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$$\ce{O2(g) ->[h\nu][\nu\,<\,240~nm]2O^{.}(?)}$$

Oxygen is split by UV light below 240 nm to form oxygen radicals, which then go on to react with oxygen molecules to form ozone.

Firstly, is the oxygen a free radical?

Oxygen has six outer shell electrons, but my teacher mentioned that by the way orbitals work, they don't actually all pair off, rather you end up with two pairs of electrons and two unpaired electrons, hence making oxygen a free radical, though I'm not too sure. But because the free radical is so highly reactive, I'm not sure whether I'm even allowed to assign it a state, because straight after being formed it pretty much immediately reacts.

Going by process of elimination, it can't be aqueous, because it isn't dissolved in water, and I don't think it is solid, because solids are structurally rigid. I am therefore inclined to say that it is a gas, but my main problem is that each reaction only creates two oxygen free radicals, which immediately disappear, so it shouldn't actually show any properties of being a gas.

So what is the state I should assign the oxygen, or does it simply not need one?

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According to the reaction $$\ce{O2\,(g) ->[h\nu][\nu\,<\,240~\mathrm{nm}] 2O}$$ photolytic cleavage of oxygen yields two radicals. These will most likely be in the high spin triplet state. And yes these are free radicals according to the definition of the IUPAC Goldbook, as they have two unpaired electrons.

I assume you are talking about state of matter and then of course, you are right. Oxygen radicals can exist neither in aqueous solution, nor liquid, nor solid. Even in gas phase the concentration of oxygen radicals will be very low as they are an incredibly short lived species. They might exist in plasma, too. Keeping that in mind it is probably unnecessary to assign a state to the oxygen radical.

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It does not need one. Short-lived reactive intermediates never form a phase of their own. There are just too few of them at any given moment.

On second thought, however, that lone oxygen atom does exist inside some phase, and that is the gas phase. So it would not be entirely inappropriate to assign it the gas state. With some other reactions, it could have been different. There are examples of reactions between gases that really occur on the solid surface of certain catalyst. In such a case, the intermediates would not fly in the air among the gas molecules, but sit on that surface. (Not that I would then call them "solid", though.)

Now, the radicals with two unpaired electrons use to be called diradicals, and that's what your atomic oxygen is.

BTW, the molecule $\mathrm O_2$, despite being quite stable by itself, is also a diradical, for unrelated reasons.

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