Carnot Cycle and Efficiency

The Carnot cycle is a fully reversible ideal heat engine cycle consisting of four stages: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. It demonstrates the maximum theoretical efficiency allowed by the second law of thermodynamics, establishing the upper limit for all real heat engines. Because the system returns to its initial state, the net work produced equals the net heat transferred.

Carnot engine efficiency is defined as the ratio of work performed to heat absorbed from the hot reservoir. It can be expressed as η = 1 − |qc|/qh, where qc is heat rejected to the cold reservoir and qh is heat absorbed from the hot reservoir. Efficiency can also be expressed solely in terms of the temperatures of the cold and hot reservoirs, showing that efficiency depends only on these temperature values.

The second law of thermodynamics states that no engine can be 100% efficient because some heat must always be rejected to a cold reservoir. Since the magnitude of rejected heat can never exceed the absorbed heat, efficiency always lies between zero and one. This fundamental limitation reflects the irreversible nature of real thermodynamic processes and energy dissipation.

Although entropy changes during individual stages of the Carnot cycle, the total change in entropy over one complete cycle is zero. During isothermal expansion, entropy increases by qh/Th; during adiabatic expansion, it remains constant. During isothermal compression, entropy decreases by |qc|/Tc; during adiabatic compression, it remains constant. These changes cancel out completely over the full cycle.

In the Carnot cycle, the relationship between heat and temperature is expressed as qh/Th − |qc|/Tc = 0, derived from energy conservation and the reversibility of the cycle. This equation shows that the ratio of heat absorbed at the hot reservoir temperature equals the ratio of heat rejected at the cold reservoir temperature. This relationship is fundamental to understanding why Carnot efficiency depends only on the two reservoir temperatures.

The net work produced by a Carnot engine equals the area enclosed by the cycle on a pressure-volume diagram. This enclosed area represents the net work output as the gas undergoes the four stages of expansion and compression. The larger the enclosed area, the more work the engine produces per cycle, making the P-V diagram a useful tool for visualizing thermodynamic efficiency.

The Carnot cycle is reversible because each of its four stages can be reversed without any net change to the system or surroundings. The isothermal and adiabatic processes are quasi-static, occurring infinitely slowly with no friction or irreversibilities. Reversibility ensures that the total entropy change over the complete cycle is zero, allowing the engine to operate at maximum theoretical efficiency and serve as the ideal benchmark for all real heat engines.