What Does Carnot’s Theorem Say About Heat Engines?

Carnot’s theorem gives a limit on how efficiently heat can be converted into work. It does not describe the performance of every real engine in detail; instead, it tells you the best possible efficiency for engines exchanging heat with the same two reservoirs.

The result depends only on the reservoirs’ absolute temperatures, not on whether the working substance is steam, air, or another material. This makes it a useful benchmark for evaluating an actual engine.

The two parts of Carnot’s theorem

The theorem is usually stated in two parts:

1. No engine can be more efficient than a reversible engine operating between the same hot and cold reservoirs.
2. All reversible engines operating between those same reservoirs have equal efficiency, whatever their working substances or detailed designs.

A reversible engine is an idealization. It must be possible to reverse its processes without leaving a net change in the engine or surroundings. Real friction, heat transfer across a finite temperature difference, and other dissipative effects make an engine irreversible.

The reservoirs are ideal bodies large enough to exchange heat without appreciably changing temperature. Let their temperatures be $T_h$ and $T_c$, where $T_h>T_c$. Temperatures in these equations must be absolute temperatures, such as kelvins.

Maximum efficiency formula

For an engine that absorbs heat $Q_h$ from the hot reservoir, rejects heat $Q_c$ to the cold reservoir, and produces work $W$, energy conservation over one complete cycle gives

$$W=Q_h-Q_c.$$

The thermal efficiency is the fraction of absorbed heat converted into work:

$$\eta=\frac{W}{Q_h}=1-\frac{Q_c}{Q_h}.$$

For a reversible engine, the entropy gained by the cold reservoir equals the entropy lost by the hot reservoir:

$$\frac{Q_c}{T_c}=\frac{Q_h}{T_h}.$$

Substituting this heat ratio into the efficiency expression gives the Carnot efficiency:

$$\eta_{\mathrm{Carnot}}=1-\frac{T_c}{T_h}.$$

The second law says the total entropy change of the reservoirs cannot be negative. With the heat quantities defined as positive magnitudes, this requires $Q_c/T_c-Q_h/T_h\geq 0$. Thus $Q_c/Q_h\geq T_c/T_h$, which leads to

$$\eta\leq 1-\frac{T_c}{T_h}.$$

Equality holds for a reversible engine; an irreversible engine falls below the bound.

Worked example

Suppose an engine operates between a hot reservoir at $600\,\mathrm{K}$ and a cold reservoir at $300\,\mathrm{K}$. Its maximum possible efficiency is

$$\eta_{\mathrm{Carnot}}=1-\frac{300}{600}=0.50.$$

So no engine between these reservoirs can convert more than 50% of its absorbed heat into work. If a reversible engine absorbs $1{,}000\,\mathrm{J}$, it can produce at most

$$W=0.50(1{,}000\,\mathrm{J})=500\,\mathrm{J},$$

and it must reject the remaining $500\,\mathrm{J}$ to the cold reservoir. An actual engine with 35% efficiency would produce $350\,\mathrm{J}$ from the same heat input and reject $650\,\mathrm{J}$, assuming the same two reservoirs.

How the Carnot cycle relates to the theorem

The ideal Carnot cycle is one way to realize the reversible limit. For a gas, it consists of two reversible isothermal processes—one at the hot-reservoir temperature and one at the cold-reservoir temperature—joined by two reversible adiabatic processes. “Adiabatic” means no heat crosses the boundary during that part of the cycle.

The Carnot theorem is broader than the particular cycle diagram: it applies to any cyclic heat engine between the same two reservoirs. A different reversible cycle may look different, but it cannot have a different efficiency if it uses the same reservoir temperatures.

Common mistakes

• Using Celsius temperatures: The ratio $T_c/T_h$ must use kelvins (or another absolute scale), not Celsius. For example, $300^\circ\mathrm{C}$ and $100^\circ\mathrm{C}$ do not give a valid Carnot ratio; convert them to $573.15\,\mathrm{K}$ and $373.15\,\mathrm{K}$ first.
• Expecting 100% efficiency: The formula gives 100% only if $T_c=0\,\mathrm{K}$. An engine must reject some heat when its cold reservoir is above absolute zero.
• Comparing engines with different reservoirs: The theorem’s direct comparison applies when both engines operate between the same two reservoir temperatures. A hotter source or colder sink changes the maximum possible efficiency.
• Treating the limit as a real-engine prediction: Carnot efficiency is an upper bound, not a claim that a practical engine can attain it. Reversible operation requires idealized conditions, including vanishingly small temperature differences during heat transfer.

The same reversible-cycle idea also applies in reverse: a reversible refrigerator or heat pump has the best possible performance between two specified temperatures. Its coefficient of performance is a different quantity from heat-engine efficiency, but it is limited by the same reservoir temperatures.