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The further away an electron is away from the nucleus, the easier it is to remove. Once the electron that is furthest away from the nucleus is removed, the next farthest would be removed. However, this electron is closer to the atom than the first electron, making it harder to remove, and increasing the ionization energy.
When you take away an electron, the electron's charge becomes positive since you have more protons than electrons. This means that the pull on those remaining electrons is higher, it wants to keep its electrons. This make it harder to lose another. This is my understanding, but anyone please correct me if I'm wrong.
Removing the second electron from an atom is more difficult due to the higher ionization energy required to remove it. This is due to the fact that the nuclear charge has a stronger pull on the remaining electrons as there is now one less electron. Thus, with a stronger attraction due to a positive charge and one less electron, the second ionization energy will be higher.
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