Ideal Gas Law for concave objects

[hkyriazi]

This is much longer than the normal post, so please bear with me.

I was wondering if anyone here knows the complete derivation of Ideal Gas Law (or can recommend a book that describes it clearly and fully), who can tell me whether its conclusions are valid for any shape of gas particle, and if so, why. Please read on, however, to see where I'm coming from.

Specifically, I'm interested in exploring the behavior of a dilute gas of uniquely shaped, infinitely hard particles that experience no forces other than impacts with other such particles, in which case they recoil instantly. They experience no friction, and by "dilute gas," I mean their occupation volume is much less than 1%, such that they normally travel many particle lengths between collisions.

They are long rods with a "reverse toothpick" shape (i.e., the ends are fatter than the middle), and thus have a long and gradually sloping concave shape throughout their midsection (when viewed from the side). Their extreme tips, however, are tapering, like those of a regular toothpick, but also have a concave shape (also viewed from the side). Let's call these particles "batons" (similar to the kind band majorettes twirl), and we'll call their major axis the z-axis. Their thin midsection can be called a "waist", and the broadened ends can be called "hips" and "shoulders." (No jokes, please!)

They also have an essentially infinite axial spin rate in relation to their linear velocities, such that in their collisions, their orientation in space does not change, and they do not precess appreciably.

I'm wondering if a huge collection of randomly oriented such batons might not experience a "self-organizing" effect, such that like-oriented batons tend to aggregate to some extent, and the equilibrium state of such a gas would thus not be one of maximal entropy as normally conceived.

The motivation for this mental exercise is the thought that convexly shaped objects naturally scatter at random angles during collisions, but that concave objects might not. And, spherical objects, even if concave on one side (i.e., if shaped like a bean-bag chair), would still be largely convex. But for extremely linear objects such as I've described, there could be a strong directional component to their mutual viscosity.

To begin with, realize that for these thin batons, their preferred direction of motion is sideways, as they're much more likely to be hit on the side than on the very end, in the z-direction. But, those rare collisions with predominant z-axis approach angles would bring the particles' gentle concave curves into alignment, and this type of collision would consist of multiple, sliding-type contacts which would thrust the particles more into an x-y plane of relative motion, like a ski-chute. The same is true for almost all collisions that have any z-axis component, due to the batons' unique shape. (I should probably post a drawing to illustrate this, but don't have one at the moment.)

For a spherical "flock" of such particles that happens to occur randomly, they might pack together more tightly, owing to their geometry (they can pack closer when lined up), and their mutual collisions would be such as to "lock" them together (i.e., waist of one baton locked with hips and shoulders of colliding, adjacent batons -- acting to prevent z-axis escape), while surrounding batons, having different orientations in space and hence much greater collisional cross-sections, would tend to keep them bunched together. I'm wondering whether space might not then become organized into regions of like-oriented batons, each region perhaps surrounded by orthogonally oriented flocks of batons.

So, what I'd like from folks here is a general discussion of the collisional dynamics in relation to Ideal Gas Law and/or statistical mechanics. Does Ideal Gas Law really tell us that such a gas would not engage, to any degree whatsoever, in such spatial partitioning? Either way, I'd like to know what the thinking is.

Thanks for reading!

 

[russ_watters]

Particles can be "self-organizing", but they still seek the highest entropy. You also don't seem to be considering how they'd spin with off-axis collisions.

 

[hkyriazi]

Particles can be "self-organizing", but they still seek the highest entropy.

I'm not sure what you mean by this. I'm not aware of any (other) type of hard moving objects that, absent attractive forces, will spontaneously cluster together. The situation I'm examining is one where, at equilibrium, rather than being randomized in terms of position, like-oriented batons would tend to be clustered together. That doesn't seem like anyone's definition of the highest entropy state, but perhaps for these unique particles, it is. Can you clarify?

You also don't seem to be considering how they'd spin with off-axis collisions.

Not sure what you mean here, either. For these batons, being "circularly axially symmetric," and experiencing no friction, there is no such thing as an off-axis collision -- every impact is directed directly toward the main axis, so there'd be no way to change their axial spin with single collisions. (It's almost like trying to cause a sphere to spin, that has a frictionless surface. There's no way grab ahold of it or hit it to cause it to spin or alter its spin.)

And, for side collisions near the tips, if we hadn't postulated near infinite axial spin, then certainly those collisions would result in precession, and in the case of zero axial spin, a pure twirling motion. But as it is, those collisions also merely cause the batons to move translationally, just as if one impacted them at the center of mass.

 

[russ_watters]

Maybe I'm not understanding the shape, but if they are dumbell-shaped and the collision is between two ends, perpendicular to the long axis, won't both spin?

 

[hkyriazi]

Maybe I'm not understanding the shape, but if they are dumbell-shaped and the collision is between two ends, perpendicular to the long axis, won't both spin?

Normally, yes, but not if there's a nearly infinite spin rate around the main axis. In that case, such a collision would produce an imperceptible amount of precession. Think about how difficult it is to turn a rapidly spinning gyroscope. If it were floating in space, tapping it on the end would be like tapping it at the center of mass.

 

[JPRitchie]

The ideal gas law is based upon atoms/molecules with no volume - point masses. This leads to one of its major flaws: zero volume is predicted at 0K.
Other equations of state, van der Waals attempt to remedy this flaw and others through the use of finite molecular volumes. But, there are no perfect equations of state, meaning no errors in comparing theory with experiment.
Still, you could write a monte carlo or molecular dynamics program to compute the thermodynamics of your favorite shape/volume molecule. But, I'd suggest a thorough literature search first, as such questions have been looked at. Sorry, I don't have a starting point at hand for you.
-Jim

 

[hkyriazi]

The ideal gas law is based upon atoms/molecules with no volume - point masses. This leads to one of its major flaws: zero volume is predicted at 0K.
Other equations of state, van der Waals attempt to remedy this flaw and others through the use of finite molecular volumes. But, there are no perfect equations of state, meaning no errors in comparing theory with experiment.
Still, you could write a monte carlo or molecular dynamics program to compute the thermodynamics of your favorite shape/volume molecule. But, I'd suggest a thorough literature search first, as such questions have been looked at. Sorry, I don't have a starting point at hand for you.
-Jim

People have looked at dumbbell shapes, but nothing this thin, and nothing with this sort of high axial spin rate. In terms of a molecular dynamics program, is there anything off the shelf that would allow me to specify the molecule shape and spin? It seems very hard, so I doubt it. As for Monte Carlo, I'm not sure how that works. Can one give it some "if then" type instructions on what the results of collisions would be?

I'd like to be able to do some sort of plane geometrical, probabilistic calculation based upon numbers of batons drifting in vs. out of a region of like-oriented batons, looking at a full range of collision angles, figure in the different cross-sections before and after collisions, etc. Any idea where I could find out how properly to set up such a calculation?

 

[JPRitchie]

hdyriazi,
I was interested in this area a few years ago, but haven't looked at it in a while. A search on amazon with "molecular dynamics" turned up at least a dozen books. Any university science/engineering library will likely have several. More specifically, Erpenbeck did some work on equations of state for hard spheres. As for what's available "off the shelf", I couldn't say. AMBER was the name of a code from Kollman's group. It's heavily oriented toward biological applications. There things one can look at, such as the autocorrelation function, to get some idea about whether clusters are forming and what they might look like.
-Jim