5.1-1. Formulate a proof that the energy minimum principle implies the entropy m

Question

5.1-1. Formulate a proof that the energy minimum principle implies the entropy maximum principle the "inverse argument" referred to after equation 5.7. That is, show that if the entropy were not maximum at constant energy then the energy 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.

Explanation / Answer

From definition we know that change of entropy dS=dQ/T , now if we consider an isolated system tthat means some process are still going on inside but nothing enters from outside. But in an equlibrium there is no pocess but then comes thee fundamental thermodynamic argument that if energy change while moving towards equilibrium then entropy tries to touch maximum. Lets consider two sources hot and cold for example. Now we remove little heeat delQ from hotbody at temperature T1 and that means removal of delQ/T1 entropy from hot body. Again if we transfer the same heat in the cold body in temperature T2 that means the entropy increase is delQ/T2 in the cold body.

Overall we can see that T1>T2 and entropy is already created. Now if we continue the heat flow in the cold body with transfer of heat it become hot and that means entropy is destroyed completely. But thermodynamics cannot approve complete destruction of entropy. So it proves that maximum entropy is at equilibrium which means heat flow is zero with no temperature difference. That means consstant energy situation. The same in reverse proves that energy could not be minimum at constant entropy.

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