Why is Ozone so high in the atmosphere?

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Ozone (O3) has a higher molecular weight than regular O2 gas. According to Gay-Lussacs law, O2and O3 should occupy the same volume, and therefore O3 will have a higher density. Higher densities sink, so why does ozone stay so high in the atmosphere?
 
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First off, higher densities don't sink. The atmosphere does not separate into by compounds. Good thing that, or the (non-existant) CO2 layer would kill us.

On occasion, limnically active lakes will release their pent up store of CO2 all at once, and this release does create a toxic layer of dense gas. The danger dissipates as the gas dissolves into the atmosphere as a whole.

So why is there an ozone layer? Why isn't it uniformly distributed throughout the atmosphere? The answer is because ozone molecules don't live very long in the atmosphere. The half life of an ozone molecule is 7 to 20 minutes. Ozone is too reactive and too unstable to last long at all. It doesn't have time to mix. Any portion of the atmosphere that contains anything but a trace of ozone is necessarily close to a source that produces ozone. For ground-level ozone, that source is us. For ozone high in the atmosphere, that source is the sunlight high in ultra violet that hits the upper atmosphere.