Talk:Van der Waals equation

Untitled

Is there any way we can find out the value of b theoretically if the value of radius of nucleus is known

and if there is any way of caluculating a plz do write

Yes, there is, but you will need heavy-duty quantum physics and a lot of computational power to get reasonably good answers. By the way, the atomic radius is considerably bigger than the size of the atomic nucleus. -- Karada 08:20, 1 Sep 2003 (UTC)

The van der Waals equation is really only an approximation valid at the macroscopi levels... not sure why you would want to bother using it at the subatomic level. yosofun 00:25, 7 Mar 2004 (PST)

Changed notation

I changed the notation for the formulas to the form more familiar to me of lower case p for pressure and lower case for per-particle properties. Hence v for volume per particle (specific volume), and V for total volume; N for total number of moles (or particles) rather than n which is used for the number density of molecules (number of particles per unit volume: n = 1/v = N/V, if N is the total number of particles)(or n = N x Na / V if N is the total number of moles, and where Na is Avogadro's number).

• Tom Lougheed 2005-09-27

The equations at the top need fixing

Those equations need cleaning - first and foremost: same variables should not have different meanings. Fresheneesz 08:21, 21 November 2005 (UTC)

Looking at it quickly, the only problem I see is lowercase p and uppercase P both being used for pressure. Did you see others? PAR 13:58, 21 November 2005 (UTC)

I think I fixed it abakharev 23:19, 21 November 2005 (UTC)

Does anyone mind if we have the convention N=number of particles in the system, V=volume of the system, and v=V/N = volume per particle, without introducing the concept of the mole and Avogadro's number? I will do the changing. PAR 03:57, 22 November 2005 (UTC)

There won't be R (gas constant) then, just NkT, if it is OK with you, guys, it is OK with me abakharev 04:50, 22 November 2005 (UTC)

Error in v_C

Should not kTc=8a/27b read NkTc=8a/27b ? Jhaagsma 09:25, 26 February 2007 (UTC)

No, I think the problem is that the equation for $v_{C}$ is wrong. I fixed it. Please check and see if you agree. Thanks for noticing the inconsistency. PAR 17:03, 26 February 2007 (UTC)

Are you sure the new equation is correct? If $v_{C}$ is supposed to be the critical specific volume, dividing $b$ by Avogadro's number gives a really tiny answer for it. Sorry, I've probably misunderstood the meaning of $v_{C}$. 84.12.252.210 15:40, 18 July 2007 (UTC)

$v_{C}=3b'\,$. Note the prime on b! This indicates a molecular (not a molar) quantity. If you divide b by N, nothing happens to v_C.--P.wormer 15:53, 18 July 2007 (UTC)

I've added a derivation of this equation which involves no calculus, and thereby show that the change from Nk to R is because it should have been $N_{Avogadro}k$ — Preceding unsigned comment added by Wpd0101 (talk • contribs) 15:37, 8 July 2015 (UTC)

Change title of article to 'Van der Waals equation of state'

'Van der Waals equation of state' is a more accurate description of this article and so I think it should be moved. Wiki me 15:47, 18 March 2007 (UTC)

The proper title should be "Van der Waals gas" 68.104.13.242 (talk) 11:27, 8 September 2022 (UTC)

1st equation?

The article states: v is the volume per particle of the fluid in the 1st equation;. Questions: (i) is the volume per particle v not the same as the total volume V? Both are the volume of the vessel, I would think. (V-b is the volume available to a particle). (ii) What is the 1st equation? (iii) Does anybody object if I start working on this article? —The preceding unsigned comment was added by P.wormer (talk • contribs) 15:51, 9 May 2007 (UTC).

Sorry for forgetting to sign previous comment. I reworked the article. Extended the derivation (actually gave two derivations), and checked the notations. These didn't match well, I hope I have the notation correct now and matching. I kept $\,a'$ and $\,b'$ as molecular variables (as they were in the original article). The corresponding molar variables are $\,a$ and $\,b$.--P.wormer 11:53, 15 May 2007 (UTC)

changes by 218.223.87.142

I don't like to get into edit wars, but the changes by this anonymous are so strange that I felt compelled to revert to my own last version. The anon introduced the following sentence in parentheses:

(The number of "edge" molecules is proportional to n/V and the number of interior molecules is proportional to n/V also. The number of pairs of interacting molecules is thus proportional to n2/V2 so that the forces attracting the edge molecules to the interior are proportional to n2/V2. These forces give an additional contribution to the pressure on the gas proportional to n2/V2. from http://www.chem.arizona.edu/~salzmanr/480a/480ants/EQNOFST/states.html)

Edge molecules are not yet defined. What is n2? What is V2? Why a link to a website in the middle of a derivation? I will accept it if somebody improves/shortens the phrasing of this proof, but this is not it.--P.wormer 07:41, 16 May 2007 (UTC)

And two more things: (i) I took great care to follow IUPAC in distinguishing n, N, and N_A and also v, V, and V_m. This went unnoticed by 218.223.87.142. (ii) The argument about the layer molecules is very subtle and took the genius of Laplace to discover it. It is not a matter of simple (N/V)² pairs. The number of pairs of interacting molecules per volume is N²/2V. --P.wormer 09:45, 16 May 2007 (UTC)

Good that it's removed. It's a direct quote from a copyrighted page. Yes, it's attributed, but text written from the ground up "organically" on Wikipedia is many times preferable to cut and paste from copyrighted material. Invoking copyright issues often makes resolving edit wars easy. --HappyCamper 03:34, 23 May 2007 (UTC)

reversal

I reverted the change of 09:12, 8 June 2007 by 212.192.238.23. This anonynomous removed about half of the article without any explanation. Such a drastic edit needs some motivation! Although risking an edit war, I decided to go one step back (to the version of 18:47, 26 May 2007 by JAnDbot).--P.wormer 07:53, 8 June 2007 (UTC)

I make my apologies

212.192.238.23 08:14, 9 June 2007 (UTC)

I' sorry for unintentional deletions. I would like only to fix the second equation, which has inconsistensy in the definitios (not all quantaties were in per-mole based units). No deletions were assumed.

Fluid?

Why is the whole article referring to fluids when the van der Waals equation is used to calculate the pressure of gases? Maybe I am just not aware that one can use this equation for liquids as well, although the fact that R (ideal gas constant) is used and that the equation is a modification of the ideal gas equation makes me doubt that there is any application to liquids. If I am correct then perhaps using just gas instead of fluid would be a good change, since fluid can easily be interpreted as liquid as well as gas. Aurimas 09:46, 29 August 2007 (UTC)

It appears I was incorrect. These equations do apply to liquids. My apologies, please ignore the above comment.

Actually an interesting question is, when exactly does a fluid change from a liquid to a gas? In the case of a first order phase transition the answer is clear--the change happens at the phase boundary. However, above the critical temperature, Tc, we can have a second order transition where there is essentially a smooth transition from a liquid to a gas with no clear boundary between the two. In general, we would just refer to this as a van der Waals fluid to avoid the difficulty. Jtepper (talk) 05:47, 18 November 2009 (UTC)

Aurimas 20:22, 10 September 2007 (UTC)

Internal Energy - monoatomic or general

Is the derivation for the internal energy (under other thermodynamic parameters) correct for a diatomic gas? I would think that the 3/2 comes from the specific heat capacity at constant volume and makes the formula applicable for monoatomic gasses only. Would it not be correct to change this to c_V? I think so, but I'm not sure I've done the math correctly. —Preceding unsigned comment added by 85.81.85.32 (talk) 11:01, 21 September 2007 (UTC)

Radius of Atom ?

I don't understand why the volume of an atom is given as 4/3*pi* d**2, because d usually means diameter, and d = 2*r, so isn't this wrong? --maye (talk) 16:19, 6 December 2010 (UTC)

ok it is later explained in 3.1, that the 'excluded' volume is not the same as the volume of a sphere. maybe one could improve the 3rd section in terms of structure? Make an introduction that is clearly independent of the different methods of derivation, explain required concepts (like the volume of a simple sphere), and then explain the derivation-dependent argumentation only in the respective sections? What do you think? --maye (talk) 16:24, 6 December 2010 (UTC)

The wikipedia page on the vdW radius defines the vdw volume = b/NA for monoatomic particles but this one accounts for excluded volume as well i.e. b/NA = 4*vdw volume. Can someone explain the difference?94.54.65.192 (talk) 10:55, 6 October 2012 (UTC)

In the In the conventional derivation section how do we go from:

$b'=4\pi d^{3}/3=8\times (4\pi r^{3}/3)\quad \rightarrow \quad b'=4\times (4\pi r^{3}/3)$,

to what they show on the wiki page for a real gas:

$b={\frac {RT_{c}}{8P_{c}}}$

S243a (talk) 14:38, 15 April 2013 (UTC)

You can't! The second part is a derivation (which has been include in text2604:2000:6069:3600:3D0E:2712:964E:2070 (talk) 19:50, 25 July 2015 (UTC)wpd0101@gma2604:2000:6069:3600:3D0E:2712:964E:2070 (talk) 19:50, 25 July 2015 (UTC)il.com.

oscillations?

I can see no oscillations in the PV diagram, contrary to what the caption says. --Yecril (talk) 09:12, 10 December 2013 (UTC)

I think the caption is referring to the dips in pressure across some isotherms about 1/3 of the way from the left. As the volume is decreased, the pressure (oddly, so) drops and then increases sharply as expected. The Moiré patterns don't help though. Tomásdearg92 (talk) 10:56, 10 December 2013 (UTC)

Confusing wording

In the conventional derivation paragraph, in the second and third sentence the "bulk of particles" are particles near to the walls, and in the third sentence the "bulk of particles" are particles in the center of the container:
"Further he assumed that the range of the attractive force is so small that the great majority of the particles do not feel that the container is of finite size. That is, the bulk of them have more attracting particles to their right than to their left when they are relatively close to the left-hand wall of the container. The same statement holds with left and right interchanged. Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, [...]"
I'm going to delete the 2nd and third sentence, since the last sentence has the same content.--Skarmenadius (talk) 18:28, 22 January 2014 (UTC)

I've added some wording to indicate that this derivation does not follow from its assumptions. Pairwise additive intermolecular interactions preclude factoring the configurational partition function into the N'th power of the single particle partition function.

Inset figure about oscillations

I don't have any relevant literature in front of me, so I feel uncomfortable editing the document to remove the offending sentence, but the caption to the figure showing vdW isotherms doesn't seem right:

 "The van der Waals isotherms: the model correctly predicts a mostly incompressible liquid phase,
 but the oscillations in the phase transition zone do not fit experimental data."

Unless there is a specific study to which someone is referring (e.g., someone actually measured data in the region between the spinodal points and got something other than what the van der Waals equation predicts), this is incorrect: between the spinodal points (the "saddle" points, or the points at which dP/dV = 0), the slope of P vs V is positive, as predicted by the equation. This means that the equation of state actually predicts that a fluid at those pressures with those densities would be unstable---stable thermodynamic systems always have d(-P)/dV > 0 or dP/dV < 0. This is a correct prediction, and is not incompatible with experimental data---experimental data can only be collected for 'stable' fluids, as unstable systems are not observable. Kaiserkarl13 (talk) 21:43, 10 December 2014 (UTC)

I concur with your objection, and will add a tag. This was the original figure legend text, and I did not want to do a hard edit of it. When you have a source in hand, please (!) make the changes. Note, I have added sever good sources to further reading, to facilitate use of good sources, but any reputable review or text would be fine for this basic point. (See also next long section.) Cheers, Le Prof 71.201.62.200 (talk) 15:39, 25 June 2015 (UTC)

In re: major ongoing edits

I began some revisions to the article, starting with the lede, after finding that the lede did not represent what the article contained. The new lede is not yet perfect, but is better than what was there before.

I also placed tags because I found the sourcing to be in egregious violation of WP:VERIFY and clearly a case that would bring academic plagiarism charges—the content presented is widely available, but this does not excuse one from indicating which actual source was used in presenting the material. No two derivations are identical, and readers deserve to know (and WP policies require that we say) which one or ones of the possible sources for the derivations were the actual one for this WP article.

In respons to the tagging, some editors have visited, and made rapid additions of sources, unfortunately, these have been to self-published or otherwise unacceptable scholarly sources (e.g., to Khan academy, an organization with whom I correspond, to address their online errors).

In response to the rush to put in sources, I argued at an editor's Talk page that ctations that go in:

• (1) should be solid, reputable, even esteemed secondary sources—graduate and better undergraduate physical chemistry text books like PW Atkins, journal review articles, and monograph review chapters, etc.
• (2) should be entered fully and completely (not abbreviated, and not URL-only), so another editor does not have to come in after, to add missing citation elements, andcal const
• (3) Critically, when sourcing an article post hoc, the content needs to be checked against the source, very carefully. (As a Prof, I rarely find that the content can remain precisely the same, when adding a source that is not the actual source from which the material was originally drawn, see next Talk section.)

This is all wikipedia policy, see WP:SCICITE, but it is also good research practice, and standard academic training to write about science. The quickly found web coverage—Khan academy, and self-published professor or student course information webpages—are not good or even acceptable science sources.

Finally, I have begun to add good sources to a Further reading section, and I urge other editors to do the same—share good finds in this Fr section, until each source is fully "tapped" (as in tapping a tree for syrup, i.e., "excerpted"), and until the article becomes a nominee for "Good Article".

Bottom line, slow down, make improvements that others will not have to fix. Cheers. Le Prof 71.201.62.200 (talk) 15:55, 25 June 2015 (UTC)

I need some clarification on the question of self-publishing. I've added the non-calculus derivation of the vdW a and b constants from the critical constants, and can reference my own writings, but the derivation stands on its own two feet (so to speak) of logic, and really needs no reference what so ever, i.e., the reader him/her self can tell that the derivation i correct is she/he knows algebra. Should it be "referenced"?2604:2000:6069:3600:3D0E:2712:964E:2070 (talk) 19:47, 25 July 2015 (UTC)wpd0101@gmail.com

Here is an example of what must change when unsourced text is reviewed against sources

The following is an example of edits to the "Validity" section of this article, initiated after I simply tried complete an incomplete citation and remove a tag.

I found the Terrell Hill (1960) source that was cited in the section, identified a modern reprint, and completed the citation, so that the information is now complete, and so that the source can now be found at an online book site with a "Look inside" feature.

I also read the corresponding pages: p. 280 appeared referencing the 1960, and I found the corresponding material on pp. 286-289 in the 2012 reprint. Critically, in reviewing the material, I also found that the original writer:

• did not draw all information from this source (so some tags have to remain), and
• did not accurately convey the cited authors views.

Specifically, Hill makes clear statements as to the the value of the equation, this was omitted in the earlier version, so this material was added to the paragraph and cited. At the same time, the earlier text's information that was more critical was located in Hill and so was retained, but changed in part to quote form, with the rest in a much closer paraphrase, without editorializing additions. (To finish the session, I also sourced another statement, that the theory was of use pedagogically, and added three text book examples.)

Here is the "diff", so one can compare before and after: [1].

This is the sort of work that needs to be done in this article, in my opinion. Bottom line, my tagging is not drive-by; the problems of this article are exhaustive—it needs being gone over sentence by unsourced sentence—and simply adding self-published sources at various points will not make it a good article. In this one professional's opinion. Le Prof 71.201.62.200 (talk) 16:05, 25 June 2015 (UTC)

Final comments and addition of expert needed tag

The call for an expert is to get someone whose first area and focus is PChem (not second or third, like mine). The reason for this tag's addition are given in the tag for all to consider, unless someone chooses to revert, and leave the article in appearance, looking acceptable, rather than moving it in actuality toward being a good article. Reporting for Duty. As a retired Physical Chemist, with some expertise in this area, I've been adding to the material and attempting to correct some of it. There is a problem in my mind, having to do with original derivations and sourcing them. I added a derivation which is, to my knowledge, original. How do I source it? An explanation of how original material is to appear here would be appreciated.

Note, before reverting the major edit on the Maxwell section, please attempt to read through what was there before, here [2]. There are incomplete sentences, and sentences that make little sense (to this practicing, professional, and teaching chemistry). This edit is not perfect, and the inline citations are annoying, but with the addition of a few references, and with review by a Stat Mech or other PChem expert, they all can go away. In the mean time, they mark the true status of the article, and indicate where editors with only a little time each day can apply effort.

Otherwise, a few final notes:

• The derivations remain all but WP:OR — no source is given for any one of them, as far as I can see. The standard is not "Well anyone can find these if they just look… [not specifying where to look]." It is to attribute the intellectual content of others, to those others, both because it is honest, and because it assists readers in arriving at a true understanding of a subject.
• I moved the Maxwell equal area section up, with its image, because it is more important than all the stray derivations that also appear.
• I grouped all the stray derivations together, under a major subheading, to follow the Maxwell discussion.

That is all, with this I pass this on to a physical chemist. Le Prof [Leprof_7272] 71.201.62.200 (talk) 18:18, 25 June 2015 (UTC)

Note, in final touchups to sources I was checking for broken URLs, and as a result came upon information that led me to delete a source. The citation attributed to Adrian Down is unacceptable. These are course notes from an academic course he is TAing, or has taken in the past. (Down was a UCal Berkeley undergrad and Duke graduate student, who puts all notes from courses taken and TA'd online.) This is not acceptable scholarly material here, especially given the fact that the title subject, the van der Waals equation, is discussed in depth by eminent senior scholars (as opposed to the next generation in training), and these discussions appear in all of the best PChem text books in use, see Further reading. This self-published material by a trainee is just not acceptable sourcing anywhere. Le Prof 71.201.62.200 (talk) 18:57, 25 June 2015 (UTC)

For these purposes, I qualify as a physical chemistry expert. The information in this article is generally accurate, including the "Maxwell construction". It merely lacks inline citations, and one section has improper tone. These problems are easy to fix! We don't need to wikilawyer the Van der Waals equation#Maxwell equal area rule to death; for example, the "Essay-like" tag alone suffices. I would recommend Griffiths (for physicists) or Sandler (for engineers) as starting points for sources. Both are worthy of inclusion in the final article. —wing gundam 17:10, 27 November 2015 (UTC)

I made a first pass at editing the Maxwell construction section. For future reference, you grossly overtagged that section with inline messages. That many inline template messages dramatically reduces readability. Many phrases you tagged as cn (like "... free energy, a state variable, ...") do not require one, since a citation for that paragraph (which you also tagged) would presumably cover such an uncontroversial claim (in this context). And feel free to make edits yourself! —wing gundam 18:50, 27 November 2015 (UTC)

Notes on images, esp. the precision of values offered in the Maxwell equal area rule image

First, with regard to the pair of oscillation images in the Equation section—cannot the upper, complex, poorly explained image be deleted?

Second, the values of parameters given in the figure legend of the image in the Maxwell equal area rule section vary tremendously in their precisions—e.g. 0.9 vs 2.3488 vs. 0.64700. ---The 0.9 is really 0.9000000etc., i.e., a reduced temperature for an isotherm chosen as 9/10. The two volumes are computed, and their precisions are arbitrarily obtained using the spreadsheet's limits. Since there are no "real" gases which obey the vdW equation, this is more numerology than reality. wpd0101 — Preceding unsigned comment added by Wpd0101 (talk • contribs) 12:53, 9 July 2015 (UTC)

How is this rigorous and physically informative? And is not stating PsubV is approximately equal to a number with 5 decimal places ending in two zeros either redundant (because the imprecision of the value is already implied by the last zero given), or just physically silly (to state something is approximate when stated to a precision of 1 part in 100,000)?

It is not clear to me how these expressions reflect or promote clarity of thought, or understanding of the subject. Le Prof 71.201.62.200 (talk) 18:26, 25 June 2015 (UTC)

And for that matter, the appearance of the image's position of VsubG is not consistent with the stated value, of ≈2.3, see figure legend for image, here, Van_der_Waals_equation#Maxwell_equal_area_rule. Le Prof 71.201.62.200 (talk) 21:37, 25 June 2015 (UTC)

Significant figures are not a concern in this context. If the image itself is wrong, you should comment on its talk page, or better yet fix it yourself! —wing gundam 18:30, 27 November 2015 (UTC)

Note on the representation of the pressure variable

In the article and images pressure is variously presented as lower p or upper case P, and in each case, sometimes not italicised. This variable representation of the same parameter should be corrected when the further careful revisions are done—understanding that making text and images consistent is the longer term, harder objective. Le Prof 71.201.62.200 (talk) 21:34, 25 June 2015 (UTC)

I've tried to make it consistent, changing all the upper-case Ps to lower-case p, and adding italics. A figure still needs to be edited to make everything consistent. Christopher King (talk) 21:17, 21 December 2015 (UTC)

Source for the Statistical thermodynamics derivation

The given demonstration is that of this paper as far as I can tell:

Terrell L. Hill. Derivation of the complete van der Waals’ equation from statistical mechanics. Journal of Chemical Education, 25(6):347, 1948 — Preceding unsigned comment added by Luc-j-bourhis (talk • contribs) 20:14, 4 August 2017 (UTC)

Article Introduction

The first sentence under section 1 seems like a better introductory sentence for the article than the current one. The current introduction seems stylistically wrong, and it's confusing that the first equation in a page about the van der Walls equation is not the van der Walls equation, but the ideal gas equation (the second equation is also not the van der Walls equation). Rather than focusing on an intuitive derivation, the introduction should explain succinctly what the vdw equation is. It's also confusing that the equations in the introduction reappear in section 2, but with the variables named differently! Note a vs a', b vs b'. I suggest that the introduction should be rewritten, ideally explaining what the new variables a and b represent, the first sentence of section 1 be moved into the introduction, and section 1 be renamed from "Overview and history" to "History".

2607:F140:400:A00A:71F7:8A48:C80C:3985 (talk) 19:28, 27 February 2019 (UTC)

Inconsistent symbol v for volume per particle or per mole

In the section "Equation" the symbol $v$ is defined as $N/V$, the number of particles divided by the volume of (available to) the gas, but in section "Derivation" $v$ appears as a synonym of $V_{\mathrm {m} }$, the volume of one mole of the gas. This usage was introduced in an edit July 8, 2015 at 17:25 where the comment expresses the intention to use $v$ for the molar volume.

This addition is unnecessary and does not clarify the issues. In the continuation of the section only $V_{\mathrm {m} }$ is used.

Given the earlier definition of $v$, the formula should have been with Boltzmann's constant, not with the gas constant. I suggest that the second equality is deleted. — Preceding unsigned comment added by Cacadril (talk • contribs) 12:41, 3 September 2019 (UTC)

Contradiction?

The regions of the isotherm from a–d and from c–e are interpreted as metastable states of super-heated liquid and super-cooled vapor, respectively.

super-heated and super-cooled on isotherm ?? Rwbest (talk) 07:31, 20 April 2020 (UTC)

No contradictions to be seen, this is a nomenclature that has been established so far. Though I would exclude those hyphens and leave them just superheated and supercooled instead. Probably the thing that's bugging you is how come they can be hot and cool at the same time. The reality is they refer to different states of matter but not different temperatures. Hence, what's considered a low-temperature vapor easily can be a high-temperature liquid, having the same absolute temperature whatsoever. Zomos 14:04, 26 January 2022

GA Review

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

---

This review is transcluded from Talk:Van der Waals equation/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Phlsph7 (talk · contribs) 10:03, 16 December 2023 (UTC)

Hello Airman72 and thanks for all the work you have put into this article. However, despite all the improvements, the current version of the article fails criterion 2b since there are too many unreferenced paragraphs, see WP:GA? and WP:QF. Examples are the paragraphs starting with Although the gap in v delimited by the, Real gases can be characterized by their, Textbooks in physical chemistry generally, and Next, we introduce a (not necessarily pairwise) attractive. Additionally, there are 3 citation-needed tags and 1 who-tag. According to criterion 2b, there should be references no later than the end of the paragraph. This does not apply to the lead if it summarizes material found elsewhere in the article. I suggest that you go through the relevant sources to add citations to all the unreferenced paragraphs and address the maintenance tags before a renomination. You could also consider a WP:Peer review if you want to get more feedback before. Phlsph7 (talk) 10:03, 16 December 2023 (UTC)

A few more observations:

• WP:EARWIG shows no copyright violations
• Having a lead image would be nice. What about a Pressure-Volume diagram of an isotherm or something similar?
• The lead section uses a lot of formulas and technical vocabulary. I would suggest giving a more general and accessible overview without relying too much on formulas and technical terms. They should all be mentioned and explained but the lead section is probably not the right place for all the technical details. It might also be a good idea to present the information in the bullet list in a regular paragraph instead.
• Some quotation marks use ``. They should be replaced with ".
• Consequently it can be regarded comma after "Consequently"
• The properties molar internal energy, $u$, and entropy, $s$, defined by the first and second laws of thermodynamics, hence all thermodynamic properties of a simple compressible substance, can be specified, up to a constant of integration, by two measurable functions, a mechanical equation of state, $p=p(v,T)$, and a constant volume specific heat, $c_{v}(v,T)$. This sentence is difficult to parse. Splitting all the information in it into smaller sentences would help the reader.
• volume for one mole of a Van der waals' gas upper case for "waals"
• avogadro number uppercase for "avogadro"

Phlsph7 (talk) 10:03, 16 December 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.