Kohlrausch's Law

The limiting molar conductance $\Lambda_m^{o}$ — the value molar conductance approaches at infinite dilution — is the foundation of Kohlrausch's law of independent migration of ions. The law states that, at infinite dilution, where ion-ion interactions have vanished entirely, the limiting molar conductivity of an electrolyte is simply the sum of the limiting molar conductivities of its constituent ions, migrating completely independently of one another — cations carrying current in one direction and anions in the opposite direction, each making its own fixed contribution regardless of what it is paired with.

For a uni-univalent electrolyte such as NaCl, this is written directly as

and for a general electrolyte $A_xB_y$, the law generalises to weight each ionic contribution by how many of that ion appear in the formula unit:

The two data tables alongside this section make the law directly visible: holding the anion fixed and swapping only the cation (K⁺ for Na⁺) changes $\Lambda_m^{o}$ by exactly 23.41 S cm² mol⁻¹ every single time, regardless of which anion (Cl⁻, Br⁻ or NO₃⁻) is present; and holding the cation fixed while swapping the anion (Cl⁻ for Br⁻) changes $\Lambda_m^{o}$ by exactly 2.06 S cm² mol⁻¹ every time, regardless of which cation (K⁺, Na⁺ or Li⁺) is present. Each ion truly does carry its own fixed, additive contribution.

Application 1 — molar conductance of a weak electrolyte at infinite dilution. This is impossible to measure directly for a weak electrolyte (its non-linear dilution curve cannot be extrapolated), but Kohlrausch's law provides an indirect route using three STRONG electrolytes that share the same ions. For acetic acid, combine the limiting molar conductances of HCl, NaCl and sodium acetate: $\Lambda_m^o(\text{HCl}) = \lambda^o(\text{H}^+) + \lambda^o(\text{Cl}^-)$, $\Lambda_m^o(\text{CH}_3\text{COONa}) = \lambda^o(\text{Na}^+) + \lambda^o(\text{CH}_3\text{COO}^-)$, and $\Lambda_m^o(\text{NaCl}) = \lambda^o(\text{Na}^+) + \lambda^o(\text{Cl}^-)$. Adding the first two and subtracting the third cancels the Na⁺ and Cl⁻ terms exactly, leaving $\Lambda_m^o(\text{CH}_3\text{COOH}) = \lambda^o(\text{H}^+) + \lambda^o(\text{CH}_3\text{COO}^-)$ — obtained entirely from strong-electrolyte data, with no direct measurement on the weak acid itself required.

Application 2 — degree of dissociation. For a weak electrolyte, the degree of dissociation α at any given concentration can be estimated as the ratio of the molar conductance actually measured at that concentration to the limiting molar conductance:

Combining this with Ostwald's dilution law, $K_a = \dfrac{C\alpha^2}{1-\alpha}$, and substituting α gives the dissociation constant directly in terms of measurable conductances:

$K_a = \dfrac{C\,\Lambda_m^{2}}{\Lambda_m^{o}\left(\Lambda_m^{o}-\Lambda_m\right)}$ …