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Let's assume there is no air movement in the refrigerator. Is this enough for sinking of nitrogen? If not in what (low) temperatures we will start to see evident distinction between the composition of air in the upper and lower parts?

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No, that's not enough, otherwise there would be industrial processes based on it. You won't have any significant separation before the main components of air start to liquefy.

Meet the barometric formula:

$$\mathfrak p=\mathfrak p_0\cdot exp\left(-{Mgh\over RT}\right)$$

Plug the height of $1\;\rm m$ (which is not unreasonable for a fridge), the oxygen's molar mass of $32\;\rm g/mol$, and the absolute temperature of $253\;\rm K$, because we are in a fridge, after all. You'll get $0.99985$. That's the pressure of oxygen near the top of the fridge, if the pressure near the bottom was $1$.

Now plug the molar mass of nitrogen, the lightest component of air (that's $28\;\rm g/mol$). You'll get $0.99987$.

So does the heavy oxygen sink to the bottom? Er, well... sort of. The gas near the top is enriched with nitrogen by a whopping factor of $1.00002$. Good luck trying to make use of it.

So it goes.

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