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When losing an electron, the nucleus's pull on the electrons grows stronger, as with less negative energy the positive charge of the nucleus has a stronger pull on the electrons surrounding it, thus decreasing the size of the atomic radius. Because of this, it will require a greater amount of energy to remove a second electron due to the nucleus's stronger pull on the electrons.
Second ionization energy is higher, because with one less electron there is greater pull from the nucleus on the remaining electron, and less electron-electron repulsion. This greater pull makes it much harder to remove the second electron.
To add on to what the others have said, it also has to do with stability in terms of whether the orbitals or subshells are full or not. For instance the reason that the second ionization energy for Na is so high is because it would have to remove an electron from its inner core, which would be its neon core. The core is already stable, and removing an electron would serve little purpose to maintain stability. However, the first ionization energy for Na is very low because the one electron to be removed is the only electron in the shell, subshell, and orbital, and it is far from the nuclear attraction. That is another fine example of how the second ionization energy level is higher than the first.
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