Kohlrausch's Law of Independent Migration of Ions

Because a weak electrolyte's $\Lambda_m^{0}$ cannot be found by extrapolating its own concentration-dependent data, Kohlrausch's law of independent migration of ions provides the indirect route needed to obtain it, using data measured entirely from strong electrolytes instead.

The law states that at infinite dilution, where inter-ionic interactions have vanished completely, each ion migrates independently of whatever counter-ion it happens to be paired with, and contributes a fixed value — its own limiting molar ionic conductivity, $\lambda^{0}$ — to the total limiting molar conductivity of any electrolyte containing it. For an electrolyte dissociating into $x$ moles of a cation and $y$ moles of an anion per formula unit, $\Lambda_m^{0} = x\,\lambda^{0}_{+} + y\,\lambda^{0}_{-}$.

Because strong electrolytes are fully dissociated at all measurable concentrations, their $\Lambda_m^{0}$ values can be measured directly, by extrapolation. Kohlrausch's law lets these directly-measured values be recombined algebraically to isolate the $\Lambda_m^{0}$ of a weak electrolyte containing the same ions: two strong electrolytes are chosen whose ions, between them, include both ions of the target weak electrolyte plus one unwanted 'spectator' ion pair; their $\Lambda_m^{0}$ values are added, and the $\Lambda_m^{0}$ of the spectator pair's own strong electrolyte is subtracted, cancelling the spectator contribution exactly. For acetic acid: $\Lambda_m^{0}(\text{CH}_3\text{COOH}) = \Lambda_m^{0}(\text{CH}_3\text{COONa}) + \Lambda_m^{0}(\text{HCl}) - \Lambda_m^{0}(\text{NaCl})$ Using $91.0$, $425.9$ and $126.4\ \text{S cm}^2\ \text{mol}^{-1}$ respectively, this gives $\Lambda_m^{0}(\text{CH}_3\text{COOH}) = 390.5\ \text{S cm}^2\ \text{mol}^{-1}$. …