[fsci19]

Why isn't the ionization energy of an electron equal to it's energy level such that:
E(electron)= -13.6(Z^2/n^2) = IP for that electron
But instead it is equal to the energy difference in energy between the atom and its ionized cation:
IP = E(A)-E(A+)

[Irlanur]

Maybe have a look at Koopman's theorem. In Hydrogen the two expressions are exactly the same. In Many-electron systems, u obviously have to think about electron-electron interactions!

[Enthalpy]

13.6*Z²/n² would apply if no other electron "screened" (compensated partially) the nucleus' charge. In usual chemistry, only the "outmost" electrons make bonds or leave an atom (leave is an exaggeration); the remaining "inner" electrons let the outmost see a smaller charge.

Take potassium for instance: 19 protons, 18 deeper electrons, the 19^th electron which is the candidate to leaving is attracted by 19 protons and repelled by 18 electrons so it sees a net charge of 1 proton only.

Except that, for more subtle people like Irlanur, all 19 electrons re-arrange among themselves, so that potassium's 19^th electron is more attracted than if it were on the same 4s shell of a hydrogen atom. Also, all shells overlap a lot, so the 4s is not geometrically around the 1s 2s 2p 3s 3p shells (even for hydrogen) and less than 18 electrons weaken the nucleus' attraction.

This above would be for a single atom in vacuum, which does happen in a sodium or mercury discharge lamp for instance. Chemistry is often done in solvents, which change everything. The electron and the cation are surrounded by polar modecules that orient themselves (or even, make new species) so that every ion sees the solvent molecule's end that carries an opposite charge. Ionization is then much easier. In fact, ionization needing several eV would not happen at room temperature without a solvent.

[Irlanur]

Except that, for more subtle people like Irlanur, all 19 electrons re-arrange among themselves