Fajan's rules

Picture a lone anion as a perfectly round cloud of negative charge. Bring a cation close, and its positive charge pulls the near side of that cloud toward itself. The cloud stops being a sphere and becomes egg-shaped, with extra electron density bunched in the region between the two ions. This distortion is called polarisation, and it is the seed of covalent bonding: shared electron density between two nuclei is precisely what a covalent bond means. So the real picture is a spectrum, not a switch — "ionic" and "covalent" are two ends of one continuum, and every bond sits somewhere along it depending on how strongly the cation can distort the anion. Two properties decide where a bond lands on that spectrum:

• Polarising power — a property of the cation — how strongly it can distort a neighbouring anion's cloud.
• Polarisability — a property of the anion — how easily its cloud gives in and gets distorted.

Fajans expressed the factors controlling both in three rules.

Rule 1 — small, highly charged cations polarise strongly. A small cation packs its positive charge into a small volume, giving it a high charge density and therefore a strong pull on nearby electrons. $\text{Al}^{3+}$, tiny and triply charged, is a far more powerful polariser than $\text{Na}^+$.

Rule 2 — large, highly charged anions are easily polarised. A large anion holds its outer electrons loosely and diffusely, so they are easier to drag out of their spherical arrangement. $\text{I}^-$, with its bulky electron cloud, is far more polarisable than the tightly held $\text{F}^-$.

Rule 3 — a pseudo-noble-gas cation polarises more than a same-sized, same-charged noble-gas cation. Cations such as $\text{Zn}^{2+}$, $\text{Cd}^{2+}$ and $\text{Ag}^+$ have a filled $(n-1)d^{10}$ shell — an "18-electron," or pseudo-noble-gas, configuration — instead of a simple $8$-electron noble-gas configuration like $\text{K}^+$ or $\text{Ca}^{2+}$. The rough measure connecting charge, size and screening to the pull felt by a neighbouring anion is effective nuclear charge:

where: $Z$ is the nuclear charge and $S$ is the shielding constant contributed by the cation's own inner electrons. A filled $d^{10}$ shell shields the nucleus noticeably less well than a filled $p^6$ octet of the same electron count, so $Z_{\text{eff}}$ — and hence the pull exerted outward on an anion's cloud — stays higher for a pseudo-noble-gas cation than for a similarly sized, similarly charged noble-gas one. Put the three rules together and the winning combination for covalent character is: a small, highly charged, pseudo-noble-gas-configured cation paired with a large, highly charged anion. Even then, the compound is usually still classed as "ionic" overall — Fajans' rules describe a matter of degree, not a hard boundary.

The goal is to explain, from first principles of electrostatics and shielding, why cation size, cation charge, anion size, anion charge and cation electronic configuration all push the ionic-to-covalent balance in the directions Fajans stated, rather than merely reciting the three rules as facts to memorise.

Step 1. Start from Coulomb's law in spirit: the force an ion exerts on a nearby electron cloud grows with the ion's charge and shrinks as the distance between charge centres grows. Packing a given positive charge into a smaller cation volume brings that charge physically closer to the anion's outer electrons, so the pull — and hence the distortion — is stronger. This is why shrinking the cation (Rule 1, first half) increases covalent character, and why raising the cation's charge for a similar size (Rule 1, second half) does the same: more charge, same distance, stronger pull.

Step 2. Now look at the anion side of the same interaction. An anion's outermost electrons are held by its own effective nuclear charge. A large anion has its outer shell farther from its own nucleus, so that shell is held more loosely; a highly charged anion has extra electrons piled on top of the same nuclear charge, so the outer electrons are, on average, held even more loosely still (more electron-electron repulsion, same pull inward). Loosely held electrons deform more easily under an external cation's field — this is exactly Rule 2.

Step 3. Electronic configuration enters through shielding rather than through charge or size directly. Consider two cations of nearly identical radius and identical $+1$ charge: $\text{K}^+$ ($[\text{Ar}]$, an $8$-electron noble-gas core) and $\text{Ag}^+$ ($[\text{Kr}]4d^{10}$, an $18$-electron pseudo-noble-gas core). Both have the same net charge as seen from far away, but the $d^{10}$ electrons in $\text{Ag}^+$ shield the nucleus less efficiently than a simple $s^2p^6$ octet does — $d$-orbitals are more diffuse and penetrate the nucleus less, so they screen an outer test charge less effectively. The result is a higher effective nuclear charge $Z_{\text{eff}}$ felt just outside $\text{Ag}^+$ than just outside $\text{K}^+$, even though their formal charges and sizes are almost the same. A higher $Z_{\text{eff}}$ means a stronger local pull on any neighbouring anion's electron cloud, so $\text{Ag}^+$ polarises more strongly than $\text{K}^+$ despite the near-identical size and charge — this is Rule 3.

Step 4. Combine the three effects. Polarising power on the cation side rises with $Z_{\text{eff}}$ close to the surface and falls with cation radius; polarisability on the anion side rises with anion radius and anion charge. Whichever factor dominates in a given comparison — size, charge, or shielding via configuration — sets how far a bond sits from purely ionic toward covalent, which is why the same three rules can be applied consistently to size-driven comparisons (like the alkali chlorides) and to configuration-driven comparisons (like $\text{Ag}^+$ versus $\text{K}^+$) alike.

Rank the covalent character of LiCl, NaCl and KCl, and then compare AgCl with KCl even though $\text{Ag}^+$ and $\text{K}^+$ have similar radius and identical charge.

Step 1. Down Group 1, ionic radius increases while charge stays fixed at $+1$: $\text{Li}^+$ ($76\ \text{pm}$) $<$ $\text{Na}^+$ ($102\ \text{pm}$) $<$ $\text{K}^+$ ($138\ \text{pm}$). By Rule 1, the smallest cation of a fixed charge has the highest charge density and therefore the strongest polarising power, so $\text{Li}^+$ distorts $\text{Cl}^-$ the most.

Step 2. Greater polarisation means more shared electron density, so covalent character in the chlorides falls in the order $\text{LiCl} > \text{NaCl} > \text{KCl}$. This matches observation: $\text{LiCl}$ dissolves noticeably in organic solvents like ethanol (a hallmark of covalent character), while $\text{NaCl}$ and $\text{KCl}$ are essentially insoluble in such solvents and behave as highly ionic salts.

Step 3. Now compare $\text{Ag}^+$ ($126\ \text{pm}$) with $\text{K}^+$ ($138\ \text{pm}$): the sizes are fairly close and the charges are identical, $+1$ each, so Rule 1 alone predicts only a small difference. The deciding factor is Rule 3: $\text{K}^+$ has the noble-gas configuration $[\text{Ar}]$ ($8$ outer electrons, good shielding), while $\text{Ag}^+$ has the pseudo-noble-gas configuration $[\text{Kr}]4d^{10}$ ($18$ electrons, poor shielding by the diffuse $d$ subshell).

Step 4. Poorer shielding gives $\text{Ag}^+$ a higher effective nuclear charge than $\text{K}^+$ at comparable size, so $\text{Ag}^+$ polarises $\text{Cl}^-$ more strongly. Answer: covalent character follows $\text{LiCl} > \text{NaCl} > \text{KCl}$ among the alkali chlorides, and separately, $\text{AgCl}$ has substantially more covalent character than $\text{KCl}$ despite their similar cation size and identical charge, because $\text{Ag}^+$'s pseudo-noble-gas configuration polarises far more strongly than $\text{K}^+$'s noble-gas configuration.

Competitive stretch

Extend the logic to a mixed comparison: which is more covalent, $\text{ZnCl}_2$ or $\text{CaCl}_2$? Both cations carry $+2$ charge and have broadly similar radii ($\text{Zn}^{2+} \approx 74\ \text{pm}$, $\text{Ca}^{2+} \approx 100\ \text{pm}$ — actually $\text{Zn}^{2+}$ is smaller too, reinforcing the same conclusion), but $\text{Zn}^{2+}$ has the pseudo-noble-gas configuration $[\text{Ar}]3d^{10}$ while $\text{Ca}^{2+}$ has the noble-gas configuration $[\text{Ar}]$. Both Rule 1 (smaller size) and Rule 3 (poorer $d^{10}$ shielding) point the same way, so $\text{ZnCl}_2$ is markedly more covalent than $\text{CaCl}_2$ — consistent with $\text{ZnCl}_2$'s known solubility in organic solvents and lower melting point relative to typical ionic chlorides.

• Applying only the size/charge rules and forgetting the electronic-configuration rule — a $d^{10}$ pseudo-noble-gas cation can out-polarise a same-sized, same-charged noble-gas cation, and this is often the deciding factor, as with $\text{Ag}^+$ versus $\text{K}^+$.
• Applying Fajans' rules only to the cation — anion size and anion charge matter equally; a highly polarisable large anion (like $\text{I}^-$) raises covalent character even paired with an unremarkable cation.
• Treating "more covalent character" as meaning the compound is no longer ionic at all — Fajans' rules describe a sliding scale, not a binary switch.
• Mixing up polarising power (a cation property, how strongly it distorts) with polarisability (an anion property, how easily it gets distorted) — they are complementary, not interchangeable.

Ask the tutor: Want a quick drill ranking five more ionic pairs by expected covalent character using all three Fajans rules?