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when reading about polar covalent bond I came across the rule called 'Fajan's Rule' but could not find the explanation nor the derivation

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Fajans' rules (note the difference) were formulated in 1923 by Kazimierz Fajans.

They are a method for predicting ionic vs. covalent that predates electronegativity (by three decades) and make use of ionic and atomic radius data that was becoming available through x-ray crystallography.

To use Fajans' Rules, assume your binary compound is ionic and identify the potential cation and anion.

By Fajans' Rules, compounds are more likely to be ionic if: there is a small positive charge on the cation, the cation is large, and the anion is small. For example, $\ce{NaCl}$ is correctly predicted to be ionic since $\ce{Na+}$ is a larger ion with a low charge and $\ce{Cl-}$ is a smaller anion.

Compounds are more likely to be covalent if: there would be a large positive charge on the cation, the cation would be small, and the anion would be large. For example, $\ce{AlI3}$ is correctly predicted to be covalent since it would have a small cation with a high charge and a large anion.

Note that Fajans' Rules have been largely displaced by Pauling's approach using electronegtivites. However, the remnants of Fajans' Rules are found in Hard-Soft Acid-Base Theory, which predicts bonding properties based on polarizability (which is based on size and charge). Binary compounds having a soft acid and/or a soft base are often covalent.

$$\begin{array}{|c|c|c|c|}\hline \mathrm{compound} & \mathrm{Fajans} & \mathrm{Pauling} & \mathrm{HSAB}\\ \hline \ce{NaCl} & \mathrm{low + charge \\ larger\ cation \\ smaller\ anion \\ionic} & \mathrm{3.16-0.93 = 2.19\\ ionic} & \mathrm{hard\ acid \\ borderline\ base \\ ionic} \\ \hline \ce{AlI3} & \mathrm{high + charge\\ smaller\ cation \\ larger anion \\covalent} & \mathrm{2.66 - 1.61 = 1.05 \\covalent} & \mathrm{hard\ acid\\ soft\ base \\covalent} \\ \hline \end{array}$$

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Fajans' rule states that a compound with low positive charge, large cation and small anion has ionic bond where as a compound with high positive charge, small cation and large anion are covalently bonded.

For high charge, small cation will have more polarizing power. Where as larger is the size of anion, more will be the polarization of anion. Because if this electron cloud of anion is more diffused. This makes the anion easily polarizable.

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