Kohlrausch's Law

When an ionic compound dissolves in water, it dissociates into cations and anions that are free to carry electric current. The ability of a solution to conduct electricity is measured by its molar conductivity (Λm), which depends on how many ions are present and how freely they move. As a solution is diluted, ions interact less with each other, and molar conductivity increases. At infinite dilution — a theoretical state where every ion is completely independent — molar conductivity reaches its maximum value, called the limiting molar conductivity, Λ°m. Friedrich Kohlrausch, a 19th-century German physicist, discovered through careful experimentation that the limiting molar conductivity of any electrolyte can be expressed as the sum of independent contributions from each type of ion present. This elegant principle, known as Kohlrausch's Law of Independent Migration of Ions, allows chemists to calculate Λ°m for any electrolyte — including weak electrolytes that never fully dissociate — by combining tabulated values of individual ionic molar conductivities. For weak electrolytes such as acetic acid (\ceCH3COOH\ce{CH3COOH}), direct measurement of Λ°m is impossible because conductance remains too low even at practical dilutions to extrapolate accurately. Kohlrausch's Law provides the workaround: Λ°m can be computed indirectly from strong electrolyte data, and once Λ°m is known, the degree of dissociation (α) and acid dissociation constant (Ka) can be calculated at any given concentration. This makes Kohlrausch's Law a cornerstone of solution electrochemistry.

Key takeaways
  • The Law of Independent Migration of Ions
  • Calculating Λ°m for a Weak Electrolyte by the Hess-Type Addition
  • Degree of Dissociation (α) from Conductance
  • Calculating the Acid Dissociation Constant Ka
  • Units and Practical Notes

The Law of Independent Migration of Ions

Kohlrausch observed that at infinite dilution, each ion migrates independently and contributes a fixed quantity to the total conductivity regardless of which counter-ion it was paired with. For an electrolyte AxBy that dissociates into x cations and y anions, Λ°m = x·λ°(cation) + y·λ°(anion), where λ° values are the limiting molar conductivities of individual ions (units: S·cm²·mol⁻¹).

Calculating Λ°m for a Weak Electrolyte by the Hess-Type Addition

Because weak electrolytes cannot be extrapolated to infinite dilution directly, we construct Λ°m from strong electrolyte data. For acetic acid: Λ°m(CH₃COOH) = Λ°m(CH₃COONa) + Λ°m(HCl) − Λ°m(NaCl). This algebraic combination cancels unwanted ions (Na⁺ and Cl⁻) and leaves only the λ°(CH₃COO⁻) + λ°(H⁺) contributions. All three electrolytes on the right are strong and can be extrapolated reliably.

Degree of Dissociation (α) from Conductance

At a finite concentration c, a weak electrolyte only partially dissociates. The fraction of molecules that have ionised is the degree of dissociation α. Since conductivity is proportional to the number of ions present, α equals the ratio of the actual molar conductivity at that concentration to the limiting molar conductivity: α = Λm / Λ°m. Here Λm is measured experimentally (from cell constant and measured conductance), while Λ°m is obtained via Kohlrausch's Law.

Calculating the Acid Dissociation Constant Ka

For a weak acid HA ⇌ H⁺ + A⁻ at initial concentration c, the equilibrium concentrations are: [H⁺] = αc, [A⁻] = αc, [HA] = c(1−α). Therefore Ka = (αc)(αc) / [c(1−α)] = α²c / (1−α). Substituting α = Λm/Λ°m gives Ka = (Λm/Λ°m)² · c / [1 − (Λm/Λ°m)], allowing Ka to be determined purely from conductance measurements.

Units and Practical Notes

Molar conductivity Λm = κ/c, where κ is the specific conductance (S·m⁻¹) and c is concentration in mol·m⁻³. In practical problems, if κ is in S·cm⁻¹ and c is in mol·L⁻¹, use Λm (S·cm²·mol⁻¹) = [κ(S·cm⁻¹) × 1000(cm³·L⁻¹)] / c(mol·L⁻¹). Always convert units consistently before substituting into the Kohlrausch expression.

The Key Equation

Λm∘=∑iνi λi∘\Lambda^\circ_m = \sum_i \nu_i \, \lambda^\circ_i
  • \Lambda^\circ_m — Limiting molar conductivity (S·cm²·mol⁻¹): Molar conductivity of the electrolyte at infinite dilution, where ion–ion interactions vanish.
  • \nu_i — Stoichiometric number of ion i (dimensionless): Number of moles of ion i produced per formula unit of electrolyte (e.g., 2 for Ca²⁺ from CaCl₂ is 1, and Cl⁻ is 2).
  • \lambda^\circ_i — Limiting ionic molar conductivity of ion i (S·cm²·mol⁻¹): The independent contribution of a single type of ion to molar conductivity at infinite dilution. Tabulated values: λ°(H⁺) = 349.6, λ°(OH⁻) = 198.0, λ°(Na⁺) = 50.1, λ°(Cl⁻) = 76.3 (all in S·cm²·mol⁻¹).

Worked Example

Problem. At 25°C, the molar conductivity of a 0.100 mol·L⁻¹ acetic acid solution is measured to be 5.20 S·cm²·mol⁻¹. Given that λ°(H⁺) = 349.6 S·cm²·mol⁻¹ and λ°(CH₃COO⁻) = 40.9 S·cm²·mol⁻¹, calculate (a) Λ°m for acetic acid, (b) the degree of dissociation α, and (c) the acid dissociation constant Ka.

Λm∘=390.5S ⋅ cm2mol−1;α=0.0133;Ka=α2c1−α=1.79×10−5mol ⋅ L−1\Lambda^\circ_m = 390.5 \text{S}\,\cdot\,\text{cm}^2\text{mol}^{-1};\quad \alpha = 0.0133;\quad K_a = \frac{\alpha^2 c}{1-\alpha} = 1.79 \times 10^{-5} \text{mol}\,\cdot\,\text{L}^{-1}

Answer. Λ°m = 390.5 S·cm²·mol⁻¹; α = 0.0133 (1.33%); Ka = 1.79 × 10⁻⁵ mol·L⁻¹

Figure
Diagram
Molar Conductivity vs. Concentration for Strong and Weak Electrolytes√c (mol·L⁻¹)^(1/2)Λm (S·cm²·mol⁻¹)4003002001000.10.20.3Strong electrolyte(e.g. NaCl)Λ°m (strong)Weak electrolyte(e.g. CH₃COOH)Λ°m (weak, from Kohlrausch)Cannot extrapolate weakelectrolyte to Λ°m directly
Origins · 1 minFriedrich Wilhelm Georg Kohlrausch and how this was discovered
FW
Friedrich Wilhelm Georg Kohlrausch
1840–1910 · German

Kohlrausch was a pioneering experimental physicist and chemist who systematically measured the electrical conductivities of many electrolyte solutions with unprecedented precision. Through meticulous work in the 1870s–1880s, he discovered that each ion in solution migrates independently of its counter-ion at infinite dilution, formulating what became Kohlrausch's Law of Independent Migration of Ions. He also established that molar conductivity of strong electrolytes varies linearly with the square root of concentration (the Kohlrausch equation: Λm = Λ°m − K√c), a relationship later explained theoretically by Debye, Hückel, and Onsager. Kohlrausch was such a perfectionist in measurement that he personally designed many of his own instruments, including specialised resistance bridges and alternating-current apparatus to eliminate electrolysis errors during conductance measurements — a problem that had plagued earlier researchers.

  • Molar Conductivity and Its Variation with Concentration — Kohlrausch's Law defines the limiting value of molar conductivity; understanding how Λm changes with concentration provides the experimental context for applying the law.
  • Electrolytic Conductance and Cell Constant — Experimental measurement of conductance (using the cell constant) yields the specific conductance κ, which is converted to Λm before applying Kohlrausch's Law.
  • Weak Acid Equilibrium and Ka — Kohlrausch's Law is used to calculate α and Ka for weak acids; a solid understanding of acid dissociation equilibria is essential to interpret these results correctly.
  • Debye–Hückel–Onsager Equation — This equation provides the theoretical explanation for the empirical Kohlrausch linear relationship (Λm = Λ°m − K√c) observed for strong electrolytes.