# Pressure and Shape

*Most recent answer: 05/23/2009*

Q:

well in the derivation of PV = 1/3 mnC^2 we have assumed the box to be cubical. when I tried to generalize the result to any shaped container then there was some error and I got PV = mnC^2. Well my question to you - Is the equation true for any shaped container (logically it should be true) and how to generalize the result..........it is too difficult for me......thanks a lot in advance.(This is not a homework qs but my own curiosity)

- luv (age 17)

india

- luv (age 17)

india

A:

I'm not sure I recognize the formula you use, since it looks like the relativistic case only written as if there were a rest mass. However, that's beside the point for your question, which involves whether the pressure is the same function of the microscopic properties of the gas regardless of the vessel shape. Your intuition that the shape shouldn't matter is entirely correct.

Look at any small patch of the container wall, small enough so we can treat it as flat. The pressure on it depends only on the particles that are very close to it. Their density and momenta determine the average force on the patch. Therefore the shape of the overall vessel can't matter. (Of course this breaks down when the size of the vessel becomes comparable to the typical quantum wavelength of the particles.)

Perhaps in the derivation where the factor of 1/3 was lost you forgot to average over all possible directions in which the nearby particles are moving.

Mike W.

Look at any small patch of the container wall, small enough so we can treat it as flat. The pressure on it depends only on the particles that are very close to it. Their density and momenta determine the average force on the patch. Therefore the shape of the overall vessel can't matter. (Of course this breaks down when the size of the vessel becomes comparable to the typical quantum wavelength of the particles.)

Perhaps in the derivation where the factor of 1/3 was lost you forgot to average over all possible directions in which the nearby particles are moving.

Mike W.

*(published on 05/23/2009)*