How to first ionization energies vary within groups and across periods

Ionisation energy increases along a period but decreases down a group

Explanation:

As you go along the period, the amount of protons in the nuclear increases. This causes nuclear attraction to increase meaning the attraction between the protons and the electrons is higher. This causes the electrons to be held more tightly to the nucleus meaning they are harder to remove. As well as that, the shielding stays the same along a period since the electrons are being added to the same shell, just different orbitals. Overall this makes it harder to remove an electron as you go along a period causing the ionisation energy to increase along a period.

On the other hand when you go down a group, the atomic radius increases. This is because the electrons are added to new shells which are further away from the nucleus, this also causes shielding to increase as there are more shells. This means the outer electrons being removed are held less tightly by the protons in the nucleus meaning they are easier to remove. This means that the ionisation energy decrease down a group.