The molar solubility of `Al(OH)_(3)` in 0.2 M NaOH solution is `x xx 10^(-22)"mol/L"`. Given that, solubility product of `Al(OH)_(3)=2.4xx10^(-24)`. What is numerical value of x?

Explanation

To solve the problem, we need to find the molar solubility of \( \text{Al(OH)}_3 \) in a 0.2 M NaOH solution and express it in the form \( x \times 10^{-22} \, \text{mol/L} \). We are also given the solubility product \( K_{sp} \) of \( \text{Al(OH)}_3 \) as \( 2.4 \times 10^{-24} \).

### Step-by-step Solution:

1. **Write the dissociation equation for \( \text{Al(OH)}_3 \)**:
\[ \text{Al(OH)}_3 (s) \rightleftharpoons \text{Al}^{3+} (aq) + 3 \text{OH}^- (aq) \]

2. **Define the solubility**:
Let the molar solubility of \( \text{Al(OH)}_3 \) be \( S \). Therefore, at equilibrium:
- The concentration of \( \text{Al}^{3+} \) will be \( S \).
- The concentration of \( \text{OH}^- \) will be \( 3S \).

3. **Consider the presence of \( \text{OH}^- \) from NaOH**:
Since we have a 0.2 M NaOH solution, the initial concentration of \( \text{OH}^- \) ions is 0.2 M. Therefore, the total concentration of \( \text{OH}^- \) ions in the solution will be:
\[ [\text{OH}^-] = 3S + 0.2 \]

4. **Apply the solubility product expression**:
The solubility product \( K_{sp} \) for \( \text{Al(OH)}_3 \) is given by:
\[ K_{sp} = [\text{Al}^{3+}][\text{OH}^-]^3 \]
Substituting the concentrations:
\[ K_{sp} = S \cdot (3S + 0.2)^3 \]

5. **Substitute the given \( K_{sp} \)**:
We know \( K_{sp} = 2.4 \times 10^{-24} \):
\[ 2.4 \times 10^{-24} = S \cdot (3S + 0.2)^3 \]

6. **Approximate \( (3S + 0.2) \)**:
Since \( S \) is expected to be very small compared to 0.2, we can approximate:
\[ 3S + 0.2 \approx 0.2 \]
Thus, we can rewrite the equation as:
\[ 2.4 \times 10^{-24} = S \cdot (0.2)^3 \]

7. **Calculate \( (0.2)^3 \)**:
\[ (0.2)^3 = 0.008 \]
Therefore, the equation becomes:
\[ 2.4 \times 10^{-24} = S \cdot 0.008 \]

8. **Solve for \( S \)**:
\[ S = \frac{2.4 \times 10^{-24}}{0.008} = 3 \times 10^{-22} \, \text{mol/L} \]

9. **Identify the value of \( x \)**:
From the expression \( S = x \times 10^{-22} \), we find that \( x = 3 \).

### Final Answer:
The numerical value of \( x \) is \( 3 \).