21.8 : Efficiency of The Carnot Cycle

The hypothetical Carnot cycle consists of an ideal gas subjected to two isothermal and two adiabatic processes. Since the internal energy of an ideal gas depends only on its temperature, which is the same before and after the completion of the Carnot cycle, there is no change in its internal energy. Hence, using the first law of thermodynamics, the total heat exchanged by the ideal gas equals the total work done. Thus, we can quantify the efficiency of the Carnot cycle via the heat exchanged between the gas and the hot and cold reservoirs.

The total work done equals the area enclosed by the cycle on the p-V diagram.

An ideal gas equation of state also helps to connect the gas’s volumes and temperatures at various steps. The relationship between temperature and volume of an ideal gas subject to an adiabatic process helps relate its volumes before and after the isothermal expansions. The efficiency of the Carnot cycle can be calculated by combining all these equations.

It is found that the efficiency depends on the difference between the temperatures of the hot and cold reservoirs. Hence, the efficiency increases if the temperature difference increases. It can approach unity only if the temperature of the cold reservoir is zero, which is another of nature’s impossibilities.