Question

Why does ionization energy increase going down a group but decreases going across a period?

Answer 1

Hint: Let’s first discuss what ionization energy is. Ionization energy is the minimum amount of energy required to remove the most loosely bound electron of an isolated neutral gaseous atom or molecule. Ionization energy exhibits periodicity on the periodic table. The general trend is for ionization energy to increase moving from left to right across an element period. Moving left to right across a period, atomic radius decreases, so electrons are more attracted to the nucleus.

Complete answer:
Ionization energy increases across a period because the number of protons increases. This means that there is an increase in nuclear charge so there'll be more attraction.
While there is more attraction, one should know that distance from the nucleus and shielding effect remains reasonably constant. This is because all the valence electrons are in the same principal quantum shell.
So the increase in nuclear charge increases attraction and makes the removal of an electron require more energy, while distance from the nucleus and shielding effect remains reasonably constant.
Going down a group, the ionization energy decreases. This is due to the shielding or screen effect of the outer electrons from the nucleus and so the attraction is weaker and they are more easily removed.

Note:
There are however some exceptions across every period where the ionization energy drops between an atom of group $A$ and group $3$ and between group $5$ and group $6$ . The first drop is due to a slight increase in distance from the nucleus as the outer electron occupies a new subshell slightly further away from the nucleus. The second drop is due to spin pair repulsion which is due to the presence of two electrons in the same p orbital. This makes it require less energy to remove.