STATEMENT-1 : `Lu(OH)_(3)` is more basic than `Ce(OH)_(3)`
and
STATEMENT-2: `Lu^(+3)` has smaller size than `Ce^(+3)`

Options

A. Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True , Statement-2 is True , Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True , Statement-2 is False
D. Statement-1 is False , Statement-2 is True

Answer

To analyze the statements regarding the basicity of \( \text{Lu(OH)}_3 \) and \( \text{Ce(OH)}_3 \), we will break down the reasoning step by step.

### Step 1: Understanding the Compounds
Both \( \text{Lu(OH)}_3 \) and \( \text{Ce(OH)}_3 \) are hydroxides of lanthanide elements, specifically lutetium (Lu) and cerium (Ce). In these compounds, lutetium and cerium exist in the +3 oxidation state as \( \text{Lu}^{3+} \) and \( \text{Ce}^{3+} \).

### Step 2: Comparing Ionic Sizes
Lutetium is located towards the end of the lanthanide series, while cerium is located towards the beginning. As we move from lanthanum (La) to lutetium (Lu), the size of the lanthanide ions decreases due to the increase in nuclear charge without a significant increase in shielding. Thus, \( \text{Lu}^{3+} \) is smaller than \( \text{Ce}^{3+} \).

### Step 3: Basicity of Hydroxides
The basicity of metal hydroxides is influenced by the size of the metal ion. Smaller metal ions tend to form more covalent bonds with hydroxide ions, leading to a higher covalent character in the hydroxide. As the covalent character increases, the basicity decreases.

### Step 4: Analyzing the Statements
- **Statement 1**: \( \text{Lu(OH)}_3 \) is more basic than \( \text{Ce(OH)}_3 \).
- This statement is **incorrect**. Since \( \text{Lu}^{3+} \) is smaller than \( \text{Ce}^{3+} \), \( \text{Lu(OH)}_3 \) will have a higher covalent character and thus lower basicity compared to \( \text{Ce(OH)}_3 \).

- **Statement 2**: \( \text{Lu}^{3+} \) has a smaller size than \( \text{Ce}^{3+} \).
- This statement is **correct**. As established, \( \text{Lu}^{3+} \) is indeed smaller than \( \text{Ce}^{3+} \).

### Conclusion
- **Statement 1** is false.
- **Statement 2** is true.

Thus, the correct answer is that Statement 1 is false and Statement 2 is true.