# Mean free path for gas of thin rods

1. Dec 18, 2009

### hkyriazi

Let's assume we have an ideal gas made up of rod-like (i.e., cylindrical) particles, which have a length (L) 100 times their width or diameter (d).

Clausius's equation for calculating mean free path is based on spherical particles. The ratio of mean free path to particle radius exists in the same proportion as the volume of space to particle volume.

I can easily calculate the rod volume and space volume, based on the given size and numerical density of the rods. But rods can present a big range of "effective radius," from d/2 (for ones moving lengthwise, without any angular momentum), to L/2 for one twirling rapidly, that has its angular momentum and linear momentum in the same (or opposite) direction. For the latter, it'd be like falling slowly down onto a helicopter blade - its effective radius is half the length of the blade.

So, my question is, how complicated would it be to attempt a reasonable estimate of the mean free path of such rods, without actually having to do a large scale, 3-D simulation?

My thinking is that if one is a point particle moving linearly at high speed through such a gas, the average particle one encounters would be oriented at 45 degrees, and appear motionless (assuming our speed is much faster than that of the rods). That'd give an average cross-section of 0.707*L*d.

The particles are twirling, however, with as much energy in their rotation as in their linear motion. That increases their effective cross-section and radius, but I don't know how to calculate how much.

My intuitive guess is that the effective radius would be about L/4 - half the radius for a sphere of diameter L.

Any thoughts?

Last edited: Dec 19, 2009