[ptryon]

Hi
I have a question about real gas behavior: The volume (V) that is assumed to be available to an ideal gas is the volume of the container; however, in a real gas the molecules themselves occupy some volume- so there is a lower volume available to the molecules of a real gas. This effect becomes significant at high pressures since the volume the molecules occupy becomes significant relative to the volume of the container. This can also be seen in the vdW equation where a term nB is subtracted from the V term. No problem so far? It follows that:

1 Vreal ≤ Videal

2  p x Vreal ≤ p x Videal

3  p x Videal / nRT = 1

Therefore: p x Vreal / nRT must be less than 1 

But actually at high pressures the pV term is actually greater than 1. The attached graph shows the deviation of a real gas from ideal behavior: An ideal gas would have a pV/nRT value = 1. However, the real gas has a pV/nRT value ≥ 1, especially at high pressures.

http://www.chemguide.co.uk/physical/kt/realgases.html


Any thoughts would be greatly appreciated.

[Corribus]

What's the question?

[ptryon]

Why pV/NRT (the compression factor) for a real gas is greater than 1 (at high pressures) when it should be less than 1?

[Corribus]

I've written a little about compressibility factors before, but didn't really address positive deviations from 1 very much. Still you might find it useful.

http://www.chemicalforums.com/index.php?topic=72293.msg261720#msg261720

Your assumption #1 is not really supported for all pressures. Can you think of a reason why the deviation might be positive at high pressures but negative at low pressures?

[ptryon]

Thank you! I will have a read and see if this helps.

Regarding your question: I think there is also an effect resulting from intermolecular forces between gas molecules. The net attractive force on molecules at the egde of the body of gas and next to the internal walls of container is acting in the opposite direction to the motion of a molecule that would otherwise collide with the wall of the container. This is much easier to draw than describe! This reduces the speed of molecules that are heading for a collision and therefore reduces force of the collision. This results in a lower pressure than you would otherise expect. It would following in the maths that the prealV/nRT would have to also be less than 1.

[Corribus]

The walls of the container don't really have much to do with it. You can use an imaginary force field rather than a physical wall if it is helpful to think of it that way. The change in pressure/volume from ideality is an intrinsic property of the gas, due to forces strictly between particles.

Here is an easy way to think about it that is mostly correct. At low pressures, attractive forces dominate, which is why you get compressibility factors less than 1. At high pressures repulsive forces dominate, which is why you get compressibility factors greater than 1. The reason repulsive forces dominate at high pressures is because at high pressures, the number of collisions increases, and when particles get close enough to collide, the nuclear-nuclear repulsion force is exceptionally strong. Since repulsive forces become very important, the net volume is greater than the ideal case (the particles are repulsed and so like to be farther apart), where there are no repulsive forces. At low pressures collisions are rare and electron-nuclear attractive forces are more important, because these operate at slightly larger length scales. Under these circumstances, the particles like to be closer together than the ideal case, and so the volume retracts.

There are some interesting temperature effects as well, but we can leave those alone unless you're really interested.

[ptryon]

Hey, That is a really useful explanation thank you! I never considered the possibility of nucleus-nucleus repulsive forces. Your explanation goes along a different line to the chemguide article (link in my initial message) which I think explains the difference purely in terms of the difference in the volume that is available to molecules of a real gas.

[Corribus]

No problem, it is my pleasure to help.

Just to be clear, there is usually more than one way to explain the same effect in thermodynamics and statistical mechanics, and even though they may seem like quite different explanations, often they are actually functionally equivalent at some level. Some explanations just click more with certain people than others... and vice-versa. I am familiar with the ChemGuide page on this topic. ChemGuide almost always does an excellent job. I know in this case the ChemGuide page discusses collisions with the walls, even though I told you to ignore them. Well, that's a fine way to look at it, too, if it helps you understand. (The ChemGuide article also talks about the neglected volume taken up by the particles themselves, which is kind of a separate issue unrelated to intermolecular forces, but certainly is something to keep in mind.) The way I explained it was the way it was explained to me way back when, and it always made sense to me when described that way - that particles at high pressure are more likely to collide, and when real nuclei collide, they experience a repulsive force that drives them farther away than if they just collided like hard, non-interacting spheres. (Or, after they collide with each other, they hit the container walls with more force on average than they otherwise would.)

In all honesty, these are all just conceptual models to help us understand the relationship between what we observe and the very abstract equations that make up much of Statistical Mechanics. Use what tidbits you can and just be happy if you find something that works for you. There's probably no conceptual model that's 100% rigorously correct, because they're all kind of hand-waving arguments.

[ptryon]

I find chemguide so useful- I think it is the best resource for A-level/AP level/IB level chemistry I have ever come accross. I like what you are saying about who various explanations are functionally equivalent on some level. It used to bother me when I was learning chemistry because I wanted to know "the correct" explanation- but I am becoming increasingly more confortable with the notion that you have to find an explanation that is credible, plausible and helpful to you.

Thanks again for your thoughts on this 
Pete

[Irlanur]

The reason repulsive forces dominate at high pressures is because at high pressures, the number of collisions increases, and when particles get close enough to collide, the nuclear-nuclear repulsion force is exceptionally strong.

Thanks for the explanation about something I haven't thought about so much. Is there a reason why you explicitly mention the nuclear-nuclear repulsion? I think you can just use a phenomenological lennard-jones potential, which often describes the potential between gas molecules pretty well.

[Corribus]

Well, the Lennard-Jones potential is a great mathematical embodiment of what I'm referring too, so yes I think this would be a fine graphical depiction to use. Thanks for bringing it up. At intermediate distances (low pressures), attractive forces dominate and atoms/molecules stick together. At really short distances (high pressures), the repulsive forces dominate, and atoms/molecules are driven apart. At very long distances (extrapolation to zero pressure), none of the interactions are strong, and so you approach an ideal behavior again. You could probably make something of the fact that the negative charge is diffuse in space and the positive charge is concentrated into a point, and this helps to rationalize the shape of L-J potential, and the effect on gas compressibility factors. At some point you run the risk of "over-classic-ing" the physics involved, but I think it's forgiveable if it gives you some insight into some of what's going on.

(To the OP, if you're unfamiliar with the L-J potential, you can read more about it here: http://en.wikipedia.org/wiki/Lennard-Jones_potential)