For which of the following electrolyte the value of `Lambda_(m)` and `Lambda_(eq)` are same?

Options

A. `Na_(2)SO_(4)`
B. `BaCl_(2)`
C. KCl
D. `Al_(2)(SO_(4))_(3)`

Explanation

To determine for which of the given electrolytes the values of molar conductance (Λ_m) and equivalent conductance (Λ_eq) are the same, we need to analyze the relationship between molarity and normality, and how they relate to the charge of the ions in the electrolyte.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Molar conductance (Λ_m) is defined as the conductance of a solution containing one mole of electrolyte in a liter of solution.
- Equivalent conductance (Λ_eq) is defined as the conductance of a solution containing one equivalent of electrolyte in a liter of solution.

2. **Formulas**:
- The formulas for molar conductance and equivalent conductance are:
\[ Λ_m = \frac{kappa \times 1000}{M} \]
\[ Λ_eq = \frac{kappa \times 1000}{N} \]
where \( kappa \) is the specific conductance, \( M \) is the molarity, and \( N \) is the normality.

3. **Condition for Equality**:
- For Λ_m to equal Λ_eq, the condition is:
\[ M = N \]
- Normality (N) is related to molarity (M) by the equation:
\[ N = M \times z \]
where \( z \) is the number of equivalents per mole, which is determined by the charge of the ions.

4. **Analyze Each Electrolyte**:
- **Na2SO4**:
- Sodium (Na) has a charge of +1, and sulfate (SO4) has a charge of -2.
- Total charge (z) = 2 (from SO4).
- Therefore, \( N = M \times 2 \). Thus, \( Λ_m \neq Λ_eq \).

- **BaCl2**:
- Barium (Ba) has a charge of +2, and each chloride (Cl) has a charge of -1 (2 Cl).
- Total charge (z) = 2 (from Ba).
- Therefore, \( N = M \times 2 \). Thus, \( Λ_m \neq Λ_eq \).

- **KCl**:
- Potassium (K) has a charge of +1, and chloride (Cl) has a charge of -1.
- Total charge (z) = 1 (from K).
- Therefore, \( N = M \times 1 \). Thus, \( Λ_m = Λ_eq \).

- **Al2(SO4)3**:
- Aluminum (Al) has a charge of +3 (2 Al gives +6), and sulfate (SO4) has a charge of -2 (3 SO4 gives -6).
- Total charge (z) = 6.
- Therefore, \( N = M \times 6 \). Thus, \( Λ_m \neq Λ_eq \).

5. **Conclusion**:
- The only electrolyte for which the values of Λ_m and Λ_eq are the same is **KCl**.

### Final Answer:
The electrolyte for which the values of Λ_m and Λ_eq are the same is **KCl**.