The shape of the box in the kinetic theory of gases

Question

One the main results of the kinetic theory of gases is that the average molecular kinetic energy is proportional to the absolute temperature. This result is usually derived assuming that the gas is contained in a cubic box https://en.wikipedia.org/wiki/Kinetic_theory_of_gases. How can it be shown that this result is independent from the shape of the box?

Answer 1

Some people might derive it that way, but walls and box really have nothing to do with it. Physics is local; if I want to know about the pressure, temperature, etc. of a 1 cm$^3$ bit of air in the center of an airplane hangar, the collisions of molecules against the walls of the hangar many meters away simply has nothing to do with it.

I'm actually not too familiar with derivations that refer to a cubical box when deriving this result; I think I've seen that more when people derive the pressure as a function of number density and average velocity. If you wanted to do that without referring to collisions against the wall of the box, just recognize pressure as a flux of momentum through a surface, define a surface in the middle of the air somewhere, and count how many particles move through it per unit time and their mean velocity normal to the surface (and thus how much momentum they carry per unit time). This momentum per unit time can define the pressure (which is momentum per unit time per unit area).

To get the relationship between mean kinetic energy and temperature, what occurs to me is first deriving the Maxwell-Boltzmann distribution for the speeds as a function of temperature. This is just the distribution that maximizes the entropy of the air molecules for a fixed energy.

The Maxwell-Boltzmann distribution gives you a probability density $P(v)$ for the speed, $v$, of the particles. You can calculate the mean kinetic energy with $\frac12 m \int_0^\infty v^2 P(v)$, and it will be proportional to temperature. This entire argument has nothing to do with any cubical containers.

If all you want to know about is whether, in an argument about collisions with container walls, the shape of the container is important, the answer is that it is not.

Suppose we have some container. Zoom in on a very small portion of area $\mathrm{d}A$ with sides much smaller than the radius of curvature of the box there (though still much larger than the spacing between molecules), so that the patch can be considered flat. Observe the patch for a short time $\mathrm{d}t$. Then the momentum imparted to the portion of wall is the number of collisions times the mean momentum imparted per collision. The number of collisions is just the number of molecules that were close enough to the wall to hit it in that short time, or $\frac12 n \mathrm{d}A \bar{v}_x \mathrm{d}t$ where $n$ is the number density of molecules and $\bar{v}_x$ is the mean velocity in the direction normal to the wall. The fraction $\frac12$ is there because only half of the molecules are headed towards the wall. (Technically we should do an integral over the distribution of molecule speeds, but this turns out not to affect the result; using the average is fine.) Each molecule imparts an average momentum of $2 m v_x$ on the wall, with $m$ the mass of the molecules, and assuming elastic collisions. So the momentum imparted to the wall is $n m \mathrm{d}A \bar{v}_x^2 \mathrm{d}t$, and the pressure is $n m \bar{v}_x^2$, in agreement with the expression on Wikipedia.

The above argument doesn't use the overall shape of the box. It just look at one little piece of the box; the box can have any shape.