MP Board · Class 12 · Chemistry · ElectrochemistryExplain Kohlrausch's Law of Independent Migration of Ions in detail. Discuss its applications with suitable examples, particularly its use in calculating the molar conductivity of weak electrolytes at infinite dilution.

Kohlrausch's Law of Independent Migration of Ions

\nKohlrausch's Law states that the limiting molar conductivity of an electrolyte can be represented as the sum of the individual contributions of the anions and cations of the electrolyte. In other words, at infinite dilution, when dissociation is complete, each ion makes a definite contribution to the total molar conductivity of the electrolyte, irrespective of the nature of the other ion with which it is associated.

Mathematical Expression:\nIf an electrolyte on dissociation gives $\nu_+$ cations and $\nu_-$ anions, the limiting molar conductivity ($\Lambda_m^0$) is given by:

$$\Lambda_m^0 = \nu_+ \lambda_+^0 + \nu_- \lambda_-^0$$\nwhere $\lambda_+^0$ and $\lambda_-^0$ are the limiting molar conductivities of the cation and anion respectively.

Applications of Kohlrausch's Law

1. Calculation of Molar Conductivity of Weak Electrolytes at Infinite Dilution: Weak electrolytes like acetic acid ($CH_3COOH$) do not dissociate completely at any concentration, and their molar conductivity cannot be obtained experimentally by extrapolation to zero concentration. Kohlrausch's law allows us to determine it indirectly using strong electrolytes. For example, the molar conductivity of acetic acid can be calculated using strong electrolytes like $CH_3COONa$, $HCl$, and $NaCl$: $$\Lambda_m^0(CH_3COOH) = \lambda^0(CH_3COO^-) + \lambda^0(H^+)$$ By adding and subtracting appropriate strong electrolyte conductivities: $$\Lambda_m^0(CH_3COOH) = \Lambda_m^0(CH_3COONa) + \Lambda_m^0(HCl) - \Lambda_m^0(NaCl)$$

2. Determination of Degree of Dissociation ($\alpha$): The degree of dissociation of a weak electrolyte at a given concentration can be calculated from the ratio of molar conductivity at that concentration ($\Lambda_m$) to the molar conductivity at infinite dilution ($\Lambda_m^0$): $$\alpha = \frac{\Lambda_m}{\Lambda_m^0}$$

3. Calculation of Dissociation Constant ($K_a$): Using the degree of dissociation ($\alpha$) and concentration ($c$), the dissociation constant of a weak electrolyte can be calculated using Ostwald's dilution law formula: $$K_a = \frac{c \alpha^2}{1 - \alpha}$$

4. Determination of Solubility of Sparingly Soluble Salts: Salts like $AgCl$, $BaSO_4$, etc., dissolve very little in water. Their saturated solutions are considered at infinite dilution. By measuring the conductivity ($k$) of the saturated solution and using the relation: $$\Lambda_m^0 = \frac{k \times 1000}{\text{Solubility}}$$ the solubility of the sparingly soluble salt can be accurately determined.