21.9 : The Carnot Cycle and the Second Law of Thermodynamics

The Carnot engine works between two heat reservoirs of fixed temperatures. The Carnot cycle begs the following question: Is it possible to devise a heat engine that is more efficient than a Carnot engine between two fixed temperatures? The answer lies in designing a Carnot refrigerator.

Since the individual steps in a Carnot cycle can be reversed, the entire cycle is, thus, reversible. If a Carnot cycle is reversed, it becomes a Carnot refrigerator. It extracts heat Q_c from a cold reservoir at temperature T_c when the ideal gas is allowed to expand isothermally. The gas is then compressed adiabatically until its temperature reaches T_h. Isothermal compression of the gas results in heat Q_h rejected to a hot reservoir at temperature T_h. The gas is then allowed to expand adiabatically, causing its temperature to drop to T_c and, thus, completing the cycle.

The relationship between the temperature and volume of the ideal gas remains the same whichever way the engine runs. Hence, the unknown quantities Q_c and Q_h can be related to the known temperatures T_c and T_h in the same way as for a forward Carnot cycle. The total work done by the gas is also similarly given by the area enclosed by the p-V curve.

Assume that a heat engine can be designed to operate between the same two temperatures but to have an efficiency greater than a Carnot engine. Then, combine this heat engine with a Carnot refrigerator. Effectively, the combined engine extracts a net amount of heat from the hot reservoir and converts it to work without any other effect. Thus, the combination violates the second law of thermodynamics.

A corollary of this is the Carnot principle. It helps us define ideal reversible systems and sets an upper limit of the maximum achievable efficiency of any heat engine operating between two fixed temperatures.