Ozone layer is heavier than oxygen and is thermodynamically unstable and there is a dynamic equilibrium in the reaction between formation and dissociation of the ozone as shown:
O2 (g) → O (g) + O (g)
O (g) + O2 (g) → O3 (g)
(reaction is in dynamic equilibrium)
Since it’s thermodynamically unstable so it does not settle down near the earth.
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