Which out of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning :

A weak electrolyte dissociates only to a small extent in solution. The fraction of the electrolyte that actually dissociates is called the degree of dissociation, denoted by $$\alpha$$.

For any weak monobasic acid or monoacidic base the Ostwald-dilution law gives $$K_a = \frac{\alpha^{2}C}{1-\alpha}$$  $$-(1)$$ where $$K_a$$ = acid (or base) dissociation constant, $$C$$ = initial molar concentration of the electrolyte.

The molar conductivity at the given concentration, $$\Lambda_m$$, is related to the molar conductivity at infinite dilution, $$\Lambda_m^{\circ}$$, through the degree of dissociation: $$\alpha = \frac{\Lambda_m}{\Lambda_m^{\circ}}$$  $$-(2)$$

Insert $$\alpha = \dfrac{\Lambda_m}{\Lambda_m^{\circ}}$$ from $$(2)$$ into $$(1)$$:

$$K_a = \frac{\left(\dfrac{\Lambda_m}{\Lambda_m^{\circ}}\right)^2 C}{1-\dfrac{\Lambda_m}{\Lambda_m^{\circ}}}$$

Simplify the denominator and then multiply numerator & denominator by $$\Lambda_m^{\circ}$$ to clear the fraction:

$$K_a = \frac{C\Lambda_m^{2}}{\Lambda_m^{\circ 2} - \Lambda_m\Lambda_m^{\circ}}$$

Cross-multiply to obtain an equation that contains only whole-number powers:

$$K_a\bigl(\Lambda_m^{\circ 2} - \Lambda_m\Lambda_m^{\circ}\bigr) = C\Lambda_m^{2}$$

Rearrange every term to the left-hand side:

$$\Lambda_m^{2}C - K_a\Lambda_m^{\circ 2} + K_a\Lambda_m\Lambda_m^{\circ} = 0$$

This is the required relation between molar conductivity and concentration for a weak electrolyte.

Comparison with the given options shows that it matches Option B.

Hence, the correct choice is Option B.