The correct order of molar conductance at infinite dilution of LiCl, and is — Electrochemistry Chemistry Question

💡 Solution & Explanation

Step 1 - Apply Kohlrausch's Law of Independent Migration of Ions According to Kohlrausch's Law, at infinite dilution, when dissociation of an electrolyte is complete, each ion makes a definite contribution toward the molar conductance of the electrolyte, irrespective of the nature of the other ion with which it is associated. Therefore, the limiting molar conductance ($\Lambda_m^\circ$) of a $1:1$ alkali metal chloride electrolyte ($\ce{MCl}$) is expressed as: $$\Lambda_m^\circ(\ce{MCl}) = \lambda^\circ(\ce{M^+}) + \lambda^\circ(\ce{Cl^-})$$ Since the chloride anion ($\ce{Cl^-}$) is identical and common to all three salts ($\ce{LiCl}$, $\ce{NaCl}$, and $\ce{KCl}$), its individual limiting molar conductance ($\lambda^\circ(\ce{Cl^-})$) is constant across all three solutions. Consequently, the variation in the limiting molar conductance of the electrolytes is entirely governed by the individual limiting ionic conductances (and thus the ionic mobilities) of the alkali metal cations: $$\Lambda_m^\circ \propto \lambda^\circ(\ce{M^+})$$ Step 2 - Analyze Gaseous Ionic Radii and Charge Density The alkali metal cations belong to Group 1 of the periodic table. As we move down the group from lithium to potassium, the number of electron shells increases, which increases the gaseous (unhydrated) ionic radius ($r_{\text{gaseous}}$): $$r_{\text{gaseous}}(\ce{Li^+}) < r_{\text{gaseous}}(\ce{Na^+}) < r_{\text{gaseous}}(\ce{K^+})$$ Since each of these cations carries a net charge of $+1$, the charge density (charge-to-size ratio) of these ions is inversely proportional to their physical volume. Therefore, the smallest cation, $\ce{Li^+}$, possesses the highest charge density, whereas the largest cation, $\ce{K^+}$, has the lowest charge density: $$\text{Charge Density:} \quad \ce{Li^+} > \ce{Na^+} > \ce{K^+}$$ Step 3 - Evaluate Hydration and Hydrated Ionic Radii When these salts are dissolved in water (a polar solvent), the positively charged cations attract the negative dipoles of water molecules. The electrostatic force of attraction is directly proportional to the charge density of the cation. * Because the $\ce{Li^+}$ ion has an extremely high charge density, it exerts a very strong electrostatic pull on water molecules, attracting a large number of them to form a thick, heavy, and extensive **hydration shell**. * Conversely, the $\ce{K^+}$ ion, with its low charge density, interacts much more weakly with water molecules, forming a significantly smaller and lighter hydration shell. As a result, the size of the hydrated cations in aqueous solution ($r_{\text{hydrated}}$) is completely reversed compared to their gaseous sizes: $$r_{\text{hydrated}}(\ce{Li^+(aq)}) > r_{\text{hydrated}}(\ce{Na^+(aq)}) > r_{\text{hydrated}}(\ce{K^+(aq)})$$ Step 4 - Relate Hydrated Radius to Ionic Mobility and Molar Conductance The ionic mobility ($\mu$) of an ion is its drift speed per unit electric field. According to Stokes' law, the viscous drag experienced by a migrating ion is directly proportional to its effective hydrodynamic (hydrated) radius. Therefore, the bulky, heavily hydrated lithium ion ($\ce{Li^+(aq)}$) experiences immense resistance and moves very slowly through the solution, whereas the smaller, less hydrated potassium ion ($\ce{K^+(aq)}$) moves much more rapidly. The order of **ionic mobility** is: $$\mu(\ce{K^+}) > \mu(\ce{Na^+}) > \mu(\ce{Li^+})$$ Since limiting ionic conductance (and thus molar conductance) is directly proportional to ionic mobility ($\lambda^\circ \propto \mu$): $$\lambda^\circ(\ce{K^+}) > \lambda^\circ(\ce{Na^+}) > \lambda^\circ(\ce{Li^+})$$ Adding the constant contribution of the common chloride ion, the overall order of molar conductance at infinite dilution is: $$\mathbf{\Lambda_m^\circ(\ce{KCl}) > \Lambda_m^\circ(\ce{NaCl}) > \Lambda_m^\circ(\ce{LiCl})}$$ Step 5 - Evaluate the Options * **Option (A) is incorrect:** This option represents the gaseous ionic size order, which is the exact opposite of the conductivity order in aqueous solutions because it completely neglects the effects of hydration. * **Option (B) is correct:** As mathematically and conceptually derived, because the hydrated size decreases from lithium to potassium, the ionic mobility and molar conductance increase in the order: $\ce{KCl > NaCl > LiCl}$. * **Option (C) is incorrect:** This represents an incorrect, non-periodic sequence. * **Option (D) is incorrect:** This represents an incorrect, non-periodic sequence. $$\text{Correct Option: } \boxed{\text{B}}$$