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Why Is It that All Gases End Up the Same Volume?

Writer's picture: Bryan LeBryan Le


Subreddit: r/ExplainLikeImFive


User: u/Successful_Box_1007



Original Post:


ELI5: Why regardless of size of molecules, do gases at same temperature pressure and volume magically end up with the same number of molecules AND the same spread-out-ness in that volume?!


My Response:


A lot of good answers here.


The other issue is that some gases have strong dipole moments, particularly gases that are asymmetrically , which also contributes to them sticking to one another. It’s the primary reason why gases condense at all when it’s colder - there’s not enough kinetic energy for them to overcome the dipole attractions, and so their volume shrinks to a fluid. Ideal gas law no longer functions for liquids or supercritical fluids, because there’s no energy to create distance between them.


Even fully symmetrical gases, like helium, will have enough of a momentary dipole moment due to the perturbations of the electron cloud to exist as a liquid, but the temperature is extremely low. The ideal gas law does not account for these, and in practice, you add corrections to the ideal gas law based on thermodynamic data.  Corrections look like this:


(P + (an²/V²))(V - nb) = nRT


This is just a first order correction that accounts for van der Waals forces, but there’s additional correction factors that accommodate the molecular and atomic volume of the gas.


There’s more nuance corrections that can be applied, such as perturbation due to quantum tunneling and amorphous phase changes at the boundary of the liquid-gas border (such as what occurs in a supercritical fluid), but these are not practical as they sometimes derived from theory and do not account for real life situations. So for most situations, the above corrected equation works well enough.


Scenarios where you need to dig deeper are when you’re conducting chemical vapor deposition, such as when you’re depositing a highly thin-layer metal atop a semiconductor in a vacuum. The dynamics there are extremely complicated and atomic volume starts to be very important - especially as two metal atoms collide, their tendency is to stick together.


Another set of complex scenarios are MALDI (Matrix-assisted laser desorption/ionization) and mass spectrometry. Basically, a large molecular weight compound, such as a protein or polymer, is rapidly heated with a laser in a vacuum, quickly vaporizing it. These compounds are massive, so their molecular size, charges and geometry actually matter quite a bit. It’s easy to foil these systems up because if their sizes in the vapor phase.


Obviously in these two above scenarios, ideal gas law cannot come close to modeling their vapor phase behavior.


User Response:


Wow that was an AMAZINGLY concise but jam packed with nugs of digestible knowledge based answer!!!!


I do have one question though: in a gas form, why simply because the molecules are very far apart relative to their size, can we assume they can be modeled as point charges of all the identical sizes ?


My Response:


You’re really pushing my physical chemistry knowledge on a Sunday here 🤓


So this is one of those fundamental issues that arises in chemistry - bulk matter versus quantum matter. When you have a massive assemble, the behavior of a material is a statistical average of all the individual behaviors of the atoms/molecules. In the other hand, single atom or molecule physics is a whole different animal and has very strange behavior.


We apply the ideal gas law assuming all the features of bulk matter - it’s continuous, it’s consistently divisible, and has measurable, constant behavior at the same conditions. When gases were discovered, we didn’t have a relatively complete understanding of the atomic model, let alone the quantum models that we have now.


So these equations were developed with the behaviors of the gases that were used at the time. Oxygen, nitrogen, carbon dioxide. Very simple. Very small. Their assemble behavior is very similar, and you can apply a neat little equation to model their behavior.


This is like Newton’s laws of motion. There are no edge cases, so a simple model is apt to cover the bases we see at the time. But as we know, reality is much more complex than three laws of motion can cover. Now you have friction, gravity, changes in phase state, air temperature, etc. when you’re talking about objects in motion. Ideal moving objects don’t exist, and neither do ideal gases.


Let’s do some math.


Let’s say we have 22.4 L of helium at standard temperature and pressure - so about one mole’s worth of gas atoms. Helium has a diameter of 2.67 angstroms, so that translates to 2.67 x 10^-10 meters.


For sake of simplicity, we can pretend these spheres pack like cubes if they were compressed together. In reality, spheres pack into little tetrahedrons with three spheres in a triangle and the fourth sphere sitting in the dimple. There’s some fun geometry there, but if I recall, it’s about a 16% difference in volume from a true cube of spheres.


Anyway. Let’s pretend these helium spheres have a volume of a cube with a side of 2.67 x 10^-10 meters - so their volume is going to be 1.90 x 10^-29 cubic meters. Now let’s pack one mole’s worth of these atoms together (6.022 x 10^23 atoms).


That’s going to be a volume of 0.0000114 cubic meters, or 0.0114 liters.


Back to STP conditions. If you have 22.4 L and 0.0114 L is taken up by atomic volume, that’s still only a difference of 0.05%. Is it accurate? No. Does it work in most cases? Yes. And it was good enough for the 18th and 19th century chemists who were working with volumetric glassware that probably had an accuracy that differed by +/-5% depending on the season, so that minuscule difference isn’t going to make much sense to even try to measure.


Even in most cases, it’s not worth the hassle of the calculation. The heaviest gas used in commercial applications, uranium hexafluoride, has a molecular diameter of 5.98 angstroms, so not much difference between gases either.


Now let’s break the model.


We’re going to look at PG5, which is the largest stable synthetic molecule ever synthesized. It has a molecular diameter of 10 nanometers. That’s 1.0 x 10^-8 meters, and again assuming cubic packing, would give you a volume of 1.0 x 10^-24 cubic meters.


A whole Avogadro’s amount of these molecules would have a molecular volume of 0.6022 cubic meters, or 602.2 L.


There’s no way in hell that number of molecules is going to fit in the standard model of 24.4 L, if somehow it manages to be vaporized. Completely destroys the model. But this is an outlier, and for most cases of the periodic table, you get good mileage out of the ideal gas law with some corrections to account for the attractive forces.


The reality is that the point charge assumption only works if your instrumentation is crude and you have more error from measurement than how much difference the atomic volume contributions. So the model is overly simplistic and only works because we don’t care about that 0.05% to 0.1% when we’re doing chemistry homework at 2am on a Sunday.


But it does matter in real life applications, and it does need to be accounted for in certain circumstances when working with gases. Otherwise, that amount of error is good enough and looks more like a rounding error for most intents and purposes.


 

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