The `K_(sp)` for `X_(2) SO_(4)` at `25^(@) (X^(+)` is a monovalent ion) is `3.2 xx 10^(-5)` The maximum concentration of `X^(+)` that could be attained in a saturated solution of this solid at `25^@C` is

Options

A. `4xx10^(-2)`M
B. `2.89xx10^(-4)`M
C. `3xx10^(-3)M`
D. `6xx10^(-3)` M

Answer

To solve the problem, we need to determine the maximum concentration of the ion \( X^+ \) in a saturated solution of \( X_2SO_4 \) given its solubility product \( K_{sp} \).

### Step-by-Step Solution:

1. **Understanding the Dissociation of the Salt**:
The salt \( X_2SO_4 \) dissociates in water as follows:
\[ X_2SO_4 (s) \rightleftharpoons 2X^+ (aq) + SO_4^{2-} (aq) \]
From this equation, we can see that for every mole of \( X_2SO_4 \) that dissolves, it produces 2 moles of \( X^+ \) ions and 1 mole of \( SO_4^{2-} \) ions.

2. **Setting Up the Expression for \( K_{sp} \)**:
The solubility product \( K_{sp} \) is given by:
\[ K_{sp} = [X^+]^2 [SO_4^{2-}] \]
If we let the solubility of \( X_2SO_4 \) be \( S \), then:
- The concentration of \( X^+ \) ions will be \( 2S \)
- The concentration of \( SO_4^{2-} \) ions will be \( S \)

Substituting these into the \( K_{sp} \) expression gives:
\[ K_{sp} = (2S)^2 (S) = 4S^2 \cdot S = 4S^3 \]

3. **Substituting the Given \( K_{sp} \)**:
We know that \( K_{sp} = 3.2 \times 10^{-5} \). Therefore, we can set up the equation:
\[ 4S^3 = 3.2 \times 10^{-5} \]

4. **Solving for \( S \)**:
Rearranging the equation gives:
\[ S^3 = \frac{3.2 \times 10^{-5}}{4} \]
\[ S^3 = 0.8 \times 10^{-5} = 8.0 \times 10^{-6} \]

Now, we take the cube root:
\[ S = (8.0 \times 10^{-6})^{1/3} \]

Calculating the cube root:
\[ S = 2.0 \times 10^{-2} \text{ M} \]

5. **Finding the Concentration of \( X^+ \)**:
Since the concentration of \( X^+ \) is \( 2S \):
\[ [X^+] = 2S = 2 \times (2.0 \times 10^{-2}) = 4.0 \times 10^{-2} \text{ M} \]

### Final Answer:
The maximum concentration of \( X^+ \) that could be attained in a saturated solution of \( X_2SO_4 \) at \( 25^\circ C \) is:
\[ \boxed{4.0 \times 10^{-2} \text{ M}} \]