Might be a silly questions but I had a thought while learning about the collision frequency of gas molecules on a single surface of a container using kinetic theory. For a 3D object in the container, can I use the same equations? If I understand correctly a single surface the collision flux of a container surface can be described by the equation:

$$J = \frac{P}{(2\pi mK_bT)^{\frac{1}{2}}}$$

and hence the number of collisions is:

$$N = \frac{P}{(2\pi mK_bT)^{\frac{1}{2}}}*A*\Delta t$$

whereby $P$ is pressure, $m$ is mass of a molecule, $t$ is time, $A$ is surface area and $T$ is temperature.

if there was a 3d object in the container e.g. a ball or statute, my intuition is that given that the equations above are for gas molecules travelling in a single direction. I could still apply the equations above to a 3D object (e.g. a ball) to calculate the number of collisions across all of its surfaces in time $t$ given the total surface area, like I would with a surface of a container wall.

However I wasn't able to find anything online and I was wondering if my assumptions are wrong?


1 Answer 1


Your idea of using the flux and number of particle collisions in more general contexts is a good one. It is actually how I teach the derivation of the ideal gas law in my physical chemistry course! The key insight is that we can take the limit that the time over which we are investigating the collisions and the area over which we are looking for them are entirely arbitrary. If they weren’t, we would get significantly different values for the pressure at all different points on an arbitrarily shaped object or over different time intervals. In the limit that both are very small, the frequency of collisions between particles (controlled mainly by the time interval) and the flexibility of our description (mainly controlled by the area of collision) are both improved. Namely, the area element becoming arbitrarily small means that we can locally treat the surface of any smooth object/surface as if it were just made up of tiny rectangles. You may think of this like a polygon model of a video game character, for instance. This allows us to expand our idea of gas pressure to bodies of arbitrary shape rather than just rectangular walls. The reason we should look at short time intervals is that we don’t want to worry about the mean free path or particle-particle scattering. In the short time limit near our surface we essentially only have the particles hitting the wall to a good approximation. Of course, the pressure is not actually perfectly uniform, but the statistical fluctuations are sufficiently small that it is irrelevant in thermodynamics.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.