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How does the Carnot cycle set the maximum possible efficiency for heat engines?

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Engraved portrait of Rudolf Julius Emanuel Clausius, a bearded man in formal attire with sideburns, facing forward. — thermodynamics Rudolf Clausius (1822 - 1888) is considered one of the central founders of the science of thermodynamics.

The Carnot cycle establishes the theoretical upper limit for the efficiency of any heat engine operating between two temperatures. It demonstrates that 100 percent efficiency is physically impossible, as some waste heat must always be rejected to the environment to satisfy the second law of thermodynamics.12

The maximum efficiency ( $η$ ) is determined solely by the absolute temperatures of the heat source ( $T_{1}$ ) and the heat sink ( $T_{2}$ ):

$η = 1 - \frac{T_{2}}{T_{1}}$

This limit arises because the net entropy change of the system and its reservoirs must be greater than or equal to zero ( $Δ S \geq 0$ ). To maximize work, one must minimize the exhausted heat ( $Q_{2}$ ); however, $Q_{2}$ cannot be zero without violating the second law. The Carnot cycle represents a reversible process where $Δ S = 0$ , providing the highest possible ratio of work output to heat input for any engine operating between those specific temperatures. While real-world engines fall short of this limit due to factors like friction, the Carnot principle remains a fundamental constraint of nature that no design can surpass.32