I think you should probably turn the question around and ask why would they? It can be difficult to visualise in your mind what is going on with gasses, but if you think about mixing together water and Ribena (for example) it comes as no great surprise that they don’t separate again afterwards. Since oxygen and nitrogen don’t react with each other they are both free to spread out independently in the atmosphere which will naturally lead to them mixing rather than separating (you could also explain this as being an increase in entropy, since the mixture is more disordered than separating the two gases)
As Ed said, it all comes down to entropy and reactivity. Oxygen and Nitrogen in the air are very stable so they won’t react so just mix together, and because of the laws of entropy this mixing will always move towards disorder. The way my science teacher described entropy was the natural tendency for things to move towards chaos.
Ed’s answer is great – basically, in the air, all the atoms are moving really quickly and bouncing off each other – nitrogen and oxygen have a similar weight/density – and so one doesn’t sink to the bottom any faster than the other, so they stay mixed up – just like Ribena.
If you had a much heavier gas, they would separate and the heavier gas would sink to the bottom in the air.
The universe is lazy, when given a choice it would rather stay messy… in other words, for those of you familiar with it, everything tends to higher entropy (disorder).
The air is no different: there is nothing pushing those 2 gases apart. On the contrary, they have similar masses and are stable at air temperatures. So they simply do not bother to use up energy to separate.
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