CBSE · Class 12 · Chemistry · ElectrochemistryState Kohlrausch's Law of Independent Migration of Ions. Explain its two important applications with suitable examples and mathematical expressions.

Kohlrausch's Law of Independent Migration of Ions

\nKohlrausch's law of independent migration of ions states that the limiting molar conductivity of an electrolyte can be represented as the sum of the individual contributions of the anion and cation of the electrolyte. In other words, at infinite dilution, each ion makes a definite contribution to the total molar conductivity of an electrolyte, regardless of the nature of the other ion with which it is associated. \nMathematically, for an electrolyte like $\text{NaCl}$ which dissociates into $n_+$ cations and $n_-$ anions: $$\Lambda_m^0 = n_+ \lambda_+^0 + n_- \lambda_-^0$$\nwhere $\Lambda_m^0$ is the limiting molar conductivity, and $\lambda_+^0$ and $\lambda_-^0$ are the limiting molar ionic conductivities of the cation and anion respectively.

Applications of Kohlrausch's Law

1. Calculation of Limiting Molar Conductivities of Weak Electrolytes: Weak electrolytes such as acetic acid ($\text{CH}_3\text{COOH}$) do not dissociate completely at any concentration, so their $\Lambda_m^0$ cannot be determined experimentally by extrapolation of $\Lambda_m$ vs $\sqrt{C}$ graph. Kohlrausch's law allows us to calculate this indirectly using strong electrolytes. For example, for acetic acid: $$\Lambda_m^0(\text{CH}_3\text{COOH}) = \lambda^0(\text{CH}_3\text{COO}^-) + \lambda^0(\text{H}^+)$$ By adding and subtracting the appropriate limiting molar conductivities of strong electrolytes like $\text{CH}_3\text{COONa}$, $\text{HCl}$, and $\text{NaCl}$: $$\Lambda_m^0(\text{CH}_3\text{COOH}) = \Lambda_m^0(\text{CH}_3\text{COONa}) + \Lambda_m^0(\text{HCl}) - \Lambda_m^0(\text{NaCl})$$

2. Determination of Degree of Dissociation of Weak Electrolytes: The degree of dissociation ($\alpha$) of a weak electrolyte at a given concentration $C$ can be calculated using the molar conductivity ($\Lambda_m$) at that concentration and the limiting molar conductivity ($\Lambda_m^0$) at infinite dilution: $$\alpha = \frac{\Lambda_m}{\Lambda_m^0}$$ Furthermore, the dissociation constant ($K_a$) of the weak electrolyte can be determined using Ostwald's dilution law: $$K_a = \frac{C \alpha^2}{1 - \alpha} = \frac{C (\Lambda_m / \Lambda_m^0)^2}{1 - (\Lambda_m / \Lambda_m^0)} = \frac{C \Lambda_m^2}{\Lambda_m^0 (\Lambda_m^0 - \Lambda_m)}$$