency and no waste. No such engine has ever existed — and the reason why sets a hard ceiling on how efficient any real engine can possibly be.

9 min read

Why the first law alone isn't enough

The first law of thermodynamics is only a bookkeeping rule: it demands that energy balances, but it says nothing about which direction energy transfers are allowed to happen in. Taken purely at face value, the first law would permit an engine that takes in heat $Q_H$ from a hot source and converts every last joule of it directly into useful work, with no waste heat rejected anywhere at all — a 100% efficient heat engine. Energy would balance perfectly; nothing in $Q = \Delta U + W$ forbids it.

And yet no engine like this has ever been built, and there is no reason to believe one ever could be. This isn't a failure of engineering — it reflects a genuine, separate law of nature that the first law simply doesn't capture, which is exactly the gap the second law of thermodynamics fills.

The second law, for engines

Applied specifically to heat engines, the second law states that a heat engine must operate between a hot source and a cooler sink: it draws in heat $Q_H$ from the hot source, converts some fraction of it into useful work $W$, and is required to reject the rest as waste heat $Q_C$ to the cold sink. An engine can never convert all of the heat it takes in into work — some heat rejection to a cooler reservoir is unavoidable.

This gives a natural definition of efficiency for any heat engine:

$$\text{efficiency} = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H}$$

where the equivalence of the two expressions follows directly from energy conservation: whatever heat isn't converted to work ($Q_H - W$) must be the rejected heat $Q_C$, so $W = Q_H - Q_C$.

A schematic showing a hot reservoir bar labelled T-hot at the top and a cold reservoir bar labelled T-cold at the bottom, with a circle labelled Engine positioned between them. An arrow labelled Q-in points down from the hot reservoir into the engine, an arrow labelled Q-out points down from the engine into the cold reservoir, and an arrow labelled W-useful points out to the right from the engine.
Every heat engine draws heat Q_H from a hot reservoir, converts part of it into useful work W, and must reject the remainder as waste heat Q_C to a cold reservoir — it can never convert all of Q_H into work.

The Carnot limit

Even granting that some heat rejection is unavoidable, a further question remains: what's the best efficiency any heat engine could possibly achieve, operating between a given hot source at temperature $T_H$ and cold sink at temperature $T_C$? The answer is the theoretical maximum, or Carnot efficiency:

Worked ExampleMaximum theoretical efficiency
1
The maximum theoretical efficiency of any heat engine operating between absolute temperatures $T_H$ and $T_C$ is given by $\eta_{\max} = \dfrac{T_H - T_C}{T_H}$.
Both temperatures must be in kelvin — this formula comes directly from the absolute (kinetic-energy) temperature scale, exactly as with $pV=nRT$.
2
A power station's boiler produces steam at $T_H = 820\ \text{K}$, and its condenser (the cold sink) operates at $T_C = 310\ \text{K}$. Find the maximum theoretical efficiency.
3
$\eta_{\max} = \dfrac{820 - 310}{820} = \dfrac{510}{820} = 0.622$.
4
The theoretical maximum efficiency is 62.2% — no engine operating between these two temperatures, however cleverly designed, could ever exceed this figure.
A real power station running between these temperatures will always achieve somewhat less than 62.2% in practice, for the reasons discussed below.
Result

The Carnot efficiency depends only on the two operating temperatures — nothing about the engine's design, fuel, or mechanism can push efficiency above this limit, though poor engineering can certainly leave an engine well short of it.

A ceiling, not a target

Real engines always fall well short of the Carnot efficiency calculated for their operating temperatures. Friction between moving parts, heat losses to the surroundings through cylinder walls and exhaust gases, and the fact that real gas processes are never perfectly reversible (unlike the idealised processes the Carnot limit assumes) all eat into the efficiency actually achieved. The Carnot efficiency should be understood as a hard theoretical ceiling set by the temperatures alone — not a realistic target that a real, physical engine could ever actually reach.

Closing the gap: combined heat and power

Since some heat rejection to a cold sink is unavoidable by the second law, one practical strategy for getting more useful benefit out of a given amount of fuel is to stop treating the rejected heat $Q_C$ as pure waste. In a combined heat and power (CHP) scheme, the heat that would otherwise be dumped into the cold sink and lost is instead captured and put to direct use — commonly for space heating or hot water in a building near the plant, rather than being discarded into the atmosphere or a river.

This doesn't change the engine's work-output efficiency, $W/Q_H$, at all — that's still governed by the same second-law limits as before. But by making productive use of both the work output $W$ and a large fraction of the heat $Q_H$ supplied (rather than letting the rejected portion go entirely to waste), the overall useful-energy yield from the fuel rises substantially. Maximising the use of both $W$ and $Q_H$, rather than treating waste heat as a fixed and unavoidable cost, is the guiding idea behind schemes like this.

Key takeaways

  • The first law alone cannot rule out a 100%-efficient engine, since it only requires energy to balance — the second law is what actually forbids this.
  • A heat engine must operate between a hot source and a cold sink, taking in $Q_H$, doing useful work $W$, and rejecting waste heat $Q_C$ — it can never convert all of $Q_H$ into work.
  • Efficiency $= W/Q_H = (Q_H - Q_C)/Q_H$.
  • The maximum theoretical (Carnot) efficiency between temperatures $T_H$ and $T_C$ (in kelvin) is $(T_H - T_C)/T_H$.
  • The Carnot efficiency depends only on the two operating temperatures, and is a theoretical ceiling, never actually achieved by a real engine.
  • Real engines fall short of this ceiling due to friction, heat losses, and irreversible (non-ideal) processes.
  • Combined heat and power schemes raise the overall useful-energy yield from fuel by capturing and using rejected heat productively, without changing the underlying work-output efficiency.
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