$Be{{(OH)}_{2}}$ ​ is insoluble in water while $Ba{{(OH)}_{2}}$ is highly soluble due to
(A) Bond order
(B) Lattice energy difference
(C) Common ion effect
(D) Hard acid

Hint: The given compounds are the hydroxides of alkaline earth metals. These are so named as the hydroxides and oxides of these metals are alkaline in nature. These metals are found in the earth’s crust. The alkaline earth metals have low ionisation energy, low electron affinity, and low electronegativity. These are good conductors of electricity.

Complete Step by Step Answer:
Both $Be$ and $Ba$are the elements of group 2. As we move down the group, the size of the cation increases. Hence, size increases from $Be$to $Ba$. As the size of the cation increases, there is a decrease in lattice energy. The hydration energy increases as the lattice energy decreases. Hence, the solubility of hydroxides increases from $Be$to$Ba$. $Be{{(OH)}_{2}}$is insoluble in water while $Ba{{(OH)}_{2}}$is highly soluble due to a lattice energy difference.
Hence, the correct answer is Correct option: (B) Lattice energy difference.

Additional Information: Lattice energy is the energy that must be supplied to one mole of an ionic solid to convert it into gaseous ionic constituents. It is also known as lattice enthalpy. Hydration energy, or hydration enthalpy, is the amount of energy released when one mole of solute is dissolved in a solvent. The more the energy required to remove an electron from the outermost shell, the greater the hydration energy.

Note: $Be{{(OH)}_{2}}$ is soluble in acids and bases. $Be{{(OH)}_{2}}$ and $Mg{{(OH)}_{2}}$ are almost insoluble, $Ca{{(OH)}_{2}}$ is sparingly soluble while $Sr{{(OH)}_{2}}$ and $Ba{{(OH)}_{2}}$ are fairly soluble. $Ba{{(OH)}_{2}}$ is a hard acid.