Kohlrausch's Law of Independent Migration

What the law actually claims

Dissolve any salt in water, apply a voltage, and the solution conducts because both ions drift — cations toward the cathode, anions toward the anode. Kohlrausch's central discovery is that when the solution is diluted all the way down to infinite dilution, the total conducting ability splits into two pieces that don't talk to each other. Each ion carries a fixed amount of current that depends only on what the ion is — not on the partner it arrived with.

The bookkeeping quantity is molar conductivity, written Λ_m. It normalizes conductivity κ (which you measure directly) by concentration, so that solutions of different strengths can be compared fairly:

    Λ_m  =  κ / c        (units: S·cm²·mol⁻¹ when κ is in S·cm⁻¹ and c in mol·cm⁻³)

Extrapolate Λ_m to zero concentration and you get the limiting molar conductivity Λ°_m. Kohlrausch's law of independent migration then says this limit is a plain sum:

    Λ°_m  =  ν₊ · λ°₊  +  ν₋ · λ°₋

  where  λ°₊, λ°₋   = limiting molar ionic conductivities of cation and anion
         ν₊, ν₋     = number of each ion per formula unit

For NaCl (ν₊ = ν₋ = 1), Λ°_m = λ°(Na⁺) + λ°(Cl⁻). For a 2:1 salt like CaCl₂, Λ°_m = λ°(Ca²⁺) + 2·λ°(Cl⁻). The λ° values are true ionic constants: λ°(Cl⁻) is the same number whether the chloride came from HCl, NaCl, or KCl.

Why the ions become independent — the physics

In a concentrated solution the ions are not independent at all. Every ion sits inside a diffuse cloud of oppositely charged ions — its ionic atmosphere. Two effects born of that cloud slow the ion down:

1. Electrophoretic effect. The atmosphere carries solvent in the opposite direction to the central ion's motion, so the ion swims upstream against a counter-current of dragged water.
2. Relaxation (asymmetry) effect. As the ion moves, its atmosphere can't keep up and re-forms lopsidedly behind it, producing a retarding electrostatic pull. The atmosphere takes a finite relaxation time to rebuild, and during that lag it drags on the ion.

Both effects scale with how dense the ionic atmosphere is — that is, with concentration. Dilute the solution and the average ion-to-ion distance grows, the atmosphere thins, and the drag fades. In the mathematical limit c → 0, the nearest counter-ion is infinitely far away, the atmosphere disappears, and each ion migrates as if it were alone in the solvent. That is why the contributions become independent and additive: the only thing left influencing an ion is the ion's own size, charge, and hydration shell interacting with the water. Debye, Hückel, and Onsager later put numbers on exactly how the drag grows with √c — Kohlrausch had already found the pattern empirically decades earlier.

Getting Λ° from data: the square-root law

You can't dilute to literally zero concentration, so Λ°_m has to be reached by extrapolation. Kohlrausch's second empirical gift is that, for a strong electrolyte, the falloff is beautifully linear in the square root of concentration:

    Λ_m  =  Λ°_m  −  K·√c

  Plot Λ_m against √c → a straight line.
  Intercept at √c = 0  →  Λ°_m
  Slope  = −K  (the Kohlrausch coefficient; depends on the ion charges and solvent)

The √c dependence is exactly what Debye-Hückel-Onsager theory predicts, because both retarding effects turn out to scale as √c. So you measure Λ_m at a handful of low concentrations (say 0.001-0.02 M for KCl), plot against √c, draw the best-fit line, and read the intercept. That intercept is Λ°_m — the number the independent-migration law then decomposes into ions.

Crucially, this trick only works for strong electrolytes, whose Λ_m-vs-√c plot is genuinely straight. Weak electrolytes curve away wildly near c = 0, which is the problem the next section solves.

Ionic conductivities: the raw numbers

Because the law is additive, all of practical electrolyte chemistry can be tabulated as single-ion λ° values (25 °C, aqueous, in S·cm²·mol⁻¹). A working subset:

Cation	λ° (S·cm²·mol⁻¹)	Anion	λ° (S·cm²·mol⁻¹)
H⁺	349.8	OH⁻	198.0
K⁺	73.5	SO₄²⁻ (½)	80.0
NH₄⁺	73.6	Br⁻	78.1
Na⁺	50.1	Cl⁻	76.3
Ag⁺	61.9	NO₃⁻	71.4
Li⁺	38.7	CH₃COO⁻	40.9

Two things jump out. First, H⁺ and OH⁻ are extraordinary — roughly five and three times faster than a typical ion. They don't shove through the water; they hop along hydrogen-bonded chains (the Grotthuss mechanism), so the charge relays from molecule to molecule without any single atom traveling far. Second, notice that Li⁺ conducts less than Na⁺, which conducts less than K⁺ — the opposite of what bare ionic radius predicts. The small lithium ion holds a big, tightly bound hydration shell and effectively drags a fat coat of water, so the hydrated Li⁺ is the bulkiest and slowest of the three.

Worked example: Λ° and α for acetic acid

Here is the calculation that made the law indispensable in the 1880s and still appears in every physical-chemistry course. Acetic acid is a weak electrolyte — its Λ_m-vs-√c plot never straightens, so you cannot extrapolate its Λ° directly. Instead, build it from three strong electrolytes whose Λ° values can be extrapolated:

    Λ°(CH₃COOH)  =  Λ°(CH₃COONa)  +  Λ°(HCl)  −  Λ°(NaCl)

  Check by cancelling ions:
      (CH₃COO⁻ + Na⁺) + (H⁺ + Cl⁻) − (Na⁺ + Cl⁻)
    =  CH₃COO⁻ + H⁺
    =  the acetic-acid ion pair  ✓  (Na⁺ and Cl⁻ cancel)

Plug in tabulated limiting conductivities:

    Λ°(CH₃COONa) =  91.0
    Λ°(HCl)      = 426.1
    Λ°(NaCl)     = 126.4
    ---------------------------------
    Λ°(CH₃COOH)  =  91.0 + 426.1 − 126.4  =  390.7  S·cm²·mol⁻¹

  Sanity check against ionic values:
    λ°(H⁺) + λ°(CH₃COO⁻) = 349.8 + 40.9 = 390.7  ✓

Now find how dissociated 0.1 M acetic acid actually is. Measure its molar conductivity — Λ_m ≈ 5.2 S·cm²·mol⁻¹ at that concentration — and take the ratio, because Λ_m tracks the number of ions actually present while Λ°_m corresponds to complete dissociation:

    α  =  Λ_m / Λ°_m  =  5.2 / 390.7  ≈  0.0133      (only 1.3% dissociated)

    K_a  =  c·α² / (1 − α)
         =  (0.1)(0.0133)² / (1 − 0.0133)
         ≈  1.8 × 10⁻⁵                                 (matches the textbook value)

That last number — the dissociation constant of acetic acid — was among the first equilibrium constants ever obtained, and it came straight out of a conductivity bridge and Kohlrausch's law. No pH meter, no titration.

Strong vs weak electrolytes at a glance

	Strong electrolyte (e.g. KCl, HCl)	Weak electrolyte (e.g. CH₃COOH, NH₃)
Degree of dissociation	~100% at all concentrations	Small; rises sharply on dilution
Λm vs √c plot	Straight line, gentle slope	Curves steeply upward near c = 0
Can you extrapolate Λ° directly?	Yes — read the √c intercept	No — plot never straightens
How to get Λ°m	Square-root-law extrapolation	Add/subtract strong-electrolyte Λ° values
Why Λm falls with c	Only ionic-atmosphere drag	Drag and shrinking dissociation
α = Λm/Λ°m meaningful?	Always ≈ 1, so uninformative	Yes — the whole point; gives α directly
Governing theory near c = 0	Debye-Hückel-Onsager	Ostwald dilution law + Kohlrausch

Transport numbers: measuring the ionic pieces

The law is only useful if you can pin down the individual λ° values, not just their sum Λ°_m. That extra piece of information comes from the transport (transference) number t, which is the fraction of the total current a given ion carries:

    t₊  =  λ°₊ / Λ°_m          t₋  =  λ°₋ / Λ°_m          t₊ + t₋ = 1

Kohlrausch and others measured t experimentally — by Hittorf's method (watching concentration changes near each electrode) or the moving-boundary method (timing a visible ionic front migrate along a tube). Once you know both Λ°_m (from conductivity) and t₊ (from a transport experiment), you can split Λ°_m into λ°₊ and λ°₋ separately. For KCl the split is nearly even (t₊ ≈ 0.49), which is exactly why KCl is the standard salt in salt bridges — the cation and anion migrate at almost equal speed, so it introduces minimal junction potential.

Who and when

Friedrich Wilhelm Georg Kohlrausch (1840-1910), a German experimental physicist, established both laws in the mid-1870s — the independent-migration statement and the √c falloff appear in his papers of roughly 1875-1876, built on a decade of painstakingly precise conductance measurements. His experimental achievement was as important as the laws themselves: to measure ionic conductivity cleanly you must avoid electrolysis at the electrodes, so Kohlrausch pioneered the use of alternating current together with a Wheatstone bridge and platinized-platinum electrodes, and he obsessively purified his water (the residual conductivity of pure water is largely due to its own H⁺ and OH⁻).

The theoretical "why" arrived a generation later. Arrhenius's 1884-1887 theory of ionic dissociation explained why weak electrolytes dissociate more on dilution; Debye and Hückel (1923) and then Onsager (1927) derived the √c drag from first principles, turning Kohlrausch's empirical slope K into a calculable quantity. Kohlrausch found the shape of the curve; theory later explained its origin.

Limitations and where it breaks

• It is an infinite-dilution law only. Perfect additivity holds strictly at c = 0. At any finite concentration the ions do interact, so λ° values only approximate real behavior; the more concentrated the solution, the worse the additive prediction.
• Ion pairing quietly removes carriers. In low-permittivity solvents (or with highly charged ions like 2:2 electrolytes such as MgSO₄), cations and anions associate into neutral or partially neutral pairs that carry less current. The apparent Λ° drops below the additive sum, and the plot deviates from the clean √c line.
• Very high fields (Wien effect). Under enormous electric fields the ionic atmosphere can't form at all and conductivity rises above the low-field value — a departure from ordinary Ohmic behavior that the simple law doesn't capture.
• High-frequency AC (Debye-Falkenhagen effect). At very high AC frequencies the relaxation drag disappears because the atmosphere never gets a chance to become asymmetric, so conductivity rises — again outside the static picture.
• Not every solvent gives independent ions. The whole framework assumes the solvent fully separates the ions. In weakly dissociating solvents the electrolyte may behave as a weak electrolyte even if it is "strong" in water.

Where the law earns its keep

• Conductometric titrations. Because H⁺ and OH⁻ conduct so much better than other ions, neutralizing a strong acid with a strong base produces a sharp V-shaped conductivity minimum at the equivalence point — a titration you can run in colored or turbid solutions where an indicator would be invisible.
• Weak-acid pKₐ and dissociation constants. The acetic-acid calculation above generalizes: conductivity plus Kohlrausch's law gives Kₐ for weak acids and bases without any indicator or electrode calibration.
• Solubility of sparingly soluble salts. For something like AgCl, the saturated solution is so dilute that Λ_m ≈ Λ°_m. Measuring κ and using the additive λ° values gives the tiny saturation concentration — and hence Ksp — directly.
• Water purity monitoring. Ultrapure water for pharmaceuticals and semiconductor fabs is graded by conductivity; the theoretical floor (~0.055 µS·cm⁻¹ at 25 °C) is set by the λ° of water's own H⁺ and OH⁻.
• Choosing salt-bridge electrolytes. KCl (and KNO₃) are picked precisely because t₊ ≈ t₋, minimizing the liquid-junction potential that would otherwise corrupt cell-voltage measurements.