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When a compound gains an electron, why are they more likely to give them up (i.e. become stronger reducing agents)?

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Simple electrostatic repulsion can be used to explain this. For example, if you have $\ce{O^2-}$, there are 8 protons in the nucleus, but 10 electrons, so there is a net negative charge of 2. Like charges repel, pushing the electrons away.

BTW, compounds rarely "gain an electron", the reactive species are usually radicals.

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