Thermal physics

Heat engines, efficiency, and the Carnot cycle

A heat engine converts part of the heat received from a hot reservoir into work and rejects the rest to a cold reservoir; the second law limits its maximum efficiency.

A heat engine runs cyclically: it absorbs heat $Q_h$ from a hot reservoir, produces net work $W$, and rejects heat $Q_c$ to a cold reservoir. After each cycle its working substance returns to its initial state.

The Carnot cycle

The Carnot cycle consists of two reversible isothermal processes at $T_h,T_c$ joined by two reversible adiabatic processes. It is an ideal limit, not a cycle fully realizable in finite time. Any engine between the same reservoirs satisfies $\eta\le1-T_c/T_h$.

Quick check

Over a complete cycle, the working substance returns to its initial state, so its internal-energy change is zero. The first law then gives $W=Q_h-Q_c$: output work is the portion of absorbed heat not rejected to the cold reservoir. Efficiency is below one whenever $Q_c>0$; rejecting heat is unavoidable for a cyclic engine operating between reservoirs. The Carnot limit depends only on the reservoirs' absolute temperatures, not on the identity of the working substance. Raising $T_h$ or lowering $T_c$ can raise the ideal limit, though materials and engineering safety constrain real designs. A refrigerator runs a heat engine in reverse: it uses work to remove heat from a cold region and release it into a warmer room. Thus distinguish engine efficiency from the coefficient of performance of a refrigerator; both are constrained by the second law but are defined differently.

If an engine absorbs 800 J and rejects 500 J each cycle, its output work is 300 J and its efficiency is $300/800=37.5$. Comparing this value with the Carnot limit checks whether the claimed performance is possible; exceeding the limit signals a sign or calculation error.