Efficiency of The Carnot Cycle

The Carnot cycle consists of two isothermal and two adiabatic processes. During isothermal expansion at the hot reservoir temperature, the ideal gas absorbs heat. During isothermal compression at the cold reservoir temperature, it rejects heat. The two adiabatic processes connect these isothermal steps, with temperature and volume relationships governing the gas behavior throughout.

Internal energy of an ideal gas depends only on temperature. Since isothermal processes occur at constant temperature, the internal energy remains unchanged during these steps. This allows the first law of thermodynamics to relate the absorbed or rejected heat directly to the work done by or on the gas.

Carnot cycle efficiency is determined by combining equations relating heat exchanged, volumes, and temperatures across all four processes. The analysis reveals that efficiency depends only on the temperatures of the hot and cold reservoirs, expressed as a function of their temperature difference. This makes the Carnot cycle the most efficient theoretical heat engine operating between two fixed temperatures.

Carnot cycle efficiency increases as the temperature difference between the hot and cold reservoirs increases. The efficiency can approach unity only if the cold reservoir temperature approaches absolute zero, which is physically impossible. This demonstrates a fundamental limitation imposed by nature on heat engine performance.

The total work done by the Carnot cycle equals the area enclosed by the cycle on the pressure-volume diagram. This graphical representation shows the net work output as the difference between work done during expansion and work done during compression. The enclosed area directly quantifies the mechanical energy produced per cycle.

Adiabatic processes connect the two isothermal steps by changing the gas temperature without heat exchange. The relationship between temperature and volume during adiabatic compression and expansion allows the gas to transition between the hot and cold reservoir temperatures. These processes are essential for completing the reversible cycle and achieving maximum theoretical efficiency.

Since internal energy returns to its initial value after one complete Carnot cycle, the first law of thermodynamics requires that total heat absorbed minus total heat rejected equals total work done. The cycle's reversible nature ensures no energy is lost to irreversible processes, making this energy balance exact and fundamental to calculating cycle efficiency.