Real gas formula

[JDbb]

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In EK Gchem, p. 27, it states that the ratio of PV/RT increases due to molecular volume and PV/RT decreases due to attractive intermolecular forces. Shouldn't the ratio decrease in both cases since it isn't ideal anymore? If there is molecular volume, then V-nb must get smaller, so the ratio must get smaller. If there are attractive forces, then the pressure of the container must decrease since the molecules are sticking together, so I guess this part is right in what is being described. But how would you explain PV/RT increasing due to molecular volume?

 

[JDbb]

Wait nvm, I think I get it. So the volume actually increases since the molecules of gas occupies space. But how does that work into the equation if it's V-nb where nb is equal to the amount of volume that the gas occupies? V would still get smaller if it deviates from ideal conditions so PV/RT should continue to decrease due to molecular volume

 

[gabdolce]

Under extremely high pressures, your volume estimation will be lower than reality because of the fact that molecules take up real space and once you reach that limit, you can't [theoretically] get much smaller than that (you begin to observe condensation and liquids/solids are for all intensive purposes incompressible). Under moderately high pressures, your volume estimation will be higher than reality because the attractive forces are keeping the molecules closer together. Under very low pressures you're approaching ideal behavior.

 

[JDbb]

If they take up space, won't my volume estimation still be higher?

 

[gabdolce]

In the moderate pressure scenario, you are no longer compressing beyond the limits of physical volume from the molecules subatomic particles. The space the *particles* take up in this scenario is irrelevant because now (to make it slightly easier to visualize) the volume we're considering is larger and now also essentially encompassing free space as well. In this case, the attractive forces can now really show their stuff directly as a result of this "free space" that is being counted in our "volume." The attractions can nullify some of this "free space volume" and consequently make the volume of the gas "smaller" than expected.

I understand why and how this may be exceptionally confusing. If you would like, I could further elaborate with figures at a later time.

 

[JDbb]

Thanks. I understand the moderate pressure one, but I still don't get the high pressure one. If you could provide diagrams for that one, I would appreciate it

 

[gabdolce]

Thanks. I understand the moderate pressure one, but I still don't get the high pressure one. If you could provide diagrams for that one, I would appreciate it

Ok, so in the left side of the attached figure, the moderate pressure example is shown.

Now let's address the right side of the attached figure:

So the black boxes signify your calculated volumes. According the the ideal equation PV=nRT, as you increase pressure (let's say it approaches infinity), your volume decreases (and this would then go to 0).

But looking closely, we've eliminated all of that "free space volume" that we talked about before. Now ideal theory specifies that the volume should approach 0. But you and I both know that's an impossibility - where would the particles with mass go? Do they just disappear? Surely this cannot be the case - we would be destroying mass, and that just does not occur in our reality.

In this situation, we've reached a limit or a bound to how small our volume can really go --> because the IG equation is based on a set of assumptions, the key one in this case being the particles have no mass, the IG equation begins to break down in the presence of an extreme violation of one of its underlying assumptions.

As I've attempted to convey in my figure, even though we've calculated a smaller volume, my liberal application of conservation of mass suggests that we've reached an absolute limit to how small we can actually go. Hence, past this threshold, our calculations will be smaller than reality.

Does that clear it up at all?

 

[JDbb]

Yes it certainly helps. I definitely understand the logic now, but I don't get how it plays into the PV/RT ratio?

 

[gabdolce]

Yes it certainly helps. I definitely understand the logic now, but I don't get how it plays into the PV/RT ratio?

So if you want to analyze this in respect to a law, the appropriate equation to use is no longer the IG law (remember there are like 4 assumptions that must be all true to use IG) because we are considering extremes.

In this case, is there an equation we can use to better approximate the behavior of gases? One such example is the Van der Waal's equation.

(P+n^2a/v^2)(V-nb)=nRT

So two easy ways to look at it:

1. If you consider when volume = nb, the P term (everything in the parentheses where the P should be) approaches infinity. I.e. it takes an infinite amount of pressure to make your volume merely EQUAL to nb (the volume occupied by your particles).
   
   Hence, you cannot go any smaller than this. If you try, you get a negative number on the left... how are you ever going to get a negative number on the other side? Can pressure [the left side] ever by negative (we're talking about absolute pressure within a vessel)? A negatory to both.
2. If you try to make the P term grow larger, you make the V term smaller. In moderate cases of moderately small volumes, the n^2a/v^2 term makes your 'effective' pressure larger. Additionally the 'effective' V is smaller because of the subtractive term.

You'll notice that when you have LARGE volumes, the -nb term is negligible compared to V and the additive term to the effective P term is also approaching zero. And in this scenario, what do you get? The Ideal Gas law!