The molar conductivity of a conductivity cell filled with $$10$$ moles of $$20 \text{ mL NaCl}$$ solution is $$\Lambda_{m1}$$ and that of $$20$$ moles of another identical cell having $$80 \text{ mL NaCl}$$ solution is $$\Lambda_{m2}$$. The conductivities exhibited by these two cells are same. The relationship between $$\Lambda_{m2}$$ and $$\Lambda_{m1}$$ is

We have two identical conductivity cells filled with NaCl solutions: Cell 1 holds 10 moles in 20 mL and Cell 2 contains 20 moles in 80 mL, both showing the same conductivity.

Recalling that molar conductivity is given by

$$\Lambda_m = \frac{\kappa}{c}$$

where $$\kappa$$ is the conductivity and $$c$$ is the molar concentration (in mol/L).

For Cell 1, the concentration is

$$c_1 = \frac{10 \text{ mol}}{0.020 \text{ L}} = 500 \text{ mol/L}$$

In a similar way, for Cell 2 we find

$$c_2 = \frac{20 \text{ mol}}{0.080 \text{ L}} = 250 \text{ mol/L}$$

Since both cells exhibit the same conductivity, we write

$$\kappa_1 = \kappa_2 = \kappa$$

Hence, the molar conductivity of Cell 1 is

$$\Lambda_{m1} = \frac{\kappa}{c_1} = \frac{\kappa}{500}$$

and that of Cell 2 is

$$\Lambda_{m2} = \frac{\kappa}{c_2} = \frac{\kappa}{250}$$

Taking their ratio yields

$$\frac{\Lambda_{m2}}{\Lambda_{m1}} = \frac{\kappa/250}{\kappa/500} = \frac{500}{250} = 2$$

It follows that

$$\therefore \Lambda_{m2} = 2\Lambda_{m1}$$

The answer is Option A.