An acidified of `0.05 M Zn^(2+)` is saturated with 0.1 M `H_2S`. What is the minimum molar concentration (M) or `H^+` required to prevent the precipitation of ZnS ? Use `K_(sp)(ZnS) = 1.25 xx10^(-22)` and overall dissociation constant of `H_2S, K_("NET") =K_1K_2=1xx10^(-21)` .

Explanation

To solve the problem, we need to determine the minimum molar concentration of \( H^+ \) required to prevent the precipitation of \( ZnS \) from a solution that is saturated with \( H_2S \) and contains \( 0.05 M \) \( Zn^{2+} \).

### Step-by-Step Solution:

1. **Identify the Equilibrium Expression for \( ZnS \)**:
The solubility product constant \( K_{sp} \) for \( ZnS \) is given by:
\[ K_{sp} = [Zn^{2+}][S^{2-}] \]
Given \( K_{sp}(ZnS) = 1.25 \times 10^{-22} \).

2. **Determine the Concentration of \( S^{2-} \)**:
Since the solution is saturated with \( H_2S \), we need to find the concentration of \( S^{2-} \) in equilibrium. We can use the \( K_{sp} \) expression:
\[ K_{sp} = [Zn^{2+}][S^{2-}] \]
Rearranging gives:
\[ [S^{2-}] = \frac{K_{sp}}{[Zn^{2+}]} \]
Substituting the known values:
\[ [S^{2-}] = \frac{1.25 \times 10^{-22}}{0.05} = 2.5 \times 10^{-21} \, M \]

3. **Use the Dissociation Constant of \( H_2S \)**:
The overall dissociation constant \( K_{NET} \) for \( H_2S \) is given as:
\[ K_{NET} = K_1 K_2 = 1 \times 10^{-21} \]
The dissociation of \( H_2S \) can be represented as:
\[ H_2S \rightleftharpoons H^+ + HS^- \]
\[ HS^- \rightleftharpoons H^+ + S^{2-} \]
The equilibrium expression for the dissociation of \( H_2S \) is:
\[ K_{NET} = \frac{[H^+][S^{2-}]}{[H_2S]} \]

4. **Substituting Known Values**:
We know \( [S^{2-}] = 2.5 \times 10^{-21} \, M \) and \( [H_2S] = 0.1 \, M \). We can substitute these into the \( K_{NET} \) expression:
\[ 1 \times 10^{-21} = \frac{[H^+](2.5 \times 10^{-21})}{0.1} \]

5. **Solving for \( [H^+] \)**:
Rearranging gives:
\[ [H^+] = \frac{1 \times 10^{-21} \times 0.1}{2.5 \times 10^{-21}} = \frac{1 \times 10^{-22}}{2.5 \times 10^{-21}} = 0.04 \, M \]

6. **Conclusion**:
The minimum molar concentration of \( H^+ \) required to prevent the precipitation of \( ZnS \) is:
\[ [H^+] = 0.04 \, M \]