Applications of Kohlrausch's Law

At infinite dilution every ion moves independently of its partner, contributing a fixed limiting ionic conductivity $\lambda^{\circ}$ that depends only on the ion's identity, charge, and the solvent — not on what counterion it started with. This is the content of Kohlrausch's law of independent migration, assumed known here. For a strong electrolyte like KCl, NaCl, or CH₃COONa, $\Lambda_m$ versus $\sqrt{c}$ is linear (Kohlrausch's empirical law), so extrapolation to $c=0$ gives $\Lambda_m^{\circ}$ cleanly. Weak electrolytes such as acetic acid behave differently. Because dissociation itself is incomplete and concentration-dependent, $\Lambda_m$ rises steeply as $c \to 0$ and the plot curves sharply, never approaching linearity. Direct extrapolation is impossible. But since $\Lambda_m^{\circ}(\text{CH}_3\text{COOH})$ is just $\lambda^{\circ}(\text{CH}_3\text{COO}^-) + \lambda^{\circ}(\text{H}^+)$, we can assemble it from three strong electrolytes whose extrapolations are accessible: HCl, CH₃COONa, and NaCl. Adding and subtracting their $\Lambda_m^{\circ}$ values cancels the unwanted ions and leaves exactly the combination we want. Once $\Lambda_m^{\circ}$ is known, a single conductivity measurement at finite concentration $c$ gives the degree of dissociation $\alpha$, because ionic mobility itself is essentially constant at low concentration while only the fraction of dissociated molecules changes. Feeding $\alpha$ into the Ostwald dilution law yields $K_a$. For sparingly soluble salts like AgCl or BaSO₄, the saturated solution is so dilute that $\Lambda_m \approx \Lambda_m^{\circ}$ directly, converting a measured conductivity into a solubility.

where $\nu_i$ are stoichiometric ion counts and $\lambda_i^{\circ}$ limiting ionic conductivities.

where each term on the right is measured by ordinary extrapolation for a strong electrolyte.

where $\Lambda_m(c)$ is the measured molar conductivity at concentration $c$.

where $c$ is the formal (analytical) concentration of the weak electrolyte.

where $s$ is molar solubility (mol dm⁻³), $\kappa_{\text{salt}}$ is the conductivity of the saturated solution corrected for water, and the 1000 converts cm³ to dm³ when $\kappa$ is in S cm⁻¹ and $\Lambda_m^{\circ}$ in S cm² mol⁻¹.

Example 1 — Λm° and Ka of acetic acid (foundational). Given limiting molar conductivities at 298 K: $\Lambda_m^{\circ}(\text{HCl}) = 426.2$ S cm² mol⁻¹, $\Lambda_m^{\circ}(\text{CH}_3\text{COONa}) = 91.0$ S cm² mol⁻¹, $\Lambda_m^{\circ}(\text{NaCl}) = 126.5$ S cm² mol⁻¹. Step 1: Kohlrausch construction.

Step 2: measured data. A 0.0100 mol dm⁻³ acetic acid solution has measured conductivity $\kappa = 4.60\times10^{-4}$ S cm⁻¹ (typical literature value). Convert to molar conductivity, remembering $c$ must be in mol cm⁻³:

Step 3: degree of dissociation.

About 11.8% of the acetic acid molecules are dissociated at this concentration — physically reasonable for a weak acid at 0.01 M (compare $pK_a \approx 4.76$ literature value). Step 4: dissociation constant.

This is close to the accepted $K_a(\text{CH}_3\text{COOH}) = 1.75\times10^{-5}$; the modest difference reflects the approximation $\Lambda_m(c) \approx \alpha \Lambda_m^{\circ}$ (ignoring the small mobility decrease with concentration) and the illustrative conductivity value chosen. Sanity check: $\alpha \ll 1$ and $K_a \sim 10^{-5}$, exactly the order of magnitude expected for a weak carboxylic acid.

Example 2 — Solubility of silver chloride from conductivity (applied, exam-level). A saturated aqueous solution of AgCl at 298 K has measured specific conductivity $\kappa_{\text{solution}} = 3.41\times10^{-6}$ S cm⁻¹. The water used has $\kappa_{\text{water}} = 1.60\times10^{-6}$ S cm⁻¹. Limiting ionic conductivities at 298 K: $\lambda^{\circ}(\text{Ag}^+) = 61.9$ S cm² mol⁻¹, $\lambda^{\circ}(\text{Cl}^-) = 76.3$ S cm² mol⁻¹. Step 1: correct conductivity for solvent.

Step 2: Kohlrausch construction of $\Lambda_m^{\circ}(\text{AgCl})$. Since AgCl is 1:1, $\nu_+ = \nu_- = 1$:

Step 3: why $\Lambda_m \approx \Lambda_m^{\circ}$ here. AgCl is sparingly soluble, so its saturated solution is extremely dilute (solubility on the order of $10^{-5}$ mol dm⁻³). At such low concentration, ionic mobilities are essentially at their infinite-dilution values and AgCl behaves as a fully dissociated strong electrolyte, so:

Step 4: solve for solubility. Using $\Lambda_m = 1000\,\kappa / c$ with $c$ in mol dm⁻³ and $\kappa$ in S cm⁻¹:

Step 5: Ksp and sanity check.

This is close to the accepted literature $K_{sp}(\text{AgCl}) \approx 1.8\times10^{-10}$ at 298 K, confirming the method. The solubility, about $1.9\ \text{mg dm}^{-3}$ ($M = 143.3$ g mol⁻¹), is exactly the tiny, near-insoluble scale expected for AgCl.

Friedrich Kohlrausch spent decades making precision conductivity measurements the gold standard of physical chemistry, engineering AC bridge techniques that avoided electrode polarization and produced data reliable to a few parts in ten thousand. His empirical square-root law for strong electrolytes, and the deeper principle that ions migrate independently and contribute additively to conductivity, together supplied the toolkit that Ostwald, Arrhenius, and later workers used to quantify dissociation equilibria before any modern spectroscopic method existed. The solubility-product and $pK_a$ determinations shown above are direct descendants of his bridge-and-cell measurements from the 1870s–1900s.

Ask the tutor: How would this method change if the weak electrolyte were diprotic, like H₂CO₃, instead of a simple monoprotic acid?