Whhich one of the following is most soluble ?

Options

A. `Bi_2S_3(K_(sp)=1xx10^(-70))`
B. `Ag_2S(K_(sp)=6 xx 10^(-51))`
C. `CuS(K_(sp)=8xx10^(-37))`
D. `MnS(K_(sp)=7xx10^(-16))`

Answer

To determine which compound is the most soluble among the given options, we will analyze the solubility products (Ksp) of each compound. The higher the solubility, the higher the Ksp value will be.

### Step-by-Step Solution:

1. **Identify the Compounds**: We have four compounds to analyze: Bi2S3, Ag2S, CuS, and MnS.

2. **Write the Dissociation Equations**:
- For **Bi2S3**:
\[ \text{Bi}_2\text{S}_3 (s) \rightleftharpoons 2 \text{Bi}^{3+} (aq) + 3 \text{S}^{2-} (aq) \]
- For **Ag2S**:
\[ \text{Ag}_2\text{S} (s) \rightleftharpoons 2 \text{Ag}^+ (aq) + \text{S}^{2-} (aq) \]
- For **CuS**:
\[ \text{CuS} (s) \rightleftharpoons \text{Cu}^{2+} (aq) + \text{S}^{2-} (aq) \]
- For **MnS**:
\[ \text{MnS} (s) \rightleftharpoons \text{Mn}^{2+} (aq) + \text{S}^{2-} (aq) \]

3. **Write the Ksp Expressions**:
- For **Bi2S3**:
\[ Ksp = [\text{Bi}^{3+}]^2 [\text{S}^{2-}]^3 = (2S)^2 (3S)^3 = 4S^2 \cdot 27S^3 = 108S^5 \]
- For **Ag2S**:
\[ Ksp = [\text{Ag}^+]^2 [\text{S}^{2-}] = (2S)^2 (S) = 4S^2 \cdot S = 4S^3 \]
- For **CuS**:
\[ Ksp = [\text{Cu}^{2+}][\text{S}^{2-}] = S \cdot S = S^2 \]
- For **MnS**:
\[ Ksp = [\text{Mn}^{2+}][\text{S}^{2-}] = S \cdot S = S^2 \]

4. **Substitute Ksp Values**:
- For **Bi2S3**, \( Ksp = 10^{-70} \)
- For **Ag2S**, \( Ksp = 6 \times 10^{-51} \)
- For **CuS**, \( Ksp = 8 \times 10^{-37} \)
- For **MnS**, \( Ksp = 7 \times 10^{-16} \)

5. **Calculate Solubility (S)**:
- For **Bi2S3**:
\[ 108S^5 = 10^{-70} \implies S \approx 10^{-14} \text{ (approx)} \]
- For **Ag2S**:
\[ 4S^3 = 6 \times 10^{-51} \implies S \approx 10^{-17} \text{ (approx)} \]
- For **CuS**:
\[ S^2 = 8 \times 10^{-37} \implies S \approx 10^{-18.5} \text{ (approx)} \]
- For **MnS**:
\[ S^2 = 7 \times 10^{-16} \implies S \approx 10^{-8} \text{ (approx)} \]

6. **Compare Solubility Values**:
- Bi2S3: \( 10^{-14} \)
- Ag2S: \( 10^{-17} \)
- CuS: \( 10^{-18.5} \)
- MnS: \( 10^{-8} \)

7. **Conclusion**: The most soluble compound is **MnS** with a solubility of approximately \( 10^{-8} \).