The relation between the energy minimum and entropy maximum

In summary, the energy minimum principle implies the entropy maximum principle by showing that, if the entropy is not maximum at constant energy, then the energy cannot be minimum at constant entropy. This is demonstrated by exploiting the permissible increase in entropy to extract heat from a reversible heat source and deposit it in a reversible work source, resulting in a decrease in energy and an increase in entropy in the system. This proves that the energy minimum principle and the entropy maximum principle are interconnected and support each other.
  • #1
Gabriel Maia
72
1
Hi. This is the problem 5.1-1 from the second edition of Callen's Thermodynamics. It says

Formulate a proof that the energy minimum principle implies the entropy maximum principle. That is, show that if the entropy were not maximum at constant energy then the enrgy could not be minimum at constant entropy.
HINT: First show that the permissible increase in entropy in the system can be exploited to extract heat from a reversible heat source (initially at the same temperature as the system) and to deposit it in a reversible work source. The reversible heat source is thereby cooled. Continue the argument.


Let's suppose that the system is at equilibrium and the energy is minimum but the entropy is not maximum. I increase the entropy of the system by extracting heat from a reversible heat source. Since the system and the heat source are at the same temperature there is no variation in the system's energy.

This heat is then deposited in a reversible work source. So the system does work on this source. How come the source is cooled and not the system?

I would think that work deposited on the reversible work source would keep the entropy the same but lower the energy of the system, and therefore, the energy would decrease even though it was supposed to be a minimum, concluding the argumentation that minimum energy goes hand-in-hand with maximum entropy. How come the source is cooled? Where is the flaw in my reasoning?

Thank you very much.
 
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  • #2
Your reasoning is not flawed, but the conclusion is incorrect. The source is cooled because the heat extracted from the reversible heat source is used to do work on the reversible work source. This means that the energy in the system has been converted into useful work, and the entropy of the system has increased. Since the energy in the system is now lower (due to the work done on the work source), the system is at a lower energy state than before, thus proving that the energy minimum principle implies the entropy maximum principle.
 

FAQ: The relation between the energy minimum and entropy maximum

What is the concept of energy minimum and entropy maximum?

The energy minimum and entropy maximum are two fundamental principles in thermodynamics that describe the behavior of a system. The energy minimum refers to the state of a system where it has the lowest possible energy, while the entropy maximum refers to the state where the system has the highest possible disorder.

How are energy minimum and entropy maximum related?

The relationship between energy minimum and entropy maximum is known as the second law of thermodynamics. This law states that in a closed system, the total entropy will always increase over time until it reaches a maximum value. As a result, the energy of the system will decrease until it reaches a minimum value.

Why is the concept of energy minimum and entropy maximum important?

The concept of energy minimum and entropy maximum is important because it helps us understand and predict the behavior of complex systems. It also explains why certain processes occur spontaneously and others do not.

How do scientists measure energy minimum and entropy maximum?

Energy minimum and entropy maximum are measured using thermodynamic parameters such as temperature, pressure, and volume. These parameters can be used to calculate the energy and entropy of a system and determine its minimum and maximum values.

What are some real-world examples of energy minimum and entropy maximum?

One example of energy minimum and entropy maximum is the diffusion of molecules in a gas. In this process, the molecules move from an area of high concentration to an area of low concentration, resulting in a decrease in energy and an increase in entropy. Another example is the melting of ice, where the solid ice has a lower entropy compared to the liquid water, which has a higher entropy.

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