What Is the Carnot Cycle and What's the Maximum Possible Efficiency of a Heat Engine?

HL only (B.4 Thermodynamics). The Carnot cycle is an ideal, completely reversible heat-engine cycle operating between hot and cold reservoirs. Its maximum possible efficiency is $\eta_{\text{Carnot}}=1-\frac{T_c}{T_h}$, where both temperatures must be in kelvin.

How the Carnot Cycle Works

A heat engine absorbs thermal energy $Q_h$ from a hot reservoir, converts part of it into useful work $W$, and transfers the remainder $Q_c$ to a cold reservoir. The Carnot cycle consists of four reversible processes involving an ideal gas:

The net work done by the engine equals the area enclosed by the cycle on a pressure-volume diagram. Conservation of energy gives $W=Q_h-Q_c$, so

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

For a reversible Carnot engine, $\frac{Q_c}{Q_h}=\frac{T_c}{T_h}$, producing the maximum theoretical efficiency.

For example, if $T_h=600\,\text{K}$ and $T_c=300\,\text{K}$,

$$\eta_{\text{Carnot}}=1-\frac{300}{600}=0.50=50\%.$$

A common misconception is that 100% efficiency is achievable. It would require $T_c=0\,\text{K}$, which is unattainable; real engines are even less efficient because their processes are irreversible and involve friction and unwanted thermal transfers.

Exam Technique

For a Calculate question, convert temperatures to kelvin before substitution and express efficiency as either a decimal or percentage. For an Explain question, state that the Carnot efficiency is an upper limit because the Carnot cycle is reversible, whereas real engines increase total entropy through irreversible processes.