Talk:Entropy of mixing

Initial discussion

"The increase in entropy comes about because each gas absorbs heat in the process of isothermal expansion which is the same whether or not the gases have mixed. Thus the idea that increases in entropy, in non-interacting systems, is somehow connected with the mixing process is false."

How would a mixture of gases in an adiabatic chamber be able to absorb any heat at all? —Preceding unsigned comment added by 190.203.49.128 (talk) 02:34, 5 January 2009 (UTC)

I strongly concur with the above criticism. In particular, an ideal gas that expands into a larger evacuated space without doing mechanical work on the environment (as is the case here) is isothermal without absorbing heat, in accordance with Joule's law. In short, I agree with the conclusion (the second sentence above) but believe the reasoning (the first sentence) is erroneous. Rather, it is simply the entropy increase associated with the larger occupied volume for each constituent that is responsible for the entropy of mixing, which is just a reiteration of Gibbs' theorem. I will therefore delete the quote in question. Gpetty (talk) 13:51, 27 September 2009 (UTC)

I agree with you. In the quote, I think the second sentence is correct, but it doesn't follow from the first sentence. At least not outside the context of the original text. I will delete it again, since I had undone that change (sorry). Jorgenumata (talk) 10:43, 27 November 2009 (UTC)

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I believe the formula given on this page is only valid for an ideal gas; for other types of substance interactions between the molecules should make it a more complicated expression. I'm not 100% sure however, otherwise I would have edited the page to point it out. (The derivation given on the page is based on a lattice gas model and obviously doesn't include interaction terms.) Nathanielvirgo (talk) 19:07, 7 March 2009 (UTC)

Having thought about it a bit more, I'm sure this is correct: for non-ideal gases the entropy increase upon expanding into the volume is not equal to $\Delta S_{i}=-nR(x_{i}\ln x_{i})\,$ so, by Gibbs' Theorem as stated in the article the entropy of mixing will be different. So I will change the article to make that clear, I hope that's ok. Nathanielvirgo (talk) 19:14, 7 March 2009 (UTC)

I think the entropy of mixing non-ideal fluids is modified using the fugacity or activity coefficients for gases or liquids, respectively, but I'm not an expert - can someone more qualified please add this to the article?--82.69.126.85 (talk) 17:50, 14 March 2009 (UTC)

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The following sentence, from the introduction, is not true: "The name entropy of mixing is misleading, since it is not the intermingling of the particles that creates the entropy change, but rather the change in the available volume per particle." In fact one can calculate the entropy due to the intermingling of the particles, as is done in the body of the article, and it comes out to the correct numerical value.

To see why the statement that the entropy change cannot be due to the change in available volume per particle, consider mixing two liquids --- not sparse ideal gases but actual, non-dilute, incompressible liquids --- the particles of which each take the same amount of volume. Then after mixing there will be no change in available volume per particle, but the entropy of mixing is certainly positive. This is easy to see: one only has to note that work would have to be done to separate the liquids again in order to realise it must be true. Thus the increase in entropy in this case is purely down to the intermingling of the particles and has nothing to do with a change in volume.

It is only in the case of ideal gases that the entropy of mixing is numerically equal to the increase in entropy due to each gas occupying a larger volume. In all other cases one has to do the full calculation based on the mixing itself.

I see that further down, someone has inserted a quote from a paper making an obviously erroneous argument about this, essentially stating that because the result is true for ideal gases it must be true in general, without actually doing the calculation, and then concluding from this that the entropy of mixing is not actually due to mixing at all. In my opinion, if this belongs anywhere it belongs on the Gibbs paradox page, since it's historically been a subject of considerable debate (although I think these days it's generally seen as having been settled).

When I have time (which won't be for a while) I will have a think about how to reorganise the article in order to clarify that there is debate about this issue and explain why the "entropy of mixing is due to change in volume" point of view is false. If I don't see a reply here in the next week or so I will probably temporarily remove the references to it, because as it stands the article simply makes false statements. (or, at best, it makes controversial statements as if they were uncontested)

Nathaniel Virgo (talk) 08:58, 18 August 2009 (UTC)

The Gibbs "paradox" is extremely relevant to the understanding of the entropy "of mixing". I would urge you to read Ben-Naim, Arieh, On the So-Called Gibbs Paradox, and on the Real Paradox, Entropy, 9, 132-136, 2007 Link. This is an interesting topic and you may contrast your current point of view to Arieh's. By talking about liquids instead of ideal gases you seem to believe that you make the discussion closer to a real experiment, but for the issue at hand it just makes things more complicated (as liquids may have non-ideal interactions such as hydrogen bonds, etc). In any case, mixing two different liquids DOES increase the volume that each particle can roam about (anthropomorphizing it, "the volume it can explore"). If you have 1/2 liter of red particles and 1/2 liter of blue particles, upon mixing each red particle will roam about in 1 whole liter. This is the source for entropy to increase (entropy=uncertainty of position). Of course this is an irreversible process, and to separate the substances again you need work. But this doesn't prove that the increase in entropy comes from the intermingling. It is in general true that any process with an increase in entropy will not spontaneously reverse. Jorgenumata (talk) 10:38, 27 November 2009 (UTC)

Putting it that way it sounds reasonable - but note that the only reason the particles can be said to "explore" a larger area is that they are intermingled! So the assertion in the article that the increase in entropy is due to the increase in available volume and not the intermingling is still just weird and wrong. The increase in volume and the intermingling are two sides of the same coin.

Also, imagine you have 1/2 litre of red particles and another 1/2 litre of red particles. Now upon mixing each particle will explore the same volume as it would if half the particles were blue, but there will be no entropy increase. It really, really isn't just the increased volume that causes the entropy increase, and you really do have to factor in the question of whether two (classically) distinguishable types of particle are intermingled. Entropy does not equal "uncertainty of position" for a single particle - it equals "log of number of states available to the whole system" and you can't define that without defining which particles you consider distinguishable from each other.

But anyway, debating this here is relatively pointless, because it's still debated in the academic literature. But that's an important point -because it's still controversial the article should not be presenting either view as fact.Nathaniel Virgo (talk) 23:18, 26 June 2010 (UTC)

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An argument in favour of Ben-Naim's conclusion is the following. What is of concern here is the identification of process factors that in themselves are mechanisms of increase total system entropy. The question is whether mixing is such a factor. The question is not whether a process that involves mixing can change the system entropy; it is whether it must do so because it involves mixing. An important test for this is to consider a pure mixing process. The existence of a pure mixing process that does not increase system entropy provides a definite answer 'no' to the question of whether it must do so because it involves mixing. If a process that involves mixing increases system entropy, then one must consider the other features of the process as possible factors of the entropy increase. It seems that all the examples so far offered involve intermolecular interaction, which is a kind of semi-chemical reaction: this, and not the mixing itself, is the factor that affects the entropy; the presence of such non-mixing process factors is a distractor that does not provide information about the question of concern. It seems that Ben-Naim's choice of the ideal gas as the test-bed for this question was right, because it isolates mixing as the solely effective factor. Chjoaygame (talk) 19:49, 8 January 2011 (UTC)

Lattice comment

This page is truly terrible. Proof "let's imagine things on a lattice" how can you start a proof with that? As far as I know the world does not exist on a lattice. However, I think the confused nature of the page just reflects a lack of consensus in general on this. —Preceding unsigned comment added by Cplowe1 (talk • contribs) 22:01, 18 June 2010 (UTC)

• Proof involves 2-dimensional lattice. Compare with edit 1 V8rik (talk) 21:43, 27 June 2010 (UTC)

New edit by Dirac66

Many years ago I actually met the real P.A.M. Dirac (he was a real person), so I suppose you might be an avatar?

Your edit would benefit from some in-line citation support in the first paragraph. Your edit would be improved by an indication that the entropy of interest here is due to intermolecular attractive or repulsive forces, or whatever. The entropy of mixing you are concerned with is really a colligative property rather than referring to the kind of "mixing" of the "Gibbs paradox", which is about change of available volume for a species, not really about mixing at all. I have tried to put something to this effect into the lead. It would be better if you indicated more precisely the partial derivative character of your quantity (d²ΔS/dx²), indicating which variables are the arguments of your function S. You could perhaps write it $\left({\frac {\partial ^{2}\Delta S}{\partial x^{2}}}\right)_{P,T}$ or something like that. Your edit does not indicate why the ideal entropy of mixing is always positive with negative curvature. I think your edit should give some indication of why this is so. This subject is not altogether simple. There are also upper critical solution temperatures, and some mixtures have both, for example m-toluidine + glycerol at atmospheric pressure.Chjoaygame (talk) 08:05, 25 June 2011 (UTC)

Thank you for your suggestions. I know who Paul Dirac was, and I chose to honor him in my choice of wikiname.

I agree with some of your points, so I will (soon) try to include citations in my first paragraph, an explicit mention of the role of intermolecular forces, and a mathematical justification of the statement about curvature as well as better notation.

I did mention UCST at the end of my first subsection on Ideal and regular solutions. The fact that the same mixture can have both UCST and LCST seems off-topic here, but is mentioned in the LCST article with the examples of nicotine-water (which I inserted in April) as well as polymer-solvent systems.

I maintain however that the term "entropy of mixing" can be used to describe the change in entropy for any mixing process, meaning any process in which two initially separate (and different) substances come to occupy the same volume. This is the usual everyday meaning of the word "mixing", and thermodynamic quantities are defined for macroscopic states without considering molecules. Of course, the usual ideal-gas formula can be explained as due to the fact that each gas expands to a larger volume, but that is an explanation and not a definition. And the point of my edit is that this explanation is incomplete to describe entropies of mixing in the general case. Dirac66 (talk) 22:41, 25 June 2011 (UTC)

Perhaps a good way to say it is that the entropy of mixing in the general case is due to two effects: 1) the increase in volume available to each component, and 2) non-random mixing in the presence of strong interactions between the components. Eventually the article intro could mention both effects for generality. Dirac66 (talk) 00:04, 26 June 2011 (UTC)

Thank you for your response. As you say, the article needed to make it clear that the "entropy of mixing" can refer to two factors, volume availability and molecular difference. I think that Fowler and Guggenheim (1939/1965 reprint, on pages 163-164) were right to focus on which are the independent variables. Careful statement should be able to get round this muddle. I think time spent by the writer, carefully stating which are the independent variables, and other formalities, are many times saved for the reader. I do not think we should let the usages of ordinary language completely over-rule the need for consistency and clarity in thermodynamics. Following Fowler and Guggenheim, I have tried to define it in the lead by referring to definite volume. I propose that the changed reference-volume factor is a recognized muddle and we should not nourish it. For thermodynamics, the changed reference-volume factor is a matter of diffusion, not a true mixing effect, I say. The molecular difference effect of interest to you is the true mixing effect, I say, and this should be signalled by including the fixed volume statement in the definition as proposed in effect, by Fowler and Guggenheim. So I say, let's put " " marks around the changed-volume version. Macroscopically, for gases, one could define mixing at fixed volume by requiring both components each separately to have the final mixed volume in the initial state, and the process to proceed by compressing them and then slowly removing the partition between them so as to let them expand slowly again into the common final volume. I suppose this would work for suitable gases, though hardly for liquids, which are the commoner materials of interest. But for all cases I think careful definition as suggested by Fowler and Guggenheim should be used to cut out the unnecessary muddle, rather than our sustaining the muddle just to comply with an over-the-top "ordinary language over-rules all" approach. The "Gibbs' paradox" story is verging on a waste of time and effort, and should be minimized. Gibbs himself did not see a paradox. In the article as it stands I see hardly any explicit statement that one is thinking in terms of controlled volume or controlled pressure or whatever. The article, as it stands, jumps straight from an apparent macroscopic thermodynamic approach into a statistical mechanical approach, without a blink. I am inclined to think that, in some respects, the statistical mechanical version actually increases confusion about Gibbs' paradox rather than dispelling it. Quite likely it will be possible to partly re-write and improve the whole article to make more sense of the physics.Chjoaygame (talk) 00:45, 27 June 2011 (UTC)

I am glad you have now added a mention of intermolecular forces in the introduction. This ties in well with the new section on non-ideal mixtures, which will make the article more useful for chemists and engineers whose interests are not confined to the ideal gases of some physics books. I intend to concentrate on improving this new section for now, since as you have pointed out, it still needs much polishing.

We do need to agree however on whether or not the the ideal (volume expansion) term is also included in the definition of entropy of mixing, or whether it should be excluded as implied by the statement that the entropy of mixing is zero in the absence of intermolecular interactions. I checked one leading physical chemistry textbook (Peter Atkins and Giulio de Paula, Atkins’ Physical Chemistry, 8th edn W.H.Freeman 2006, p.143) which does describe the ideal-gas formula –nR(x₁ ln x₁ + x₂ ln x₂) as an entropy of mixing. The independent variables are T and p which is the usual case in chemical thermodynamics. I’m pretty sure this is standard practice in recent physical chemistry texts, so readers would find confusing a claim that the entropy of mixing is really zero for the mixing of ideal gases. I understand the point you are trying to make, but I think it would be clearer to say something like "For the mixing of ideal gases (at constant T and p), the entropy change is not due to any interaction between the gases which are mixed, but only due to the fact that each gas expands to occupy a greater volume".

As for the Gibbs paradox, I agree that this section is not very useful since everyone knows that mixing two identical gases is not real mixing and the entropy change is zero. I would suggest just deleting this text and adding at the end a See also section with a link to Gibbs paradox. Dirac66 (talk) 02:22, 29 June 2011 (UTC)

Thank you for this, Dirac66. I am in favour of following the line suggested by Fowler and Guggenheim, who emphasize the benefits of specifying which variables are dependent and which independent; I am in favour of letting this idea over-rule the path taken by Atkins and de Paula on this point. The point here is that the variables need to be specified for three systems: the two initial separate systems and the combined final system. I think that that it is not good to mix two distinct physical concepts (diffusion and colligation) in a single word. I propose that the Fowler and Guggenheim resolution of this problem is preferable. Prigogine and Defay (Chemical thermodynamics, translated by Everett 1954) are quite happy on page to say: "The fact that the entropy of mixing is zero, that is to say that the internal energy is unaffected by the mixing process, means that the interactions between the different molecules in the solution must be equal to the arithmetic mean of those between the molecules in the two pure liquids (cf. chap. III, §9)." They still accept the ideal gas formula that you mention.

As explained by Truesdell, fixed volume has important merits for basic thermodynamics and if it is stated clearly I think it is preferable as a base case, from which specialized branches can grow. Though the Gibbs free energy approach is very convenient for practitioners, it is in a sense too convenient for an introductory article, because it hides the more fundamental and generally safer fixed volume aspect of the matter. It would be best, I think, to start with the fixed volume approach and to derive the Gibbs free energy approach. This may call for reading a range of basic texts to get the information.

You mention that there seems to be a standard practice to assume a particular set of independent variables. This is not the best practice for a general reference like the Wikipedia, I suggest, which would do better to state the choice of independent variables explicitly rather than tacitly accepting a specialized custom. I think it seriously confusing to have a definition that requires one to add a rider such as "For the mixing of ideal gases (at constant T and p), the entropy change is not due to any interaction between the gases which are mixed, but only due to the fact that each gas expands to occupy a greater volume". We are not obliged to follow the most recent mode, but should prefer the best authority, and if necessary, as seems to be the case here, to mention other points of view, and to compare them. We do not have to make out that only the currently customary point of view exists.

The section on the Effect of removing a partition is all about the volume change and makes no mention of intermolecular interactions. Thus it focuses attention on the minor aspect, the "Gibbs paradox" for gases, and ignores the major aspect, the intermolecular interactions which appears in the mixing of liquids, which is your main concern. Thus it appears that in the past the minor aspect has dominated the article to the exclusion of the major aspect. This seems to mean that not everyone, at least not every Wikipedia editor, really "knows that mixing two ideal gases is not real mixing". I think it worth stating, even though experts know it without needing to be told. The article should be for general readers, including textbook-reading physicists, not only specialists such as chemists and engineers. The largest section of the article is now about the "Gibbs paradox" aspect, and this section should be somehow de-emphasized, quite how I am not sure at this moment.

If you put in more material that shows and clarifies the importance of the subject for liquid mixtures, I am very happy for the "Gibbs paradox" story to be moved to the end of the article. I have now done just this. I do not think it needs to be deleted altogether, because it offers an explanation of the difference between the textbooks stories about the "Gibbs paradox" and the more important aspects on which you are now adding material. I would right now further shift the order of the sections, and put your section on Temperature dependence of miscibility right up as the first section after the lead, but the wording would need some adjustment because at present it says things like "The above theory". So I will leave that re-arrangement to you. The citation from Callen of what he calls "Gibbs' theorem" could well be removed. I have now done that.

At the risk of being unnecessarily repetitive, I would say that I think that it is not good to rely on custom which the reader must guess if he is not familiar with it. Most especially for thermodynamics, time spent by the writer, carefully stating which are the independent variables, and other formalities, are many times saved for the reader.Chjoaygame (talk) 22:00, 29 June 2011 (UTC)

I would like to see Dirac66 make a structural change to the present article, putting his current work first after the lead, adding some general account of the entropy of mixing near the top of the article, and moving the calculation about the effect of removing a dividing partition to the end or near the end of the article. I am not proposing to make such restructuring myself because I think Dirac66 has the matter in hand and does not need help or interference. Probably to make the article read consistently and logically it may perhaps be convenient also to edit or partly re-construct the section on the effect of removing a dividing partition. I am hoping that Dirac66 will at some stage find it convenient to do something along such lines.Chjoaygame (talk) 09:12, 6 July 2011 (UTC)

Thank you for the vote of confidence. I have now revised the new section on "Temperature dependence on miscibility" as per your previous suggestions. I do plan to consider further changes to the rest of the article, but I will first take some time to consider the points you have raised and to read more. Dirac66 (talk) 15:39, 6 July 2011 (UTC)

P.S. I do agree with your placement of a shortened Gibbs paradox section at the end.Dirac66 (talk) 16:16, 6 July 2011 (UTC)

Just a little about what textbooks mention in relation to Gibbs' paradox in terms of (A) removing a partition between two distinct volumes of gas, (B) use of semipermeable membranes or other unspecified means to mix initially separate volumes by merging into the same volume rather than by adjoining by removal of partition, (C) entropy of mixing, (D) Gibbs' paradox in relation to the foregoing, and (E) Gibbs' theorem in relation to the foregoing. I have looked up the indexes for 'entropy of mixing' and 'Gibbs' paradox' with only very slight further search in several texts. Here are my results. The books mention the items as follows:

A B C D E (m denotes a mention, o denotes no mention)

m m m m o Adkins 1968/1975

m m m m o Bailyn 1994

m m m o m Callen 1960/1985

m m m o o Fowler Guggenheim 1939/1965

m o m m o Grandy 2008

o m m o m Iribarne Godson 1973/1981

m o m m o Kondepudi 2008

m o o m o Landsberg 1978

o o m o o Prigogine Defay 1950/1954

m m m m o Tolman 1938

From this I infer that the Wikipedia article on entropy of mixing has to say something about Gibbs' paradox.

To clarify about this I find particularly helpful the following. Apparently writing on the assumption that the "system" has fixed mass composition, Prigogine and Defay 1950/1954 write on pages 85, 86:

"In general an intensive thermodynamic quantity P in a uniform system:

P = P(T, p, x₁, x₂ ... x_c)

can always be split up into the sum of two functions, the first of which is a function of T and p only, while the second depends also on the mole fractions:

P = P^©(T, p) + P^M(T, p, x₁, x₂ ... x_c)

The first of the terms on the right hand side is called the standard function, while the second which represents the effect of composition is often called the function of mixing, e.g., heat of mixing or entropy of mixing.

This subdivision is not unique, since we can always add any function of T and p to P^© and subtract the same function from P^M. We must always therefore state precisely how this division is made."

More detail follows; the discussion of these details was emended (with the agreement of the authors) by the translator Everett because the book was for English speakers who use English and American conventions.Chjoaygame (talk) 23:10, 6 July 2011 (UTC)

The present article on the Gibbs paradox is about statistical mechanics and not about thermodynamics. It does not deal with the relation between entropy of mixing for gases and for liquids. I am not optimistic about fixing this within the article on the Gibbs paradox.Chjoaygame (talk) 00:21, 7 July 2011 (UTC)

Much improved

Today's version by Chjoaygame is much improved. We now have a much clearer intro without the very confusing statement that ΔS_m is zero if there are no intermolecular interactions, which is not true in general. Instead this case has been moved to a later paragraph where it is now made clear that this only applies when the volume available to each component is unchanged. Also the semipermeable membrane process has been clarified and moved into the same later paragraph. Much better. Some rewording would still help, but we are on the right track now.

I will change one word now which is "colligative". Colligative properties are properties of a solution (such as freezing point lowering), which depend on the solute concentration but not on the specific nature of the solute. However in this article the decrease in entropy of mixing in some nonideal liquid solutions (in the section I have added) is due to specific properties of some components, such as the formation of hydrogen bonds. I will replace "colligative" by "specific".

As for the plan of the article, I think we should leave the sections in order of increasing complexity: ideal gases first (with only the term for increased volume of each component), then ideal and regular solutions, then nonideal solutions. However I will think about integrating sections 4.Solutions and 5.Temperature dependence of miscibility. The sections on mixing with and without volume change and Gibbs paradox can remain at the end. Dirac66 (talk) 18:33, 10 July 2011 (UTC)

Thank you Dirac66 for your helpful correction of "colligative". I was a vague in my mind when I wrote "colligative", and now I learn that I was not just vague, but was mistaken.

I am not quite happy with a straight rule requiring an order of increasing complexity. The only interesting aspect of entropy of mixing is in the intermolecular forces, an aspect that was entirely absent from the article till Dirac66 came along and did something about it. It seems to me not good, to fill the early parts of the article with long-winded statistical mechanical proofs of the merely nuisance element that is in the expansion of available volume aspect, which is obvious enough without thought of mixing as such. I think that from the start it should be clear that the intermolecular aspect is the interesting one, and the the expansion of available volume aspect is a mere nuisance distraction. By all means tell about the ideal cases early on, but let it be very clear from the start that they are just formal preliminaries to the really interesting stuff.Chjoaygame (talk) 00:47, 12 July 2011 (UTC)

Well, I agree that the initial ideal-gas section is too long-winded. Why don't we move the two statistical mechanical proofs down, either to the end or else just before the Gibbs paradox section? I think there should be a short initial section with the formula for the ideal gas case, but we can say that the proof(s) can be found further down. Then we will get to the interesting stuff much sooner than we do now. Dirac66 (talk) 01:26, 12 July 2011 (UTC)

Yes I agree that the statistical mechanical proof would probably be better moved away from the very beginning. I am not saying it is too long-winded, just that it is long-winded and not really very much to the heart of the matter. I would like to see it made clear when the ideal gas story is first introduced that it is just a kind of framework or background on which the interesting stuff is supported or contrasted with. I have no problem with introducing the ideal gas case early provided its merely supportive purpose is made clear. There is not too much harm in structuring a story in three parts: tell what you intend to say, say it, and tell what you said. (Otherwise introduction, body, and summary.) This way the naive reader has some over-all guidance into a presentation of possibly complicated material.

I think here we may be looking at a difference of approach between conceptual simplicity and laboratory convenience? The fixed volume case is conceptually simple while the fixed presssure case is experimentally convenient. The really interesting stuff is about deviations from the ideal, because the deviations tell about the intermolecular effects.

The entropy (in the established usage) of the ideal case is not zero, but it is conceptually a kind of reference case, in some sense in conceptual effect or virtually a zero reference, so to speak. This is made explicit in the unconventional fixed volume version of "mixing", but takes a lot of explaining in the conventional established usage, where it is in a sense only implicit.

Though a rather theoretical point, there is in general some degree of conceptual merit in the use of fixed volume processes, or in this case fixed 'partial density' or fixed concentration processes.

The discussion of entropy of mixing is most usually conducted in terms of the Gibbs free energy.

This may be viewed as a focus on fixed pressure, but alternatively as a focus on controlled chemical concentration. The Gibbs potential is mainly used by chemists because they are mainly interested in chemical concentration, I think? Experimentally I suppose they use a vessel of nearly fixed volume and put varying amounts of various liquids into the mixture; this is done under more or less controlled pressure conditions; but what is greatly varying is the chemical concentration more than the final common volume, I think? With gases, gross experimental manipulation of volume is quite feasible, with large changes of the volume of a fixed mass of gas being easy enough, but hardly with liquids. Many textbook phase diagrams of liquid mixtures have mole fraction on the abscissa and temperature on the ordinate. One might argue that the chemist is interested in mole fraction rather than pressure as an independent variable, except that fixed pressure is a convenient framework for definiteness and laboratory practice. I suggest that selecting volumes of liquid for the mixture is conceptually like the semipermeable membrane method for gases, when it comes to looking at the finally constituted mixture. So I am suggesting that conceptually the fixed volume approach is not really too foreign or excessively mysterious for chemists, and does not need complete expulsion from the article.

I think it better that I leave the reconstruction mostly to you because matters of continuity arise: 'the above remarks ...', 'it is shown below in this article that ...', etc., etc..Chjoaygame (talk) 03:42, 12 July 2011 (UTC)

OK, I will try to reconstruct the article and give more weight to the nonideal part, but this will take some time. I envisage a plan of three (groups of) sections.

1) Brief presentation of the ideal case first, enough to show what the nonideal case is deviating from. Here we can include a simple thermodynamic justification - ΔS_m is just a sum of ΔS for the (isothermal, irreversible) expansion of two or more ideal gases. My intro phys chem classes have understood this quite well without any statistical mechanics.

2) Nonideal case = ideal case + effects of intermolecular interactions

3) More theory for ideal case: "proofs" relabelled as stat-mech significance of ΔS_m, reversible ΔS_m (with semi-permeable membranes), Gibbs paradox etc.Dirac66 (talk) 15:47, 12 July 2011 (UTC)

Your careful work and plan are most admirable.Chjoaygame (talk) 00:55, 13 July 2011 (UTC)

I have now reordered and regrouped the sections as above, with what I think is the minimum on ideal gases before getting into Solutions which include the effects of intermolecular interactions. Eventual rewording of some sections is planned in order to address questions such as continuity which Chjoaygame mentioned above. Dirac66 (talk) 00:36, 16 July 2011 (UTC)

Irreversible and reversible mixing

After rereading the "Mixing with and without change of available volume", I realize that the two processes would be better described as irreversible and reversible mixing. The "established customary usage" refers to irreversible mixing with ΔS_m > 0, as in the usual practical mixing of near ideal gases. The process using semipermeable membranes (now described as "mixing" in quotes) is a reversible mixing process with ΔS_m = 0 proposed (by Gibbs?) to justify Gibbs' theorem. An equivalent formulation of this theorem is now in the ideal gas article and states that the entropy of a multicomponent system is equal to the sum of the entropies of each chemical species (in the common volume). Dirac66 (talk) 01:24, 11 July 2011 (UTC)

Thank you Dirac66 for this comment. I do not agree that the difference is in reversible or irreversible mixing. I think that Fowler and Guggenheim are right that it is in the specification of the dependent and independent variables. I think to call it a difference between reversible and irreversible mixing hides the point that what matters is the hypothesis of some separation technique such as the use of putative semipermeable membranes. To relegate this to the ideal gas article is to miss the point, that it is not to do with ideality, but is to do with change of available volume. It just happens to be easy to prove it by argument for the ideal gas case, but that doesn't tell about its being fully significant for all materials, ideal or not. Perhaps my logic is astray here, do you think?Chjoaygame

(talk) 00:56, 12 July 2011 (UTC)

I think the mixing process using semipermeable membranes was only invented to provide an example of reversibility. It is not a practical process - a lab chemist who wants to combine (gas A at volume V) and (gas B at volume V) to produce a mixture at combined volume V (at higher pressure) can proceed in two steps: remove the partition to produce a mixture at volume 2V, and then compress back to volume V with the performance of external work. But this lab process increases the entropy in the first step, and decreases the entropy in the second step which is only possible because the system is not isolated. Since the total ΔS is zero for ideal gases, Gibbs (or someone) wanted to show that the process could in principle be carried out reversibly and devised the semipermeable scheme (which you have explained well in the article).

For non-ideal systems this semipermeable membrane process would be less interesting since the entropy of the two steps does not cancel so the process is not exactly reversible. To take an extreme case [for discussion only as I have no source], if triethylamine and water were mixed this way in the gas phase (so the available volume can remain constant), then the formation of hydrogen bonds should lead to an appreciable decrease in entropy. If the process were carried out in an adiabatic calorimeter, there would be an increase in temperature to provide enough entropy increase to compensate. Dirac66 (talk) 18:59, 12 July 2011 (UTC)

Thank you Dirac66. I do not know the history but I have no problem with the idea that the semipermeable membrane scheme was invented for purely theoretical reasons, and that it is hardly likely to be a practical procedure. I have no problem with your hypothetical method of "mixing" at constant available volume for each gas. I have no problem with your hypothetical triethylamine and water experiment.

I should have written not that "I do not agree that the difference is in reversible or irreversible mixing." I should have written that 'I do not agree that the interesting part of the difference is in the reversibility or irreversibility of the mixing.' I expressed myself better in my next sentence: "I think to call it a difference between reversible and irreversible mixing hides the point that what matters is the hypothesis of some separation technique such as the use of putative semipermeable membranes." My concern is with the idea that what is interesting about the entropy of mixing is the part that is left after the ideal case moiety of the process has been subtracted away, and that this should be clear to the reader from the outset, and not entirely delayed in the exposition till after an excursion into the uninteresting effects of volume expansion for the ideal case.

The entropy of mixing is in a sense an artificially defined quantity: it is defined as a difference between two actual entropies, at least according to Callen 1960/1985: "It represents the difference in entropies between that of a mixture of gases and that of a collection of separate gases each at the same temperature and density as the original mixture N_j / V_j = N / V, (and hence at the same pressure as the original mixture); see Problem 3.4-15. The close similarity, and the important distinction between Gibbs's theorem and the interpretation of the entropy of mixing should be noted carefully by the reader." Even here Callen himself is apparently letting the ideal gas case intrude itself when it is not really quite to the point. Fowler and Guggenheim also wave a red flag here. I think the warnings of these authorities should find expression in the Wikipedia article. They are theoreticians and perhaps do not quite have exactly the point of view of practitioners of chemistry that I suppose informs the established customary usage of chemists, which, like any definition of entropy of mixing, is in a sense artificial, at least to some extent. It would be a pity to let the artificial convention do a snow job on the interesting physics. For this problem, I find talk of reversibility or irreversibility opaque. I am saying not that it is wrong, just that it doesn't speak lucidly to the question of interest here. It is not the question of reversibility or irreversibility that is most relevant here: it is the question of how to subtract out the less interesting volume expansion effect to leave the more interesting intermolecular interaction effect standing out for the reader to see; this is a matter of which variables are treated as dependent or independent, as proposed by Fowler and Guggenheim, I think.Chjoaygame (talk) 22:49, 12 July 2011 (UTC)

OK, I think we are sufficiently in agreement for the moment, so I will stop discussing the theory here and start trying to reconstruct the article as suggested above. Dirac66 (talk) 23:49, 12 July 2011 (UTC)

Yes.Chjoaygame (talk) 00:50, 13 July 2011 (UTC)

Notation for publication dates

One small question of style. What does it mean when two dates are given for the same reference, as Callen 1960/1985 and several other examples? My guess is first edition 1960, edition consulted 1985. However I was taught to just identify the edition consulted (e.g. 4th edition 1985) and forget the original edition. Dirac66 (talk) 23:49, 12 July 2011 (UTC)

You are probably right for the context in which you were taught, and perhaps also for the present context. Perhaps it is a bad habit that I like to briefly indicate to the reader that there is a history to the publication. I don't see it as very harmful here, even if not according to the rules and regulations.Chjoaygame (talk) 00:48, 13 July 2011 (UTC)

Ideal liquid solutions do have intermolecular forces

To Chjoaygame: I agree with most of your latest edit, but the statement that "Ideal materials do not have intermolecular forces" requires correction. It is true that ideal gases and mixtures of ideal gases do not have intermolecular forces. However all liquid (and solid) solutions must have intermolecular forces in order to provide cohesion as a liquid (or solid) - with no forces a liquid solution would just evaporate. A liquid solution will be ideal if all the intermolecular forces (A-B, A-A, B-B) are the same, so that a given molecule "sees" no difference between its A neighbours and its B neighbours. This is explained correctly in the ideal solution article. If a molecule does see a difference between its neighbours, then the interactions are "specific" and the solution is not ideal. Dirac66 (talk) 00:44, 19 July 2011 (UTC)

Thank you. Advice followed.Chjoaygame (talk) 03:41, 20 July 2011 (UTC)

This point is correct now. However I am not happy with the phrase "ideal material", which is added several more times in today's edit. I have never seen it used to mean "ideal gas or ideal solution". Googling "ideal material" leads instead to many solid materials which are supposedly ideal for constructing various objects. I would rather use the standard terms and say "ideal gas (mixture) or ideal solution" where appropriate. Dirac66 (talk) 23:27, 26 July 2011 (UTC)

Thank you. Advice followed.Chjoaygame (talk) 23:43, 26 July 2011 (UTC)

On second thoughts, I spoke too soon. It seems too rigid to rely upon a Google survey to dictate what words one should use. The words 'ideal' and 'material' are not copyright or proprietary trademarks, but are words of ordinary language. I will try a compromise, because to repeat the phrase "ideal gas (mixture) or ideal solution" too often would attract a criticsm of clumsy anaphora.Chjoaygame (talk) 23:50, 26 July 2011 (UTC)

OK. At least now we have defined our non-standard vocabulary. (And I had to look up anaphora.) Dirac66 (talk) 00:23, 27 July 2011 (UTC)

Gibbs free energy of mixing

The appearance here of the Gibbs free energy is like a flash of magical insight coming out of nowhere. I think it needs explanation.

The article on Gibbs free energy contains the miraculous statement

"Therefore the Gibbs free energy of an isolated system is:

$\Delta G=-T\Delta S\,$"

which looks like nonsense to me, and is accordingly unhelpful to me. I am not interested in the article on the Gibbs free energy.Chjoaygame (talk) 22:57, 26 July 2011 (UTC)

The importance of $\Delta G\,$ is stated in the first sentence of this section - it determines whether mixing at constant (T,p) is spontaneous. It is referred to in the discussion on UCST and LCST so it has to be introduced first.

As for the math, the general equation $\Delta G=\Delta H-T\Delta S\,$ is found in the article Gibbs free energy, and is a good starting point because it is well known to chemists (and many physicists too). The shortened equation $\Delta G=-T\Delta S\,$ is merely the result of setting $\Delta H=0\,$ Dirac66 (talk) 23:53, 26 July 2011 (UTC)

Thank you Dirac66. The section says that the Gibbs free energy is important but gives no hint as to why this and some other quantity is the important one. Chemists and physicists mostly do not come to the Wikipedia to find out such things as that the Gibbs free energy is useful for discussion of mixing. As you say, they know that already. I think the Wikipedia aims to speak to naive readers. The article as it stands does not do that. The equation $\Delta G=\Delta H-T\Delta S\,$ is indeed found in the other article as you say, but that article is very poorly set out, and is not a good foundation for another article. As I wrote above, the statement

"Therefore the Gibbs free energy of an isolated system is:

$\Delta G=-T\Delta S\,$"

looks like nonsense to me, and is accordingly unhelpful to me. It gives no hint of the physical nature of mixing. This is because it is just a mathematical formula wanting an account of its physical meaning. A change of Gibbs free energy is a character of a process, not a property of an isolated system. We do not here have to follow the slipshod ways of the article on the Gibbs free energy. You write: "As for the math, ...". I like to think of this article as about physics and chemistry, not as a mathematical exercise. In a mathematical exercise, mathematicians like to know which are the dependent and which the independent variables. To make the article about physics and chemistry, one needs to state the physical and chemical reasons for the choices of the variables. Switching from the enthalpy to the Gibbs free energy is not a very clear physical or chemical account.Chjoaygame (talk) 00:55, 27 July 2011 (UTC)

The reason one wants to work with the Gibbs free energy is that the present customarily accepted definition of entropy of mixing makes it visible most directly in terms of the Gibbs free energy, because of the variables which are the arguments of the Gibbs free energy. Probably the reason for the choice of presently customary definition is that chemists like to work with the Gibbs free energy, and they are the people who are most interested in intermolecular forces as revealed by mixing. Really, the enthalpy of mixing and the entropy of mixing are not physical effects, they are ways of describing a physical effect. The real physical effect is in the differences of intermolecular interaction.Chjoaygame (talk) 10:45, 27 July 2011 (UTC)

Notational confusion and LaTeX problems

Stirling's approximation is written as

$\ln n!=\sum _{k}\ln k\approx \int _{1}^{n}dk\ln k=n\ln n-n$

which is correct, except that n is already used in this article for total number of moles. And it should even be so used in the same section to explain that N k_B = n R so that the entropy of mixing obtained is the same as that given earlier.

Simple solution: change n in Stirling's formula to m throughout. However ... when I try I keep obtaining a Failed to parse Error. Obviously I do not understand the LaTeX language for writing these equations. Can someone (Chjoaygame?) figure out how to change all the n in this equation to m? [Computers are wonderful - except when they stubbornly refuse to do what we want.] Dirac66 (talk) 00:33, 27 July 2011 (UTC)

I too tried to edit another expression and ran into a problem like this that seemed to be driven by a mysterious demon. I do not know the answer. Perhaps we are the witnesses of an improvement, according to the principle that every improvement makes things worse?Chjoaygame (talk) 01:26, 27 July 2011 (UTC)

That does help, because I did not realize it was a general problem. After trying similar edits of a few randomly chosen equations elsewhere, I see that it is. So after consulting one Wikipedia administrator for advice, I have reported the problem at Wikipedia:Village pump (technical)#LaTeX equation problems. Dirac66 (talk) 02:39, 27 July 2011 (UTC)

I have just discovered that the problem is browser-dependent! I get a Failed to parse error with Internet Explorer (version 7.0.5730.11), but I can edit the equation correctly with Mozilla Firefox (version 3.6.3). It's late here, so I'll report this on Village pump now, and edit the article tomorrow. It still should be fixed to work with any browser of course. Dirac66 (talk) 03:13, 27 July 2011 (UTC)

Thank you for this. Perhaps the problem may be due to a recent improvement in Internet Explorer? It is a subtle problem because it comes in way that is hard to predict.Chjoaygame (talk) 10:33, 27 July 2011 (UTC)

You don't explicitly say so, but I suspect that you are using some sort of gadget or browser add-on to edit these equations. I don't think it's necessary to do this when all you need to do is change the five instances of "n" to "m". You can use the normal editing window. In this, the equation shows as <math>\ln n! = \sum_k \ln k \approx \int_{1}^{n}dk \ln k = n\ln n - n</math>. The letter "n" occurs ten times here: four times in the name of the \ln (natural logarithm) function, and once in the name of the \int function (definite integral) so clearly those should be left alone; and the other five just need changing from "n" to "m", giving <math>\ln m! = \sum_k \ln k \approx \int_{1}^{m}dk \ln k = m\ln m - m</math>. Click "Show preview", and this is

$\ln m!=\sum _{k}\ln k\approx \int _{1}^{m}dk\ln k=m\ln m-m$

and if it displays correctly (which it does for me in Google Chrome 12.0.742.122, Microsoft Internet Explorer 7.0.5730.13, Mozilla Firefox 3.6.18, Opera 11.50 (Build 1074) and Safari 5.1 (7534.50)), click "Save page". --Redrose64 (talk) 12:43, 27 July 2011 (UTC)

Thanks. I think I did do this using the edit window, and yes I changed the same n's as you though I described it wrong here, but yesterday it didn't work. However today I tried again with Explorer (7.0.5730.11) and it did work in the preview! Perhaps I did something wrong that I can no longer reproduce. Anyway I am more interested in chemistry than computers, so I'll fix the equation and get back to editing. Dirac66 (talk) 13:00, 27 July 2011 (UTC)

Notation - clearest subscript is "mix"

The article now freely interchanges $\Delta S\,$ and $\Delta S_{m}\,$ to mean the same thing. For consistency and clarity, I propose to change to $\Delta S_{mix}\,$ throughout. This is in agreement with the referenced textbooks by Atkins and de Paula, and by Rock, as well as with the IUPAC Green Book. It is clearer than $\Delta S\,$ which could refer to any process, and clearer than $\Delta S_{m}\,$ where the m could mean molar. Dirac66 (talk) 18:10, 27 July 2011 (UTC)

Total entropy of a system

Good point, necessary correction. Thank you, Dirac66.Chjoaygame (talk) 00:53, 28 July 2011 (UTC)

"Random mixing"

Call me old-fashioned if you like, but it makes me very unhappy to see the word 'random' introduced in this way. The notion of randomness belongs to statistics and is not in the home territory of macroscopic thermodynamics. If you want to put something about statistical mechanical explanations of thermodynamics, that is to say, some statistical thermodynamics, into the lead, I would prefer it to be done explicitly in a new paragraph rather than see it done by slipping in a phrase that I feel is out of place.

The concept of 'randomness' is not as obvious as perhaps one might be led to think by some of its users. It is often used, I think, as a substitute for clear thinking and expression. It should be used carefully.

I am not about to overwrite your present edit but I would like to ask you to revise it.Chjoaygame (talk) 21:06, 4 August 2011 (UTC)

I don’t agree that there is a problem here. I juxtaposed the mentions of randomness with the mentions of intermolecular forces, which are also not part of macroscopic thermodynamics. Historically intermolecular forces were proposed by van der Waals as an add-on to the kinetic theory of ideal gases, which is the simplest version of statistical mechanics. Their juxtaposition in the introduction with the mention of randomness is intended to indicate briefly just why the differences in intermolecular forces affect the entropy.

As for the definition of randomness, in the context of the article it means that the probability of finding a given component at any lattice site (or position) is independent of the environment of the lattice site. This is not true for example in the triethylamine-water system, where the triethylamine molecules strongly favor positions next to water molecules to which they can hydrogen-bond. I don’t think this point should be explained in detail in the introduction which is long enough already, but it could be added to the section on nonideal solutions. Dirac66 (talk) 03:25, 5 August 2011 (UTC)

I accept that talk of intermolecular forces is not part of macroscopic thermodynamics. For that reason perhaps they should be put into a separate paragraph, like talk of randomness. I have made an attempt along these lines.Chjoaygame (talk) 21:41, 5 August 2011 (UTC)

OK, the two levels of the discussion are now separated. I have just added the words "On a moleculsr level" as a clear signal to the reader that this is the start of the molecular discussion. Dirac66 (talk) 22:07, 5 August 2011 (UTC)

Yes.Chjoaygame (talk) 22:24, 5 August 2011 (UTC)

Chemical substances or material components?

The new intro sentence was overly long so I have removed a few words which seemed less essential. I am still trying to understand the phrase separate systems ... of different and chemically non-reacting chemical substances or material components, are mixed .... What is the difference here between a chemical substance and a material component? Could we say instead separate systems ... of different composition are mixed without chemical reaction ...? This seems not only clearer but also more general as it is includes the case where (some or all) substances are in common but in different proportion, for example the mixing of air with pure oxygen gas. Dirac66 (talk) 20:07, 18 June 2013 (UTC)

Yes, I agree. Well said. I suggest you go ahead and make the changes.Chjoaygame (talk) 22:37, 18 June 2013 (UTC)

a few occupied cells

I have edited over the previous edit which read: "Almost everywhere we look, we find empty lattice cells. But we do find molecules in those few occupied cells."

That previous version does not make sense. The word 'those' in it has no naturally defined reference. One has to infer that it refers to the non-empty cells implied by its first sentence, though they are not explicitly indicated in that first sentence. Such an expression is sometimes called unresolved anaphora. Use of the word 'those' presupposes that some determinate reference object is immediately present. In unresolved anaphora, that is not so, and the reader has to guess to resolve the anaphora. Mostly this will constitute a fault of style or even of grammar.

Since this point has arisen once already just recently with the same editor, and the meaning was not clear to the editor 75.0.185.27, and my edit to correct it apparently did not convey the message, it may seem that the new extension of wording is perhaps necessary. Alternatively, in the previous version, that editor 75.0.185.27 edited over, the word 'those' referred to cells identified in the same sentence; thus it was acceptable in style and grammar, so that the edit was unnecessary.

It would be good if editor 75.0.185.27 would give himself a user name. So far as I know it is safe to do so. And it makes it happier to talk with him.Chjoaygame (talk) 00:13, 1 December 2013 (UTC)Chjoaygame (talk) 00:21, 1 December 2013 (UTC)

Serious contradictions in introduction

There are some serious contradictions in this page's introduction. I think the worse one is: "The final volume need not be the sum of the initially separate volumes..." (second paragraph) "In a process of mixing of ideal materials, the final common volume is the sum of the initial separate compartment volumes." (fifth paragraph)

I believe the first one is the correct one - the thermodynamic process of mixing materials is not constrained for a defined volume. You can mix materials at constant pressure, temperature or volume, each process will give a different result. But you can't say that mixing always yields the volume of the sum on initial volumes - it depends on the specific thermodynamic process made. However, entropy of mixing is defined as the added entropy when materials are mixed at constant pressure and temperature, so that is the correct "mixing interpretation" that should appear in this page.

There are some more contradiction close to this one, like saying the internal energy is always the sum of the initial internal energy in one paragraph, and saying it is not in another. The correct answer is (for the appropriate interpretation of mixing - const T,P) that internal energy can of course change. — Preceding unsigned comment added by 5.102.216.5 (talk) 15:54, 22 December 2017 (UTC)

The second paragraph refers to the general case (ideal or non-ideal) for which the final volume is not necessarily the sum of the initial volumes. The fifth paragraph refers to IDEAL solutions or IDEAL gas mixtures, which are a special case for which the final volume is the sum. However the sixth paragraph refers to NON-IDEAL solutions for which this is not true.

I have changed a few words to emphasize what refers to ideal systems and what to non-ideal. Dirac66 (talk) 02:48, 23 December 2017 (UTC)

Editorial opinions in the article

The last two sections of this article, "Mixing under other constraints" and "Gibbs' paradox: 'mixing' of identical species versus mixing of closely similar but non-identical species" appear to contain a lot of forceful editorializing. They seem to be extensively-referenced opinion pieces. In particular, they seem to avoid any re-interpretations provided by Jaynes (in the 1950's! A long time ago!) and seem to adamantly argue for conclusions that have not obviously withstood the test of time (despite the dozen citations and references given!) The article on Gibbs paradox does give Jaynes the time of day (even though that article is problematic, but in a different way), and the article on entropic force gives at least some hints where the work comes from/goes to (even though this is also a controversial topic.). It doesn't help at all that these sections aren't decorated with any diagrams showing where these permeable membranes are located, what direction they move in, how far they move, what forces are applied, and what the temperature gradients and heat flows might be. We've got P,V,S,T and maybe a chemical potential or maybe not, some enthalpy or maybe not, but they're all tangled up in the prose. Overall, it comes off as an opaque opinion piece, rather than trustworthy information. I'm not happy about that. It doesn't smell right. FWIW, exploring the article history, it appears that these sections were added by User:Chjoaygame between 2011 and 2014. He/she has clearly made prolific contributions to this article. Failing to address Jaynes is perhaps the most serious issue. Opaqueness due to missing diagrams is also problematic. I can't figure out what the deal is. 67.198.37.16 (talk) 02:00, 20 May 2024 (UTC)

Thank you for your care for this article. I am a keen admirer of Jaynes, and am sorry that I didn't give him adequate notice. Great to add some diagrams. I am not a fan of the term 'entropic force'. I always feel more comfortable with Wikipedia editors who select and use a user name, which may preferably be chosen to preserve anonymity or security.Chjoaygame (talk) 02:56, 20 May 2024 (UTC)

You are welcome to call me 'Linas'. I'll reiterate: as currently written, it is not obvious that the content in those sections is actually correct. It's also not obvious that a sentence-by-sentence dissection would be a good use of time. So, I'm not sure how to proceed, other than to call attention to it. 67.198.37.16 (talk) 04:53, 21 May 2024 (UTC)

Thank you for your response. I was suggesting that you give yourself a Wikipedia user name. Then it is obvious to all that they are conversing with a definite person, as distinct from a web address. I suggest you choose one that doesn't risk your privacy or security. Of course, you are free to improve the article as you please.Chjoaygame (talk) 19:47, 21 May 2024 (UTC)

My wikipedia user name is Linas. Editing from an IP address offers strong advantages over editing from a WP login. This is generally understood by all. Nothing to do with anonimity nor privacy. If you wish to have personal conversations, we can have them on my personal talk page. Doing this in article space is not appropriate. 67.198.37.16 (talk) 20:58, 21 May 2024 (UTC)

I am in complete agreement with Linas. The discussion of entropy of mixing is quite long, confusing to an outsider, and presents a biased perspective on the issue, disregarding the clarification provided by Jaynes (which is that entropy is not absolute, but defined relative operationally in terms of available transformations/distinctions). One suggestion might be to shorten this section by a lot, and move deep-dive into relevant issues of distinguishable/indistinguishable particles to the Gibbs paradox page. 188.26.214.197 (talk) 15:41, 21 June 2024 (UTC)

Thank you for your care for this article. I always feel more comfortable with Wikipedia editors who select and use a user name, which may preferably be chosen to preserve anonymity or security.

I am a keen admirer of Jaynes, and am sorry that I didn't give him adequate notice. Jaynes does a great job on the statistics of distinguishable and indistinguishable particles, but there are other aspects to the entropy of mixing. The distinguishable/indistinguishable particle thing is mainly about non-interacting particles. Physical entropy of mixing takes into account also the interactions of particles. Of course, you are free to improve the article as you please.Chjoaygame (talk) 10:47, 22 June 2024 (UTC)