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I got this question in school:

Explain, based on the properties of an ideal gas, why the ideal gas law only gives good results for hydrogen.

We know that the ideal gas law is $$P\cdot V=n\cdot R\cdot T$$ with $P$ being the pressure, $V$ the volume, $n$ the amount of substance, $R$ the gas constant and $T$ the temperature (Source: Wikipedia - "Ideal gas").

An ideal gas must fulfill the following:

  • The particles do have an infinitely small volume (or no volume),

  • The particles do not interact with each other through attraction or repulsion,

  • The particles can interact through elastic collisions.

Now, why does only hydrogen sufficiently fulfill these conditions? I initially assumed that the reason is that it has the smallest volume possible as its nucleus only consists of a single proton. However, two things confuse me:

  • (Let's first assume that my first idea was correct and the reason is the nucleus' scale/volume) helium's nucleus consists of two protons and two neutrons. It is therefore four times as large than hydrogen's nucleus. However, hydrogen's nucleus is infinitely times larger than an ideal gas molecule (which would have no volume), so why does the difference of $4$ significantly affect the accuracy of the ideal gas law, while the difference of an infinitely times larger hydrogen (nucleus) doesn't?

  • My first idea is not even true, as atoms do not only consist of their nucleus. In fact, most of their volume comes from their electrons. In both hydrogen and helium, the electrons are in the same atomic orbital, so the volume of the atoms is identical.

Other possibilities to explain that the ideal gas law only work for hydrogen and therefore only leave the collisions or interactions. For both of these, I do not see why they should be any different for hydrogen and helium (or at least not in such a rate that it would significantly affect the validity of the ideal gas law).

So where am I wrong here?

Note: I do not consider this a homework question. The question is not directly related to the actual problem, but I rather question whether the initial statement of the task is correct (as I tested every possible explanation and found none to be sufficient).


Update

I asked my teacher and told them my doubts. They agreed with my (and yours from the answers, of course!) points but still were of the opinion that Hydrogen is the closest to an ideal gas (apparently, they were taught so in university). They also claimed that the mass of the gas is relevant (which would be the lowest for hydrogen; but I doubt that since there is no $m$ in the ideal gas equation) and that apparently, when measuring, hydrogen is closest to an ideal gas.

As I cannot do any such measurements by myself, I would need some reliable sources (some research paper would be best: Wikipedia and some Q&A site including SE - although I do not doubt that you know what you are talking about - are not considered serious or reliable sources). While I believe that asking for specific sources is outside the scope of Stack Exchange, I still would be grateful if you could provide some soruces. I believe it is in this case okay to ask for reference material since it is not the main point of my question.

Update 2

I asked a new question regarding the role of mass for the elasticity of two objects. Also, I'd like to mention that I do not want to talk bad about my teacher since I like their lessons a lot and they would never tell us something wrong on purpose. This is probably just a misconception.

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    $\begingroup$ It would be interesting to read what the expected answer is, and check how wrong it is. Please come back and tell us! $\endgroup$ – Eric Duminil Jul 16 at 8:12
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    $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$ – David Z Jul 16 at 8:37
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    $\begingroup$ It's interesting that no one mentioned the fact that hydrogen molecules are consisted of two atoms, and hydrogen in its single-atom form is highly unstable. $\endgroup$ – polfosol Jul 17 at 7:54
  • $\begingroup$ @EricDuminil I know that it's been a while but I did not see my teacher until today. I've added their response to my question. Unfortunately, I did not have time to talk a lot since it was the last lesson before summer break. $\endgroup$ – Jonas Jul 24 at 18:43
  • $\begingroup$ @EricDuminil oups yes, thanks for pointing that out! $\endgroup$ – Jonas Jul 24 at 19:05
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The school question is wrong. What were they thinking? (My guess is that it was a simple slip-up and they meant helium.)

The ideal gas equation of state works for any gas in the limit of low density. In order to give a quantitative estimate of how well the equation models a gas, one can compare it with measurements or with other equations which do a somewhat better job of modelling the gas. An equation often used in the design of chemical processing plants is named after Peng and Robinson. But for the present question a simpler one called the van der Waals equation will do. This equation is $$ \left( p + a \frac{n^2}{V^2} \right) \left( V - n b \right) = n R T $$ where $n$ is the number of moles and $a$ and $b$ are constants which depend on the gas. This equation is not perfectly accurate, but it helps us see the accuracy of the ideal gas equation. The ideal gas is obtained in the limit where $$ a \frac{n^2}{V^2}\ll p, \;\;\; \mbox{ and } \;\;\; nb \ll V $$ The constant $a$ is owing to inter-particle attractive forces; the constant $b$ is owing to the finite size of the particles (atoms or molecules). You can look up values of $a$ and $b$ for many common gases, and thus find out how well they are approximated by the ideal gas equation at any given pressure and temperature. That is enough to answer your question.

Here are the values for hydrogen and helium and a couple of other gases: $$ \begin{array}{lcc} & a & b \\ & (L^2 bar/mol^2) & (L/mol) \\ \mbox{helium} & 0.0346 & 0.0238 \\ \mbox{hydrogen} & 0.2476 & 0.02661 \\ \mbox{neon} & 0.2135 & 0.01709 \\ \mbox{nitrogen} & 1.370 & 0.0387 \end{array} $$

You see from this that helium is closest to ideal at any given pressure and temperature. This is because its inter-atomic interactions are small compared with other elements, and helium atoms are smaller than other atoms (and molecules).

There is another very interesting point that is worth a mention here. It is a notable fact that all ordinary$^1$ gases behave alike once you scale the pressure and temperature in the right way. It follows that they are all equally well approximated by the ideal gas equation, if you express the pressure as a multiple of the critical pressure and the temperature as a multiple of the critical temperature. (The critical pressure and temperature correspond to the point on the liquid to vapour transition line called the critical point.)

$^1$ By 'ordinary' here I am just ruling out some highly reactive gases, or some with very complicated molecules or something like that.

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  • $\begingroup$ Re "helium atoms are smaller than other atoms": Can you quantify that? And by what measure? Covalent radius? $\endgroup$ – Peter Mortensen Jul 18 at 18:30
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    $\begingroup$ @PeterMortensen most likely it's the radius of outer-most electron orbital. Going downwards on the periodic table adds layers = bigger radius. Going rightward adds protons = smaller radius. That way Helium is the smallest $\endgroup$ – Noone AtAll Jul 19 at 2:41
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The short answer is ideal gas behavior is NOT only valid for hydrogen. The statement you were given in school is wrong. If anything, helium acts more like an ideal gas than any other real gas.

There are no truly ideal gases. Only those that sufficiently approach ideal gas behavior to enable the application of the ideal gas law. Generally, a gas behaves more like an ideal gas at higher temperatures and lower pressures. This is because the internal potential energy due to intermolecular forces becomes less significant compared to the internal kinetic energy of the gas as the size of the molecules is much much less than their separation.

Hope this helps.

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    $\begingroup$ Thanks for the explanation! you said that helium might be the closest to an ideal gas. Is this simply derived from observations, or is there any explanation regarding the structure of the helium atom? $\endgroup$ – Jonas Jul 15 at 20:26
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    $\begingroup$ I suspect it has to do with the fact that helium is one of the chemical elements that are stable single atom (monatomic) molecules (at standard temperature and pressure (STP) along with the other noble gases. It is also the lightest monatomic gas. $\endgroup$ – Bob D Jul 15 at 20:39
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    $\begingroup$ You could also look at the fact that Helium has a filled valence shell. In general, the noble gasses have smaller atomic radii than the other elements in their respective periods (and Helium is no exception to this). It will thus experience weaker van der Waals forces than any other element, regardless of molecular structure. $\endgroup$ – Kevin Jul 16 at 7:27
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    $\begingroup$ @cmaster-reinstatemonica This is to correct the comment from cmaster-reinstate monica. Boyle's law and the fact that internal energy depends only on temperature (Joule's law) together define the thermodynamic ideal gas. It is NOT required that heat capacity be independent of temperature. When rotations and vibrations are excited, the gas REMAINS an ideal gas (obeying Boyle's law and Joule's law). $\endgroup$ – Andrew Steane Jul 16 at 11:02
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    $\begingroup$ @Jonas Two other things about helium. One requirement for ideal gas behavior is that any collisions involving the gas molecules are perfectly elastic. This assumes the molecules approximate hard shells. To quote from the Hyperphysics web site on mean free path: “For noble gases, the collisions are probably close to being perfectly elastic, so the hard sphere approximation is probably a good one”. $\endgroup$ – Bob D Jul 16 at 13:26
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The ideal gas law is routinely used in engineering for calculations regarding air, natural gas, water or other vapor, ICE exhaust gases and almost everything that is sufficiently away from condensing pressure/temperature and some other conditions like the molar volume not being too low.

It works.

The condition "sufficiently away from condensing pressure/temperature" is different for different gases. That's where helium and hydrogen rule - they need only a few K temperature in order to behave. Water vapor may need some 800 K in order to be an ideal-ish gas no matter of the pressure.

PS: The ideal gas law is also applicable in some pretty unexpected places, like osmotic pressure (where dissolved substance behaves like it is an ideal gas in the volume of the solution).

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