There are a lot to say about this, but I will skip majority of that to say just two three things:

1) The reaction vessel could be purged by N2. Usually even more times. Or you could work on vacuum-inert line, where you repeatedly exchange atmosphere inside the vessel from air to vacuum, from vacuum to N2, than from N2 to vacuum, than to N2. Usually repeated three times. In a very nice way you could calculate how much of O2 and water remains after three cycles if you go, say from 1000 mbar to 0.1 mbar in each cycle. Repeating this bring much better results than just switching from N2 to Ar. Also check Schlenk techniques. And you can also bubble your samples or work in a freeze-pump-thaw regime to expel all gaseous molecules dissolved in you solvent.

2) For many reaction you do not need super-super-hyper dry conditions. Or you are using substantial excess of water/oxygen sensitive reagent which will finish drying of the reaction media, hence "just putting ballon on a flask" is enough in many cases. (Hint: synthetic organic chemists are well known to do as little of unnecessary manipulations as needed = Laziness is mother of progressivenes)

3) N2 is way cheaper than Ar. We also use Ar, but only in situations where it really pays off (e.g. bubbling of sensitive small samples, like in samples for some spectroscopy or in similar situations).

There are whole books on laboratory techniques, so in case you need some more specialized details consult them (also all good chemical university curricula do have some lecture on this topic, although usually it is optional)

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Nitrogen reactivity is quite low compared to air and oxygen, also cheaper tan argon.

@Yesim Karademir -  So you mean that we flush out all the air by making N2 pass through the vessel and then sealing the vessel? I guess this makes sense. But what I have seen people do in lab, is they just put Nitrogen balloons on the round bottom flask. They don't 'flush out the air'. When i saw the densities of these gases, I was more curious as to how N2 provides inert atmosphere.

@Teodoro S Kaufman - Sir, i wanted to know how Nitrogen is providing atmosphere in the round bottom flask. If in a round bottom flask, i have air and add nitrogen, nitrogen should remain towards the upper part of the RB flask while Oxygen should go down(deduction made on the basis of densities of gases). This was the problem i had in mind. I saw that i had made an error in writing the question. I edited it. :) I hope the problem is clear now.

@Victor H Ponce- Ya i get that point you are making. But my question is different. Its how N2 provides inert atmosphere.

@Yesim Karademir - Hey, It seems that what you said is right. I consulted some organic chemists, and they told me that they flush out air using Nitrogen. You are right. 

Thanks.

Nitrogen is inert but still more reactive then Ar, but N2 is much much cheaper then Ar... We pay one quarter per liter of liquid N2... Argon is about 80 $ per large bottle refill.

There are a lot to say about this, but I will skip majority of that to say just two three things:

1) The reaction vessel could be purged by N2. Usually even more times. Or you could work on vacuum-inert line, where you repeatedly exchange atmosphere inside the vessel from air to vacuum, from vacuum to N2, than from N2 to vacuum, than to N2. Usually repeated three times. In a very nice way you could calculate how much of O2 and water remains after three cycles if you go, say from 1000 mbar to 0.1 mbar in each cycle. Repeating this bring much better results than just switching from N2 to Ar. Also check Schlenk techniques. And you can also bubble your samples or work in a freeze-pump-thaw regime to expel all gaseous molecules dissolved in you solvent.

2) For many reaction you do not need super-super-hyper dry conditions. Or you are using substantial excess of water/oxygen sensitive reagent which will finish drying of the reaction media, hence "just putting ballon on a flask" is enough in many cases. (Hint: synthetic organic chemists are well known to do as little of unnecessary manipulations as needed = Laziness is mother of progressivenes)

3) N2 is way cheaper than Ar. We also use Ar, but only in situations where it really pays off (e.g. bubbling of sensitive small samples, like in samples for some spectroscopy or in similar situations).

There are whole books on laboratory techniques, so in case you need some more specialized details consult them (also all good chemical university curricula do have some lecture on this topic, although usually it is optional)

Thanks a lot Peter Sebej.

Peter seems to have covered it here. It is also worth noting that oxygen and nitrogen are freely diffusible and that gases do not commonly separate by density. Air is a complex mixture and does not separate out into layers of oxygen and nitrogen. most gases diffuse freely down their concentration gradient. when using nitrogen as an inert atmosphere it is usually used under a positive pressure with positive flow. If the flow rate of the pure nitrogen gas is faster than the diffusion rate of oxygen into the system then the system will remain under an inert atmosphere.

If you for example have a schlenk flask filled with air attached to a nitrogen line and open the tap under a significant flow of nitrogen then take off the stopper, nitrogen will flood into the flask mix with the air and flow out the top. any of the components of air in the flask will be diluted by the continuous in flow and outflow of gas as the air and fresh nitrogen mix.