Problem 16 - A Complete Resource Book in Chemistry for JEE Main

Question

Amongst the following hydroxides, the one which has the lowest value of \(\mathrm{K}_{\mathrm{s}}\) at ordinary temperature (about \(25^{\circ} \mathrm{C}\) ) is (a) \(\mathrm{Mg}(\mathrm{OH})_{2}\) (b) \(\mathrm{Ca}(\mathrm{OH})_{2}\) (c) \(\mathrm{Ba}(\mathrm{OH})_{2}\) (d) \(\mathrm{Be}(\mathrm{OH})_{2}\)

Step-by-Step Solution

Answer

(d) \( \mathrm{Be(OH)}_2 \) has the lowest \( K_s \) at ordinary temperature.

Step 1: Understanding Solubility Product Constant (Ks)

The solubility product constant, denoted as \( K_s \), is a measure of the solubility of a compound. It is the highest value for the compound that is most soluble in water at a given temperature. Conversely, the lower the \( K_s \) value, the less soluble the compound is. We are asked to identify the hydroxide with the lowest \( K_s \) value among the given options.

Step 2: Comparing Solubility of Hydroxides

Among alkaline earth metal hydroxides, \( \mathrm{Be(OH)}_2 \) and \( \mathrm{Mg(OH)}_2 \) are known to be much less soluble in water compared to \( \mathrm{Ca(OH)}_2 \) and \( \mathrm{Ba(OH)}_2 \). Furthermore, due to the small ionic size and high charge density of \( \mathrm{Be}^{2+} \), \( \mathrm{Be(OH)}_2 \) is less soluble than \( \mathrm{Mg(OH)}_2 \). Thus, \( \mathrm{Be(OH)}_2 \) has the lowest solubility and the smallest \( K_s \) value.

Step 3: Conclusion

Given the options, \( \mathrm{Be(OH)}_2 \) has the lowest \( K_s \) value among the hydroxides listed. This is because it is the least soluble due to the smaller size and higher charge density of \( \mathrm{Be}^{2+} \) leading to stronger lattice energy.

Key Concepts

Solubility of Hydroxides

In the context of hydroxides, solubility plays a crucial role in understanding how compounds dissolve in water. Solubility refers to the ability of a substance to dissolve, forming a homogeneous solution at a given temperature and pressure.
This is usually quantified with the Solubility Product Constant, often represented as \( K_s \). The solubility of hydroxides declines as you move from lighter to heavier alkaline earth metals. For instance, \( \mathrm{Be(OH)}_2 \) and \( \mathrm{Mg(OH)}_2 \) are significantly less soluble compared to \( \mathrm{Ca(OH)}_2 \) and \( \mathrm{Ba(OH)}_2 \).

Several reasons can explain this trend:

• Alkaline earth metals have ions with varying ionic sizes and charge densities, impacting solubility.
• Smaller ions with high charge density, like \( \mathrm{Be}^{2+} \), form strong interactions, reducing their solubility.
• Conversely, larger ions with lower charge density tend to dissolve more, increasing solubility.

Thus, the solubility of hydroxides correlates with the size and the charge density of the metal ions.

Alkaline Earth Metals

Alkaline earth metals belong to Group 2 of the periodic table and include beryllium, magnesium, calcium, strontium, barium, and radium. These metals play an integral role in forming several important compounds, including hydroxides.
Each metal has a specific set of properties that affects its hydroxide's solubility and related characteristics.

Key points about alkaline earth metals include:

• Their hydroxides generally increase in solubility as you move down the group. This is due to decreasing lattice energies and increasing ionic sizes.
• These metals have a charge of \( +2 \), leading to the formation of compounds that can have significant interactions between the cations and anions.
• The example of \( \mathrm{Be(OH)}_2 \), which has the lowest solubility (smallest \( K_s \)) among the hydroxides given, primarily due to the small ionic size and high charge density of \( \mathrm{Be}^{2+} \).

As one moves down the group, these trends help predict behaviors observed in hydroxide compounds, such as solubility changes and lattice energies.

Lattice Energy in Hydroxides

Lattice energy is an important concept when examining the solubility of hydroxides. It represents the energy required to separate one mole of solid ionic compound into its gaseous ions. High lattice energy indicates a strong interaction between ions, thereby reducing solubility.
When studying the solubility of hydroxides, it is essential to consider the lattice energy, which typically varies with the ionic size and charge.

Factors impacting lattice energy include:

• Ionic size: Smaller ions will often have higher lattice energies because they can pack more closely, leading to stronger interactions.
• Charge density: Ions with higher charge densities can form stronger ionic bonds, elevating the lattice energy.
• Beryllium hydroxide, \( \mathrm{Be(OH)}_2 \), demonstrates this concept as it has strong lattice energy due to \( \mathrm{Be}^{2+} \)'s small size and high charge density.

Thus, lattice energy impacts a compound's overall stability and dissolution in water.