To determine in which case BaSO₄ will precipitate, we need to compare the ionic product (IP) of the solutions with the solubility product constant (Ksp) of BaSO₄. The Ksp of BaSO₄ is given as 1.1 × 10⁻¹⁰. A precipitate will form when the ionic product exceeds the Ksp.
### Step-by-Step Solution:
1. **Understand the Concept of Ksp and Ionic Product:**
- Ksp (Solubility Product Constant) is the equilibrium constant for a sparingly soluble salt. For BaSO₄, it is represented as:
\[
K_{sp} = [Ba^{2+}][SO_4^{2-}]
\]
- A precipitate forms when the ionic product (IP) of the ions in solution exceeds Ksp:
\[
IP = [Ba^{2+}][SO_4^{2-}] > K_{sp}
\]
2. **Calculate Ionic Product for Each Case:**
- **Case A:** 100 mL of 6.0 × 10⁻³ M BaCl₂ and 300 mL of 6.0 × 10⁻⁴ M Na₂SO₄.
- Calculate the concentrations after mixing:
- Total volume = 100 mL + 300 mL = 400 mL
- Concentration of Ba²⁺:
\[
[Ba^{2+}] = \frac{(6.0 \times 10^{-3} \, M) \times (100 \, mL)}{400 \, mL} = 1.5 \times 10^{-3} \, M
\]
- Concentration of SO₄²⁻:
\[
[SO_4^{2-}] = \frac{(6.0 \times 10^{-4} \, M) \times (300 \, mL)}{400 \, mL} = 4.5 \times 10^{-4} \, M
\]
- Calculate the ionic product:
\[
IP = [Ba^{2+}][SO_4^{2-}] = (1.5 \times 10^{-3})(4.5 \times 10^{-4}) = 6.75 \times 10^{-7}
\]
- **Case B:** 100 mL of 4.0 × 10⁻⁴ M BaCl₂ and 300 mL of 6.0 × 10⁻⁸ M Na₂SO₄.
- Total volume = 400 mL
- Concentration of Ba²⁺:
\[
[Ba^{2+}] = \frac{(4.0 \times 10^{-4} \, M) \times (100 \, mL)}{400 \, mL} = 1.0 \times 10^{-4} \, M
\]
- Concentration of SO₄²⁻:
\[
[SO_4^{2-}] = \frac{(6.0 \times 10^{-8} \, M) \times (300 \, mL)}{400 \, mL} = 4.5 \times 10^{-8} \, M
\]
- Calculate the ionic product:
\[
IP = (1.0 \times 10^{-4})(4.5 \times 10^{-8}) = 4.5 \times 10^{-12}
\]
- **Case C:** 300 mL of 4.0 × 10⁻³ M BaCl₂ and 100 mL of 6.0 × 10⁻⁴ M Na₂SO₄.
- Total volume = 400 mL
- Concentration of Ba²⁺:
\[
[Ba^{2+}] = \frac{(4.0 \times 10^{-3} \, M) \times (300 \, mL)}{400 \, mL} = 3.0 \times 10^{-3} \, M
\]
- Concentration of SO₄²⁻:
\[
[SO_4^{2-}] = \frac{(6.0 \times 10^{-4} \, M) \times (100 \, mL)}{400 \, mL} = 1.5 \times 10^{-4} \, M
\]
- Calculate the ionic product:
\[
IP = (3.0 \times 10^{-3})(1.5 \times 10^{-4}) = 4.5 \times 10^{-7}
\]
3. **Compare Ionic Products with Ksp:**
- Ksp of BaSO₄ = 1.1 × 10⁻¹⁰
- Case A: IP = 6.75 × 10⁻⁷ (greater than Ksp, precipitate forms)
- Case B: IP = 4.5 × 10⁻¹² (less than Ksp, no precipitate)
- Case C: IP = 4.5 × 10⁻⁷ (greater than Ksp, precipitate forms)
### Conclusion:
BaSO₄ will precipitate in **Case A** and **Case C**.
