Talk:Helmholtz free energy

Gibbs free energy

In the second paragraph on Gibbs free energy, I think the word "where" is misleading. Please review my correction: In contrast, the Gibbs free energy or free enthalpy, is most commonly used as a measure of thermodynamic potential, especially in chemistry, when chemical reactions occur at constant pressure. Note, however, that even in chemistry, the Gibbs free energy can be inconvenient when chemical reaction do not occur at constant pressure. For example, in explosives research, Helmholtz free energy is often used, since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental equations of state of pure substances. Benjamin.friedrich (talk)

Link Missing

THE FOLLOWING LINK IS MISSING

Application to fundamental equations of state

The Helmholtz free energy function for a pure substance (together with its partial derivatives) can be used to determine all other thermodynamic properties for the substance. See, for example, the equations of state for water, as given by the IAPWS in their IAPWS-95 release.

Comment

Does this article need to be marked as a stub? It seems at least somewhat comprehensive. If things are missing, how about the attention-tag? O. Prytz 21:54, 22 November 2005 (UTC)

I think the stub tag can be removed. I will do it and see what happens. PAR 22:32, 22 November 2005 (UTC)

IUPAC compliance

The page urgently needs to have the word 'free' eliminated to comply with IUPAC recommendations. Ref. http://goldbook.iupac.org/goldbook/H02772.html Need to set up new page and convert this one into a REDIRECT. DIV, 2006-05-31 (Melbourne)

The discussion was moved to Talk:Gibbs free energy, and can now be found in Talk:Gibbs free energy/Archive1.

Outcome after extensive discussion was strong consensus for No change -- continue with the title as Helmholtz free energy.

Further points should be raised at Talk:Gibbs free energy.

Jheald 11:25, 6 March 2007 (UTC)

Link

Under see also, there's no link to anything other than general wikipedia pages where it says "This link details the Helmholtz energy...". I'd remove it, but I'm hoping someone know where it's supposed to go to, since I'd like to see it. —Preceding unsigned comment added by 70.110.240.11 (talk • contribs) 03:24, 20 September 2007

Changing A to F

I did this because most physics books still use F. Certain equations are so common with F used in it (like F = - k T Log(Z)) that replacing F by A makes them look weird. But I'm not going to insist on this change and will let others decide on what they prefer.

I'm now going to write up the derivation of "F = -k T Log(Z)" and of the Bogoliubov inequality for the free energy. Count Iblis (talk) 16:10, 19 May 2008 (UTC)

I'm going to change F back to A pending further discussions on this matter. Count Iblis (talk) 16:21, 19 May 2008 (UTC)

Physical chemists tend to use A, and IUPAC uses A.[1] This may be one of those cultural differences between chemists and physicists... Are there any equivalent standard recommendations or style guides for physicists that recommend F that could be cited? In any case, if it is shown that there are conflicting standards, I don't really mind which one is followed as long as the article mentions the existence of the other (which it already does). --Itub (talk) 09:30, 21 May 2008 (UTC)

I don't think I've ever seen a style guide that spells this out. The physics style guides are not rigorously enforced anyway, at least that is my experience. I think there would be no problem in using the symbol X for the free energy in a Physical Review journal article, e.g. if F was already used for something else. Would a referee or editor object? I don't think so.

So, the "cultural differences" could be that in physics we don't really stick to such detailed mandated rules. The authors will use the symbols as he/she sees fit. Because it is in their interest to make what they write as easily understandable as is possible, they'll then stick to the usual convention. Count Iblis (talk) 15:02, 21 May 2008 (UTC)

An Introduction to Thermal Physics by Daniel V. Schroeder uses F but notes that A is often used. —Preceding unsigned comment added by 76.69.61.97 (talk) 16:07, 13 December 2009 (UTC)

A Modern Course in Statistical Physics by L. E. Reichl (University of Texas Press, Austin) uses A, rather than F. — Preceding unsigned comment added by 2603:6000:AA4D:C5B8:0:3361:EAF8:97B7 (talk) 06:01, 5 April 2022 (UTC)

I propose to return again to F designation as A occupied by the surface. This creates problems, for example in the canonical ensemble, the partition function of which contains the surface, but also connected with a free energy.Luksaz (talk) 19:26, 15 February 2015 (UTC)

Constant Volume?

This is wrong. Constant volume is not necessary. Read Fermi's book on thermodynamics: "If a system suffers a revesible transformation from an initial state A to a final state B both of which states have a temperature equal to that of the environment only, during the transformation, the work done by the system during the transformation is equal to the decrease in the free energy of the system."

Fermi's derivation for the free energy is simple, and never assumes constant volume. —Preceding unsigned comment added by 160.39.186.143 (talk) 21:52, 21 October 2008 (UTC)

The work will then include volume pressure work done by the system on the heat bath, which is not useful work that can be extracted from the system. E.g. if you keep the pressure constant, then W in the formula will contain a term P delta V, but this is work done on the heat bath. The work that can leave the system+heat bath is then W minus that term which can then be equal to the drop in the Gibbs energy at most. Count Iblis (talk) 02:16, 22 October 2008 (UTC)

Why is the work done on the heat bath by the system unusable? In particular, why is the system in a heat bath? Couldn't I have a piston submerged in a heat bath, with the rod extending far above? (160.39.152.17 (talk) 05:07, 22 October 2008 (UTC))

Yes, you can do that. But then your system boundary is at some other point. What then matters is what boundary conditions are imposed there. E.g. suppose you have some ammount of fuel. Given some engine you can then extract some ammount of work froim the fuel. But this depends on the efficiency of the engine. You can ask what the maximum possible amount of work is that you can extract from the fuel. This depends on the circumstances under which you run the engine. So, you have to specify that. Suppose you put the whole system in a room that is at constant volume. The room is in thermal contact with the environment, so we can assume that some time after the engine has stoped running the temperature will be back at what is was before the engine started.

The exhaust gasses will stay in the room, so the pressure will rise. The initial state is a state of thermal equilibrium, and the final state will all the exhaust gasses in the room and the room having cooled down again is again a a state of thermal equilibrium at the same tamperature and volume. Then, as shown in the article, the maximum work that you could have extracted is given by the drop in the Helmholtz free energy when the fuel in converted to exhaust gasses (assuming that the engine after running has returned to the same state as it was before it wa started)

Similarly, you can put the engine in some open space and ask the same question. Then, you are imposing constant pressure boundary conditions. What then happens is that the volume pressure work done on the environment (atmosphere) is lost. However you look at it, the exhaust gasses will expand and do work against the atmosphere and that work comes at the expense of the work you can extract. In this case, then, the drop in the Gibbs energy will give you the maximum amount of work you can extract from the fuel. Count Iblis (talk) 14:33, 22 October 2008 (UTC)

I get skeptical when arguments get so concrete and complicated - that's when mistakes happen. But I'll think about this. Thank you. (160.39.186.143 (talk) 01:43, 23 October 2008 (UTC))

Well, the formal argument in the article itself is quite simple, just a few lines of equations. There are certainly no mistakes there, as it is very transparant. The questions raised by you have to do with the relevance of the seting: Constant volume, or if the volume is not kept constant, why not also count the volume-pressure work, and what is the heat bath doing there? This is because you want to look at a system that is interacting in some way with the environment and you are only interested in the system.

In a constant volume the drop in the Helmholtz free energy gives you the maximum amount of work that you can extract from the system. If, instead, the presure is kept constant, then the maximum work is the drop in the Gibbs energy. In that case the drop in the Helmholtz energy contains volume pressure work that is then done on the environment (which is modeled as a heat bath). If you then say: Yes, but we could extract that work too, then you need to contain that heat bath in a larger volume which is then kept constant. So, then you are again looking at the case of constant volume. Count Iblis (talk) 02:08, 23 October 2008 (UTC)

The nagging point that bugs me is that, for total entropy to increase, the system plus heat bath must be isolated. So work done by the system has to be work done to the heat bath. This type of argument was used by Gibbs in one of his treatises. Maybe it would be easier to use Gibbs' argument to derive the Clausius inequality, as done here: https://en.wikipedia.org/wiki/Clausius_theorem, then use that to prove the work theorem for a system held at constant T by the heat bath. But starting with Gibbs' construction, the work by/on the bath needs to be counted somehow in order to keep the combination system+bath closed. Gibb's notation was painful in that his symbols were all different. But his argument is recognizable in his 1873 paper.... https://en.wikisource.org/wiki/Scientific_Papers_of_Josiah_Willard_Gibbs,_Volume_1 DanP4522874 (talk) 15:48, 6 May 2023 (UTC)

We take it for granted that the thermal contact with the heat bath and the mechanical contact with the work load are separate from one another.

Then I think the difference between Gibbs and Helmholtz is whether you can extract the energy done by volume expansion. Gibbs does the correct counting when you can't extract the volume energy, like when it just lifts the atmosphere. Helmholtz does the correct counting when you can. Which case you're actually in depends on the setup of your experiment, engine, or phenomenon. Of course there are hybrid cases, where neither type of free energy is right. Then it has to be supplemented by other terms.

An example for Helmholtz is when there's no ambient atmosphere, but instead the volume changes push against some force that you provide, such as a piston with a spring behind it. If there is a net volume change, then there is a net amount of energy added to (or subtracted from) the potential energy of the spring. You also get a variable pressure. The work done is ∫p(t) (dV/dt) dt.

Now here is a good question: how much work can you extract? This is at least *limited* by the fact that (for an exothermic reaction liberating an amount -∆U>0 of energy) at least some energy (namely -T∆S, presumably positive) must go to feed the heat bath (to equilibrate the temperature, or equivalently, to maximize the joint probability of macrostates between system and bath, subject to other contributions and constraints). An upper bound on the energy available for work, including the pressure work ∫p dV, is given by the change in the Helmholtz energy, namely -∆H=-(∆U-T∆S).

I'm a little puzzled by the term ∫p dV, since it seems variable and uncontrolled. If we know more about this term, maybe we can get a better estimate than Helmholtz. But can we achieve Helmholtz? Here is an approach. Assume it really is a spring, like above, and the volume changes from V_1 to V_2. Suppose you design the spring so that p is a chosen function p=p(V). Then the volume work is ∫p(V) dV. If we optimize the choice of the function p(V), and other parts of the system are efficient (reversible, ideal, etc), can we achieve the Helmholtz free energy exactly?

Is F(or A) really the Work which can be used at constant temperature AND Volume? this makes no sense as Work is defined as p*dV? With dF = -SdT -PdV = 0 (For constant N) According to me F is the work usable in isothermic processes. —Preceding unsigned comment added by 129.206.196.190 (talk) 10:30, 18 July 2008 (UTC)

Ok I just read the argument given on the issue i just raised. According to me the resolution is wrong. Since F(T,V,N) = E - TS = -p(T,V,N)V + mu(T,V,N)*N. with the chemical Potential mu. since T, V and N are constant p and mu donot change. so F cannot change. This equation is valid for the 1 state and the second state. (As both states are in equilibrium). This would mean F1 - F2 = 0. —Preceding unsigned comment added by 129.206.196.190 (talk) 10:42, 18 July 2008 (UTC)

The conclusion that F1 = F2 is invalid because in general the system cannot go from state 1 to state 2 while remaining in thermal equilibrium. If the changeis slow enough, you can actually go from state 1 to 2 while remaining in equilibrium, but then you need to add additional external variables to the volume, so the thermodynamical state space is then enlarged. This is already explained in the article. You assumption about a single chemical potential and 1 type of molecules is actually not warranted, in the article we don't make any such assumption.

Practical example: Consider a battery kept at constant temperature and volume. Clearly, this system can perform work. We can consider the initial state of the battery and the final state in which it is completely used up. The difference in the free energies is the maximum amount of work the battery can perform. Since the chemical reactions in the battery proceed quite slowly, we can actually describe the change using equilibrium thermodynamics, but then we need to add additional external variables and their conjugate general forces to describe the change. In the case of the battery, that would simply be the numbers of molecules of each type and their chemical potentials.

In the general case the reaction may proceed so fast that equilibrium thermodynamics is invalid. Even in that case the differnce between theintitial and the final free energies gives the maximum work that can be extracted from the system. Also, such changes can happen for a system consisting of only one type of molecule. Consider e.g. superheated water thst is in a metastable equilibrium. It can boil explosively when disturbed a bit. The free energy for superheated water has a larger free energy than the free energy for the water/steam mixture that is in thermal equilibrium.

How can we extract work if the volume is constant? Clearly if we are keeping the volume constant, we are not going to extract pressure volume work done against the heat bath that kees the system at constant volume and temperature. But then, why would we required to extract work in exactly that way?

Example: consider a big room in which there is air, some gasoline and an engine that runs on gasoline. The whole room is kept at constant temperature and volume. Can we extract work from such a system? Of course we can, just let the engine run! You may object and say that we cannot keep the temperature constant in the room while the engine is running. But then, this is only assumed in the initial and the final state. So, in the initial state, the room had some temperature and volume. We require that in the final state, say, long after the gasoline is used up and the engine has stopped running, the room is again in thermal equilbrium with the heat bath at the same temperature.

Again, the difference in the free energy gives you the maximum amount of work that can be extraced from the system. In the case of the enigine in the room, that would be volume pressure work that could be converted to electrical work and then extracted from the system. Also, since the processes in the engine are not so extremely violent that thermodynamics completely breaks down, you can describe the intermediary states inbetween the two equilibrium states using thermodynamics, but then you need to add additional external variables and their generalized forces besides the total volume and the temperature of the system. In this case you would eed a complete description of the engine, the temperature and pressure distributions etc. etc. Count Iblis (talk) 14:03, 18 July 2008 (UTC)

The best we can do is to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium.

If we wish to describe phenomena like chemical reactions, then the best we can do is to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium.

This sentence is necessary otherwise people may ask why we don't just use the expression of the Helmholtz energy in infinitessimal form and then you get the paradox discussed in the previous section on this talk page. Note that that was exactly what was wrong with the version of this article of a few months ago with this article. Count Iblis (talk) 13:41, 1 September 2008 (UTC)

NPOV/Accuracy

See [2]. The tags should remain until this is clarified. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 01:28, 22 February 2010 (UTC)

Are you crazy? Count Iblis (talk) 01:30, 22 February 2010 (UTC)

Formatting

The article lacks consistency of variable formatting. All variables should be italic. Subscript labels that are not variables themselves should be roman. This is generally adhered to in the display equations, but not throughout the body text.
See IUPAC / SI / ISO guidelines on Use of Italic and Roman Fonts for Symbols in Scientific Text and Value and numerical value of a quantity, and the use of quantity calculus and ISO 31-0 Typographic conventions.
—DIV (138.194.12.32 (talk) 09:30, 12 April 2010 (UTC))

"This result seems to contradict the equation dA = − SdT − PdV, as keeping T and V constant seems to imply dA = 0 and hence A = constant. In reality there is no contradiction. After the spontaneous change the system, as described by thermodynamics, is a different system with a different free energy function than it was before the spontaneous change. We can thus say that where the Ai are different thermodynamic functions of state."

This paragraph is not very clear. Actually, dA = - SdT - PdV is only good for a single phase and single component. Based on Gibbs phase rule, f = c-p+2 ( where f is the freedom for a thermodynamic system, c is the number of components and p is the number of phases), if both c and p are one, then f = 2. Therefore, for such system, there are only two freedoms and those two are often chosen as T and V for state function A. (They are called natural variables of A). Under the constraints of constant T and V, A does not change indeed. Image if the V and T of a gas system are fixed, every thermodynamic functions, U, H, S, A, G, etc, are fixed. However, Helmholtz Energy (A) has a wider application in phase changes and chemical reactions. In this scenario, dA = -SdT - PdV + ∑μ_idn_i, where μ_i and n_i are the chemical potential and the amount of each individual component and phase. In this case, even if T and V are constrained, A is still seeking for the minimum value in a spontaneous process. —Preceding unsigned comment added by 134.250.247.76 (talk) 19:28, 18 March 2011 (UTC)

$\Delta A\leq 0$ not a contradiction

I think the note that $\Delta A\leq 0$ does not contradict the equation $dA=-SdT-PdV$, should be more precise as to why it doesn't.

Possible mistake

I think there is an important mistake because constant volume is not necessary, only constant temperature is. — Preceding unsigned comment added by 200.108.136.163 (talk) 12:02, 22 July 2011 (UTC)

If you don't keep the volume constant, then some results are still valid, but with the work replaced by total work, which then includes the work done by the system on the heat bath which is not useful work one can extract from the system. If you subtract that latter part, then you get an inequality for the maximum amount of useful work, which in case of constant pressure involves the change in Gibbs energy. In that case, at equilibrium, the Gibbs energy is minimized. Count Iblis (talk) 15:16, 22 July 2011 (UTC)

Why does the system have to do mechanical work on its own heat bath? The two functions are separate. You could have rigid, but heat-conducting contact with the heat bath along part of the boundary, and mechanical, (flexible, volume changing) insulated contact with a work load along another part of the boundary. The whole contraption could be located on the moon.

The mechanical forces acting from outside on the flexible part of the boundary, as well as the shape of the boundary (and the enclosed volume), are unknown functions. They are coupled by certain constraints that also couple them with forces coming from the interior. But they also retain some remaining freedom, which can be viewed as control functions. The system should be quasi-static for best results.

Any pressure work along this boundary is captured, by hypothesis; there are practical limitations, but some idealized springs could capture and store it all. The point is that it is not thermalized. Instead, it is put into an idealized elastic mechanical system. It is not clear that the Helmholtz free energy gives the last word on how much energy can be extracted, because one might get better bounds depending on the details of the setup. And for minimization, one must add terms for these boundary forces and boundary shape. But Gibbs is definitely not the right potential function, since it throws away the pressure work.

178.38.67.158 (talk) 00:15, 10 March 2015 (UTC)

Modification of Section on Minimum free energy and maximum work principles

I was not happy with the Section on “Minimum free energy and maximum work principles” for the following reasons:

1. A system of fixed temperature and volume can only do work on its surroundings if it is inhomogeneous. (On second thought there are homogeneous systems which can interact with their surroundings even if their volume is constant such as magnetic systems (Kittel, Thermal Physics, p.69).(Adwaele (talk) 14:31, 4 July 2012 (UTC)).) This is an essential feature of the concept of free energy which was not stressed in the article. In fact, the opening sentence of this Section: “The laws of thermodynamics are only directly applicable to systems in thermal equilibrium.” is misleading. Most of the time thermodynamics deals with systems, like heat engines, refrigerators, etc. which are far from thermal equilibrium. The only thing that is required is that the thermodynamic parameters, such as temperature and pressure are well-defined.
2. The article does not distinguish between pV work and other forms of work, which is, again, essential for understanding the minimum free energy and maximum work principles.
3. There is a serious mistake in the expression ΔU+ΔS+W=0. A factor T in front of ΔS is missing, the sign is wrong, and it is not so that, in general, Q =TΔS since there can be irreversible processes. In practice there usually are!
4. As a result of all this the derivation is hard to follow.
5. The end of the Section gives expressions for dA when it is a function of other variables than T and p. These relations only apply to homogeneous systems (i.e. T and p have to be uniform otherwise the relations are meaningless). It has nothing to do with maximum work and minimum free energy principles and can better be moved to the Section on Mathematical development.

I must admit that I feel uneasy about this but I have written a new text, which, I sincerely hope, is better. I also added a practical example which should clarify the issue for readers who are not familiar with the thermodynamic jargon. It should also clarify many of the issues addressed in this talk page.(Adwaele (talk) 19:55, 20 June 2012 (UTC))

While I'm sure you know a lot about this subject, I think you misread/misunderstood the original text (which can be found in some textbooks). The system under consideration was assumed to be at constant volume and kept at thermal equilibrium with a heat bath. Then we considered a change in the system from one state of (global) thermal equilibrium to another. There is no issue with having to consider inhomegenous systems, because we only look at the initial and final states in which you have global thermal equilibrium with the heat bath. Then the relation Q = T Delta S was applied to the heat bath to consider its entropy change, which is valid as an equality because no matter what the system does, the heat bath remains in internal thermal equilibrium.

Now, you did do a lot of work in rewriting the text, but I think the original argument should be brought back in some way, because it is completely rigorous and doesn't assume anything about the way the change happens. If you invoke inhomogenesous systems and attempt to give a thermodynamic description of the change as it is progressing, then this only works approximately and under the assumption that you do still have an approximate thermodynamic description. So, then you lose the rigorous result that says that in any arbitray change from the initial to the final state (no matter how far away form internal thermal equilibrium it went) the work is bounded by minus the free energy difference.

An example that would fit in the old text could have been an initial state where you have a hydrogen oxygen mixture at some temperature and volue, and a final state where you have water vapor, also in thermal equilibrium at the same volume and temperature. Count Iblis (talk) 22:12, 20 June 2012 (UTC)

Unfortunately, having read the new text, I found that there are fundamental errors. E.g. if you write down the entropy increase, you cannot just look at the entrop increase of the system, you have to ad up the entropy change of the environment to that. This makes any argument along your lines to be invalid. Note that no textbook I've seen uses any other argument than given in the original text, so that casts doubt on the possibility of making the simpler argument work. Count Iblis (talk) 22:50, 20 June 2012 (UTC)

Actualy, I now see how you can justify dS >= dQ/T but then you are doing the same as was done before. The inequality dS >= dQ/T is actually never used in rigorous arguments, because it only applies in some assumed context. In general, the second law is formulated by saying that the total entropy can only increase and that for quasistatic infinitesimal processes, you have dS = dQ/T. This then implies that dS > dQ/T in some cases, like here, but it is better to show that from the standard frmulation of the second law explicitely. Count Iblis (talk) 01:37, 21 June 2012 (UTC)

Replies to the points raised:

I agree about point 3, an eror had slipped in due to an IP edit.

1) I don't agree here; first of all we don't consider work done on the environment here, the volume has been fixed. We only consider work of any other sort than presssure volume work that can be extracted from the system. The system does not have to be assumed to be inhomegeneous, regardless of whether one considers work done against the environment.

Then we do talk about being able to apply the laws of thermodynamics directly, but perhaps we should add "rigorously". In practice thermodynamics is aplied when it srictly speaking doesn't apply. But when a system s far from thermdynamic equilibrium, you cannot apply the laws of thermodynamics in a rigorous way, even if you can define temperature in some parts of the system.

2) I agree that this should be made clear better. Thing is that we only consider non-PV work in this article when discussing maximum work, as the volume is kept fixed.

Then in general, we should not invoke inhomogenous systems, as they can't be treated rigorously within thermodynamics. If you invoke systems with varying T, then that's a system that is not in thermal equilibrium and thermodynamics doesn't apply in a rigorous way. That doesn't mean that people in practise use thermodynamics in such situations, but in any rigorous argument you have to bypass having to deal with such situations, like considering suitable initial and final states in which you do have global thermodynamic equilibrium. Count Iblis (talk) 01:53, 21 June 2012 (UTC)

Response

You removed my text as if it is a dangerous virus. I hope you saved my text and that you study it carefully. Sometimes one needs a little time to digest a new approach. We both have the ambition to improve the quality of the articles on thermodynamics in Wikipedia. Now follows my response to your reaction:

In thermodynamics one must always clearly define the system in consideration. From this point of view the present text should be improved:

The Section "Definition" gives a definition of the free energy. The free energy of what? Implicitly it is assumed that we are dealing with a system with a well-defined temperature, otherwise it makes no sense to take the product of T and S. This must be expressed in the onset. And then it should be explained what happens if the system has no uniform temperature. Would any harm be done if we would say that A is an extensive variable and that the total free energy of a composed system is the sum of the free energies of the components just like the internal energy and the entropy?

The Section "Mathematical Development" is again unclear which system is considered. From the context one gets the impression that one is dealing with a homogeneous system since it is apparently assumed that the pressure and temperature are well-defined. Furthermore it puts δQ=TdS, which is a property of homogeneous systems in which, by definition, only reversible processes can take place. It would be better to say that this section applies to homogeneous systems. No reason to keep the reader in the dark.

In the Section "Minimum...." the system, and how it interacts with the surroundings, is never defined. It talks about a system, but at the same time it applies the first and second law to a combination of the "system" and a thermal bath. In this way one can never get a clean discussion about the implications of the laws of thermodynamics. Many readers are familiar only with pV work and will wonder how a system can do work if the volume is constant.

I am not proposing to extend the sections by a large amount. A few words here and there can do miracles.

An advantage of my approach is that there is no need to include a thermal bath. The thermal bath is an unnecessary complication. One only needs that the temperature of the surface of the system is T and that its volume is constant (both are system properties). Furthermore the derivation is short. The whole concept is explained in six, rather trivial, steps:

$\delta Q=\mathrm {d} U+\delta W$

$\mathrm {d} S\geq \delta Q/T$

$\mathrm {d} (TS)\geq \mathrm {d} U+\delta W$

$\mathrm {d} A+\delta W\leq 0$

${\text{if }}\delta W=0{\text{ then }}\mathrm {d} A\leq 0$

$W\leq A_{i}-A_{f}.$

(By the way: it is interesting to note that the condition that the volume is constant is only needed for dA<=0. The statement for max work also holds in case V is not constant.)

If the free energy depends only on T and V and dA=-SdT-pdV one runs into trouble. If T and V are constant dA=0 whatever you do. In reality the total free energy is a function of (at least) one more variable (in my example it is Va) and, in equilibrium, A is a minimum with respect to this third variable.

A nice example of such a third parameter can be found in Kittel, p.69, where the 3rd parameter is the spin excess s, which is equivalent with the magnetization. In this case the interaction with the surroundings is via the magnetic field.(Adwaele (talk) 08:08, 26 June 2012 (UTC))

The argument, that one is dealing with a different system after a spontaneous process (what is that?) has taken place, leaves me speechless.

You know I have problems with the statement: "After the spontaneous change, the system, as described by thermodynamics, is a different system with a different free energy function than it was before the spontaneous change." I would appreciate one or two references which support the statement.(Adwaele (talk) 20:45, 25 June 2012 (UTC))

That statement was included after talk page coments here a long time ago. I'm not sure I can source that directly, because it is rather trivial. You can distinguish two cases, one where you have at least one more parameter, like the case you cite above with the spin nexcess and the magnetic field. There for fixed spin excess, you have a different free eenergy function for each value of that spin excess. Here you can consider describing the change in terms of the change in that parameter, you can then also consider changing that parameter continuously.

However, in the most general case, you may not have not have such a continuus description that interpolates from the intial to the final state. You then simply have two systems and somehow a transition takes place from the initial and final state. How that happens doesn't per se have to have a formulation that is (approximately) describable in terms of thermodynamics, or it may simply be too compicated and completely useless.

E.g. take the inital state to be an engine and some fuel, this is all inside a closed room at some temperature. Then the engine starts, it burns fuel, the exhaust gasses stay in the room. Then the engine stops, it cools down end eventually everything in the room comes into thermal equilibrium with the heat bath at temperature T. In this case, we can bound the work the engine could have done by the drop in the free energy, and that statement doen't depend on being able to give a desscription of how the engine actually works, that during the running of the engine you could have described the situation approximately by introducing local temperatures, chemical potententials for the various compounds etc. etc. Count Iblis (talk) 17:04, 26 June 2012 (UTC)

The simple answer to the question, addressed in this paragraph, is that the free energy can be a function of more variables than just the temperature and the volume (e.g. magnetization, pressure distribution, chemical composition). In equilibrium the free energy is a minimum with respect to these variables.(Adwaele (talk) 20:27, 28 June 2012 (UTC))

The second law for closed systems, formulated as dS≥δQ/T, is generally valid. The equality sign holds if the internal processes during the heat supply are reversible. It is not sufficient that the process is infinitesimal and slow. There are processes that are irreversible even if they are infinitesimal and slow. Heat flow due to a finite temperature difference is the best-known example. No matter how slow the heat flows, the process remains irreversible.

Finally: you claim that thermodynamics can only be used if the system is static. This is confirming a widespread misunderstanding and denies the dynamic aspect of thermodynamics.(Adwaele (talk) 08:31, 22 June 2012 (UTC))

While I agree with some of your points about improving the text, like making it clear that there can be work done besides pressure volume work, that the more general form for DA should be given in the earlier section etc. etc.,

Why don't you start improving the text as you describe. Perhaps it turns out that we agree more than we think.(Adwaele (talk) 20:45, 25 June 2012 (UTC))

Ok., I'll improve the text based in the points you have raised here. Count Iblis (talk) 16:24, 26 June 2012 (UTC)

I strongly disagree with your presentation of the topic. You end up introducing some of the same elements in a hidden way when you arrive at a similar result. E.g., you don't introduce a thermal bath, but you have to make sure the surface of the system is at temperature T and this is kept constant.

Your derivation may be shorter, but you skip over some nontrivial steps when you use dS >= dQ/T, which is not a mere statement of the second Law, and the conclusion you arrive at comes with extra baggage which isn't present in the textbook statement of the result.

The formulation of the second law that I use is, or should be, common knowledge for over 140 years. The expression dS>=dQ/T was first formulated by R. Clausius in 1867. Nowadays, you can find the expression dS>=dQ/T in many books such as: K. Huang, Introduction to Statistical Physics, 2001, p. 23, Eq.(2.41); D.R. Olander, General Thermodynamics p.30, Eq.(1.11); R.E. Sonntag, C. Borgnakke, and G.J. Van Wylen, Fundamentals of Thermodynamics, 6th edition, p.266; Landau and Lifshitz, Statistical Physics 3rd edition, part I, p.47, Eqs.(13.7 and 8); K. Denbigh, The Principles of Chemical Equilibrium, 3rd edition, p.40, Eq.(1.16); E.A. Guggenheim, Thermodynamics, an advanced treatment for chemists and physicists, 4th edition, page 17, Eq.(1.17.5); S.R. de Groot and P. Mazur, Non-equilibrium thermodynamics, 1969, page 21, Eq.(5); Kondepudi, Introduction to modern Thermodynamics, 2008, p.118 (even refers to Clausius!); C.J. Adkins, Equilibrium thermodynamics, p. 78. F. Reif in Fundamentals of statistical and thermal physics, 1965 p.122-123 splits the second law in two: one relation for a thermally isolated system and one for a not isolated system that undergoes a quasi-static process (he means: reversible process) but Reif gives no relation for a nonisolated system with internal irreversible processes.

Please go to a library and check for yourself.(Adwaele (talk) 15:00, 23 June 2012 (UTC))

I don't dispute the validity of dS >= dQ/T in this process for an infinitesimal change, but you apply it in the wrong way here, which has the effect of assuming what you set out to prove. The problem with your derivation is that the entropy increase doesn't yield the change in free energy for the system, because away from thermal equilibrium, you don't have a rigorously well defined temperature for the system. You invoke some "temperature of the boundary", which gives you the formally correct result, but it isn't the correct argument. The formulation of F. Reif of the second law is entirely consistent with the classical formulation of the Second law given in the other books. If the heat bath loses an amount of heat of Q, its entropy drops by Q/T (equality here because this process is quasistatic (and reversible!)) and so whatever happens in the system, its entropy must increase by more than Q/T. Count Iblis (talk) 16:25, 23 June 2012 (UTC)

It is true that the equation dS = dQ/T can only be used for "quasistatic processes", but this does apply for the heat bath, it remains in thermal equilibrium, no matter what the system does. In fact, if you were to actually proof that dS >=dQ/T applies to the system, this is an essential step that you skip over in your presentation.

For dS = dQ/T "Quasistatic" is not enough. Reif makes the same mistake. It must be reversible, which, in most cases, also means quasistatic. If I bring ice and boiling water in thermal contact the total amount of entropy that is produced in the end is the same even in the limit of a very small heat flow. As to my supposed skipping a step: see my remark after the previous paragraph.(Adwaele (talk) 15:00, 23 June 2012 (UTC))

If your argument has to depend on "Reif is wrong", then that implies that you are wrong (e.g. by misreading Reif and drawing conclusions that are not implied by his definitions), not Reif. Reif defines quasistatic in such a way that it is a limiting prcess that approaches reversibility in that quasistatic limit. E.g. in heat flow, that means that the slowness of the process involves the two systes that exchange heat being closer and closer to mutual thermal equilibrium. Ice in boiling water is never a quasistatic process (as defined by Reif) for the ice-water system, no matter how slow the ice melts. Count Iblis (talk) 15:43, 23 June 2012 (UTC)

In your presentation, you end up using T as the temperature of the system even though what you consider is a system that is not in thermal equilibrium. For such systems, you can't define a temperature at all. If you want to describe a system that is in local thermal equilibrium, you have to use far more sophisticated techniques and you end up with weaker results.

No. As you have seen in the figure of the practical example (I hope you have it somewhere) T is the temperature of the surface of the tank. This is in accordance with the second law where for T one must take the temperature at which the heat enters the system.(Adwaele (talk) 15:00, 23 June 2012 (UTC))

But the free energy of systems not in thermal equilibrium cannot be defined by invoking some ill defined "temperature at the surface". Count Iblis (talk) 16:25, 23 June 2012 (UTC)

And thermodynamics does only directly apply to systems in equilibrium that are therefore static, this is made clear in most of the standard textbooks on the subject. E.g. F. Reif says that "thermodynamics" is a misnomer, the topic should have been called "thermostatics".

I assume this is in the book of 1965. I wanted to study the context, but I could not find it. I would appreciate if you could give the page number of this statement by Reif.(Adwaele (talk) 20:45, 25 June 2012 (UTC))

The way you can nevertheless draw conclusions about systems that undergo changes is by clever reasoning about initial and final states which are states of thermal quilibrium. That is exactly what we do here, the system can undergo a rapid change during which none of the equations of thermodynamics apply. This actually makes the derivation far more general than your presentation. Count Iblis (talk) 16:25, 22 June 2012 (UTC)

Thermodynamics as an engineering science dealing with real systems in operation and not only with systems in equlibrium. The science of thermodynamics of irreversible processes is entirely based on nonequilibrium situations.(Adwaele (talk) 15:00, 23 June 2012 (UTC))

That's true, but that requires more work, not less. If the rigorous aproach where you have thermal equilibrium at the start and finish requires what experts have written about it, then if you lift this constraint and want to describe the system as it is in non-equilibrium states, that cannot make the derivation of the same result simpler. Count Iblis (talk) 16:25, 23 June 2012 (UTC)

Final remarks by Adwaele

Unfortunately it looks like we cannot reach consensus. It makes no sense to continue the discussion endlessly. I end with giving my opinion one more time:

• The Article, in its present form, does not clearly define the systems it is talking about.
• The Article, in its present form, does not clearly define the processes it is talking about. In particular it does not explain how a system, with a constant volume, can do work.
• The author has a very limited perception on the applicability of thermodynamics which is not shared by the vast majority of scientists and engineers active in the field of thermodynamics.
• As a result of all this the Article is asking for trouble and trouble it gets. If indeed dA = -SdT-pdV and dT=0 and dV=0, then dA=0. So how can A change nevertheless? The statement “After the spontaneous change, the system, as described by thermodynamics, is a different system with a different free energy function than it was before the spontaneous change.“ makes no sense. It looks like it is the private invention of the author to repair the unwanted outcome of a false line of reasoning. If I am wrong and if the cited text is common knowledge some references to the literature confirming this point of view (as is demanded by an Article in an encyclopedia) are in order. If no references are found the present text cannot be maintained.
• One can only hope that no students stumble on this Article in its present form, since it would be bad for their education if they would copy this approach.(Adwaele (talk) 14:31, 4 July 2012 (UTC))

I am actually working on a rewrite based on some your comments, but I'm doing this offline, as I can't do everything at once (due to lack of time). Things will be clarified, however from your comments, it won't be to your liking. E.g., in general, the spontaneous change is not described by integrating the differential dA expressed in terms of infinitemal changes of external parameters, because during the change a thermodynamic description does not have to exist. You simply refuse to accept this, but this is precisely why equilibrium thermodynamics is such a powerful tool. It can be applied even when systems during the change go out of equilibrium. This means that certain conclusions are not conditional on being able to give a description during the change. Non-equilibrium methods also exists that can approximately capture what goes on during the change provided it doesn't go too far out of equilibrium is neither here nor there.

Another thing is that you can also describe the system during the change assuming it is quasistatic, while at a temperature ddifferent from the heat bath. This is similar to what you did, and Reif actually also treats it this way. However, then all conclusions are based on a non-standard free energy function A0 = E - T0 S, where E and S are the energy and entropy of the system, but T0 is the temperature of the reservoir, not of the system.

Count Iblis (talk) 16:00, 4 July 2012 (UTC)

Helmholtz versus Gibbs

Some people reading your article are new at thermodynamics (such as yours truly).

Would it help to clarify the difference between Helmholtz Free Energy and Gibb's Free Energy as follows?

H = U + PV

A = U - TS

G = H - TS

TS = U - A

TS = H - G

U - A = H - G

U - A = U + PV - G

A = G - PV

A < G

Or maybe it is easier to say

H = U + PV, so U < H by the amount PV

A = U - TS

G = H - TS

so A < G by the same amount that U < H, i.e., by the amount PV

Perhaps I am all wrong. These equations are all new to me. P times V is always positive, right? Or perhaps the article already contains this information -- maybe I missed it when I was reading the article.

— Preceding unsigned comment added by Marvin W. Hile (talk • contribs) 15:10, 26 October 2012 (UTC)

No, this misses an essential part of the discussion - which potentials go with which set independent variables ... which is the major talking point of the potentials. This is better captured, not by noting the algebraic relations, by the differential relations: dU = T dS - P dV, dA = -S dT - P dV, dG = -S dT + V dP and dH = T dS + V dP; and then noting how this generalizes with the inclusion of number and chemical potential, and with more or other canonical (intensive, extensive) pairs of variables (X, x) in place of or in addition to (-P, V). You could then note, afterwards, that A = U - ST (or, more properly: A(T,V) = U(S(T,V),V) - S(T,V) T, when S can be expressed in terms of T and V), G = U - ST + PV and H = U + VP (with similar qualifications about their respective functional dependencies), which can almost be read off directly from the differential relations. — Preceding unsigned comment added by 2603:6000:AA4D:C5B8:0:3361:EAF8:97B7 (talk) 06:13, 5 April 2022 (UTC)

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Why Xdx?

The derivation in Helmholtz_free_energy#Relation_to_the_canonical_partition_function works the same without introducing, then cancelling the variable X and the change Xdx ? CyreJ (talk) 08:35, 20 August 2020 (UTC)