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Heat engine efficiency is η = W_net / Q_in. The Carnot engine, operating between two thermal reservoirs, achieves maximum efficiency: η_Carnot = 1 − T_C/T_H. No real engine can exceed this; it is an upper bound set by the second law. This shows the fundamental limit imposed by thermodynamics.
A heat engine is any device that converts thermal energy into mechanical work by operating cyclically between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. From your study of the Carnot cycle, you know one specific example: the ideal engine that runs through two isothermal and two adiabatic steps. Engine efficiency is defined as the fraction of the heat absorbed from the hot reservoir that gets converted to useful work: η = W_net / Q_H. Since energy is conserved over one cycle (ΔU = 0), we have W_net = Q_H − Q_C, so η = 1 − Q_C / Q_H. The question is: how small can Q_C / Q_H be?
The answer comes from the second law of thermodynamics. In any cyclic process, the total entropy change of the universe must be non-negative. The engine absorbs Q_H from the hot reservoir (lowering its entropy by Q_H / T_H) and dumps Q_C into the cold reservoir (raising its entropy by Q_C / T_C). The second law requires: Q_C / T_C − Q_H / T_H ≥ 0, which means Q_C / Q_H ≥ T_C / T_H. Substituting into the efficiency formula: η ≤ 1 − T_C / T_H. The Carnot efficiency η_C = 1 − T_C / T_H is the upper bound — achieved only when the entropy inequality is an equality, which happens only for a reversible engine.
Carnot's theorem states this more directly: no engine operating between two thermal reservoirs can be more efficient than a reversible engine operating between those same two reservoirs. All reversible engines operating between T_H and T_C achieve exactly η_C, regardless of their working substance or cycle details. Any irreversible engine achieves less. This is not an engineering limitation waiting to be overcome with better materials — it is a consequence of the second law of thermodynamics, as fundamental as energy conservation.
The practical implications are sobering. A coal power plant operating between a flame at roughly 800 K and the environment at 300 K faces a Carnot limit of 1 − 300/800 ≈ 62.5%. Real plants achieve 35–45% due to irreversibilities. A car engine operating between combustion temperatures (~2000 K) and ambient (~300 K) has a Carnot limit near 85%, but real engines achieve 25–35%. The gap between ideal and actual efficiency is the engineer's challenge, but the ceiling itself is set by T_C / T_H — which is why industrial processes burn fuel as hot as possible and exhaust heat as cold as possible. The efficiency formula η_C = 1 − T_C / T_H also shows that η_C → 1 only when T_C → 0 (absolute zero) or T_H → ∞, neither of which is achievable in practice.