Why do ionisation energies increase across a period?

It's experimentally observed that ionisation energies have a general increase as one moves across the period: the first ionisation energy of lithium is ~500 kJ mol‑1, whereas that of neon is over 2000 kJ mol‑1. What causes this difference?

We invoke the idea of electronegativity to explain this. This essentially asserts that certain element 'want' electrons more than others. Electronegativity increases as you move across a period and up a group. So Ne is more electronegative than Li, but Li is more electronegative than K. The more electronegative an atom, the higher its ionisation energy.

Why would some atoms 'want' electrons more than others? Well, it depends on the strength of the electrostatic interaction between the nucleus and the (outermost) electron. It turn this depends on the magnitude of the nuclear charge felt by the electron (let's call it the effective nuclear charge) and the distance between them*, as dictated by Coulomb's Law.

So how might the strength of this interaction change as we move across a period? Comprehending this requires an understanding of the concept of shielding. Take hydrogen for example, with one proton and one electron. The electron 'feels' a +1 charge from the nucleus, as there are no other electrons around. However, when we move to helium we have 2 protons and 2 electrons. Electrons repel each other, and this repulsion messes things up.

The mutual repulsion felt by each electron subtracts from the attraction between each electron and the nucleus. So the nuclear charge of helium is +2, but each electron ends up 'feeling' an effective nuclear charge of around +1.4, because the presence of the other electron shields some of the nuclear charge from reaching it. Nonetheless we note that +1.4 is greater than +1, and so predict the ionisation energy of helium is greater than that of hydrogen, which is experimentally observed. Nice!

Since one extra electron can never fully shield the charge from an extra proton, it follows that this effective nuclear charge must increase as you move across a period, which means the strength (or magnitude) of interaction between the outermost electron and the nucleus increases across a period. This means the amount of energy required to remove the electron from its bound state increases across the period, and this is just another way of saying that the ionisation energy increases across a period.

We must note that there is often finer structure to the ionisation energy trends that are due to other effects, such as exchange energy, but the overall trend is always an increase across the period.

*we must note that the distance between nucleus and electron is also dependent on the effective nuclear charge, but also depends on the period that the atom is in. As you move down a group, atomic radius increases due to the increasing prinicipal quantum number, n. For a 1s orbital, n = 1, for 2s and 2p, n = 2, and so on.