I know there are similar questions, but I have some arguments which seem to explain that temperature of an ideal gas in a gravitational field will be lower at higher altitudes. I am assuming that:
1.Molecules of the gas are point sized.
2.They interact only during a collision.
3.All collisions are elastic.
4.Time of collision is negligibly small.
During an elastic collision of 2 equal masses, say A & B, velocities of A and B are just exchanged. If A and B are molecules of the same gas, then they would be indistinguishable in their appearance. Even if they exchanged their velocities, using their indistinguishability and their zero size, one could say that particles passed right through each other because in reality there is no label A or B on molecules. One can't distinguish whether they collided elastically or passed through each other unaffected.
In a lump of ideal gas in a box made up of rigid walls, it is like every molecule is moving freely as if it is alone in the box. So treating each molecule as isolated, I could assert that it will slow down when moving upwards (against the gravitational field) & this applies to all the molecules in that box. Hence, the average kinetic energy at higher altitudes will be lesser than those of the lower altitudes & so the temperature will be lesser at higher altitudes. So am I wrong in concluding this? Or are the assumptions that I made too impractical?