Heat Engines and Thermodynamic Cycles
A heat engine takes in heat from a hot reservoir, converts part of it to useful work, and dumps the rest to a cold reservoir — no engine can convert 100% of input heat to work, a direct consequence of the second law, not just an engineering limitation. The Carnot cycle is the theoretical best-case engine operating between two temperatures, and its efficiency, e=1−Tc/Th, is a hard ceiling that no real engine operating between those same two temperatures can ever exceed, however cleverly designed.
What you'll learn
- Calculate the efficiency of a heat engine from the heat absorbed and work output (or heat rejected).
- Calculate the theoretical maximum (Carnot) efficiency for an engine operating between two given temperatures.
- Explain why no real engine can exceed the Carnot efficiency for its operating temperatures.
- Calculate the coefficient of performance (COP) of a refrigerator or heat pump.
- Relate a heat engine's efficiency to the heat absorbed, heat rejected, and work output using energy conservation.
- Interpret a PV diagram for a thermodynamic cycle, including identifying the net work done per cycle.
1. Heat Engines and Efficiency
A heat engine operates in a cycle, absorbing heat Qh from a hot reservoir, converting some of it to useful work W, and rejecting the remainder Qc to a cold reservoir. Since the engine returns to its starting state every cycle (ΔU=0), energy conservation requires Qh=W+Qc, giving efficiency e=W/Qh=1−Qc/Qh.
No real engine converts 100% of absorbed heat to work — some waste heat rejection is unavoidable, a direct requirement of the second law, not merely a practical engineering shortfall.
2. The Carnot Limit
The Carnot cycle describes an idealized, perfectly reversible engine — the theoretical best possible performance for any engine operating between two fixed temperatures. Its efficiency, e_Carnot=1−Tc/Th (temperatures in Kelvin), is a hard ceiling: the second law guarantees no real engine, however cleverly built, can exceed it. Real engines always have some irreversibility (friction, rapid uncontrolled heat transfer), which is exactly what keeps their actual efficiency below this Carnot limit.
3. Refrigerators and Heat Pumps
A refrigerator or heat pump runs a heat engine's cycle in reverse: work input is used to move heat from a cold region to a hot one, rather than letting heat flow spontaneously (and using some of that flow to do work). Performance is measured by coefficient of performance rather than efficiency — COP=Qc/W for a refrigerator (heat removed from the cold space per unit work) or COP=Qh/W for a heat pump (heat delivered to the warm space per unit work) — and unlike engine efficiency, COP can and typically does exceed 1, since it compares heat moved (not heat created) to work spent.
Key equations
- e = W/Qh = 1 − Qc/Qh — The efficiency of any heat engine — the fraction of absorbed heat converted to useful work rather than dumped as waste heat.
- e_Carnot = 1 − Tc/Th — The theoretical maximum efficiency for any engine operating between temperatures Th and Tc — a hard upper limit set by the second law, achieved only by an idealized, perfectly reversible Carnot cycle.
- COP_refrigerator = Qc/W, COP_heat pump = Qh/W — Coefficient of performance — how much heat is moved per unit of work input, for a refrigerator (heat removed from the cold space) or a heat pump (heat delivered to the warm space). Unlike engine efficiency, COP can exceed 1.