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In an oxygen radical (produced by the break down of O3 for instance), there are 2 unpaired electrons in the outermost subshell which make the molecule unstable. Why don't these electrons pair up which would make the atom more stable (I think)?

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Pairing up of electrons in this case is unfavourable due to increased inter-electron repulsion; the average spatial separation of the electrons is higher in the triplet state where the two electrons are unpaired. Regular oxygen molecules also exist in a triplet state and have unpaired valence electrons; they are said to be biradicals.

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