# Talk:Ideal gas law/Draft

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To learn how to update the categories for this article, see here. To update categories, edit the metadata template.
 Definition:  Relates pressure, volume and temperature for hypothetical gases of atoms or molecules with negligible intermolecular forces. [d] [e]
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 Workgroup categories:  Chemistry, Physics and Engineering [Categories OK] Subgroup category:  Chemical Engineering Talk Archive:  none English language variant:  American English

1. IUPAC prescribes lowercase p for pressure.
2. Is it necessary to give all forms? In other words can't we expect the reader to know some elementary algebra? (When I went to highschool it was first grade material, student's age around 12 years)
3. Molar gas constant R in SI units: 8.314 472 J mol−1K−1 see: http://physics.nist.gov/cgi-bin/cuu/Value?r%7Csearch_for=R Don't you also want to give: R = NA kB ?
4. The word "constant" in roman: \mathrm{constant}. Same for mol, K, and atm.
5. The name of the law varies. I learned in school "law of Boyle and Gay-Lussac".
6. You give these worked-out problems, that is OK for a textbook, but for an encyclopedia?

Hope these comments are useful. --Paul Wormer 05:19, 4 October 2007 (CDT)

## ideal gas and ideal gas law

Maybe we should merge the two into one article? ideal gas doesn't exist yet, so no real merging is necessary. What do you think? Yuval Langer 06:46, 4 October 2007 (CDT)

## Gas laws developed in 1660s

We need to change this back, because only 1 law was started in 1660s. The rest were developed later. David E. Volk 02:14, 2 November 2007 (CDT)

This article needs lots of work on explanations, particularly on what the formulas mean what why they're used, and more elaboration on some concepts. I think it needs some narrative work. --Robert W King 14:58, 8 January 2008 (CST)

David Volk: Would you please explain why this equation ${\displaystyle V=\left(nV_{\mathrm {m} }\right).}$ (at fixed temperature and pressure) has a period in it? Also please explain Vm . - Milton Beychok 12:52, 4 February 2008 (CST)

## Special cases

For the various special-cases, rather than writing "= constant" or giving the constant a name such as ${\displaystyle V_{\mathrm {m} }}$, how about using the proportional-to symbol? IE:

• Amonton's law: ${\displaystyle p\propto T}$ (constant V, n)
• Boyle's law: ${\displaystyle p\propto 1/V}$ (constant T, n)
• ${\displaystyle p\propto n}$ (constant V, T)
• Charles's law: ${\displaystyle V\propto T}$ (constant p, n)
• Avagadro's law: ${\displaystyle V\propto n}$ (constant p, T)

Downside: people might not know the meaning of the "${\displaystyle \propto }$" symbol. How about in English for the benefit of the notationally phobic?

 Amonton's law: pressure is proportional to temperature (constant quantity and volume). Boyle's law: pressure is inversely proportional to volume (constant quantity and temperature).

Warren Schudy 20:23, 5 February 2008 (CST)

## Explaining edits made to Intro section.

The wording was re-arranged and reformatted so that the discussion of real gas deviation from ideal gas behavior is now in a separate paragraph.

Revised the wording "... one of the real gas equations, such as the van der Waals equation must be used" to state that there are many equations of state (EOS) that can be used for real gases, of which the van der Waals equation is the simplest. The words "must be used" made it sound as if the use of the van der Waals equation was somehow mandatory.

The ideal gas equation parameters were changed to bold font to make them more obvious. The words "absolute temperature in degrees Kelvin" were changed to "absolute temperature" because the equation applies for other absolute temperature scales as well ... such as degrees Rankine that is still used by many USA engineers. The word "pressure" was changed to "absolute pressure" because the ideal gas equation requires absolute pressure (as differentiated from gauge pressure).

Added a reference to document the quoted numerical value of R. Also added a table of various values of R in other units (using the same table as in the molar gas constant article, for consistency).

To the best of my knowledge, I did not significantly change any of the content ... just made it more readable (I hope) and a wee bit more correct. - Milton Beychok 00:43, 6 May 2008 (CDT)

## Nice changes Milton

Milton, thanks for the changes. I still don't what to do with the "Special cases" subsection. I have repeatedly considered deleting the whole thing, writing it out in paragraph form rather than equation form, or rewriting the equations as suggested above with the proportional symbol. Any thoughts on this section? David E. Volk 10:08, 16 May 2008 (CDT)

Personally, I would delete that entire section as you have previously considered. It really doesn't add anything substantive to the article. Regards, Milton Beychok 10:45, 16 May 2008 (CDT)
How about moving it to an /Advanced subpage? J. Noel Chiappa 11:53, 16 May 2008 (CDT)

David, I wanted to create a definition for the Definition subpage, but could not think of one that was consistent with the current lead-in section. So I thought, why not change the lead-in? This is what I came up with:

"The ideal gas law applies to hypothetical gases consisting of atoms or molecules with no intermolecular forces that are in constant random motion undergoing perfectly elastic collisions with the gas container walls and each other.
The ideal gas law is useful for calculating temperatures, volumes, pressures or number of moles ..... (and the rest of the current lead-in section)."

The definition for the Definition subpage could then be: "A gas law that applies to hypothetical gases consisting of atoms or molecules with no intermolecular forces that are in constant random motion undergoing perfectly elastic collisions with the gas container walls and each other."

What do you think? Milton Beychok 11:50, 16 May 2008 (CDT)

Ooooh, I like that! Although in the def, to keep it short, I'd substitute 'particles', or something like that, for "atoms or molecules"; you'd have to substitute 'interactive' for "intermolecular" (and how do you have "intermolecular forces" if you have a gas which isn't molecules, but I digress :-). But given the nature of the theory, which treats the entities as featureless particles, I think some wording like that would be OK. They can get the full story in the article... J. Noel Chiappa 12:20, 16 May 2008 (CDT)

## My definition

I went with this definition (below), which shortens the ideas of others above. Note I left the name out, because I think that is the rule for definitions.

Relates pressure, volume and temperature for hypothetical gases of inelastic particles with negligible intermolecular forces. David E. Volk 12:35, 16 May 2008 (CDT)

Very good, I like that better than my suggested definition. But I still think the article might be improved by the intro section having a first paragraph that states it is about hypothetical gases such as I proposed above:
"The ideal gas law applies to hypothetical gases consisting of atoms or molecules with no intermolecular forces that are in constant random motion undergoing perfectly elastic collisions with the gas container walls and each other."
Perhaps "particles" could replace "atoms or molecules" and perhaps "interactive" could replace "intermolecular". Milton Beychok 12:59, 16 May 2008 (CDT)

## Inelastic particles?

In my understanding an inelastic particle is a particle with internal structure that can take up energy. A tennis ball that can be squeezed in, or a molecule that can vibrate. An elastic collision transfers only momentum (translational kinetic energy). So, in my view most ideal gases consist of elastic particles. In any case an ideal gas of "mono-atomic molecules" (term used by T.L. Hill, an outstanding thermodynamicist) consists of elastic particles.

For diatomic and polyatomic molecules the term is debatable, because collisions can excite rotations. Although rotations are seen as external motions, one could argue that a collision exciting a rotation is inelastic and hence a polyatomic molecule could be seen as an inelastic particle even if it is so rigid that its vibrations are not excited. Personally, I would call a polyatomic molecule only inelastic if its vibrations (or electrons) were excited, but as I say, this is debatable. --Paul Wormer 05:10, 17 June 2008 (CDT)

## Towards approval

Following a request by Milton, I had a quick look at the cluster and don't think it is ready for approval yet. In brief:

1. I share Paul's concerns with respect to the (in)elasticity
2. The sections "Background" and "Special cases" largely overlap, while the article as a whole appears incomplete to me without some derivation from first principles (which could go into an Advanced subpage) and without some specific hints to non-specialists on why this law is important
3. The Bibliography should contain some of the classic papers
4. There should be some useful websites (e.g. with applets) to add to the External links

Hope this helps. --Daniel Mietchen 01:25, 4 January 2009 (UTC)

The typography needs some improvement, too. Don't use = in a sentence and don't use bold letters like: p = absolute pressure (and what is absolute pressure?) I would write p is the pressure of the gas contained in a vessel of volume V, or something similar. I could do this editing, but then I cannot approve the article. --Paul Wormer 11:17, 4 January 2009 (UTC)
Hi, Paul. CZ:Approval process states "Group approval. If there are three editors, all of whom are expert in the topic of an article, and all of which have been at work on an article as authors, then any one of them may approve of an article with the concurrence of the other two." In other words, you may have worked on the article but you can still approve it if you are one of three editors joining to approve the article. That is why I invited Daniel Mietchen, David Volk and you to join in the approval. In that way, everyone can contribute in improving the article and yet join in the approval. So, Daniel and Paul, please make whatever changes you think are appropriate.
As for the meaning of absolute pressure, see Pressure#Absolute pressure versus gauge pressure. In my opinion, it is always important to make that distinction. Milton Beychok 16:59, 4 January 2009 (UTC)
Point taken about absolute pressure. Tomorrow I will probably do some editing on the article. About inelastic scattering: I believe that the present text is incorrect, only the translational partition function depends on volume, and hence the ideal gas law is not influenced by inelastic rotational and/or vibrational excitation by scattering, (I will enter this tomorrow).
It occurred to me that the ideal gas law has a built-in contradiction: the molecules don't interact and have no size, so from the point of view of scattering theory they cannot exchange energy. Yet there is thermal equilibrium. I checked this in some stat mech books and some of them talk about collisions with the wall and others say that there are just enough collisons to thermalize the gas, but that nevertheless the interactions can be neglected. --Paul Wormer 17:16, 4 January 2009 (UTC)
Points 1 and 2 of my list are now largely OK (though I haven't looked at the details of the derivation yet); 3 and 4 remain. --Daniel Mietchen 05:38, 11 January 2009 (UTC)
3 and 4 done now. I am ready to approve. --Daniel Mietchen 17:58, 16 January 2009 (UTC)
Daniel, thanks for all of your work on this article. I see that you have already signed as an approval nominator and that you have updated the version to be approved. I will now ask Paul to sign as an approval nominator. I also moved the final approval date from Jan. 26th to Jan. 21st ... which still leaves 4-5 days for others to review the article. Milton Beychok 18:26, 16 January 2009 (UTC)

## bold letters

Milt I saw you made some changes but let p, etc. bold. These are scalar (not vector) quantities, so I would write p, etc. --Paul Wormer 17:20, 4 January 2009 (UTC)

Paul, feel free to change them. I just wanted them to stand out ... no other reason. Milton Beychok 17:58, 4 January 2009 (UTC)

## Overhaul

I did a fairly major rewrite:

1. Removed reference to collisions and sizes of molecules.
2. Added a stat mech derivation that shows that neither sizes nor collisions play a role.
3. Removed section about "when ideal gas law fails". In intro already reference to other equations of state; that seemed sufficient to me.
4. Commented out last section, because it is repetitive, but it is still there, if somebody disagrees it can easily be restored.

--Paul Wormer 11:31, 5 January 2009 (UTC)

## Minor edits to Paul's rewrite

Paul, as per your post on my Talk page, I have gone over your rewrite. Most of my edits were very minor rewords. I also removed periods amd commas from the end of equations. In my opinion, they are distracting and they really are not needed.

I mus admit that the derivation is over my head. However, I added one more equation at the end of the derivation in order to finsh the derivation. Rather than make the reader have to mentally substitute N = nNA and NAkB = R into your last equation, I did it for the reader. Milton Beychok 22:41, 5 January 2009 (UTC)

Ah, the punctuation of mathematical equations, the great divide between chemists and physicists. You know that the Journal of Physical Chemistry (an ACS journal) does not punctuate and that the Journal of Chemical Physics (an APS journal) does punctuate. Apparently chemical engineers are on the chemistry side of the divide :-) . --Paul Wormer 08:27, 6 January 2009 (UTC)

## Some further streamlining

I removed an historical sentence from the intro (and changed the section title "Background" to "Historical background"). I entered into the intro the fact that a gas consisting of polyatomic molecules may be ideal, in contrast to what our influential competitors have to say about it. And I inserted a few more steps into the stat mech derivation to help the reader, following the example set by Milton. --Paul Wormer 09:31, 6 January 2009 (UTC)

## Notation

The canonical partition function is sometimes denoted by Q and sometimes by Z (from Zustandssumme). The IUPAC "Green Book" [1] (pdf page 56) allows both. --Paul Wormer 10:47, 6 January 2009 (UTC)

## Example problems

I looked at the history of this page and saw that at one point it contained example problems. Would it not be good, now that we have clustered pages, to have a subpage in the cluster with these problems revived? --Paul Wormer 08:22, 7 January 2009 (UTC)

I think this would be a good idea. I revived them on a tutorials subpage. They could probably use some context. Chris Day 08:28, 7 January 2009 (UTC)
Sounds good to me. Chris, does one just create another subpage or what? How would Paul go about implementing his idea? Could you spell that out for us? Milton Beychok 08:55, 7 January 2009 (UTC)
See the tutorials tab at the top? To create a similar tab on other articles all you have to do is create the subpage. Be warned the name has to be exactly right. Best way to create such a subpage is to go to the talk page. You can see a list of unused subpage options in the header (you will have to click the show link). If you click on one of the subpage names it will start the correct subpage for you. Chris Day 16:44, 7 January 2009 (UTC)

## Revising the order of the sections does help with "digesting" the article

I agree with User:Daniel Mietchen that switching the order of the sections does make it easier to digest the article. Milton Beychok 04:53, 11 January 2009 (UTC)

As far as I'm concerned the article is now ready to approve. I did not check the historical facts, though; I assume that they are correct. Did somebody check my work on the tutorial page? That would make me feel better. Problem 3 and 4 are old (dug up by Chris) and reformated by me, I added problems 1, 2, and 5.--Paul Wormer 09:52, 11 January 2009 (UTC)

Paul, I checked the tutorial and found one very minor error in Problem 2. I also revised the very last calculation in Problem 5 for better clarity.
I will wait a few more days to see if Daniel has any further edits and then I will change the Approval nomination so that the nominated article is the latest one. At that time, Paul Wormer and Daniel Mietchen should also please sign the Metadata template as Approval nominators since we need three nominators. Milton Beychok 17:27, 11 January 2009 (UTC)
At the moment, the article seems says that ${\displaystyle Q=\cdots =q^{N}}$ just above where it says ${\displaystyle Q=q^{N}/N!}$. Perhaps this can be clarified? Furthermore, in the first formula for Q, it says that I labels the different energies; should there be a factor corresponding to the probability that the system has energy ${\displaystyle {\mathcal {E}}_{I}}$, or (more likely) is this just my misunderstand of statistical mechanics? It would be nice if the history section said who dreamt up this statistical mechanics derivation first. Similarly, it would be nice to have the computation of qtransl somewhere; perhaps a suitable red link could be added somewhere? -- Jitse Niesen 17:10, 17 January 2009 (UTC)

1. Should there be a factor corresponding to the probability that the system has energy ${\displaystyle {\mathcal {E}}_{I}}$?

Often the partition function contains a sum over energy levels. An energy level can be degenerate, which means that there is more than one linearly independent eigenstate of the Hamiltonian with this energy. The degree of degeneracy is usually indicated by g and one writes the sum over levels as
${\displaystyle Q=\sum _{I}g_{I}e^{-\beta {\mathcal {E}}_{I}}\qquad \beta \equiv 1/(kT).}$
In a more advanced formalism one writes Q as a trace
${\displaystyle Q=\mathrm {Tr} \left(e^{-\beta H}\right)}$
This makes clear that Q can be expanded in any basis -- not necessarily eigenstates of H -- and that a sum over all states spanning state space (Hilbert space) is required. That is why in German Q is called a Zustandssumme (sum over states). In the main article I implied a sum over states -- not levels -- but maybe I must be more explicit on this.

2. The article seems says that ${\displaystyle Q=\cdots =q^{N}}$ just above where it says ${\displaystyle Q=q^{N}/N!}$.

Yes, that is right, the counting factor N! is introduced ad hoc. (For the ideal gas law it is not necessary, it drops out by the differentiation with respect to V). Historically the factor was introduced by the Dutchman Tetrode (1912), who noticed that the equation for the entropy derived by people as Boltzmann, Planck, Nernst, Gibbs (I don't know who exactly did what) was not size-extensive. Size-extensivity means linearity in the amount of substance. Tetrode fixed it ad hoc by multiplying the integral over phase space (the classical form of the partition function) by 1/N! The same year the German physicist Sackur multiplied the volume element of this integral by h−3, where h is Planck's constant. The correction 1/(N! h3) is often called the Sackur-Tetrode correction. Later (1924) Einstein and Bose discovered the Bose-Einstein statistics. Basically the trace is taken over boson states, which are symmetrized states (symmetric in the permutations of the N gas molecules). These states are multiplied by counting factors containing N! and occupation numbers of different states. For "high" temperatures (above 1 K) the occupation numbers of different states are all 0 or 1 and the Bose-Einstein statistics goes over into Boltzmann statistics, i.e., only the factor 1/N! remains.

3. It would be nice if the history section said who dreamt up this statistical mechanics derivation first.

I don't know this history. I believe the term "Zustandssumme" is due to Planck and that Fowler gave it the name partition function.

4. Similarly, it would be nice to have the computation of qtransl somewhere.

I know two derivations, both come down to an integral over a Gaussian function, because the energy of the free particle is quadratic in the momentum,
${\displaystyle H=-{\frac {\hbar ^{2}}{2M}}p^{2}}$
Both derivations contain an approximation that is not serious. The derivation I like most is simply computing the trace in the x representation, where the approximation is that the x states are taken to be continuous and yet the integral
${\displaystyle \int d\mathbf {x} =V<\infty }$
The textbook derivation integrates over quantum numbers of the particle in a box, and makes implicitly the same approximation. Further one transforms from Cartesian to polar coordinates and then one neglects the "corners" of the box.

--Paul Wormer 09:20, 18 January 2009 (UTC)

After I wrote the above answer to Jitse, I saw that Daniel introduced a new Q (with a tilde on top). This is confusing, so I decided to enter a few extra lines on the problem of statistics (Bose-Einstein and Boltzmann). If I would write even more on it, the section would become almost an introduction to statistical thermodynamics, which was not my intention to write, especially not in an article called ideal gas law. But I do hope that the reason for inserting N! is clearer in the present version.

I also entered a line about summing over states, not levels. --Paul Wormer 12:49, 18 January 2009 (UTC)

I agree with the changes Paul made, including removal of the tildes. --Daniel Mietchen 09:22, 19 January 2009 (UTC)
I just updated the version to be approved (on the metadata template) so as to reflect the edits that Paul made. Milton Beychok 18:55, 19 January 2009 (UTC)
Thanks, Paul (and also the rest of you, of course). And sorry for making you write a whole essay; that was not my intention. I agree that the whole N! business is not really relevant to this article, so it's good you refrained from writing even more. -- Jitse Niesen 11:40, 21 January 2009 (UTC)

## Approved Version 1.0

Congratulations to the three of you. You have figured it out! Keep going. D. Matt Innis 00:51, 22 January 2009 (UTC)