Which of the following species is more soluble in water ?

Options

A. `M(OH)_(3)`,`(k_(sp)=10^(-35))`
B. `M(OH)_(2)`,`(k_(sp)=10^(-30))`
C. `MOH`,`(k_(sp)=10^(-28))`
D. `MOH`,`(k_(sp)=10^(-26))`

Explanation

To determine which of the given species is more soluble in water, we will analyze the solubility of each compound based on their dissociation and the solubility product constant (Ksp).

### Step-by-Step Solution:

1. **Identify the Compounds**:
We have four compounds to analyze:
- Option 1: \( M(OH)_3 \)
- Option 2: \( M(OH)_2 \)
- Option 3: \( MOH \)
- Option 4: \( M(OH) \)

2. **Dissociation of Each Compound**:
- **For \( M(OH)_3 \)**:
\[ M(OH)_3 \rightarrow M^{3+} + 3OH^{-} \]
Let the solubility be \( S_1 \). The concentration of ions will be:
- \( [M^{3+}] = S_1 \)
- \( [OH^{-}] = 3S_1 \)

- **For \( M(OH)_2 \)**:
\[ M(OH)_2 \rightarrow M^{2+} + 2OH^{-} \]
Let the solubility be \( S_2 \). The concentration of ions will be:
- \( [M^{2+}] = S_2 \)
- \( [OH^{-}] = 2S_2 \)

- **For \( MOH \)**:
\[ MOH \rightarrow M^{+} + OH^{-} \]
Let the solubility be \( S_3 \). The concentration of ions will be:
- \( [M^{+}] = S_3 \)
- \( [OH^{-}] = S_3 \)

- **For \( M(OH) \)**:
\[ M(OH) \rightarrow M^{+} + OH^{-} \]
Let the solubility be \( S_4 \). The concentration of ions will be:
- \( [M^{+}] = S_4 \)
- \( [OH^{-}] = S_4 \)

3. **Write the Ksp Expressions**:
- **For \( M(OH)_3 \)**:
\[ K_{sp} = [M^{3+}][OH^{-}]^3 = S_1(3S_1)^3 = 27S_1^4 \]
Given \( K_{sp} = 10^{-35} \):
\[ 27S_1^4 = 10^{-35} \implies S_1 = \left(\frac{10^{-35}}{27}\right)^{1/4} \approx 7.8 \times 10^{-10} \]

- **For \( M(OH)_2 \)**:
\[ K_{sp} = [M^{2+}][OH^{-}]^2 = S_2(2S_2)^2 = 4S_2^3 \]
Given \( K_{sp} = 10^{-30} \):
\[ 4S_2^3 = 10^{-30} \implies S_2 = \left(\frac{10^{-30}}{4}\right)^{1/3} \approx 1.58 \times 10^{-10} \]

- **For \( MOH \)**:
\[ K_{sp} = [M^{+}][OH^{-}] = S_3S_3 = S_3^2 \]
Given \( K_{sp} = 10^{-28} \):
\[ S_3^2 = 10^{-28} \implies S_3 = 10^{-14} \]

- **For \( M(OH) \)**:
\[ K_{sp} = [M^{+}][OH^{-}] = S_4S_4 = S_4^2 \]
Given \( K_{sp} = 10^{-26} \):
\[ S_4^2 = 10^{-26} \implies S_4 = 10^{-13} \]

4. **Compare Solubility Values**:
- \( S_1 \approx 7.8 \times 10^{-10} \)
- \( S_2 \approx 1.58 \times 10^{-10} \)
- \( S_3 = 10^{-14} \)
- \( S_4 = 10^{-13} \)

From the calculated values, we can see that:
\[ S_1 > S_2 > S_4 > S_3 \]

5. **Conclusion**:
The species that is more soluble in water among the given options is \( M(OH)_3 \) (Option 1).