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Think of polarizability as "How easily can this atom lose an electron". If it has high polarizability, it'll be really easy for it to lose an electron and vice versa. Bigger atoms have higher polarizability because the bigger an atom is, the more electrons they, and the more electrons they have, the more shielding the valence electrons experience of all of the neighboring electrons from the nucleus's pull. The smaller the atom, the less shielding and therefore the closer it is to a nucleus (that's why the smaller atoms have low polarizability).
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