21.8:
Efficiency of The Carnot Cycle
In a Carnot cycle, the known quantities are the temperature of the hot reservoir, T–h, and the temperature of the cold reservoir, T–c.
The ideal gas absorbs heat Q–h during its isothermal expansion at T–h and rejects heat Q–c during its isothermal compression at T–c.
Since the internal energy of an ideal gas depends only on its temperature, it remains constant during these two steps. Using the first law of thermodynamics, the absorbed heat can be calculated. While Q–h depends on the gas’s volumes before and after the expansion, Q–c depends on the gas’s volumes before and after the contraction.
For the adiabatic processes in the Carnot cycle, the relationship between the temperature and volume of gases can be used. The four equations can be combined to relate the ratio of heat exchanged and the temperatures of the reservoirs.
When combined with the expression of the efficiency of a heat engine, it implies that the efficiency of a Carnot cycle depends only on the temperatures T–c and T–h.
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.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 20.6; page 663.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 4.5; page 155.