Read the passage given below and answer the question:
The cell constant is usually determined by measuring the resistance of the cell containing a solution whose conductivity is already known. For this purpose, we generally use KCl solutions whose conductivity is known accurately at various concentrations and at different temperatures. consider the resistance of a conductivity cell filled with 0.1 M KCl solution is 200 Ohm. if the resistance of the same cell when filled with 0.02 M KCl solution is 420 Ohm.
(Conductivity of 0.1 M KCl solution is 1.29 S `m^(-1)`).
Q. What will happen to the conductivity of the cell with the dilution ?

Options

A. First decreases then increases
B. Increases
C. First increases then decreases
D. Decreases

Explanation

To determine what happens to the conductivity of the cell with dilution, we need to analyze the relationship between resistance, conductivity, and concentration.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The conductivity (σ) of a solution is related to its conductance (G) and the cell constant (k) by the formula:
\[ \sigma = G \times k \]
Where conductance (G) is the reciprocal of resistance (R):
\[ G = \frac{1}{R} \]

2. **Calculate Conductance for 0.1 M KCl**:
Given the resistance (R) of the 0.1 M KCl solution is 200 Ohm:
\[ G_{0.1} = \frac{1}{R_{0.1}} = \frac{1}{200} = 0.005 \, S \]

3. **Calculate Conductivity for 0.1 M KCl**:
Using the known conductivity of the 0.1 M KCl solution:
\[ \sigma_{0.1} = 1.29 \, S/m \]

4. **Calculate Cell Constant (k)**:
Rearranging the conductivity formula:
\[ k = \frac{\sigma_{0.1}}{G_{0.1}} = \frac{1.29}{0.005} = 258 \, m^{-1} \]

5. **Calculate Conductance for 0.02 M KCl**:
Given the resistance (R) of the 0.02 M KCl solution is 420 Ohm:
\[ G_{0.02} = \frac{1}{R_{0.02}} = \frac{1}{420} \approx 0.00238 \, S \]

6. **Calculate Conductivity for 0.02 M KCl**:
Using the cell constant calculated earlier:
\[ \sigma_{0.02} = G_{0.02} \times k = 0.00238 \times 258 \approx 0.613 \, S/m \]

7. **Conclusion**:
As the concentration of KCl decreases from 0.1 M to 0.02 M, the conductivity also decreases from 1.29 S/m to approximately 0.613 S/m. Therefore, the conductivity of the cell decreases with dilution.

### Final Answer:
The conductivity of the cell decreases with dilution.