Amongst the following hydroxides, the one which has the lowest value of `K_(sp)` is:

Options

A. `Mg(OH)_(2)`
B. `Ca(OH)_(2)`
C. `Ba(OH)_(2)`
D. `Be(OH)_(2)`

Explanation

To determine which hydroxide has the lowest value of \( K_{sp} \) among the given options, we can follow these steps:

### Step 1: Identify the Hydroxides
Let's assume the hydroxides mentioned in the question are:
- \( \text{Be(OH)}_2 \) (Beryllium hydroxide)
- \( \text{Mg(OH)}_2 \) (Magnesium hydroxide)
- \( \text{Ca(OH)}_2 \) (Calcium hydroxide)
- \( \text{Ba(OH)}_2 \) (Barium hydroxide)

### Step 2: Understand the Trend in Solubility
The solubility of hydroxides of alkaline earth metals (Group 2) generally increases as we move down the group. This is due to the following reasons:
- The size of the metal ion increases down the group.
- The bond length between the metal ion and hydroxide ion increases, leading to a decrease in bond energy.
- As a result, the solubility product (\( K_{sp} \)) increases down the group.

### Step 3: Compare the Hydroxides
- **Beryllium Hydroxide (\( \text{Be(OH)}_2 \))**: Has the lowest solubility among the group due to the small size of \( \text{Be}^{2+} \) ion, which forms a stronger bond with \( \text{OH}^- \).
- **Magnesium Hydroxide (\( \text{Mg(OH)}_2 \))**: More soluble than beryllium hydroxide but less soluble than calcium and barium hydroxides.
- **Calcium Hydroxide (\( \text{Ca(OH)}_2 \))**: More soluble than magnesium hydroxide.
- **Barium Hydroxide (\( \text{Ba(OH)}_2 \))**: The most soluble among these hydroxides.

### Step 4: Conclusion
Based on the trend in solubility and the reasoning above, we can conclude that:
- **Beryllium hydroxide (\( \text{Be(OH)}_2 \)) has the lowest value of \( K_{sp} \)** among the given hydroxides.

### Final Answer
The hydroxide with the lowest value of \( K_{sp} \) is \( \text{Be(OH)}_2 \).

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