Many electrons can be added to the empty space around a nucleus without dramatically affecting the atom's size. Larger atoms have more protons which attract their electrons in more closely.
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There are two primary explanations. First, the number of protons in the nucleus increases along with the number of electrons. The outer electrons are more attracted to the more positive nucleus, and are pulled in tighter. (Note: there is some so-called electron screening caused by the core electrons, which diminishes the effect somewhat.) Second, as electrons are added, they can reside in the higher-n shells, which hold more electrons at approximately the same radius.