Foundations — Fajan's rules — covalent character in ionic compounds

Before we can reason about when pulling is strong, we must own every word and symbol the parent note throws at you. We build them one at a time, each from a picture, each earned before use.

1. Charge — the thing that pulls and is pulled

The picture: think of two dots. Two $+$ dots or two $-$ dots flee from each other. A $+$ dot and a $-$ dot rush together. That "rush together" is the entire engine of Fajan's rules.

• A cation is an atom that has lost electrons, so it is net positive: written like ${\text{Na}}^{+}$ (lost 1), ${\text{Al}}^{3+}$ (lost 3).
• An anion is an atom that has gained electrons, so it is net negative: written like ${\text{Cl}}^{-}$ (gained 1), ${\text{S}}^{2-}$ (gained 2).

The little raised number is the size of the charge. We give it the symbol $q$.

Why the topic needs it: Rule 2 says "high cation charge → more covalent." That rule is literally a statement about $q$. No $q$, no Rule 2.

2. Radius $r$ — how big the ion is

The picture: a ball. The radius is the arrow from the middle to the surface. A marble has a small $r$; a beach ball has a large $r$.

We measure $r$ in picometres (pm): $1\text{pm}=10^{-12}\text{m}$, a trillionth of a metre. From the parent note: ${\text{Li}}^{+}\approx 76\text{pm}$, ${\text{K}}^{+}\approx 138\text{pm}$.

Why the topic needs it: Rule 1 ("small cation → more covalent") and Rule 3 ("large anion → more covalent") are both statements about $r$. We will see the field depends on $r$ — and dramatically so.

3. Electric field $E$ — how hard a charge pulls at a distance

Here is the first real tool. Charge pulls — but how hard, and how does that change with distance? We need a number that answers "if I stood here, how strong is the pull from that charge over there?"

Why this tool and not just "charge"? Because charge alone doesn't tell you the pull at the anion's location. The anion sits some distance away, and pull weakens with distance. The field $E$ bakes in both the charge and the distance into one number. That is exactly the question Fajan asks: how hard does the cation tug the anion's cloud?

See Coulomb's law and electric fields for the full build of this law.

The picture: imagine ripples of pull spreading out from the charge like light from a bulb. Close to the bulb it's blinding; far away it's faint. Because the ripples spread over a sphere whose surface grows as $r^2$, the intensity drops as $1/r^2$ — the inverse-square law.

Reading the proportionality signs:

• $E\propto q$ means "$E$ grows in step with $q$" — double $q$, double $E$.
• $E\propto 1/r^2$ means "$E$ shrinks as $r^2$ grows" — double $r$, quarter the $E$.

4. The electron cloud — the squishy thing being pulled

The picture: the water balloon from the parent note. The nucleus is the knot; the water is the electron fog. Squeeze one side and it bulges out the other. A big, floppy balloon (large anion) distorts easily; a small, tight balloon (small anion) barely budges.

See Covalent bonding — electron sharing for how a shared cloud between two nuclei IS a covalent bond.

5. Polarisation, polarising power, polarisability

Now the three words join hands.

The picture: a strong magnet (high polarising power) next to a soft balloon (high polarisability) → big bulge (lots of polarisation).

Why the topic needs it: covalent character is the electron density in the bulge between the nuclei. More bulge = more covalent. Every one of Fajan's four rules is just "what makes the bulge bigger?" See Electronegativity and bond polarity for the complementary view of unequal sharing.

6. Effective nuclear charge $Z_{\text{eff}}$ — the hidden pull (Rule 4)

Rule 4 is about $d$-electrons that "shield poorly." To get it, we need one more idea.

The picture: a bright lamp (nucleus) behind a few layers of frosted glass (inner electrons). From outside you see a dimmer lamp than the true bulb. Well-packed spherical inner electrons frost the glass thoroughly; diffuse, lumpy $d$-electrons frost it badly, letting more of the nucleus's pull leak through.

So two cations of the same charge and size can pull differently: the one whose inner shells shield poorly (like ${\text{Cu}}^{+}$ with its $3d^{10}$ core) has a higher $Z_{\text{eff}}$, hence stronger real pull, hence more covalent character. That's Rule 4. See Effective nuclear charge and shielding.

7. Ionic potential $\varphi$ — packing charge and size into one number

Why this combination? Fajan's field really goes as $q/(r_{\text{cation}}+r_{\text{anion}}{)}^2$. The ionic potential $\varphi =q/r$ is a rough proxy that captures "high charge, small cation" in one glance — but it drops the anion's radius and uses $1/r$ instead of $1/r^2$. So treat $\varphi$ as a ranking hint, never a law. (The parent's third "mistake" is exactly this trap.)

Prerequisite map

Every arrow says "you need this before that." Charge and radius feed the field; the field plus a poorly-shielding core give polarising power; a soft cloud gives polarisability; together they make the bulge, which is covalent character — the heart of Fajan's rules.

Equipment checklist

Test yourself — cover the right side and answer before revealing.