CBSE · Class 12 · Chemistry · ElectrochemistryExplain Kohlrausch's Law of Independent Migration of Ions in detail. Discuss its various applications, particularly how it is used to calculate the limiting molar conductivity of weak electrolytes like acetic acid, with proper mathematical expressions.
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 molar conductivity of the electrolyte irrespective of the nature of the other ion with which it is associated.
Mathematical Formulation\nIf an electrolyte on dissociation gives $\nu_+$ cations and $\nu_-$ anions, then the limiting molar conductivity ($\Lambda^0_m$) is given by:
$$\Lambda^0_m = \nu_+ \lambda^0_+ + \nu_- \lambda^0_-$$\nwhere $\lambda^0_+$ and $\lambda^0_-$ are the limiting molar conductivities of the cation and anion respectively.
Applications of Kohlrausch's Law
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Calculation of Limiting Molar Conductivities of Weak Electrolytes:\nWeak electrolytes such as acetic acid ($CH_3COOH$) do not dissociate completely at any concentration. Therefore, their limiting molar conductivity cannot be obtained by direct extrapolation of experimental conductance data at infinite dilution. Kohlrausch's law makes it possible to determine $\Lambda^0_m$ for weak electrolytes indirectly using strong electrolytes.
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Step-by-Step Calculation for Acetic Acid:\nTo determine the $\Lambda^0_m$ of $CH_3COOH$, we measure the limiting molar conductivities of three strong electrolytes, such as $HCl$, $CH_3COONa$, and $NaCl$:
- $\Lambda^0_{m(HCl)} = \lambda^0_{H^+} + \lambda^0_{Cl^-}$
- $\Lambda^0_{m(CH_3COONa)} = \lambda^0_{CH_3COO^-} + \lambda^0_{Na^+}$
- $\Lambda^0_{m(NaCl)} = \lambda^0_{Na^+} + \lambda^0_{Cl^-}$ \nAdding equation (1) and (2) and subtracting equation (3), we get: $$\Lambda^0_{m(HCl)} + \Lambda^0_{m(CH_3COONa)} - \Lambda^0_{m(NaCl)} = (\lambda^0_{H^+} + \lambda^0_{Cl^-}) + (\lambda^0_{CH_3COO^-} + \lambda^0_{Na^+}) - (\lambda^0_{Na^+} + \lambda^0_{Cl^-})$$ $$\Lambda^0_{m(CH_3COOH)} = \lambda^0_{H^+} + \lambda^0_{CH_3COO^-}$$
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Determination of Degree of Dissociation:\nThe degree of dissociation ($\alpha$) of a weak electrolyte at any concentration $c$ can be calculated using the ratio of molar conductivity at that concentration ($\Lambda_m$) to the limiting molar conductivity ($\Lambda^0_m$): $$\alpha = \frac{\Lambda_m}{\Lambda^0_m}$$
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Determination of Dissociation Constant ($K_a$):\nKnowing the degree of dissociation ($\alpha$) and concentration ($c$), the dissociation constant can be calculated using Ostwald's dilution law formula: $$K_a = \frac{c \alpha^2}{1 - \alpha} = \frac{c (\Lambda_m / \Lambda^0_m)^2}{1 - (\Lambda_m / \Lambda^0_m)} = \frac{c \Lambda_m^2}{\Lambda^0_m (\Lambda^0_m - \Lambda_m)}$$