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Ionization energy is the energy required to remove an electron from the valence shell. By removing one electron, there are more protons than electrons now, and thus the electrons are more tightly bound to the nucleus. Thus, the second ionization energy will be higher than the first now that it has to overcome the slight positive charge of the atom.
The second ionization energy is higher than the first ionization energy because, in short, it takes more energy to remove an electron from a positively-charged atom (cation) than it does to remove an electron from a neutral atom. When an electron is removed, that means there's more protons than electrons, which means that the effective nuclear charge is technically greater. Due to this higher effective nuclear charge, the electrons become more difficult to remove because the nucleus is holding more tightly onto these remaining electrons, thus making the second ionization energy higher.
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