Q.How would you explain the following observations?

(i) BeO almost insoluble but BeSO$_4$ soluble

BeO consists of two small, similarly-sized ions (Be$^{2+}$ and O$^{2-}$), which gives it an exceptionally high lattice enthalpy that hydration enthalpy cannot overcome, so BeO is essentially insoluble (this also mirrors why alkaline earth oxide/hydroxide solubility rises down the group — Be being the smallest cation gives the least soluble oxide/hydroxide). BeSO$_4$, however, pairs the very small Be$^{2+}$ with the much larger sulphate ion; when a small cation is paired with a large anion, the lattice enthalpy is comparatively lower (dominated by the large anion, similar to a large-cation/small-anion mismatch) and the very high hydration enthalpy of the small Be$^{2+}$ ion is enough to overcome it, making BeSO$_4$ soluble.

(ii) BaO soluble but BaSO$_4$ insoluble

With the small oxide ion, lattice enthalpy is controlled mainly by the (comparatively small) anion, and as we saw for the hydroxides (Example 4), solubility of the Group 2 oxide/hydroxide series increases down the group, so the large Ba$^{2+}$ pairs favourably enough with the small O$^{2-}$ for BaO to be soluble. In BaSO$_4$, however, the anion itself is large, so lattice enthalpy stays high regardless of cation size, while Ba$^{2+}$'s hydration enthalpy (already low, being the largest Group 2 cation) is too small to compensate — so BaSO$_4$ is insoluble (this is exactly the carbonate/sulphate trend of Example 5, where solubility falls down the group).

(iii) LiI more soluble than KI in ethanol …

For (i)/(ii), apply the lattice-enthalpy-vs-hydration-enthalpy solubility rule separately for each cation/anion size combination (small/small, small/large, large/small, large/large), matching the logic already used in Examples 4 and 5. For (iii), apply Fajans' rules ( …

• Applying a single blanket rule ('smaller ions are always less soluble') to all four salts instead of separately tracking how cation size and anion size interact in each specific pairing.
• For (iii), trying to use the lattice/hydration-enthalpy argument (meant for aqueous solubility) instead of the covalent-character/Fajans'-rules argument, which is what actually explains solubility in an organic solvent like ethanol. …