Kohlrausch Law of Independent Migration

Dip two platinum electrodes into a weak acid solution and try to measure its "complete" conductivity. You will fail. The acid is only partly ionized, so you can never dilute it enough to reach the limiting value directly, since it never stops dissociating further. Friedrich Kohlrausch solved this puzzle in 1875 with a strikingly simple idea. He proposed that at infinite dilution, each ion moves independently, carrying its own fixed share of the current. This means limiting molar conductivities can be added and subtracted like building blocks. Strong electrolyte data, which are easy to extrapolate, can then predict the unreachable limiting conductivity of any weak electrolyte. This article builds the law from the concentration dependence you already know for strong electrolytes. You will see the additivity relation derived conceptually. Then you will compute real limiting conductivities for acetic acid and ammonia. Finally, you will see how this law opens the door to ionic mobilities and conductometric titrations.

Key takeaways

Kohlrausch's law: the limiting molar conductivity of an electrolyte is the sum of independent limiting ionic conductivities.
Each ion contributes a fixed value regardless of its partner ion, valid strictly only at infinite dilution.
Strong electrolytes give straight-line Kohlrausch plots against √c, allowing easy extrapolation to $c \to 0$ .
Weak electrolytes curve sharply at low concentration, so their limiting conductivity cannot be measured directly.
Combining limiting conductivities of related strong electrolytes lets you calculate the limiting conductivity of any weak electrolyte.
Key idea: At infinite dilution, ions move independently, so their conductivities simply add.

The core insight

At infinite dilution, ion–ion interactions vanish, so each ion migrates as if no other ionic species existed. That means the limiting molar conductivity of any electrolyte is exactly the sum of the limiting conductivities contributed independently by its cation and anion, weighted by their stoichiometric numbers.

The physical picture

Molar conductivity, $\Lambda_m$ , measures how effectively one mole of dissolved electrolyte conducts electricity, once you have already normalized for concentration and cell geometry (see Equivalent Conductivity). For strong electrolytes such as KCl or NaCl, $\Lambda_m$ decreases smoothly and mildly as concentration rises. It follows the Kohlrausch square-root law,

\Lambda_m = \Lambda_m^{\circ} - K\sqrt{c}

a linear relationship in $\sqrt{c}$ that you met when studying the concentration dependence of molar conductivity. Because this line is straight, extrapolating to $c=0$ gives a reliable value of $\Lambda_m^{\circ}$ , the limiting molar conductivity. Weak electrolytes such as acetic acid behave completely differently. Only a small, concentration-dependent fraction of molecules ionize. As you dilute the solution, more molecules dissociate and $\Lambda_m$ rises steeply, especially at low concentration, following a curve that bends sharply upward rather than approaching linearly. No straight-line extrapolation is possible because the curve's slope diverges as $c\to0$ . You cannot experimentally reach a concentration low enough to read off $\Lambda_m^{\circ}$ directly. Kohlrausch's insight was to look at ions rather than whole electrolytes. At infinite dilution, ions are so far apart that they no longer feel each other's electric fields. Each ion type—say Na⁺, or Cl⁻, or CH₃COO⁻—drifts under the applied field with a mobility that depends only on its own size, charge, and solvation shell, never on which counter-ion happens to be present. If that is true, the total limiting conductivity of any salt must be built additively from independent ionic pieces: a fixed number for the cation, a fixed number for the anion, scaled by how many of each ion the formula unit produces.

Figure

Sharp divergence for a weak acid versus the gentle line for a strong electrolyte

The core relations

\Lambda_m^{\circ} = \nu_+ \lambda_+^{\circ} + \nu_- \lambda_-^{\circ}

where: $\nu_+, \nu_-$ are the number of cations and anions per formula unit, and $\lambda_+^{\circ}, \lambda_-^{\circ}$ are the limiting molar ionic conductivities.

\Lambda_m^{\circ}(\text{weak electrolyte}) = \Lambda_m^{\circ}(\text{strong salt 1}) + \Lambda_m^{\circ}(\text{strong salt 2}) - \Lambda_m^{\circ}(\text{strong salt 3})

where: the three strong electrolytes are chosen so that unwanted ions cancel, leaving exactly the ions of the target weak electrolyte.

\lambda_+^{\circ} = z_+ F u_+^{\circ}, \qquad \lambda_-^{\circ} = z_- F u_-^{\circ}

where: $z_\pm$ are ionic charge numbers, $F$ is the Faraday constant, and $u_\pm^{\circ}$ are limiting ionic mobilities (foreshadowing the next topic, Ionic Mobility).

Derivation ABuilding the additivity law from independent ion transport

Goal. Show that $\Lambda_m^{\circ}$ must split additively into ionic contributions, and use that to justify combining strong-electrolyte data for a weak electrolyte.

Step 1. Define conductivity in terms of ion flux. The conductivity $\kappa$ of an electrolyte solution arises from the drift of both cations and anions under an applied electric field $E$ . Each ion type $i$ contributes a current density proportional to its concentration $c_i$ , its charge $z_i e$ , and its drift velocity $v_i = u_i E$ , where $u_i$ is the ionic mobility. Summing contributions from both ion types gives the total conductivity as a sum of independent terms, one per ion.

Step 2. Convert to molar quantities. Dividing $\kappa$ by the electrolyte's molar concentration $c$ defines the molar conductivity $\Lambda_m = \kappa/c$ . Because each ionic current term already appeared additively in Step 1, and because $c_+ = \nu_+ c$ and $c_- = \nu_- c$ for full dissociation, the molar conductivity itself splits as

\Lambda_m = \nu_+ \lambda_+ + \nu_- \lambda_-

where $\lambda_\pm$ are molar ionic conductivities that still depend on concentration through inter-ionic forces at finite $c$ .

Step 3. Take the infinite-dilution limit. As $c \to 0$ , the average distance between ions grows without bound, and the electrostatic drag that one ion exerts on another's motion (via the ionic atmosphere) vanishes. In this limit each $\lambda_\pm$ approaches a value $\lambda_\pm^{\circ}$ that depends only on the identity of that ion, its charge, size, and solvation—never on the identity of its counter-ion. This is the physical content of Kohlrausch's hypothesis of independent ionic migration.

Step 4. Write the limiting law. Taking the limit of the Step 2 expression and inserting the ion-only values from Step 3 gives

\Lambda_m^{\circ} = \nu_+ \lambda_+^{\circ} + \nu_- \lambda_-^{\circ}

This is exact at infinite dilution for any electrolyte, strong or weak, because it is a statement about non-interacting ions, not about the degree of dissociation.

Step 5. Exploit additivity across compounds. Since $\lambda_+^{\circ}$ and $\lambda_-^{\circ}$ are properties of the ion alone, they are transferable between compounds. If you know $\Lambda_m^{\circ}$ for three strong electrolytes chosen so that adding two and subtracting one cancels every unwanted ion, the ionic contributions of the unwanted ions cancel exactly, algebraically, leaving only the ions of interest. A weak electrolyte's inaccessible limiting conductivity is thereby reconstructed purely from strong-electrolyte extrapolations, which are experimentally easy because their Kohlrausch plots are linear in $\sqrt c$ .

Boxed result.

\boxed{\Lambda_m^{\circ}(\text{electrolyte}) = \sum_i \nu_i \lambda_i^{\circ}, \quad \text{transferable ion by ion between compounds}}

Worked examples

Example 1 — Limiting molar conductivity of acetic acid (foundational). Given limiting molar conductivities at 298 K (standard tabulated values, S cm² mol⁻¹): $\Lambda_m^{\circ}(\text{HCl}) = 426.2$ $\Lambda_m^{\circ}(\text{CH}_3\text{COONa}) = 91.0$ $\Lambda_m^{\circ}(\text{NaCl}) = 126.5$ Choose the combination that leaves H⁺ and CH₃COO⁻ while canceling Na⁺ and Cl⁻:

\Lambda_m^{\circ}(\text{CH}_3\text{COOH}) = \Lambda_m^{\circ}(\text{HCl}) + \Lambda_m^{\circ}(\text{CH}_3\text{COONa}) - \Lambda_m^{\circ}(\text{NaCl})

Substituting:

\Lambda_m^{\circ}(\text{CH}_3\text{COOH}) = 426.2 + 91.0 - 126.5 = 390.7 \text{ S cm}^2\text{mol}^{-1}

Sanity check: this matches the accepted tabulated value of $\Lambda_m^{\circ}(\text{CH}_3\text{COOH}) \approx 390.7\ \text{S cm}^2\text{mol}^{-1}$ , well within experimental agreement, and is dominated by the very high H⁺ contribution (H⁺ conducts anomalously fast via the Grotthuss proton-hopping mechanism, so its ionic conductivity, $\lambda^{\circ}(\text{H}^+) \approx 349.6\ \text{S cm}^2\text{mol}^{-1}$ , is roughly seven times larger than a typical alkali cation).

Example 2 — Predicting the degree of dissociation of a weak base from conductivity data (applied). Ammonia, NH₃, forms the weak base NH₄OH (equivalently NH₃·H₂O) in water. You are given: $\Lambda_m^{\circ}(\text{NH}_4\text{Cl}) = 149.7\ \text{S cm}^2\text{mol}^{-1}$ $\Lambda_m^{\circ}(\text{NaOH}) = 248.1\ \text{S cm}^2\text{mol}^{-1}$ $\Lambda_m^{\circ}(\text{NaCl}) = 126.5\ \text{S cm}^2\text{mol}^{-1}$ A conductivity cell measures $\Lambda_m$ for a $0.010\ \text{mol dm}^{-3}$ NH₄OH solution as $9.5\ \text{S cm}^2\text{mol}^{-1}$ .

Step A — build $\Lambda_m^{\circ}(\text{NH}_4\text{OH})$ from strong electrolytes.

\Lambda_m^{\circ}(\text{NH}_4\text{OH}) = \Lambda_m^{\circ}(\text{NH}_4\text{Cl}) + \Lambda_m^{\circ}(\text{NaOH}) - \Lambda_m^{\circ}(\text{NaCl})

\Lambda_m^{\circ}(\text{NH}_4\text{OH}) = 149.7 + 248.1 - 126.5 = 271.3\ \text{S cm}^2\text{mol}^{-1}

Step B — Ostwald dilution law relates $\Lambda_m$ to the degree of dissociation $\alpha$ . For a weak electrolyte, $\alpha \approx \Lambda_m / \Lambda_m^{\circ}$ (valid because conductivity is approximately proportional to the concentration of free ions at low $\alpha$ ):

\alpha = \frac{\Lambda_m}{\Lambda_m^{\circ}} = \frac{9.5}{271.3} = 0.0350

So only 3.5% of NH₄OH molecules are ionized at this concentration.

Step C — sanity check via $K_b$ . With $c = 0.010\ \text{mol dm}^{-3}$ and $\alpha = 0.0350$ :

K_b = \frac{c\alpha^2}{1-\alpha} = \frac{(0.010)(0.0350)^2}{1 - 0.0350} = 1.27 \times 10^{-5}

This is close to the accepted literature value of $K_b(\text{NH}_3) \approx 1.8 \times 10^{-5}$ , with the modest discrepancy coming from the approximation $\alpha \approx \Lambda_m/\Lambda_m^{\circ}$ neglecting ionic-strength corrections — reasonable for a hand calculation and confirming the method's internal consistency.

Limiting cases & common traps

Situation	Correct treatment	Common error
Strong electrolyte, dilute	$\Lambda_m^{\circ}$ read directly from linear extrapolation	Trying to extrapolate a weak electrolyte's curve the same way
Weak electrolyte	Build $\Lambda_m^{\circ}$ from three strong-electrolyte values	Assuming Kohlrausch's law only applies to strong electrolytes
Polyvalent ions	Weight $\lambda^{\circ}$ by stoichiometric number, e.g. $\Lambda_m^{\circ}(\text{MgCl}_2)=\lambda^{\circ}(\text{Mg}^{2+})+2\lambda^{\circ}(\text{Cl}^-)$	Forgetting the factor of 2 for the anion
H⁺ and OH⁻	Recognize anomalously high mobility via Grotthuss mechanism	Treating H⁺ as an "ordinary" cation of similar size to Na⁺
Using $\alpha \approx \Lambda_m/\Lambda_m^{\circ}$	Valid approximation for weak electrolytes only	Applying it to strong electrolytes, where $\alpha$ is already essentially 1 and the ratio instead reflects ionic-atmosphere effects

The chemist behind it

Friedrich Kohlrausch was a German physicist whose meticulous conductivity measurements, made using an early alternating-current bridge method he helped perfect, established conductivity as a precise quantitative tool rather than a qualitative curiosity. His 1875 recognition that limiting molar conductivities are additive over ions transformed electrolyte chemistry, letting chemists calculate unreachable limits for weak acids and bases from easily measured strong-electrolyte data. He also produced some of the first tables of highly purified "conductivity water," recognizing that trace ionic impurities corrupted conductivity measurements at the levels needed for this precision work.

Origins · 1 minFriedrich Kohlrausch and how this was discovered

Friedrich Kohlrausch

1840–1910 · German physicist who established quantitative electrolytic conductivity and formulated the law of independent ionic migration

Ask the tutor: How do transport numbers let us split a measured Λm° into separate cation and anion contributions?

Why does Λm diverge sharply for weak electrolytes as concentration approaches zero?

Molar conductivity depends on both how many ions are present and how fast they move. For a weak electrolyte, the degree of dissociation α itself increases as concentration decreases, since Le Chatelier's principle favors more dissociation at higher dilution. Near c = 0, α approaches 1 rapidly, so Λm shoots upward steeply rather than leveling off linearly the way a strong electrolyte's Λm does. This steep, non-linear approach makes direct extrapolation experimentally impractical.

How were tables of limiting ionic conductivities like λ°(H⁺) or λ°(Cl⁻) actually built?

Chemists cannot measure a single ion's conductivity directly, since any real solution needs both a cation and an anion. Instead, one ion's transport number is measured (from Hittorf or moving-boundary experiments), which splits a known Λm° between cation and anion. Once one ionic value is anchored this way, Kohlrausch's additivity propagates consistent values to every other ion, building the complete self-consistent tables found in reference works today.

Is the relation α ≈ Λm/Λm° always accurate for weak electrolytes?

It is a good working approximation, especially at low concentrations where ionic-strength effects on the ions actually present are small. It assumes that once ions are formed, they move at essentially the limiting mobility, which slightly overestimates their real speed at finite concentration. More rigorous treatments correct this with an activity-based approach, but the simple ratio is standard for introductory calculations and typically matches equilibrium constants within about 20–30%.

How does this law connect to what comes next: ionic mobility and conductometric titrations?

Once $\lambda_\pm^{\circ}$ values are tabulated for many ions, you can predict $\Lambda_m^{\circ}$ for any new combination without ever measuring it, which is exactly what Applications of Kohlrausch's Law explores. The same ionic conductivities also encode each ion's limiting mobility, $u^{\circ} = \lambda^{\circ}/(zF)$ , connecting macroscopic conductivity to microscopic ion speed, the subject of Ionic Mobility. Conductometric titrations exploit how differently mobile ions (like fast H⁺ replaced by slower Na⁺) shift Λm during a reaction.