The equivalent conductance of NaCl at concentration, C at infinite dilution are `Lambda_(c)` and `Lambda_(oo)` respectively. The correct relationship between `Lambda_(c)` and `Lambda_(oo)` is given by (where the constant B is positive)

Options

A. `Lambda_(c) = Lambda_(oo) +(B)C`
B. `Lambda_(c) = Lambda_(oo) - (B)C`
C. `Lambda_(c) = Lambda_(oo)-(B) sqrt(C)`
D. `Lambda_(c) = Lambda_(oo) + (B)sqrt(C)`

Answer

To derive the relationship between the equivalent conductance at concentration \( C \) (denoted as \( \Lambda_c \)) and the equivalent conductance at infinite dilution (denoted as \( \Lambda_{\infty} \)), we can use the Debye-Hückel theory. Here are the steps to arrive at the correct relationship:

### Step-by-Step Solution:

1. **Understanding Equivalent Conductance**:
- Equivalent conductance (\( \Lambda \)) is defined as the conductance of an electrolyte solution divided by the number of equivalents of solute in the solution. It is a measure of the ability of ions to conduct electricity in a solution.

2. **Defining the Conductance at Infinite Dilution**:
- At infinite dilution (\( \Lambda_{\infty} \)), the ions are far apart, and there is minimal interaction between them. This is the maximum conductance that can be achieved by the electrolyte.

3. **Conductance at Concentration \( C \)**:
- As the concentration of the electrolyte increases, the equivalent conductance (\( \Lambda_c \)) decreases due to increased ion interactions and decreased mobility.

4. **Debye-Hückel Equation**:
- According to the Debye-Hückel theory, the relationship between the equivalent conductance at concentration \( C \) and at infinite dilution is given by:
\[ \Lambda_c = \Lambda_{\infty} - B \sqrt{C} \]
- Here, \( B \) is a positive constant that depends on the nature of the electrolyte and the solvent.

5. **Rearranging the Equation**:
- Rearranging the equation gives us:
\[ \Lambda_{\infty} = \Lambda_c + B \sqrt{C} \]
- This shows that the equivalent conductance at infinite dilution is equal to the equivalent conductance at concentration \( C \) plus a term that accounts for the concentration.

6. **Conclusion**:
- The relationship indicates that as the concentration \( C \) increases, the term \( B \sqrt{C} \) becomes significant, causing \( \Lambda_c \) to be less than \( \Lambda_{\infty} \).

### Final Relationship:
Thus, the correct relationship between \( \Lambda_c \) and \( \Lambda_{\infty} \) is:
\[ \Lambda_{\infty} = \Lambda_c + B \sqrt{C} \]