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The number of valence electrons greatly affects the electron affinity of an atom. An atom is much more likely to gain an electron from another atom when their outer valence shell is almost filled because they are looking to reach the ideal state of having a full outer shell. Because the nuclear charge of ions relates directly to the number of electrons in their valence shell, this also has an affect on the electron affinity of an ion.
Nuclear charge plays a role because higher nuclear charge will have a greater pull on electrons which means higher electron affinity, which is why electron affinity increases going across a row. The almost-full valence shell is another way of thinking about it, because atoms want to have noble gas configurations.
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