In the structure of oxygen molecule, the two oxygen atoms are held by a double bond while in ozone, three oxygen atoms are held by two equivalent bonds of intermediate bond lengths to a central oxygen atom,
In the molecule O2, we have two oxygen atoms connected to each other by a double bond. The bond order in the this molecule is 2 as shown in the image attached to this answer. The bond lengths in this bond correspond exactly to that of an O=O double bond.
In the molecule O3, we can see that there are three oxygen atoms. The central oxygen atom has two other oxygen atoms attached to it. The bond lengths observed in ozone are intermediate between those of a pure double or pure single bond.
The reason behind this is resonance involving the oxygen atoms.
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