What will be the molar conductivity of Al 3+ ions at infinite dilution if molar conductivity of `Al^(2) (SO_(4))_(3)` is 858 S `cm^(2)` `"mol" ^(-1)` and ionic conductance of `SO_(4)^(2-)` is 160 S `cm^(2) "mol" ^(-1)` at infinite dilution ?

Options

A. 189 S `"cm"^2 "mol"^(-1)`
B. 698 S `"cm"^2 "mol"^(-1)`
C. 1018 S `"cm"^2 "mol"^(-1)`
D. 429 S `"cm"^2 "mol"^(-1)`

Explanation

To find the molar conductivity of Al³⁺ ions at infinite dilution, we can use Kohlrausch's law of independent migration of ions. According to this law, the molar conductivity of an electrolyte at infinite dilution is equal to the sum of the molar conductivities of its constituent ions.

### Step-by-Step Solution:

1. **Identify the given values:**
- Molar conductivity of Al₂(SO₄)₃ (aluminum sulfate), \( \Lambda_{m}^0 \) = 858 S cm² mol⁻¹
- Ionic conductivity of SO₄²⁻ (sulfate ion), \( \Lambda_{SO₄}^{0} \) = 160 S cm² mol⁻¹

2. **Write the dissociation equation:**
The dissociation of aluminum sulfate in water can be represented as:
\[ \text{Al}_2(\text{SO}_4)_3 \rightarrow 2 \text{Al}^{3+} + 3 \text{SO}_4^{2-} \]
From this equation, we can see that:
- 2 moles of Al³⁺ ions are produced for every mole of Al₂(SO₄)₃.
- 3 moles of SO₄²⁻ ions are produced for every mole of Al₂(SO₄)₃.

3. **Apply Kohlrausch's law:**
According to Kohlrausch's law:
\[ \Lambda_{m}^0 (\text{Al}_2(\text{SO}_4)_3) = 2 \Lambda_{m}^0 (\text{Al}^{3+}) + 3 \Lambda_{m}^0 (\text{SO}_4^{2-}) \]

4. **Rearranging the equation:**
We can rearrange the equation to solve for the molar conductivity of Al³⁺:
\[ \Lambda_{m}^0 (\text{Al}^{3+}) = \frac{\Lambda_{m}^0 (\text{Al}_2(\text{SO}_4)_3) - 3 \Lambda_{m}^0 (\text{SO}_4^{2-})}{2} \]

5. **Substituting the known values:**
Now, substituting the known values into the equation:
\[ \Lambda_{m}^0 (\text{Al}^{3+}) = \frac{858 - 3 \times 160}{2} \]

6. **Calculating the result:**
\[ \Lambda_{m}^0 (\text{Al}^{3+}) = \frac{858 - 480}{2} = \frac{378}{2} = 189 \text{ S cm}^2 \text{ mol}^{-1} \]

### Final Answer:
The molar conductivity of Al³⁺ ions at infinite dilution is **189 S cm² mol⁻¹**.