In Ionization Energy you learned the trends: ionization energy rises across a period and falls down a group. Here is the reason, and it is the same force you met in Attraction and Repulsion: opposite charges attract, more strongly when the charges are bigger and closer.
Key idea
Coulomb's law: the force between two charges is $$F = k\frac{q_1 q_2}{r^2},$$ and the energy it takes to pull them apart grows as $q_1 q_2 / r$. For an electron in an atom: more nuclear charge holds it more tightly, and more distance holds it less tightly. Nearly every periodic trend is one of these two effects winning.
Why it works
Removing an electron means pulling it away from the positive nucleus, so the energy needed tracks the attraction. The force is an inverse-square variation: double the distance and the force drops to a quarter; double one charge and it doubles. Down a group, each element puts its outer electron in a new shell, much farther out, while the extra protons are mostly hidden behind the extra inner electrons (the next topic makes this exact). Distance wins, so ionization energy falls. Across a period, electrons go into the same shell while protons pile up, so charge wins and ionization energy rises.
Worked example: Scaling the force
Two charges attract with force $F$. If one charge is doubled and the distance between them is tripled, what is the new force?
- $F \propto \dfrac{q_1 q_2}{r^2}$. Doubling one charge multiplies $F$ by 2.
- Tripling $r$ multiplies $r^2$ by 9, which divides $F$ by 9.
- New force: $F \times \dfrac{2}{9} = \tfrac{2}{9}F$.
Answer: $\tfrac{2}{9}F$, about 22% of the original.
Worked example: Explaining a trend with charge and distance
Why is the first ionization energy of magnesium (738 kJ/mol) larger than that of sodium (496 kJ/mol)?
- Both outer electrons are in shell 3, at a similar distance from the nucleus.
- Magnesium has 12 protons to sodium's 11, and its extra electron is in the same shell, so it hides little of the extra charge.
- More charge at about the same distance: a stronger pull, so more energy to remove the electron.
Answer: Same shell, more nuclear charge: stronger attraction.
Worked example: The energy of one ion pair
A $\mathrm{Na^+}$ ion and a $\mathrm{Cl^-}$ ion touch, $283\ \text{pm}$ apart. Find their Coulomb energy, $E = kq_1q_2/r$, with $k = 8.99 \times 10^{9}\ \text{N·m}^2/\text{C}^2$ and $e = 1.602 \times 10^{-19}\ \text{C}$.
- $$q_1q_2 = (+e)(-e) = -(1.602 \times 10^{-19})^2 = -2.566 \times 10^{-38}\ \text{C}^2$$
- $$E = \dfrac{(8.99 \times 10^{9})(-2.566 \times 10^{-38})}{283 \times 10^{-12}} = -8.15 \times 10^{-19}\ \text{J}$$
- Negative: the pair has less energy than the separate ions, so pulling them apart takes energy. Per mole of pairs that is about $-491\ \text{kJ/mol}$.
Answer: $-8.15 \times 10^{-19}\ \text{J}$
Common mistake
Explaining trends by what atoms "want": "sodium wants to lose an electron to get an octet." No atom wants anything, and removing sodium's electron costs 496 kJ/mol; it never happens on its own. Explanations should name a charge and a distance: "potassium's outer electron is farther from the nucleus, and the core hides about the same charge, so it is held less tightly." Check: if your explanation would work just as well for the opposite trend, it isn't an explanation.