Assumptions

1. There are a very large number of molecules (N)
2. Molecules have negligible volume compared to the container
3. The molecules show random motion (ranges of speeds and directions)
4. Newton’s Laws of Motion can be applied to the molecules
5. Collisions are elastic and happen quickly compared to the time between collisions
6. There are no intermolecular forces acting other than when they collide

The Big, Bad Derivation

The molecules move in all directions. Let us start with one molecule of mass m travelling with velocity v_x. It collides with the walls of the container, each wall has a length of L.