What is the minimum pH required to prevent the precipitation of ZnS in a solution that is 0.01 M `ZnCl_(2)` and saturated with 0.10 M `H_(2)S` ?
`["Given" :K_(sp)=10^(-21),K_(a_1)xxK_(a_2)=10^(-20)]`

Options

A. 0
B. 1
C. 2
D. 4

Answer

To determine the minimum pH required to prevent the precipitation of ZnS in a solution that is 0.01 M ZnCl₂ and saturated with 0.10 M H₂S, we can follow these steps:

### Step 1: Understand the Dissociation of H₂S
H₂S dissociates in two steps:
1. H₂S ⇌ H⁺ + HS⁻ (with equilibrium constant K₁)
2. HS⁻ ⇌ H⁺ + S²⁻ (with equilibrium constant K₂)

The product of these two dissociation constants is given as:
\[ K_a1 \times K_a2 = 10^{-20} \]

### Step 2: Write the Solubility Product Expression for ZnS
The solubility product (Ksp) for ZnS is given by:
\[ K_{sp} = [Zn^{2+}][S^{2-}] \]
Where:
- [Zn²⁺] is the concentration of zinc ions, which comes from the dissociation of ZnCl₂.
- [S²⁻] is the concentration of sulfide ions.

### Step 3: Calculate the Concentration of Zn²⁺
From the dissociation of ZnCl₂:
\[ ZnCl₂ \rightarrow Zn^{2+} + 2Cl^{-} \]
Since the concentration of ZnCl₂ is 0.01 M, the concentration of Zn²⁺ is also:
\[ [Zn^{2+}] = 0.01 \, M \]

### Step 4: Relate [S²⁻] to Ksp
Using the Ksp value for ZnS:
\[ K_{sp} = [Zn^{2+}][S^{2-}] = 10^{-21} \]
Substituting the known concentration of Zn²⁺:
\[ 10^{-21} = (0.01)[S^{2-}] \]
From this, we can solve for [S²⁻]:
\[ [S^{2-}] = \frac{10^{-21}}{0.01} = 10^{-19} \, M \]

### Step 5: Use the Relationship Between [H⁺], [HS⁻], and [S²⁻]
From the earlier step, we know:
\[ K_a1 \times K_a2 = \frac{[H^+]^2 [S^{2-}]}{[H_2S]} \]
Substituting the known values:
\[ 10^{-20} = \frac{[H^+]^2 (10^{-19})}{0.1} \]

### Step 6: Solve for [H⁺]
Rearranging the equation gives:
\[ [H^+]^2 = 10^{-20} \times 0.1 \times 10^{19} \]
\[ [H^+]^2 = 10^{-21} \]
Taking the square root:
\[ [H^+] = 10^{-10.5} \]

### Step 7: Calculate the pH
The pH is calculated using the formula:
\[ pH = -\log[H^+] \]
Substituting the value:
\[ pH = -\log(10^{-1}) = 1 \]

### Conclusion
The minimum pH required to prevent the precipitation of ZnS is:
\[ \text{Minimum pH} = 1 \]