21.9:
The Carnot Cycle and the Second Law of Thermodynamics
In the Carnot cycle, since all the steps are reversible, the cycle itself is reversible. A Carnot engine that operates in reverse converts work into heat. It is called a Carnot refrigerator.
Let a Carnot refrigerator operate between hot and cold reservoirs of temperatures T–h and T–c, respectively.
The relationship between the heat rejected to the hot reservoir, Q–h, the heat extracted from the cold reservoir, Q–c, and the known temperatures T–h and T–c, is the same as a Carnot engine.
Suppose it was possible to have a heat engine working between the same temperatures but having greater efficiency than a Carnot engine. Then, it could be combined with a Carnot refrigerator to show that the combined engine is equivalent to a perfect heat engine which violates the second law of thermodynamics.
It motivates the Carnot Principle, which states that no engine working between two reservoirs can have greater efficiency than a reversible engine.
The principle helps us define ideal, reversible processes as those that maximize the efficiency of a heat engine operating between two temperatures.
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 Qc from a cold reservoir at temperature Tc when the ideal gas is allowed to expand isothermally. The gas is then compressed adiabatically until its temperature reaches Th. Isothermal compression of the gas results in heat Qh rejected to a hot reservoir at temperature Th. The gas is then allowed to expand adiabatically, causing its temperature to drop to Tc 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 Qc and Qh can be related to the known temperatures Tc and Th 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.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 20.6; page 667.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 4.5; page 157.