Equivalent conductivity of `BaCl_2,H_2SO_4` and HCl, are `x_1,x_2"and"x_3Scm^(-1)eq^(-1)` at infinite dilution. If conductivity of saturated `BaSo_4` solution is x`Scm^(-1)`, then `K_(sp)` of `BaSO_4` is :

Options

A. `(500x)/((x_1+x_2-x_3)^2)`
B. `(10^6x^2)/((x_1+x_2-2x_3)^3)`
C. `(2.5xx10^5x^2)/((x_1+x_2-2x_3)^2)`
D. `(0.25x^2)/((x_1+x_2-2x_3)^2)`

Answer

To solve the problem, we need to find the solubility product (Ksp) of BaSO4 using the given equivalent conductivities of BaCl2, H2SO4, and HCl at infinite dilution, as well as the conductivity of a saturated BaSO4 solution.

### Step-by-Step Solution:

1. **Identify the Equivalent Conductivities**:
- Let the equivalent conductivities at infinite dilution be:
- \( \Lambda_{BaCl_2} = x_1 \)
- \( \Lambda_{H_2SO_4} = x_2 \)
- \( \Lambda_{HCl} = x_3 \)

2. **Calculate the Equivalent Conductivity of BaSO4**:
- The equivalent conductivity of BaSO4 can be expressed in terms of the equivalent conductivities of the other electrolytes:
\[ \Lambda_{BaSO_4} = \Lambda_{BaCl_2} + \Lambda_{H_2SO_4} - 2\Lambda_{HCl} \]
- Substituting the values:
\[ \Lambda_{BaSO_4} = x_1 + x_2 - 2x_3 \]

3. **Determine the Molar Conductivity**:
- The molar conductivity (\( \Lambda_m \)) of BaSO4 is related to its equivalent conductivity:
\[ \Lambda_m = n \cdot \Lambda_{BaSO_4} \]
- Here, \( n \) is the number of ions produced per formula unit of BaSO4. Since BaSO4 dissociates into one Ba²⁺ and one SO4²⁻ ion, \( n = 2 \):
\[ \Lambda_m = 2 \cdot (x_1 + x_2 - 2x_3) \]

4. **Relate Molar Conductivity to Conductivity**:
- The relationship between molar conductivity, conductivity (\( \kappa \)), and concentration (\( C \)) is given by:
\[ \Lambda_m = \frac{\kappa \cdot 1000}{C} \]
- Rearranging gives:
\[ C = \frac{\kappa \cdot 1000}{\Lambda_m} \]
- Substituting \( \kappa = x \) (the conductivity of saturated BaSO4):
\[ C = \frac{x \cdot 1000}{2(x_1 + x_2 - 2x_3)} \]

5. **Calculate the Solubility Product (Ksp)**:
- The solubility product \( K_{sp} \) of BaSO4 is given by:
\[ K_{sp} = [Ba^{2+}][SO_4^{2-}] = C^2 \]
- Substituting for \( C \):
\[ K_{sp} = \left(\frac{x \cdot 1000}{2(x_1 + x_2 - 2x_3)}\right)^2 \]
- Simplifying gives:
\[ K_{sp} = \frac{(1000x)^2}{4(x_1 + x_2 - 2x_3)^2} \]
- This can be further simplified to:
\[ K_{sp} = \frac{250000x^2}{(x_1 + x_2 - 2x_3)^2} \]

### Final Expression for Ksp:
\[ K_{sp} = \frac{250000x^2}{(x_1 + x_2 - 2x_3)^2} \]