Question

Why the temperature of the hot reservoir has to be minimally higher than the temperature of the hot working fluid during the isothermal expansion phase of the Carnot cycle (to limit new entropy being produced in the working fluid that we have to get rid of). But during the isothermal compression phase why do we need the cold reservoir to be only minimally cooler than the cold working fluid? The working fluid looses the same entropy independent of the temperature of the cold reservoir, so why do we need to minimize the new entropy created in the cold reservoir? What is wrong with just letting the cold reservoir get more and more entropy as long as the working fluid returns to the same state as before the cycle?

Answer 1

During the isothermal expansion phase of the Carnot cycle, the hot reservoir needs to be minimally higher in temperature than the hot working fluid to limit the new entropy produced. In the isothermal compression phase, the cold reservoir needs to be minimally cooler than the cold working fluid to minimize new entropy and maximize efficiency.

Step-by-step explanation:

During the isothermal expansion phase of the Carnot cycle, the temperature of the hot reservoir needs to be minimally higher than the temperature of the hot working fluid to limit the new entropy being produced in the working fluid. This is because, according to the second law of thermodynamics, the total entropy of a system either increases or remains constant in any process.

Therefore, if the hot reservoir and the hot working fluid were at the same temperature, no work could be done and there would be no isothermal expansion phase. By having the hot reservoir at a higher temperature, there is a loss of entropy in the hot working fluid, allowing it to do work.

On the other hand, during the isothermal compression phase, the cold reservoir needs to be minimally cooler than the cold working fluid. While the working fluid loses the same entropy regardless of the temperature of the cold reservoir, the goal is to minimize the new entropy created in the cold reservoir.

This is because the efficiency of the heat engine depends on the temperature difference between the hot and cold reservoirs. The smaller the increase in entropy in the cold reservoir, the more heat energy is available to do work.

In summary, the temperature difference between the hot and cold reservoirs is crucial in the Carnot cycle to limit the production of new entropy in the working fluid and to maximize the efficiency of the heat engine.