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As you move across a period, atoms have higher electron affinity because they want to fill their valence shell. For example, Lithium is less likely to gain an electron because they want to "get rid of" the one they already have in their 2s shell versus Fluorine that wants an electron to complete its octet.
The electron affinity trend increases from left to right and upward for groups because atoms towards the right side of the periodic table have more valance electrons, therefore they want to complete their valance shell by accepting electrons rather than losing them. For example, F and Cl, both on the right side of the periodic table, really want to gain an e- to complete their valance shell. That's why are you approach the righthand side of the periodic table, electron affinity increases.
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