# Helmholtz free energy related query.

by A Dhingra
Tags: energy, free, helmholtz, query
P: 51
 Quote by A Dhingra So the derivatives used by Maxwell have actually nothing to do with the equivalence principle of entropy maximum and energy minimum, right?
Yes.The Maxwell derivatives dont have anything to do with minimum energy or maximum entropy principle.Also,do realize that this section of the text doesnt want to say anything about maxwell derivatives or them being equal. It tries to explain that " The energy minimum principle with entropy held constant,or the entropy maximum principle with energy held constant,both lead to the same equilibrium point".I think Callen chapter 5,the initial sections elucidate this points very well.
P: 196
 Quote by Studiot Since you did not respond to the bulk of my post #11 I'm not really quite sure where you are having difficulty. You asked several questions mixed together.
I did not mean to offend you in any may. Sorry that i did not say a word about your earlier replies because I thought i knew the formulas for the potentials; After i read the portion again i had a few queries so asked.

 Quote by Studiot Do you fully appreciate the distinction between differences (Δx) , differentials, dx, partial differentials ∂x and small variations δx ? The distinction comes into play when we derive Maxwells relations. Have you seen the derivation?.
Yes. I have seen the proof using the definition of the potentials and using the idea of exact differential.

 Quote by Studiot Are you aware that whilst each partial specifies one constant variable, two are in total specified as I said before? Can you apply say $${\left( {\frac{{\partial S}}{{\partial V}}} \right)_T} = {\left( {\frac{{\partial P}}{{\partial T}}} \right)_V}$$ to the piston in cylinder example?
I know how to write the Maxwell relations, but i think the process described by Callen can't be written in any such form. This relation being not equal to the situation is what both of you have been mentioning.
I just want to see the equality of the two sides of this or any one of the relation of maxwell through a process (actually two processes). Can you help me with that? I thought that each process which should brings about this equality will hold the minimum energy or maximum entropy principle. But that's not the case like pabloenigma said.
Probably this is not what the text said, but this is what i wanted to see and hence interpreted this way (as in the first post).
P: 5,462
 and if it is not wrong i can provide a link to pdf and the page i am referring to.
 P: 196 The text that i quoted is in chapter-5 pg no135.
P: 5,462
OK I have now read the passage and I have to say that I cannot recommend this book as a result of this analysis.

 "Any equilibrium state can be characterized either as a state of maximum entropy for given energy or as a state of minimum energy for given entropy. But these two criteria nevertheless suggest two different ways of attaining equilibrium. Let us consider a piston originally fixed at some point in a closed cylinder. We are interested in bringing the system to equilibrium without the constraint on the position of the piston. We can simply remove the constraint and allow the equilibrium to establish itself spontaneously. Here the entropy increases and the energy is maintained constant by the closure condition. This process is suggested by the maximum entropy principle. Alternatively, we can permit the piston to move very slowly, reversibly doing work on an external agent until it has moved to the position that equalises the pressure on the two sides. During this process energy is withdrawn from the system and entropy remains a constant (the process is reversible and no heat flows).This is the process suggested by the minimum energy principle. Independent of whether the equilibrium is brought about by either of these two processes, or by any other process, the final equilibrium state in each case satisfies both extrema conditions."
I have emboldened the crucial part of the description where Callen changes the system without warning or consideration of the implications, which are crucial to the argument.

Callen has introduced a second system (the "external agent"), but fails to consider the entropy and energy of this system.

In other words he has made the classic failure to consider changes in the surroundings.

In post#10 I proved that the entropy for the original system is constant call it So

Now consider adding the "external agent" with entropy Se

If the two systems are isolated then the total entropy is So + Se = St

Now we have something we can maximise viz the total entropy St

Since So is constant maximising St comes down to maximising Se.

Also since the total system is isolated the total internal energy Ut is constant.

So we are maximising St subject to constant internal energy, which was one of Callen's conditions.

But Ue is an increasing function of Se so maximising Se means maximising Ue. This is another way if saying that the more work we do on our external agent the more we increase its internal energy, U.

Further since Ut = Uo+Ue

Maximising Ue means minimising Uo, which is what we wanted to prove.

So Callen is not wrong but he only tells half the truth, since he misses out half the subject ie the surroundings.
 P: 196 Then which book would you recommend for a thorough self study of thermodynamic potential and equivalence principle,particularly ?
 P: 5,462 Did you understand the chain of reasoning in my post #23?
 P: 196 Honestly, i didn't get that. (Reason being: I have my semester exam coming in a few days(2 days later is mathematical physics) so couldn't take time reading it a number of times to be able to grasp, i read it once but couldn't make it out, so gave up. Would you mind if I come back to this when my exams are done so that I can focus and try to understand, and ask you if i have something to ask.... )
 P: 5,462 Good luck with your exams. Post again when you have more time. If you really want book recommendations, include some information about the viewpoint. I originally learned thermo as part of Industrial Chemistry and later from a more general engineering point of view. Also do you have access to a university library? This would give wider choice especially for some very good older books.

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