The volume of water needed to dissolve `1mg` of `PbSO_(4) (K_(sp) = 1.44 xx 10^(-8), Mw of PbSO_(4) = 303g)` at `25^(@)C` is

Options

A. `80mL`
B. `43mL`
C. `27.5mL`
D. `10mL`

Explanation

To find the volume of water needed to dissolve 1 mg of PbSO₄, we can follow these steps:

### Step 1: Write the Dissolution Reaction
The dissolution of lead(II) sulfate (PbSO₄) in water can be represented as:
\[ \text{PbSO}_4 (s) \rightleftharpoons \text{Pb}^{2+} (aq) + \text{SO}_4^{2-} (aq) \]

### Step 2: Set Up the Ksp Expression
The solubility product constant (Ksp) expression for the dissolution reaction is:
\[ K_{sp} = [\text{Pb}^{2+}][\text{SO}_4^{2-}] \]

If we let \( s \) be the solubility of PbSO₄ in moles per liter, then:
\[ [\text{Pb}^{2+}] = s \]
\[ [\text{SO}_4^{2-}] = s \]

Thus, the Ksp expression becomes:
\[ K_{sp} = s \cdot s = s^2 \]

### Step 3: Substitute Ksp Value
Given \( K_{sp} = 1.44 \times 10^{-8} \), we can set up the equation:
\[ s^2 = 1.44 \times 10^{-8} \]

### Step 4: Solve for s
To find \( s \), take the square root of both sides:
\[ s = \sqrt{1.44 \times 10^{-8}} \]
\[ s = 1.2 \times 10^{-4} \, \text{mol/L} \]

### Step 5: Convert Moles to Mass
Now, we need to convert the solubility in moles to mass. The molecular weight (Mw) of PbSO₄ is given as 303 g/mol. The mass of PbSO₄ that can dissolve in 1 L of water is:
\[ \text{mass} = s \times \text{Mw} \]
\[ \text{mass} = (1.2 \times 10^{-4} \, \text{mol/L}) \times (303 \, \text{g/mol}) \]
\[ \text{mass} = 0.03636 \, \text{g/L} \]

### Step 6: Convert to mg/L
To convert grams to milligrams:
\[ 0.03636 \, \text{g/L} = 36.36 \, \text{mg/L} \]

### Step 7: Calculate Volume for 1 mg of PbSO₄
Now, we need to find out how much water is needed to dissolve 1 mg of PbSO₄. We know that 36.36 mg of PbSO₄ requires 1 L of water. Therefore, for 1 mg:
\[ \text{Volume} = \frac{1 \, \text{mg}}{36.36 \, \text{mg/L}} \]
\[ \text{Volume} = \frac{1000 \, \text{mL}}{36.36} \]
\[ \text{Volume} \approx 27.5 \, \text{mL} \]

### Final Answer
The volume of water needed to dissolve 1 mg of PbSO₄ is approximately **27.5 mL**.

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