Talk:Chemical potential

About chemical potential of a system of electron
Here is a quote from Solid State Physics, Ashcroft and Mermin(1976),Chapter 2, p.43 eq.(2.53): "[...]for metals the chemical potential remains equal to the Fermi energy to a high degree of precision, all the way up to room temperature. As a result, people frequently fail to make any distinction between the two when dealing with metals. This, however, can be dangerously misleading. In precise calculations it is essential to keep track of the extent to which μ, the chemical potential, differs from its zero temperature value, εF[the Fermi energy]". This would be a precision of "The chemical potential of a system of electrons is also called the Fermi level" from the article. —Preceding unsigned comment added by 216.144.116.202 (talk) 21:11, 30 January 2011 (UTC)


 * The article text "The chemical potential of a system of electrons is also called the Fermi level" is exactly correct. "Fermi level" as used in semiconductor physics is not the same as "Fermi energy" as used by some solid-state physicists. "Fermi level" in semiconductor physics is an exact synonym of "chemical potential". The "Fermi level" (as the term is used in semiconductor physics) does vary with temperature, sometimes substantially, and its temperature-dependence is common knowledge and it is discussed frequently and openly as an important topic. It is not defined as a zero-temperature limit! This confused claim comes up so often that I made a little webpage about it,.


 * Perhaps the article text could be clarified, "In semiconductor physics, the chemical potential of a system of electrons is called the Fermi level." --Steve (talk) 21:39, 30 January 2011 (UTC)

I think this statement is plainly wrong. The chemical potential is only equal to the Fermi energy at zero kelvin. The fact that a lot of people are confused and that in several cases the chemical potential has a very small temperature dependence does not change this. I would suggest the following statement:

In physics the chemical potential for a system at 0 Kelvin is known as the Fermi level. — Preceding unsigned comment added by 92.224.192.89 (talk) 19:34, 7 September 2013 (UTC)

on the simplified introductory sentence
Chemical potentials are not a per-system phenomenon. One system can have multiple chemical potentials. For instance, in a non-equilibrium system, at any location, the chemical potentials of different species are not guaranteed to be the same. Furthermore, even in an equilibrium system (such as one in a steady state steady flow condition), the chemical potential would be different at different locations. Finally, the change in energy is not just a change in Gibbs free energy, it is a change in all the energy potentials of the system including internal energy (excepting energy potentials that are not functions of the chemical potential). ChrisChiasson (talk) 03:35, 20 November 2007 (UTC)
 * OK, I over-simplified. Under the "precise definition" section (Chemical_potential) it does state that "the chemical potential is the partial derivative of the Gibbs free energy with respect to number of particles" "under conditions of constant temperature and pressure" (typical laboratory conditions). I do think that a simpler introductory sentence would be helpful for general readers than the existing introduction -- without compromising accuracy. --Ben Best (talk) 07:40, 20 November 2007 (UTC)
 * Hi. I find the introductory section to be rather poorly worded. I, for one, did not understand it. Perhaps we should update it Paskari (talk) 13:42, 7 October 2009 (UTC).
 * I concur with Paskari. The introduction should begin with a definition, not a history.  --BBUCommander (talk) 21:40, 6 February 2012 (UTC)
 * I've moved the cited definition into the first sentence. It's still not the best definition, but it's better than just stating the origin of the term, I think.Forbes72 (talk) 01:44, 28 February 2012 (UTC)

Random comments
1) Chemical Potential is a noun, it should not be described as a verb, such as "change". Velocity is a noun and defined as the change in position with time. ...... So the article is fine.

2) Constituent: an artifact that is one of the individual parts of which a composite entity is made up; especially a part that can be separated from or attached to a system

A particle is added to a group of particles. The energy required defines the chemical potential. I ommited the word "constituent particle" because the additional particle need not be "one of the individual [particles] of which the [group of particles]" is added to. A chemical potential is still defined if I take a particle that is not a member of the group of particles and add it to the group of particles.

3) Point 1) above is not right. Change can be a noun as well as a verb. E.g., if I change (v.) x to x+1, then the change (n.) in x is 1.

4) The fundamental concept of the chemical potential (1st sentence of the article) as being the change in energy on adding a particle, is often (in many books and papers) taken literally to mean a finite difference F(N+1)-F(N), where N=number of particles and F is either the internal energy or the Helmholtz free energy (the difference is taken at constant entropy and volume in the former case, at constant temperature and volume for the latter). That this can be essentially incorrect for finite systems (even for some very large finite systems) has been discussed recently (see The Chemical Potential, by T. A. Kaplan, J. Stat. Phys. 122, 1237 (2006)).--Tomkaplan 19:16, 3 January 2007 (UTC)

5) Appreciation of unity in physics where it exists is clearly desirable. Making a distinction between `thermodynamic chemical potential' and `electronic chemical potential' seems to introduce a fundamental distinction where it, in fact, does not exist: the electronic version introduced in the main page of this subject is actually the zero-temperature limit of the thermodynamic chemical potential. For this see J. P. Perdew in Density-functional methods in Physics}, edited by R. M. Dreizler and J. da Providencia, Plenum, New York, 1985, and T. A. Kaplan, J. Stat. Phys. 122,1237 (2006). Thus the distinction, as presented in the fundamental definitions section of the article on the chemical potential, is highly undesirable.--Tomkaplan 19:16, 3 January 2007 (UTC)

The article states: "In the case of photons, photons are bosons and can very easily and rapidly appear or disappear. Therefore, at thermodynamic equilibrium, the chemical potential of photons is always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero. Since this process occurs extremely rapidly (at least, it occurs rapidly in the presence of dense charged matter), it is safe to assume that the photon chemical potential is never different from zero."

It should be made clearer that the photons do not 'disappear'... photons are persistent perturbations in the ambient EM field. Raise the energy density of that ambient EM field, and the photons are 'subsumed' in the ambient EM field, no longer persistent. Remember, photons are energy... energy cannot just 'disappear'. — Preceding unsigned comment added by 71.135.41.65 (talk) 02:08, 28 January 2021 (UTC)

Other Contexts
Chemical potential plays a huge role in the theories of chemical equilibria. But it also plays a role in chemical species diffusion, bose-einstein condensation, etc.. There's a lot of room to expand.

Should this article be split?
I believe this article should eventually be split into 2 articles and then have a disambiguation page. Theoretical chemists use the term "chemical potential" to refer to the electronic chemical potential. All other chemists use the term "chemical potential" to refer to the thermodynamic chemical potential. It seems totally obvious to me, but this confuses nearly all chemists (because the electronic chemical potential is rarely discussed in undergraduate course work - and even in graduate course work).

For the time being, what is written might suffice; however, if/when more content is added to describe the electronic chemical potential more accurately and completely (including the principle of chemical potential equalization arising from density functional theory - here refering to the electronic chemical potential), the amount of content and the mixing of the terminology on the same page may prove far too confusing. --anon


 * I broke it up into sections and added a description of the electronic chemical potential. It seems to work fine, in my opinion.  I don't think splitting it up into several pages is necessary. I need to include some references, but I am awfully tired right now. --anon 06:00, 2 December 2005 (UTC)


 * I think its a bad idea to split up the article. In physics, more precisely statistical physics, the chemical potential is usually defined as a rather abstract lagrange multiplier connected to the avarage number of particles. Thermodynamic and electronic chemical potentials, used in chemistry, are rather special cases of the more general concept of chemical potential! —Preceding unsigned comment added by 82.211.200.170 (talk) 20:32, 15 January 2009 (UTC)


 * As seen in this article, the physicist's definition of chemical potential is applicable to the interaction of any system of particles. It is so general it is being applied to nucleus physics and I suspect we will soon hear of the chemical potential of a galaxie. The article is on chemical potential. The article should focus on all the definitions of chemical potential that are used to calculate, measure, or predict chemical reactions (and the physical changes of phase, etc. which influence the course of chemical reactions.) Of course the article should include any underlying physical principles and especially clarifying examples thereof, but they should be put in subsidiary sections, not in the principal section. BTW, I give the idea of electronic chemical potential to my Advanced Organic Chemistry students (undergraduates also) because it is so tied in with the concepts of electronegativity, HOMO, LUMO, etc. that they use to describe the potential for an organic reaction to take place. I also use chemical potential in my course on organic photovoltaics, with much more physical theory. I also told both not to spend too much time wading this article (as of this date). There are much clearer sources. But, bravo to those who are trying to make this article clearer. Laburke (talk) 16:52, 13 September 2011 (UTC)


 * The term chemical potential is used in chemistry with a very specialized definition. In physics, the exact same term is used for a more general idea which can be specialized into the chemist's definition.  While I agree it would be preferable for chemical to refer to chemistry, the current usage in the literature suggests it does not.  Since this is an encyclopedia, the current usage should be used.  Thus splitting the article would be akin to original research as it would imply an attempt to adopt a new language not commonly used.  My suggestion for resolving this dilemma is to mention in the introductory paragraph that chemical potential is used in both a broad and specific sense.  The initial definition should use the broad sense, and then the specific use in chemistry should be given to introduce the specific sense.  That way any point of confusion is headed off immediately, especially for readers new to the subject who are likely to run into this dual-usage fairly quickly.  Thoughts? --BBUCommander (talk) 19:34, 5 February 2013 (UTC)

Relativistic chemical potential?
I think that the section titled relativistic chemical potential should either be clarified by an expert or removed. It is totally unclear to me what the relativistic chemical potential is. It is stated as being related to symmetries and charges, but some sort of definition (preferably an equation) seems necessary (I mean, there's all sorts of things related to symmetries and charges, so it is quite a vague description).

I don't do any relativistic-stuff, but I am fairly well-versed in atomic physics, in general, and if it's not clear to me I highly doubt it's clear to anyone other than the original author. I'm going let it sit for a day-or-so, and then just remove it unless there is any discussion or changes. DrF 03:43, 24 May 2006 (UTC)


 * Added Expert request tag. I'll let this section sit a little while longer before I delete it.  DrF 04:20, 26 May 2006 (UTC)


 * This reads about right I clean it a bit and add a ref link for those who want to read more.--Sadi Carnot 15:12, 25 July 2006 (UTC)

Suggestions
A curve would be a clearer illustration than the 2D plot. Chemical formulæ are not all correctly subscripted. Refers to non-existent "web-links". —DIV (128.250.80.15 (talk) 04:36, 1 April 2008 (UTC))

The example figure
I think the example figure needs to be expanded( such as how the particles flow downhill, in more detail). Or it is too dry to give any useful information. At this stage it does not help people to understand the concept of chemical potential. —Preceding unsigned comment added by 128.61.29.249 (talk) 18:20, 27 September 2008 (UTC)

Two items

 * 1) The opening paragraph has a Gibbs quotation prominently displayed.  If it's a direct quotation then it needs a reference.  My temptation is to remove that quotation until a citation can be added to show that it really comes from Gibbs.  It should be possible to track the quotation down.
 * 2) The example figure is, well, I'm not sure what it is.  The x and y axes appear to refer to chemical potential with, I assume, typical units of something like energy per mole.  However, the text says "Particles will tend to move from regions of ..." and I'm not sure what that movement means at all with reference to the figure's axes and units.  Again my tendency is just to delete the figure completely.  A previous entry on this page suggested that a curve might be clearer.  I agree, particularly since this will be the first example the reader meets.

I'm not a regular contributor to this article, and don't have time right now to become one. I trust that my comments will be taken in the helpful spirit in which they are intended. - Astrochemist (talk) 23:54, 17 December 2008 (UTC)

False distinction?
I think the article is misleading when it separates "electronic chemical potential" from "thermodynamic chemical potential". Any mobile species in local equilibrium has a chemical potential. When the species is, say, a hydrogen ion or a water molecule, the article calls it a "thermodynamic chemical potential". When the species is an electron, the article calls it an "electronic chemical potential", as if that was totally different. It's not different at all, it's the exact same definition, just electrons instead of protons, neutrons, or whatever. Would anyone object to me rewriting on that basis? :-) --Steve (talk) 07:49, 29 March 2009 (UTC)

Units
The units used for chemical potential should be discussed in the article. This seems like an important missing detail. Thanks! David Hollman (Talk) 21:40, 1 September 2010 (UTC)
 * Done. PAR (talk) 23:24, 2 September 2010 (UTC)
 * Wow, awesome, cheers! David Hollman (Talk) 11:58, 3 September 2010 (UTC)

Split the article
I concur with the previous suggestion that the article should be split, etc. There is no scientific connection between thermodynamic and electronic potentials, only the use of the same words. The first is about chemical reactions, the second is physics. I can do the chemistry. Can anyone help with the physics? Petergans (talk) 07:54, 26 April 2012 (UTC)


 * I disagree. Not only is there a scientific connection, it's the exact same concept: Change in free energy upon removing (object) from (system). I don't even think it should be called "electronic chemical potential", I think it should be called "chemical potential of electrons".
 * The electron section was obviously written by someone who had learned how, mathematically, to calculate the chemical potential of electrons using density functional theory, but basically had no conceptual idea what they were calculating. So they presented the mathematical calculation technique as being the same as the conceptual definition.
 * I want to make the exact opposite change: Rewrite the electron section (and the intro) to emphasize that electrons have a chemical potential in the exact same way (and with the exact same definition) as how sodium ions have a chemical potential. --Steve (talk) 12:30, 26 April 2012 (UTC)
 * You make a good point, but I'm looking at it as if I were a chemistry student. Such a person would have no idea what the chemical potential of an electron is. For chemists, the chemical potential is a property of a molecular species, its partial molar, or molal, free energy. This is important in understanding the free energy change in a chemical reaction. Chemical potential is a function of the activity (concentration) of the species in a mixture. What is the concentration of an electron?


 * If you don't want the article to be split, I suggest that the two aspects be completely separated apart from an explanation of the relationship somewhere near the beginning. I particularly don't like the lead-in: the first sentence is waffle and won't mean anything at all to a chemist. Petergans (talk) 09:48, 27 April 2012 (UTC)


 * The concentration of electrons is the number of electrons per volume! Isn't that obvious?...
 * No, it isn't obvious. There are no free electrons in solution. I assume the electron in question is attached to an atom, ion or molecule. Petergans (talk) 09:42, 28 April 2012 (UTC)
 * I shall do some rewriting and let's see how it turns out :-P --Steve (talk) 13:47, 27 April 2012 (UTC)
 * Done. I deleted the density functional theory part among other things. I have advertised at Talk:Density functional theory and elsewhere for that text to be found a new home in another article, because (for all I know) it was well-written and correct. --Steve (talk) 15:34, 27 April 2012 (UTC)

I'm very unhappy about this article in its present form. I came to it because of a link in equilibrium constant, which was mostly written by me. In that context this article is not helpful at all and for that reason I had not originally linked it. Now another editor has made the link. In a chemical equilibrium the sum of chemical potentials of reactants and products is equal to zero, that is, there is no tendency for the reaction to proceed in either forward or backward direction because no energy would be released in the process.

Here's my (thermodynamic) definition: the chemical potential of a chemical species in a chemical mixture is the partial molar free energy of that species. It is the partial derivative (slope) of the free energy with respect to the number of moles of the species, that is, the infinitesimal amount by which the free energy changes divided by the infinitesimal change in the amount of that species. In this definition the amounts of the other chemical substances in the mixture are assumed to be constant, hence it is a partial derivative. Temperature is also assumed to be constant. At constant pressure it related to Gibbs free energy, at constant volume it related to Helmholtz free energy. If you can, I suggest you look at Atkins, Physical chemistry, section 5.1, "Partial molar quantities", for a full explanation of the significance of chemical potential in thermodynamics. None of that material appears in the present article; it should!

Also In Atkins, Physical Chemistry, the term chemical potential is related to the Fermi level in molecular orbital bands (section 20.9(b), p 725 in the 8th edition). Petergans (talk) 09:42, 28 April 2012 (UTC)


 * Again, the exact words you say are a wonderful description of how the chemical potential of an electron in a solid is defined. (Besides the trivial exception that for electrons in solids, it's usually expressed in eV rather than kcal/mol.) The chemical potential is the partial derivative of the free energy with respect to the number of electrons in the solid. In fact I wrote in wording just like that at the start of the "electrons in solids" section.


 * Your main complaint now seems to be the introduction, not the two physics sections that I just rewrote. I did not write the introduction and I don't like it any more than you do. I hope you rewrite it, or else maybe I will when I get a chance. (I did, however, write the "overview" section.) :-) --Steve (talk) 20:04, 28 April 2012 (UTC)
 * OK, I'll give it a try. At the same time I'll put things in a different order, hopefully to clarify the presentation. I'll do it in my sandbox and will let you know when it is ready. One final point: I don't think that the bit on particle physics belong here. Sub-atomic particles don't have properties of relevance to chemistry so applying the term "chemical potential" to them seems like a misnomer to me. I understand the concept of a partial derivative of energy with respect to quantity can apply in a plasma etc., but what energy is involved here?Petergans (talk) 08:51, 29 April 2012 (UTC)
 * I've just clocked the fact that there are two sections called "Overview". Yours is the first and gives good examples of phase equilibria, but it is not linked in the index, so I didn't see it when I clicked on overview! I'll add another exmple for chemical equilibria and merge in the second, where possible. Petergans (talk) 09:05, 29 April 2012 (UTC)


 * Please don't split the article. The broad application of the concept is useful for its understanding. Everyone who claims to have understood it in one field but is annoyed by application to other fields has yet to climb to a higher level of abstraction. --Rainald62 (talk) 23:24, 3 May 2012 (UTC)
 * I agree with the preceding comment. The notion of chemical potential is universal for all physical systems in thermodynamic equilibrium.Xxanthippe (talk) 21:42, 4 March 2018 (UTC).

Explanation

 * I would like to have an explanation for the (approximate) logarithmic dependence of the intrinsic chemical potential on concentration. Is it only for the Boltzman distribution of particle energies? --Rainald62 (talk) 23:24, 3 May 2012 (UTC)
 * See E.T.Jaynes, 1957. --Rainald62 (talk) 21:30, 4 March 2018 (UTC)

Ashcroft/Mermin and the (electro)chemical potential
As far as I understand them Ashcroft/Mermin don't say that the "electrochemical potential" is the internal chemical potential (the article says they do). Instead they call the external chemical potential the "electrochemical potential" (p. 257). Imho they use the terms exactly like Kittel/Kroemer do, but please feel free to correct me if I'm wrong. Anybody mind me correcting this in the article?

They - as well as many others - sometimes add for clarification: "a voltmeter measures [...] the electrochemical potential". Anybody mind me incorporating this into the article? --Mest (talk) 14:25, 4 May 2012 (UTC)
 * This is outside my area of expertise, so go ahead. However, I have rewritten the article, draft is in User:Petergans/sandbox so we need to be careful about synchronizing versions. Petergans (talk) 15:14, 4 May 2012 (UTC)


 * I put that in but maybe I was misreading the book. Or maybe you have a different version than me and they corrected the "mistake"? I will check it ASAP. (I don't have the book on-hand right now.)
 * I hope you're right, that would make everything much simpler!!
 * I definitely agree that "a voltmeter measures the electrochemical potential of electrons". --Steve (talk) 15:18, 4 May 2012 (UTC)
 * I have that book (and I love it), so I could double check what exactly it says. Headbomb {talk / contribs / physics / books} 15:28, 4 May 2012 (UTC)

Ashcroft and Mermin write... (p.256–257) When a temperature gradient is maintained in a metal and no electric current is allowed to flow, there will be a steady-state electrostatic potential difference between the high- and low-temperature region of the specimen. Measuring this potential drop is not completely straightforward for several reasons: The thermoelectric power (or thermopower) of a metal, Q, is defined as the proportionality constant between the contribution of the metal to the reading of such a voltmeter, and the temperature change: $$-\int \mathbf{\mathcal{E}} \cdot d\mathbf{\ell} = Q ~ \Delta T$$ or $$\mathbf{\mathcal{E}}= Q ~ \nabla T$$. Headbomb {talk / contribs / physics / books} 16:03, 4 May 2012 (UTC)
 * 1) To measure voltages accurately enough to detect a thermoelectric voltage, it is essential that the voltmeter connects points of the specimen at the same temperature. Otherwise, since the leads to the meter are in thermal equilibrium with the specimen at the contact point, there would be a temperature gradient within the circuitry of the meter itself, accompanied by an additational thermoelectric voltage. Since no thermoelectric voltage develops between points of a single metal at the same temperature, one must use a circuit of two different metals (Figure 13.1 [diagram of a thermocouple ]) connected sothat one junction is at a temperature T1 and the other (bridged only by the voltmeter) at a temperature T0 ≠ T1. Such a measurement yields the difference in the thermoelectric voltages developed in the two metals.
 * 2) To measure the absolute thermoelectric voltage in a metal, one can exploit the fact that no thermoelectric voltage develops across a superconducting metal. Hence when one of the metals in the bimetallic circuit is superconducting, the measurement yields directly the thermoelectric voltage across the other.
 * 3) The points in the circuit joined by the voltmeter have different electrostatic potentials and different chemical potentials. If, as in most such devices, the voltmeter reading is actually IR, where I is the small current flowing through a large resistance R, then it is essential to realize that the current is driven not just by the electric field E, but by $$\scriptstyle{ \mathbf{\mathcal{E}} ~=~ \mathbf{E} ~+~ \left(1/e\right)\nabla \mu}$$. This is because the chemical potential gradient leads to a diffusion current, in addition to teh current driven mechanically by the electric field. As a result, the voltmeter reading will not be $$\scriptstyle{-\int \mathbf{E} ~\cdot~ d\mathbf{\ell}}$$ but $$\scriptstyle{-\int \mathbf{\mathcal{E}} ~\cdot~ d\mathbf{\ell}}$$.

Yes, page 257 is 100% clear: "a voltmeter measures not the electric potential but the electrochemical potential". BUT page 593 [and following pages] is equally clear that "the chemical potential [on the two sides of a p-n junction in thermal equilibrium] does NOT vary with position" but the electrochemical potential DOES. Look at Eq. (29.9) (phi is electric potential not measured voltage). Even more explicit is Fig. 29.2 on p594: The top shows the electrochemical potential being different on the two sides; the bottom shows the chemical potential being the same on the two sides.

Well, since writing that section a couple years ago, I've seen more and more---even in solid-state physics books---the expected definition that electrochemical potential is constant in equilibrium. I don't think I've seen the "backwards" definition anywhere except Ashcroft and Mermin page 593, and now I know that that even includes earlier text in the same book! Therefore I propose that Ashcroft and Mermin page 593 is simply a mistake and we should ignore it.

UPDATE: I edited the article accordingly. When Petergans posts the new version I'm happy to re-edit that section. --Steve (talk) 14:56, 5 May 2012 (UTC)

Major revision
I have done a major revision, principally to clarify the thermodynamics and to give chemical potential its chemical context. This was based mainly on Atkins, Physical Chemistry. The ordering of some material has been changed. For example, history was moved from the lead-in to the history section.

The questions regarding electrochemistry are still to be resolved (see above). Atkins, Sections 7.5-7.9 devotes 15 pages to electrode potentials, but there is no mention of the internal/external issue, nor is there any reference to chemical potentials on those pages; the whole treatment is in terms of the free energy and reaction quotient. Petergans (talk) 09:08, 5 May 2012 (UTC)


 * The first two sentences still seem problematic to me. The energy released in a chemical reaction is more properly the internal energy or enthalpy, right?  Also, why is chemical potential assigned as "potential" energy.  In reality, some of the Gibbs energy is "kinetic" (molecules moving) and some is "potential" (e.g. intermolecular forces).  Finally, the second sentence would seem to be confusing at minimum with a key property about the chemical potential, which is that the chemical potential of a substance in the a phase is equal to the chemical potential of the substance in another phase if the phases are in equilibrium.  What exactly is changing during a phase transition?Stieltjes (talk) 17:43, 29 August 2012 (UTC)
 * The energy released in a chemical reaction is free energy not enthalpy. In fact a reaction may be endothermic, ΔH=<0, and still occur spontaneously because of the entropy contribution. Some of the free energy released can indeed be turned into kinetic energy, for example in an explosive decomposition, which is why the term "potential" is apt.Petergans (talk) 07:49, 30 August 2012 (UTC)

Request: define variables in equations
All: I'm not qualified to edit this article, but have a small request for those who are working on it. It would be helpful for those not versed in the field if the terms in at least the important equations (e.g. the first one in the "Thermodynamic Definitions" section) were defined. The article on Helmholtz Free Energy for example, does a good job of this.

It also might be helpful to make the reference to "Gibbs energy" an inline link to that article.

Thanks to all for your effort! Ma-Ma-Max Headroom (talk) 18:05, 10 September 2012 (UTC)

Recent changes and reversions
Recent reversions by User:Nanite and User:Xxanthippe are a matter of concern.

User:Nanite claims that his changes are improvements. They are not. For instance, in Thermodynamic definitions it is incorrect to include $$\mu_i=\left(\frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}$$ because chemical potential is defined as the partial derivative of free energy. Whatevever the rights and wrongs, this reversion war has to stop. The onus is on User:Nanite to justify his edits, with citations, here, on the talk page before introducing them into the body of the article. My previous edits were based on "Physical Chemistry" by Peter Atkins. Petergans (talk) 10:14, 10 February 2014 (UTC)


 * Don't worry friend, it's a bit hasty to hail this as a reversion war. I undid an unexplained reversion of Xxan presuming that he/she mistook my edits for vandalism (Xxan seems to be a busy editor, it would be understandable for such editors to snag false positives every once in a while. No harm done.). As far as I can tell, that's all that happened.
 * A perfect citation for $$\mu_i=\left(\frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}$$ would be the first ever description of chemical potential, just as presently described in the article's History section. Allow me to quote: If to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered. Gibbs p.149, freely available here
 * In other words, the definition in terms of U is the "original definition", if that counts for anything. Regardless, it does not matter anyways since that definition is completely equivalent to the one in terms of G. The proof of this is elementary thermodynamics and I can elaborate if it's not obvious. (Actually I'm surprised your textbook doesn't mention this fact. I'm not familiar with the book, but perhaps it is oriented to a specific application such as chemistry, rather than general thermodynamic fundamentals.) Anyway, the point is that any of the equivalent definitions are equally valid, though for convenience one or the other might be preferred depending on the system in question.
 * You have seen in my edits that you reverted, that I had added the missing citation to Gibbs' book, and moreover (in relation to the previous point) I had expanded the quote to include Gibbs' remark on the equivalence of defining chemical potential in terms of U, A, or G. I really do think these and all of my edits were improvements and I'm sad to hear you disagree.
 * Anyway, I'm not interested in arguing and wasting time getting prior permission to edit articles as that is totally not the point of wikipedia, so I'll leave you to improve this page and I'll focus my efforts elsewhere. Let me know if you're interested in my input, as there are a few glaring problems with the article that I was planning to fix. Bye for now. Nanite (talk) 19:59, 10 February 2014 (UTC)
 * Hold on Nanite! I suggest you go ahead with your edits in your sandbox and let's see what they produce. When an editor starts to make changes to an important and well-established article it helps if he describes what he is up to on the talk page first. There are many equivalent expressions for chemical potential and yours looks a correct one. @ Petergans: I don't think that "Physical Chemistry" by Peter Atkins is a good source for a Wikipedia as it is a broad and often superficial undergrad text. Better sources in that genre are Kittel+Kromer, Reif and Mandl. Xxanthippe (talk) 21:56, 10 February 2014 (UTC).
 * From the point of view of a chemist, the definition in terms of free energy is the most important because chemical reactions involve free energy changes. Historically and in other contexts chemical potential can be defined in terms of internal energy. I am not aware of any applications of this concept. Is constant entropy physically realizable? I think we should be careful about using too broad definitions without reference to specifics. When I did the major revision of this article I had equilibrium chemistry very much in mind. Petergans (talk) 11:13, 11 February 2014 (UTC)
 * The notion of chemical potential applies to any equilibrium system with a variable number of particles and has a much broader application than equilibrium chemistry so the article should not be written from that point of view only. The basic expression for chemical potential given above by Nanite in this thread drops out naturally from the fundamental equation of thermodynamics dU = T*dS - P*dV + mu*dN. Chemical potential may be expressed as a derivative of any of the thermodynamic potentials. These other expressions are not alternative definitions but follow logically from the equation for dU and the definitions of the potentials H, F and G. The only reason why the Gibbs Free Energy G is of interest is because it is easier to maintain a chemical system at constant P and T than at S and V. I would like to see the article start out with the definition based on U and not that based on G. Xxanthippe (talk) 05:03, 12 February 2014 (UTC).
 * I'm not against the more general approach, but each definition should be related to real-world phenomena, not just as a mathematical construct. It's important to explain the context in which a definition is useful. Constancy of V and T can be important in atmospheric and oceanic contexts, where pressure is variable with depth. In what physical/chemical circumstances can S and V be held constant? Petergans (talk) 10:41, 12 February 2014 (UTC)

Definitions
The statement "The definitions given above are mathematically equivalent" appears to me to be meaningless. The three possible definitions
 * $$\mu_i=\left(\frac{\partial G}{\partial N_i}\right)_{T,P,N_{j\neq i}}$$
 * $$\mu_i = \left(\frac{\partial H}{\partial N_i} \right)_{S,P, N_{j \ne i}}$$.
 * $$ \mu_i = \left(\frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}$$

correspond to three different ways of defining partial molar free energy. There is no way in which the three definitions can be considered to be "equivalent" as there is no way in which Gibbs free energy (G), Helmholtz free energy (H) and internal energy (U) are equivalent.

The second and third definitions are of little practical value because they apply at constant entropy (S), a condition which is virtually impossible to achieve in practice; I have included them for the sake of completeness because they can be found in texts on thermodynamics. Petergans (talk) 09:33, 13 May 2015 (UTC)


 * The thermodynamic potentials are all related by Legendre transformations, a fact so well known that there is even a Wikipedia section on it . Xxanthippe (talk) 03:48, 20 May 2015 (UTC).
 * This is very misleading. Although there is a mathematical relationship, the definitions are not "equivalent" because they relate to different physical quantities.Petergans (talk) 08:28, 20 May 2015 (UTC)
 * They are equivalent in the sense that they are all equal to each other. Do you have a better term to use? PAR (talk) 21:10, 20 May 2015 (UTC)

A Legendre transformation is not the same as either equality or equivalence. It is clearer to state the nature of the relationship, ,as in my last edit, than to merely state that they are "equivalent" without explaining why this is so. Petergans (talk) 10:46, 21 May 2015 (UTC)

I have 39 years' experience as a lecturer in chemistry at the University of Leeds. I think that this qualifies me as an expert. Please remove the offensive "expert" tag. It appears to refer to one disputed sentence, so the tag is not relevant to the article as a whole. Petergans (talk) 07:27, 23 May 2015 (UTC)

Standard state
User:Chanwu claims that the following statement in chemical potential is wrong


 * $$\mu_i = \mu_i^{std} + RT \ln a_i,$$

The equation is correct. It is given by Atkins & De Paula, 8th. edition, Equation 5.53 on page 161 (Section 5.7(c), Activities in terms of molalities). In the text I used the term "concentration" rather than "molality". This is also correct as molality is a measure of concentration. In fact the equation is valid for all concentration units as the standard state will be defined in the same unit as the solute concentration. Petergans (talk) 15:55, 31 May 2015 (UTC)
 * This defines the relative activity then. ✅

Free energy not Potential energy
Chemical potential is the partial free energy in a species that gives the appearance of causing a system to move spontaneously toward lower free energies / chemical potentials, just like total free energy and spontaneous reactions (eg, Gibbs free energy).

In macroscopic physics like gravitational and electrical, there is such little entropy of the system that merely the potential energy itself is what primarily governs spontaneous motion / state transition. So it makes a kind of sense to call free energies in entropy-dominated systems "potentials", but let's be clear that they are most certainly not the difference between the total energy that never changes and kinetic/thermal energies, which are Potential energies.

This is extremely confusing since Chemical potential energy is a very real and important quantity: among other things, it is the latent energy reserve otherwise-inert matter has that exothermic chemical reactions draw from. Whereas Chemical Potential / Free Energy is how much of this energy is available to do useful work (electrically or mechanically) and which governs how the system will spontaneously move as it endeavors to increase total entropy.

Free energy is of the 2nd Law, and Potential energy is of the 1st Law.

(I've updated the article and am working on the article Chemical energy now. Please feel free to disagree!, but let's put any discussion about chemical potential vs. chemical potential energy on this article if you don't mind just for consistency (and since the naming of Chemical potential energy isn't cause for concern). )

--RProgrammer (talk) 10:45, 1 March 2016 (UTC)


 * I just found something related that's troubling to me: in the section "Electrochemical, internal, external, and total chemical potential"
 * It seems that Free energies are being added to Potential energies. At first this struck me as concerning, but then I realized a possible explanation.  I didn't finish polishing it and put it in the article because I feel it constitutes original research, and I lack the societal credentials to say it somewhere in public and cite myself XD
 * So I'll just post the rough draft here:


 * The reason why potential energies such as qV or mgh can be directly added to thermodynamic free energies such as the internal chemical potential $$\mu_{\mathrm{int}}$$ to produce a total free energy is that these potential energies have such little entropy all of their energy is available to do useful work. (This can be thought of as the entropy term in some FE = E - TS equation becoming zero as entropy does, yielding FE = E, and thus the lack of a distinction between Free Energy and Potential Energy in non-thermodynamic (low-entropy) physics)


 * (not part of the draft, but another way to word it is that you are using the electrical or gravitational free energy when summing with thermodynamic free energies, and the potential energy with thermodynamic potential energies—it's simply that electrical/gravitational/etc. free energy exactly = electrical/gravitational/etc. potential energy due to the negligible entropy / all of it being available to do useful work, and thus no distinction is made)
 * RProgrammer (talk) 11:10, 1 March 2016 (UTC)
 * You are corect. This is OR. Xxanthippe (talk) 21:17, 1 March 2016 (UTC).


 * To me the clearest situation which shows the distinction (internal/total chemical potential, and internal/total energy) is to imagine an enormously tall vertical test tube containing water, in a gravitational field, and in equilibrium (at room temperature). Quantities such as pressure and internal chemical potential will vary with height: in the middle of the test tube you'll find liquid water, at the bottom you will find water that has solidified from ultra high pressure, and at the top you will find a low pressure atmosphere of water vapor. Yet, the temperature and total chemical potential will be uniform throughout. When you examine total-energy quantities (Hamiltonian energy, free energy, etc.) you will find the usual thermodynamic relations hold just fine (in terms of total chemical potential), although the system is not extensive and so you can't invoke, e.g., Gibbs-Duhem equations, and you cannot define a uniform pressure variable (it becomes important where the volume is varied).
 * On this basis, I am not sure how total/internal µ distinction relates to availability of work as has suggested. And indeed it seems like OR to interpret too much into it. --Nanite (talk) 09:40, 2 March 2016 (UTC)

Thermodynamic definition
I have rewritten this section, based on the monograph of Mandl, to make it more logical. The previous version started by defining the two quantities G and $$\mu$$ in one equation! The proper logical progression is that $$\mu$$ appears first in the fundamental equation for U, as in the definition of Gibbs given in the History section. The other equivalent expressions for it are obtained mathematically by Legendre transformations. There is further discussion above on this talk page. Xxanthippe (talk) 03:59, 27 April 2016 (UTC).
 * I find this re-write lacks the clarity of the previous version. In writing the the old one I followed Atkins (8th. edition, p138) in defining chemical potential as the partial derivative of the Gibbs free energy. Atkins adds, on the next page, a section on "the wider significance of the chemical potential". This is a much clearer exposition than that of the current article. The constant pressure definition is by far the most important in practice for chemical reactions. What do people think? I will revert to the previous text unless there is a consensus that the current version is better. Petergans (talk) 07:41, 27 April 2016 (UTC)
 * As the editor has a WP:COI in the matter, being the author of the previous version, it might be better, after debate has taken place according to WP:BRD, to leave it to another editor to revert to the previous version, if consensus finds that is superior. Xxanthippe (talk) 03:26, 28 April 2016 (UTC).
 * There is no conflict of interest. We simply disagree on how best to present the material. Petergans (talk) 10:01, 28 April 2016 (UTC)
 * An actual COI is a pretty different concept from a potential COI (that anyone with an employment has when editing anything). From COI, Conflict of interest (COI) editing involves contributing to Wikipedia about yourself, family, friends, clients, employers, or your financial or other relationships. Any external relationship can trigger a conflict of interest. (Extra emphasis mine.) YohanN7 (talk) 11:07, 28 April 2016 (UTC)
 * You are correct. A better policy to refer to would have been WP:Own. Xxanthippe (talk) 01:33, 3 May 2016 (UTC).
 * I prefer the current version. Starting with the fundamental equation for U makes a lot of sense presentation wise, since it emphasizes the similarity with more familiar concepts such as temperature and pressure. The current version also makes much clearer why definition in terms of Gibbs free energy is so much more practical. TR 15:20, 28 April 2016 (UTC)
 * I also prefer Special:Permalink/717436442, which I believe is also called the "current version" for the same reason as stated above (U is more fundamental then G). When I took Thermo as an undergraduate (circa 1973), we did not mention Gibbs free energy until near the end of the course. (FYI - we used Kittel, which is probably not the best "authority" one could cite)---Guy vandegrift (talk) 15:40, 28 April 2016 (UTC)
 * Just to clarify. The new version, corrected for links and typos, is . The previous version is . Incidentally, I think Kittel is an excellent book for this subject, as are Mandl and Reif, not to forget Gibb's classic work Xxanthippe (talk) 10:56, 29 April 2016 (UTC).

This is all very well, but the article is about chemical potential, not fundamental thermodynamics. The definitions
 * $$\mu_i = \left(\frac{\partial G}{\partial N_i} \right)_{T,P, N_{j \ne i}} $$, $$\mu_i = \left(\frac{\partial H}{\partial N_i} \right)_{S,P, N_{j \ne i}} \text{ and  } \ \mu_i = \left(\frac{\partial F}{\partial N_i} \right)_{S,V, N_{j \ne i}}$$

do not refer to U, so why bring it in? It's an unneccesary distraction in this article. Incidentally, please can we use either the symbol G or F consistently for Gibbs energy. Petergans (talk) 10:09, 29 April 2016 (UTC)


 * The new version is better, but the fact that the expression for dU defines the chemical potentials should be explained better. For us this is a trivial step, but not necessarily for the readers. So, you just explain that you can consider U as a function of S, V, and the $N_j$ and that this function will have some partial derivative w.r.t. the $N_j$, one then defines the chemical potentials as these partial derivatives.


 * At the end, what is missing is that the Gibbs energy for an extensive system will be given by $$G = \sum_j \mu_i N_i$$. This is easy to explain to the readers after the relation $$dG = \sum_i \mu_i dN_i$$ at constant T and P is given. You just note that with the intensive variables kept fixed, G is a linear function of the particle numbers. The Gibbs-Duhem relation mentioned the next section follows directly from this. Count Iblis (talk) 23:22, 7 May 2016 (UTC)
 * Thanks for your comment. In such a short section one cannot develop thermodynamics from its foundations so I started from the fundamental equation of thermodynamics, which a new reader will have to refer back to. This equation already incorporates the notions of entropy and thermodynamic temperature, which are the greatest conceptual achievements of the subject and probably the most difficult to understand. So a great deal of the work has already been done. Xxanthippe (talk) 03:41, 8 May 2016 (UTC).


 * I lack the expertise to know Count Iblis is right, but I certainly think he might be right. I once looked carefully at entropy and realized that defining entropy through the differential, $$dU = T dS - P dV$$, is incomplete because it fails to establish that entropy can be integrated (i.e., that it is a state function and not a process function or equivalently, not an Inexact differential.  If this is what Count Iblis is talking about, then I agree: we need to provide, or better yet, point to an explanation.--Guy vandegrift (talk) 01:22, 8 May 2016 (UTC)

Disputed Introduction
See http://www.icsm.fr/Local/icsm/files/286/JFD_Chemical-potential.pdf --129.69.120.91 (talk) 12:47, 22 December 2017 (UTC)
 * At chemical equilibrium added stoichiometric coefficients. The rest is still not right. --Biggerj1 (talk) 12:53, 22 December 2017 (UTC)
 * Atkins & de Paula, Physical Chemistry O.U.P. 8th. edn. p 138, quote: "For a substance in a mixture, the chemical potential is defined (Atkins italics) as the partial molar Gibbs energy
 * $$\mu_i=\left( \frac{\partial G}{\partial n_i}\right) _{p,T,n'}$$".
 * Edits reverted. Petergans (talk) 15:15, 23 December 2017 (UTC)
 * Your Point is unrelated and not touched! What I e.g. improved is that at chemical equilibrium $$\Sigma _i \nu _i \mu_i =0 $$ and not as stated before $$\Sigma_i \mu_i=0$$ which is just wrong! Biggerj1 (talk) 16:57, 23 December 2017 (UTC)
 * adressed the points ✅

About the function in the section of "Chemical Potential in Ideal vs. Non-Ideal Solutions"
I am now translating this page into Chinese. I notice something not understandable to me. In the section of "Chemical Potential in Ideal vs. Non-Ideal Solutions", there is a function:

𝜇𝑖=𝜇𝑖0(𝑇,𝑃)+𝑅𝑇𝑙𝑛(𝑥𝑖)

Here it seems that 𝑥𝑖 is the amount of certain species. However, if the amount is zero, then this logarithm diverges. Is this well?

Besides, is there any special name for the term 𝜇𝑖0(𝑇,𝑃)? It looks like the chemical potential when 𝑙𝑛(𝑥𝑖)=0. — Preceding unsigned comment added by Swenly (talk • contribs) 18:58, 30 April 2018 (UTC)
 * Hi Swenly, Consider the asbolute ideal chemical potential $$ \mu_i^\text{ideal} = kT \ln(\lambda_i^3 c_i) $$ (this you get also via the difference quotient $$\mu=(F(N)-F(N-1))/1$$. For the concentration $$c_i=0$$ the ideal chemical potential approaches $$-\infty kT$$. However consider that the free energy of such a system goes like $$F(N,V,T)=TS-pV+\sum_i \mu_i N_i $$ This means that the free energy contriubtuion of the chemical potentials is given by $$ \sum_i \underbrace{c_i V}_{N_i} \underbrace{kT ln(c \lambda_i^3)}_{\mu_i}$$ which goes to zero and does not diverge in the limit (all) $$c_i$$ to zero.--Biggerj1 (talk) 11:41, 3 May 2018 (UTC)

Chemical potential
Recommend changing the confusing wording of this statement in the article: "Therefore, at thermodynamic equilibrium, the chemical potential of photons is always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero."

- to -

"Therefore, at thermodynamic equilibrium, the chemical potential of photons is always and everywhere zero. If the chemical potential of a region is higher than the chemical potential of the photons in or entering that region (chemical potential greater than zero), photons would spontaneously disappear from that region until the chemical potential went back to zero; likewise, if the chemical potential of a region was less than zero, photons would spontaneously appear until the chemical potential went back to zero."

Or similar such language to explicate that it's a comparison between the chemical potential of the region, and the chemical potential of the photons. 71.135.39.238 (talk) 07:29, 9 June 2021 (UTC)

Recent changes
Editors who want to change the article substantially should follow WP:BRD and gain consensus for their changes here. I have reverted some unneeded edits from a user who puts citations to his own papers into the article. Other editors may care to comment. Xxanthippe (talk) 03:49, 7 February 2022 (UTC).


 * Thank you for letting at least some of my corrections stand. Nevertheless, the reversion of other edits has reintroduced several scientific errors:


 * 1) For an open system that can exchange particles with the surroundings, the Gibbs free energy is not at a minimum in chemical equilibrium. This applies only if the system cannot exchange particles - the derivation relies on the laws of thermodynamics for closed systems. For instance, if the system is open, heat cannot be uniquely defined. Consider a volume of water as an open system. Now you place it in contact with a crystal of copper sulfate, which partially dissolves in the water, increasing the Gibbs free energy of the open system.


 * 2) The fundamental equation of thermodynamics does not apply to irreversible processes in general. It contains T and P, which are obviously not defined if the irreversible process is an explosion, for instance. So the equation cannot apply to uncontrolled processes like that. Quoting textbooks here probably won't do, since they routinely get the equations for irreversible processes wrong (for instance, adding a subscript ext on the pressure in infinitesimal work, leaving out the subscript surr on the temperature in the inequality of Clausius, etc.). You had asked me explicitly to add sources. When I did that, you turned around and criticized me for one out of multiple sources that I added. That's not fair. But of course I don't object if you replace this source with your own important reference that points out that the fundamental equation of thermodynamics does not apply to irreversible processes in general.


 * 3) To discuss ice cubes in liquid water above 0 °C is a bad idea. Students learn already in middle school, correctly, that ice water is at 0 °C. Almost as soon as an ice cube is dropped into warmer water, it is surrounded by water at 0 °C.


 * 4) In many chemistry texts, the chemical potential is never written exactly as on this page. The derivative is taken with respect to the amount of substance (unit: mol), not with respect to the particle number. I had added a brief sentence pointing this out, but it was deleted. It is not acceptable for an editor to suppress input from a field where the topic is important, and chemical potential is obviously important in chemistry.


 * Please correct these four errors.


 * Klaus Schmidt-Rohr (talk) 13:34, 7 February 2022 (UTC)
 * I have checked the 4 assertions and confirm that they are correct. Petergans (talk) 15:37, 7 February 2022 (UTC)
 * Indeed. I made a few minor edits to address 4 before I saw this discussion and I hope they will serve. Chemical potential has common usage both ways (per mole or per particle) depending on the context. In statistical mechanics derivations, per particle is often simplest; in applications to thermodynamics, per mole is often best, at least for me. KeeYou Flib (talk) 22:50, 11 February 2022 (UTC)
 * The use of the talk page of an article is to improve the article by reference to reliable sources. It is not to discuss the topic itself, which should be done on the numerous forums that exist for this purpose. Nonetheless, I comment briefly on the four points raised that are relevant to the second section of the article: 1 and 3 are unsourced statements of opinion. For 2, it should be understood that the mathematical symbol "d" that appears in the first equation of the article is an infinitesimal; in other words, the changes described are infinitesimally small, so, for example, dU is an infinitesimally small change in U. An explosion is not an infinitesimally small change. Point 4. has been well addressed by User:Qflib. Xxanthippe (talk) 04:23, 12 February 2022 (UTC).

The claim above that "(textbooks)..routinely get the equations for irreversible processes wrong" is questionable and unsupported. It would be better to cite a book that gets them right. Note that Atkins & De Paula (8th edn. p.76) does not use the term "irreversible". Instead they state that "No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work". Obviously, such a process is not reversible in the thermodynamic sense. Petergans (talk) 11:01, 12 February 2022 (UTC)


 * Comment. There is a Thread that refers to this article at Conflict of interest/Noticeboard. Xxanthippe (talk) 02:50, 22 February 2022 (UTC).

Links
@Petergans. I see that you have restored a link that I recently deleted. The reason that I deleted the link is that the source looked a bit idiosyncratic and flaky. However, if you think that the data is good, then your change is OK with me. Xxanthippe (talk) 02:46, 13 February 2022 (UTC).
 * There is a link to the list of sources of the data on the main page of the site.
 * "Ag-Au	|	B-Br	|	C	|	Ca-Cd	|	Cl-Cu	|	D-He	|	Hg-Kr	|	La-Mo	|	N-Nb	|	Ne-P	|	Pb-S	|	Sb-Sr	|	Te-U	|	V-Zr	|	Sources"
 * Looks good to me.Petergans (talk) 07:55, 13 February 2022 (UTC)

The chemical potential of Van der Waals gases


Hello, I submitted a section about the chemical potential of Van der Waals gases by showing how the chemical potential of Van der Waals gas depends on the Van der Waals constants $$a$$ and $$b$$ as well as on the entropy, and how these dependencies can be explained. The submitted section was deleted by user Xxanthippe. The plots, in my opinion help to understand the basic dependencies of the chemical potential from the constants of a pure gas. — Preceding unsigned comment added by E-gabrielyan (talk • contribs) 04:07, 4 July 2022 (UTC)
 * The opinions of individual editors are not enough for inclusion in Wikipedia. Original research WP:OR is not acceptable. Reliable sources WP:RS are needed to obtain consensus of other editors. Xxanthippe (talk) 04:28, 4 July 2022 (UTC).
 * I was committed to provide a high quality and useful contribution. The section which I added was removed while I was inserting the references into the text. The entire section was removed within only minutes from the submission, completely, without specific requests, and without discussion. Not having seen any interest and transparency, all desire to invest time into the improvement of articles vanished. E-gabrielyan (talk) 06:52, 4 July 2022 (UTC)
 * See my comment on your talk page. Xxanthippe (talk) 08:06, 4 July 2022 (UTC).

This is a very specialized development for one gas model on a page discussing the general concept and I don’t think it’s a great match. I think that you should consider developing this on the page for the van der Waals gas instead. Qflib, aka KeeYou Flib (talk) 16:00, 4 July 2022 (UTC)

Equations
Hi Xxanthippe, the punctuations are not part of the equation and I think the formulas look much cleaner without such punctuation or at least not in the same font style as the equation. Without punctuation is how textbooks normally show these too. What do you think? I was looking for MOS on this, but tried visual first. &#32;Ramos1990 (talk) 00:55, 5 February 2024 (UTC)