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Take an atom of any element. When you remove the one electron from that atom it becomes a positively charged cation. The nuclear charge is now stronger as we lost an electron and now the electrons are more closely attracted towards the nucleus. This stronger nuclear charge makes it harder to remove a second electron. When we have an atom with no net charge the outermost electrons aka the valence electrons are shielded from the nuclear charge by the inner electrons. This makes it so that removing the first electron isn't as hard as the second. The second electron, however, will now be more strongly attracted to the nucleus. Thus, it requires much for energy input to be removed.
The ionization energy to remove the second electron is always higher than removing the first because without 1 extra electron there is less electron repulsion and electrons are more attracted to the positive charge of the nucleus.
Given that there are usually the same amount of electrons (e-) as protons (p+) in an element, if you remove one electron the proton will want more an e- more and thus there more energy is generated due to the pull of p+. However, once you remove two e- but maintain the same amount of p+, now you have protons that want e- even more and thus the energy is higher.
Also, electron repulsion before the first electron was removed helps keep the ionization energy lower. Once the first electron is removed, there will be less electron repulsion, so the electron will be even harder to remove.
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