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This question already has an answer here:

The density of nitrogen is greater than the density of oxygen, hence the denser nitrogen would be present in the lower atmosphere. If so it would be deprived of oxygen as it would be pushed to the higher atmosphere due to its low density. But we live due to the intake of oxygen from the lower atmosphere. How is this possible?

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marked as duplicate by Mithoron, Mathew Mahindaratne, Jon Custer, DrMoishe Pippik, Tyberius Jul 25 at 14:42

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Gas molecules in the air continuously collide with each, shoot off in different directions etc. This constant motion is like stirring a fluid. You wouldn't observe perfect density separation in a container with oil and water in it if you keep shaking/stirring it. The situation is similar here, the atmosphere is a fluid that is continuously being agitated. I believe this to be the reason, at least off the top of my head.

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Although you mixed up the densities of $\ce{N2}$ and $\ce{O2}$, $\ce{O2}$ being more dense than $\ce{N2}$ rather than the other way around, the meaning of you question is still clear and valid. To paraphrase: "Why aren't the Earth's gases separated based on density, such that the most dense lie at the bottom of the atmosphere and the less dense gases rise to the top?"

According to this Wikipedia article:

The homosphere and heterosphere are defined by whether the atmospheric gases are well mixed. The surface-based homosphere includes the troposphere, stratosphere, mesosphere, and the lowest part of the thermosphere, where the chemical composition of the atmosphere does not depend on molecular weight because the gases are mixed by turbulence.

A more thorough discussion of the nature of the homosphere and heterosphere with respect to you question can be found in my answer here.

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The density of oxygen is greater than that of nitrogen. At a certain set temperature and pressure, all gases occupy the same volume. Since oxygen has a higher molar mass, it is denser.

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