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Why can singlet oxygen participate in cycloaddition reactions while triplet oxygen, on the other hand, can't?

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Normal dioxygen ($\ce{O2}$) exists as a ground state, triplet biradical. This is an example of a molecule that, thermodynamically should be quite reactive, yet is kinetically unreactive - once again, a case of kinetics vs. thermodynamics.

Because spin must be conserved in a reaction, if ground state, triplet $\ce{O2}$ were to react with something, it would have to produce a triplet product. For most compounds, the triplet state is considerably higher in energy than the singlet state. Hence reaction with ground state, triplet $\ce{O2}$ would produce a higher energy, triplet product. Since ground state, triplet dioxygen doesn't have any excess energy, the activation energy required to produce a high energy triplet product just isn't there - hence, no reaction.

On the other hand, singlet $\ce{O2}$ is a spin-paired excited state with plenty of excess energy (the first singlet lies some 95 kJ/mol above the ground state triplet). Basically, singlet $\ce{O2}$ behaves like any other electrophilic (due to the electronegativity of oxygen), reactive (due to its excess energy), compound with a double bond. It readily undergoes the Diels-Alder, ene and even 2+2 cycloaddition reactions - again, just like any other reactive olefin.

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