Why is aluminium's second ionization energy greater than silicon's?

Answer 1

This explanation covers a lot of mid-level chemistry, for those who don't know a lot about it and may need some background.
The electrons in an atom are arranged into orbitals, then further divided into sublevels. The order of sublevels is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d... The cause of the discrepancy in 4s/3d being reversed is due to electrons in 4s having less energy than those in 3d; it is close, though, so they can easily flip back and forth when excited (the science word for 'given lots more energy').
Each type of sublevel holds a fixed maximum amount of electrons: s-levels hold 2, p-levels hold 6, d-levels hold 10 (and f-levels hold fourteen, but that's another unnecessary, unrelated story). The electrons in each sublevel are then "organised" into pairs; one "up" spin, one "down" spin - their spins then cancel (another related, but unnecessary story to that is how iron is magnetic).
Now, silicon has 14 electrons while aluminium has 13. Their arrangements are:(for silicon) 1s2, 2s2, 2p6, 3s2, 3p2 and (for aluminium) 1s2, 2s2, 2p6, 3s2, 3p1. Note that aluminium has only one electron in its 3p sublevel and that silicon has two. Because 3p has a maximum size of six, neither is full.
When talking about ionisation energies (IEs), we refer to the energy needed to remove an electron only. So for the first IE, we remove one electron from each.
The arrangements are now 1s2, 2s2, 2p6, 3s2, 3p1 for silicon and 1s2, 2s2, 2p6, 3s2 for aluminium. Again, note the number in the 3p sublevel: silicon has only one, but aluminium now has zero. Since that sublevel is empty, we go to the next one down, aluminium's 3s.
When we do the second IE, we remove a second electron. This is quite easy to do to silicon, because it has only one electron in its valence (outermost) sublevel and it is unpaired. But because aluminium's valence sublevel is now 3s, which is full and has both its electrons paired, we must break up the pair and break up a full sublevel. This takes more energy than just one electron from 3p; so aluminium's second IE is higher than silicon's.