# Gas considered as lots of molecules

1. Oct 29, 2009

### WiFO215

I've read in quite a few textbooks, chemistry and physics, that we can approximate a gas as many molecules which are floating in space and bouncing off each other. I want to know on what basis we make this approximation. What forces do we ignore and why?

Here are some of the questions I asked my prof:
Somehow, you approximate these molecules to be beach balls simply floating around and bouncing, or at least, this was the picture I got. How and why is this justified?

When these things are "bouncing off", what forces of interaction come into play? The teacher replied that Coulomb(static force law) Force is the only force we consider. How can we do this when the charges are in motion. I know the obvious answer is "the other part that comes out of the motion of the charges is negligible". Can someone justify?

Apart from Coulomb attraction, he also said that we don't consider any other forces. Why not?

As far as I understand, we cannot be sitting and calculating 1023 force equations which give you the motion of these particles. This is why I have to study this subject of statistical mechanics, wherein we use probability to describe our system. But how is this beach balls picture justified?

2. Oct 29, 2009

### Andrew Mason

What makes you think that this is an approximation. It is an accurate description of what is actually occurring. The molecules interact through coulomb repulsion in which energy is conserved - so the collisions are elastic.

Monatomic gases may be viewed this way. Diatomic and polyatomic molecules have more complicated behaviour so the beach ball analogy is not quite correct.

Well, there are only two forces that act outside the nucleus. Gravity and the electro-magnetic force. The nuclear forces are not involved as they act only within a very short range in and near the nucleus.

The em force is about 40 some orders of magnitude greater than the gravitational force so there is really no other force than the em force to consider.

Where do you see a conflict between the "beach ball" model and the statistical approach? Both are used in the kinetic theory.
See: http://en.wikipedia.org/wiki/Kinetic_theory

AM

3. Oct 29, 2009

### Redbelly98

Staff Emeritus
The interaction potential between two neutral atoms looks like the following:

For sufficiently high temperatures, the kinetic energy of the atoms is high enough to ignore the little well in the graph. To a good approximation, the potential is zero when the atoms are far apart (i.e. no appreciable force), and behaves as a rigid wall (the nearly vertical part of the curve) at a certain distance apart. This is akin to rigid spheres that will bounce off each other when their centers are separated by 1 ball diameter, but otherwise do not exert a force on each other.

Last edited by a moderator: May 4, 2017
4. Oct 30, 2009

### WiFO215

I was thinking of perhaps some magnetic field being created due to the motion of the charges. Will that not happen?

5. Oct 30, 2009

### WiFO215

This is the potential due to what field? Electromagnetic? Or is there some separate field that I don't know of? I ask this, as it seems like a very funny graph. I've seen this before in textbooks but never understood what field would produce such a type of potential. Does this take into account that there will be magnetic forces on the two interacting molecs?

What happens to this "wall" diagram if the two molecs happen to interact and form some new product? Is there a different diagram for that case?

Last edited by a moderator: May 4, 2017
6. Oct 30, 2009

### Staff: Mentor

Wikipedia has some information on the origin and significance of the shape of the graph:

http://en.wikipedia.org/wiki/Lennard-Jones_potential

It's important to note that this is an empirical formula, i.e. a "guesswork" made up to fit experimental results. It can be justified only in a qualitative way by appealing to the fundamental underlying interactions.

Last edited: Oct 30, 2009
7. Oct 30, 2009

### Redbelly98

Staff Emeritus
It is electromagnetic, as far as I know magnetic forces play an insignificant role. Electrostatic forces and, as the wiki article jtbell linked to mentioned, Pauli exclusion of the electrons in the different atoms, are the dominant mechanisms here.

It's the same diagram, but for this to happen the temperature (i.e. kinetic energy of the atoms) must be low enough so that the system becomes trapped in the small potential well.

8. Oct 30, 2009

### HallsofIvy

But I suspect the only "force" you are to consider here is the force the wall imparts on a molecule bouncing off it. Imagine a molecule of mass m and velocity v (perpendicular to the wall) bouncing off the wall. Since the collision is "perfectly elastic", the molecule now has velocity -v and so momentum -v. The total change in momentum, from mv to -mv is -2mv.

Now, if that molecule is in a box and the distance from that wall to the opposite one is L, the molecule must go a distance 2L to hit the other wall, bounce back and hit the first wall again. A speed v, that takes 2L/v seconds and so the molecule hits the wall v/2L times every second. (-2mv change in momentum per hit)(v/2L hits per second)= -mv2/L "change in momentum per second" and that is force. The wall exerts a force of -mv2/L on the molecule and the molecule exerts a force of mv2/L on the wall.

We can carry this further. With N molecules in random motion, we can treat it as if N/3 were moving along each of the three axes so all of the molecules will be exerting a total force of mNv2/3L on the wall. Since pressure is "force divided by area", if that wall has height and width H and W, its area is HW, and the pressure on that wall is P= mNv2/3LHW= mNv2/3V where "V" here is the Volume of the box. You might recognize that in the form PV= mNv2/3 as the "ideal gas law" PV= NRT where P is pressure, V is volume, N is the number of molecules, R is the "gas constant" and t is the temperature (in Kelvins). That also shows that the temperature is proportional to v2- that is to the internal kinetic energy of the gas.

By the way, in 1905, Albert Einstein wrote four massively important papers. In addition to papers on the special theory of relativity, the general theory of relativity, and the photoelectric effect, his paper analyzing Brownian Motion of small objects as being due to their being struck by molecules was considered by many to be the first clear evidence of the "molecular theory".

And he did that while working as a clerk in a patent office!

9. Oct 30, 2009

Staff Emeritus
The interatomic forces are complicated and practically impossible to derive from first principles. However, the biggest correction to the ideal gas law is not that Redbelly's curve doesn't look exactly like a right angle: vertical at r=1 and horizontal at 0 for r>0. It's that the gas molecules have non-zero volume, and for cold and/or dense gasses, this is important:

$$PV = nRT$$

becomes

$$P \left(V - nb\right) = nRT$$

where b is the volume occupied by the molecules in a mole of gas. For air at STP, it's about a 0.2% effect.

The next level of approximation is to consider that gas molecules weakly attract each other (Redbelly's curve). If I parameterize this attraction by a constant a, with units Nm4, it serves to "suck the gas in", reducing its pressure, so I get:

$$\left(P + \frac{n^2a}{V^2} \right) \left(V - nb\right) = nRT$$

This is called the van der Waals gas equation. Like I said, it works better than the ideal gas equation for low temperatures or high pressures.

10. Oct 30, 2009

### WiFO215

Why do you want to go through so much trouble Halls? Since Redbelly's curve gave us the potential, why not just take it's gradient, and we will land up with the force. Wouldn't that do? Seems like you are computing something in a harder way to me and are making more approximations (all are moving with V when they hit the wall, elastic collision, exactly 1/3 going in each direction (shouldn't this be 1/6 considering that they are moving in positive and negative directions??) etc.)

11. Oct 30, 2009

### WiFO215

Huh? I don't understand the logic you used to derive the volume correction. 'b' is the volume, okayyyy. Soooooo......?

Weakly attract? Weakly attract with what force? Didn't someone just say EM force is all we need to consider? If not, doesn't the Lennard Jones curve take care of that bit?

Last edited: Oct 30, 2009
12. Oct 30, 2009

### WiFO215

Oh boy. So after reading that article, I take it that we don't build the potential diagram using any of the known forces, but try to fit the experiment as best as possible.

Following which, I have a question.

How do they do this? Is there some sort of potential measuring experiment for this set up? If there are really 1023 molecs, what sort of experiment would they use to isolate 2 molecs and study them?

13. Oct 30, 2009

Staff Emeritus
b is the volume per mole, and n is the number of moles.

These forces are ultimately electromagnetic, but the actual calculation is (as I said) very difficult to calculate. For example, two neutral spherical atoms, say neon, will attract via London dispersion forces. Furthermore, the Lennard-Jones potential might be more accurately described as a family of potentials. Each gas has a different curve.

14. Oct 30, 2009

### WiFO215

But sir, when we "made" the equation along the lines of how HallsofIvy did, we took V to be volume of the box and the volume of the molec itself was ignored. From where do we bring it into the derivation? Is there a similar derivation to get to what you are saying?

15. Oct 31, 2009

### WiFO215

Anyone?

16. Oct 31, 2009

### Staff: Mentor

17. Oct 31, 2009

### WiFO215

Thanks a lot jtbell! Now if someone could answer post #12, I'm done (for now).

18. Oct 31, 2009

### Redbelly98

Staff Emeritus
Good question. I don't know definitely, but can hazard a guess that the measured vibrational spectrum on a collection of dimers (2-atom molecules) would yield details of the potential curve's shape, along with the theoretically justified 1/r6 behavior at long range.

I.e., one would measure the transition energies between quantized states within the potential well. This is just an educated guess on my part, maybe somebody else knows more definitively.

19. Oct 31, 2009

Staff Emeritus
I don't know how it's done, but if I wanted to study the left hand side of that potential, I wouldn't be working with gasses - I'd want to be working with liquids. The idea would be to get as much of the material as I could to the left hand side of the plot.

20. Oct 31, 2009

### Redbelly98

Staff Emeritus
Are you sure that would help? The potential is for the interaction between two and only two atoms. Wouldn't having many atoms close together, as in a liquid, hopelessly distort the measured potential?

Anirudh, I did some googling, found this image:

http://images.absoluteastronomy.com/images/encyclopediaimages/m/mo/morse-potential.png [Broken]

The spacing of the energy levels becomes closer as one goes higher in energy, this is what I was talking about two posts ago.

Last edited by a moderator: May 4, 2017