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For the first ionization energy you are removing an electron from a neutral atom, however, for the second ionization energy you are removing an electron from a cation. It is harder to remove an electron from a positively charged atom than a neutral atom because the positive charge holds on to the remaining electrons more tightly.
When a neutral atom loses one electron, it will have a net positive charge. The nuclear charge pulling the electrons in is stronger than in the neutral atom, meaning that the electrons more tightly. It requires more energy to remove electrons that are tightly held. Because more energy is required to move an electron from an atom that has a net positive charge than from a neutral atom, the second ionization energy is greater.
When an atom loses its first electron, the effective nuclear charge has a greater effect on the other electrons. Thus, they are held more tightly to the atom, and they require more energy than the first energy to be removed from the electron
A cation pulls in an electron much more and therefore it requires more energy for it to be removed. After the first ionization, the atom loses a negative charge and therefore it becomes positive making the nucleus stronger requiring more energy for the next removal.
The second ionization energy is greater than the first ionization energy because removing a 2nd electron requires more energy than removing the 1st electron. This is mainly due to the fact that a neutral atom does not have as tight of a grip on the electrons as a cation (which is formed after you remove the first electrons) because losing one electron increases the effect of the nuclear charge on the rest of the electrons. Since the electrons are held more tightly due to the nuclear charge, the second ionization energy goes up.
The second ionization energy is greater than the first ionization energy because there are less electrons in the element. With one less electron, the remaining electron(s) will face a greater attraction/pull from the nucleus of the element. In order to remove the second electron that is experiencing a greater pull, more energy is needed.
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