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Why can no heat engine convert all of its heat input into useful work?

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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.

No heat engine can convert all of its heat input into useful work because doing so would violate the second law of thermodynamics.12

A heat engine operates by absorbing heat ( $Q_{1}$ ) from a high-temperature reservoir, converting a portion into work ( $W$ ), and exhausting the remaining waste heat ( $Q_{2}$ ) to a lower-temperature reservoir. According to the conservation of energy, the work done per cycle is $W = Q_{1} - Q_{2}$ .12

If an engine were to convert all heat into work, $Q_{2}$ would be zero. However, the second law requires that the net entropy change of the system and its surroundings must be greater than or equal to zero ( $Δ S \geq 0$ ). If $Q_{2}$ were zero, the entropy change would be negative, which is physically impossible. Consequently, a portion of the input heat must always be exhausted as waste heat to maintain or increase the total entropy of the universe. This fundamental limitation means that even a perfectly designed engine can never reach 100 percent efficiency.132