Which one of the following is most soluble ?

Options

A. `CuS(K_("sp") = 8 xx 10^(-37))`
B. `MnS (K_("sp") = 7 xx 10^(-16))`
C. `Bi_(2)S_(3)(K_("sp") = 1 xx 10^(-70))`
D. `Ag_(2)S(K_("sp") = 6 xx 10^(-51))`

Answer

To determine which of the given salts (CuS, MnS, Bi2S3, Ag2S) is the most soluble, we will calculate the solubility of each salt based on their solubility products (Ksp).

### Step-by-Step Solution:

1. **Identify the dissociation of each salt and write the expression for Ksp**:
- **CuS**: Dissociates into Cu²⁺ and S²⁻.
\[ \text{CuS} \rightleftharpoons \text{Cu}^{2+} + \text{S}^{2-} \]
If the solubility is \( S \), then:
\[ [\text{Cu}^{2+}] = S \quad \text{and} \quad [\text{S}^{2-}] = S \]
Thus, \( K_{sp} = S \cdot S = S^2 \).

- **MnS**: Dissociates into Mn²⁺ and S²⁻.
\[ \text{MnS} \rightleftharpoons \text{Mn}^{2+} + \text{S}^{2-} \]
If the solubility is \( S \), then:
\[ [\text{Mn}^{2+}] = S \quad \text{and} \quad [\text{S}^{2-}] = S \]
Thus, \( K_{sp} = S \cdot S = S^2 \).

- **Bi2S3**: Dissociates into 2 Bi³⁺ and 3 S²⁻.
\[ \text{Bi}_2\text{S}_3 \rightleftharpoons 2 \text{Bi}^{3+} + 3 \text{S}^{2-} \]
If the solubility is \( S \), then:
\[ [\text{Bi}^{3+}] = 2S \quad \text{and} \quad [\text{S}^{2-}] = 3S \]
Thus, \( K_{sp} = (2S)^2 \cdot (3S)^3 = 4S^2 \cdot 27S^3 = 108S^5 \).

- **Ag2S**: Dissociates into 2 Ag⁺ and S²⁻.
\[ \text{Ag}_2\text{S} \rightleftharpoons 2 \text{Ag}^{+} + \text{S}^{2-} \]
If the solubility is \( S \), then:
\[ [\text{Ag}^{+}] = 2S \quad \text{and} \quad [\text{S}^{2-}] = S \]
Thus, \( K_{sp} = (2S)^2 \cdot S = 4S^3 \).

2. **Calculate the solubility for each salt using the given Ksp values**:
- **For CuS**: Given \( K_{sp} = 8 \times 10^{-37} \)
\[ S^2 = 8 \times 10^{-37} \implies S = \sqrt{8 \times 10^{-37}} = 8.9 \times 10^{-19} \, \text{M} \]

- **For MnS**: Given \( K_{sp} = 7 \times 10^{-16} \)
\[ S^2 = 7 \times 10^{-16} \implies S = \sqrt{7 \times 10^{-16}} \approx 2.6 \times 10^{-8} \, \text{M} \]

- **For Bi2S3**: Given \( K_{sp} = 10^{-70} \)
\[ 108S^5 = 10^{-70} \implies S^5 = \frac{10^{-70}}{108} \implies S \approx 3.9 \times 10^{-15} \, \text{M} \]

- **For Ag2S**: Given \( K_{sp} = 10^{-51} \)
\[ 4S^3 = 10^{-51} \implies S^3 = \frac{10^{-51}}{4} \implies S \approx 1.14 \times 10^{-17} \, \text{M} \]

3. **Compare the solubility values**:
- CuS: \( 8.9 \times 10^{-19} \, \text{M} \)
- MnS: \( 2.6 \times 10^{-8} \, \text{M} \)
- Bi2S3: \( 3.9 \times 10^{-15} \, \text{M} \)
- Ag2S: \( 1.14 \times 10^{-17} \, \text{M} \)

4. **Conclusion**:
The salt with the highest solubility is **MnS** with a solubility of \( 2.6 \times 10^{-8} \, \text{M} \).

### Final Answer:
**MnS is the most soluble salt among the given options.**