Modeling of an ideal gas from a Classical Mechanics perspective

Question

Say we have a box with a (large enough) side $L$ in which there is are $N$ indistinguishable particles, each having a speed $\vec{v}_i$. Let us also say these particles don't interact with eachother in any way but for elastic collisions (this, of course, also applies for when a particle bumps against a wall) and all of them move freely in space (there is no gravity or any other force acting on the system). My question is, would this (over)simplified model behave as an ideal gas? I'd say it wouldn't, because I haven't imposed any conditions on the speeds of the particles (i.e: they should follow a Maxwell-Boltzmann distribution). On the other hand, if the system I have described does not behave as an ideal gas, I think it's safe to assume it would behave as something quite similar.

If my hypothesis is not correct, then what would this system behave like? Would the particles' speeds follow some sort of distribution? Having stated all the conditions my system should have, how could I model it? My goal is to understand the transition from classical mechanics to classical thermodynamics modelling the simplest thermodynamical system I know, which is the ideal gas. Thanks in advance!

Answer 1

It will behave like a gas, but not necessarily an ideal gas. You obtain the equation of state of an ideal gas if the interaction is between point particles, and the potential between these particles is a delta dirac, but you obtain different equations of states for other elastic interactions. For instance, if you assume molecules are billiard balls of volume $b$, the equation of state needs to be modified to $P(V-bN)=NkT$.

Answer 2

The situation you describe is covered by the kinetic theory of gases

The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas.