Kohlrausch's Law Calculator

Kohlrausch’s law is what made chemists realize that ions act independently in solution. In 1874, Friedrich Kohlrausch measured the conductivity of dozens of strong-electrolyte solutions at varying concentrations and noticed two things that were impossible to explain without ions: every salt solution’s molar conductivity decreased linearly with the square root of concentration, and the limiting value at infinite dilution could be predicted as a simple sum of contributions from the constituent ions. The same lithium ion contributes the same amount whether it is paired with chloride or with nitrate. That sum-of-independent-parts behavior is the law.

The two parts of Kohlrausch’s law:

1. Concentration dependence (for strong electrolytes):

Λ_m = Λ_m° − K · √C

Where Λ_m is molar conductivity at concentration C, Λ_m° is the limiting molar conductivity at infinite dilution, and K is an empirical slope that depends on the electrolyte and the solvent.

2. Independent migration of ions:

Λ_m° = ν₊ · λ₊° + ν₋ · λ₋°

Where ν₊ and ν₋ are the number of cations and anions in the formula unit, and λ₊° and λ₋° are the limiting ionic conductivities of the individual ions. So for NaCl: Λ_m°(NaCl) = λ°(Na⁺) + λ°(Cl⁻). For BaCl₂: Λ_m°(BaCl₂) = λ°(Ba²⁺) + 2·λ°(Cl⁻).

This second part is the deeper result. It means each ion has a characteristic conductivity that depends only on the ion itself and the solvent temperature. You can tabulate λ°(Na⁺) once and use it for every sodium salt in the periodic table.

Reference values at 25 °C in water (S·cm²/mol):

Cations:

• H⁺ ≈ 349.8 (anomalously high; proton hopping via Grotthuss mechanism)
• K⁺ ≈ 73.5
• Na⁺ ≈ 50.1
• NH₄⁺ ≈ 73.5
• Ca²⁺ ≈ 119.0 (for the full divalent ion: λ°/2 = 59.5 per equivalent)
• Mg²⁺ ≈ 106.0
• Cu²⁺ ≈ 107.2

Anions:

• OH⁻ ≈ 198.6 (also anomalously high; proton-hole hopping)
• Cl⁻ ≈ 76.3
• NO₃⁻ ≈ 71.4
• SO₄²⁻ ≈ 160.0
• HCO₃⁻ ≈ 44.5
• Acetate (CH₃COO⁻) ≈ 40.9

The unusually high λ° of H⁺ and OH⁻ is one of the most famous results in physical chemistry: in water, these ions move by a chain of hydrogen bonds rearranging, not by the ion itself migrating like a beach ball through molasses. That “Grotthuss mechanism” makes acid-base titration and pH measurement way more sensitive than ordinary cation/anion conduction would predict.

Worked example, KCl at 0.01 M:

Λ_m°(KCl) = λ°(K⁺) + λ°(Cl⁻) = 73.5 + 76.3 = 149.8 S·cm²/mol

Measuring at C = 0.01 M, suppose Λ_m = 142.0 S·cm²/mol. The Kohlrausch slope is:

K = (Λ_m° − Λ_m) / √C = (149.8 − 142.0) / √0.01 = 7.8 / 0.1 = 78 S·cm²·mol⁻¹·M^(-1/2)

That K value is in the typical range for 1:1 strong electrolytes (60-100). For 2:1 or 2:2 electrolytes, K is larger because there are more ions per formula unit creating ionic atmosphere drag.

Why √C, not C?

Debye and Hückel showed (1923) that the linear-in-√C dependence comes from the long-range electrostatic interaction between ions in solution. At concentration C, the average distance between ions scales as C^(-1/3), and the Debye screening length scales as C^(-1/2). The ionic atmosphere that develops around each ion slows it down (the relaxation effect) and produces an opposing flow (the electrophoretic effect). Both effects depend on the Debye length, hence the √C dependence. Onsager later (1927) derived the K constant from first principles for dilute solutions.

Degree of dissociation for weak electrolytes:

For a weak acid HA (only partially dissociated):

α = Λ_m / Λ_m°

So at concentration C, if Λ_m° is known from Kohlrausch’s law (summing ionic conductivities), measuring Λ_m gives the degree of dissociation directly. This is one of the most elegant experiments in physical chemistry: conductometric titration of acetic acid yields its dissociation degree and from there Ka (using Ostwald’s dilution law).

Worked example, acetic acid:

Λ_m°(acetic acid) = λ°(H⁺) + λ°(CH₃COO⁻) = 349.8 + 40.9 = 390.7 S·cm²/mol

At 0.01 M, suppose measured Λ_m = 16.3 S·cm²/mol.

α = 16.3 / 390.7 = 0.0417 = 4.17% dissociated

From Ostwald: Ka = α²·C / (1 − α) ≈ (0.0417)² · 0.01 / 0.96 = 1.81 × 10⁻⁵, matching the literature Ka of acetic acid.

Practical uses:

• Water purity check: deionized water has conductivity below 1 μS/cm; tap water is 100-1500 μS/cm. The values combine all dissolved ions weighted by their λ° values.
• Titration end-point: conductometric titration follows Λ_m changes as ions appear or disappear. Strong-acid/strong-base titrations show a sharp V-shape at equivalence.
• Ion-selective electrode design: knowing λ° for the target ion lets you predict response time and sensitivity.
• Pharmaceutical formulation: conductivity measurements help characterize ionic content in saline solutions and intravenous fluids.
• Industrial process control: in-line conductivity sensors monitor electrolyte concentration in plating baths, battery electrolytes, and reverse-osmosis systems.

Limits and modern view:

Kohlrausch’s law applies cleanly to strong electrolytes at low concentration (typically below 0.1 M for 1:1 salts). At high concentrations, ion pairing, activity coefficients well below 1, and viscosity changes break the linear √C behavior. Modern computational chemistry handles these regimes with molecular dynamics rather than the closed-form Onsager limiting law, but Kohlrausch’s basic insight (ions act independently at infinite dilution) remains the framework for everything in solution electrochemistry.