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If something is highly polarizable, that means that its electrons are more easily distorted. Large anions are larger atoms because extra electrons are added to the atom causing electron-electron repulsion, causing the atom to be bigger. This causes the electrons to be farther away from the positively charged nucleus. The outer electrons are therefore not as tightly held by the atom and are more easily distorted. Therefore, the larger the atom, the more polarizable it is.
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