Kohlrausch’s Law: Limiting Molar Conductivity & Applications | Full Guide

⚡ Kohlrausch’s Law of Independent Migration of Ions

Limiting Molar Conductivity · Ionic Mobilities · Solubility of Sparingly Soluble Salts · Dissociation Constants

Kohlrausch’s law states that at infinite dilution, each ion migrates independently and contributes a fixed amount to the total molar (or equivalent) conductivity, irrespective of the counter-ion. The total conductivity of an electrolyte is the simple sum of the contributions of its cations and anions. This principle enables the determination of limiting conductivities for weak electrolytes, absolute ionic mobilities, transport numbers, solubility products, and dissociation constants.

Statement & Fundamental Equation

Λ_m^∞ = ν₊·λ₊^∞ + ν_–·λ_–^∞

where λ₊^∞ and λ_–^∞ are the limiting ionic conductivities (S·cm²·mol⁻¹), and ν₊, ν_– are the numbers of cations and anions per formula unit.

📊 Experimental verification (NaCl at 25°C)

Ion / Electrolyte	λ^∞ or Λ^∞ (S·cm²·mol⁻¹)
Na⁺	50.11
Cl^–	76.34
NaCl (Λ_m^∞)	126.45

Sum of ionic contributions = 50.11 + 76.34 = 126.45 S·cm²·mol⁻¹, proving additivity.

📋 Table of Limiting Ionic Conductivities (25°C, aqueous)

Cation	λ₊^∞	Anion	λ_–^∞
H⁺	349.8	OH^–	198.6
Li⁺	38.69	F^–	55.4
Na⁺	50.11	Cl^–	76.34
K⁺	73.52	Br^–	78.14
Rb⁺	77.8	I^–	76.8
Cs⁺	77.3	NO₃^–	71.44
Ag⁺	61.92	CH₃COO^–	40.9
Ca²⁺	119.0	SO₄^2-	160.0
Ba²⁺	127.2	CO₃^2-	138.6
Mg²⁺	106.1	PO₄^3-	240.0*

Note Values in S·cm²·mol⁻¹ (molar scale for the ion as written). For polyvalent ions, these refer to 1 mole of ions of that charge.

📐 DIAGRAM 1: Independent Migration of Ions at Infinite Dilution

[Space for illustration: cations (blue) and anions (red) moving freely under electric field without interionic interactions — demonstrating additivity of conductivities]

⬤ ⬤ Cations → | Anions ← ⬤ ⬤

Applications of Kohlrausch’s Law

1️⃣ Limiting Molar Conductivity of Weak Electrolytes

Weak electrolytes (e.g., CH₃COOH, NH₄OH) do not fully dissociate even at infinite dilution; direct extrapolation to Λ_m^∞ is impossible. Using Kohlrausch’s law, Λ_m^∞ = λ_cation^∞ + λ_anion^∞.

Acetic acid (CH₃COOH): Λ_m^∞ = λ(H⁺) + λ(CH₃COO^–)
= 349.8 + 40.9 = 390.7 S·cm²·mol⁻¹

2️⃣ Absolute Ionic Mobilities (u_i)

Absolute mobility is the ion’s velocity (cm/s) under a potential gradient of 1 V/cm. Relation:

λ_i^∞ = z_i·F·u_i ⇒ u_i = λ_i^∞ / (z_i·F)
F = 96485 C·mol⁻¹

Example (Na⁺): u_Na+ = 50.11 / (1 × 96485) ≈ 5.19×10^-4 cm²·V⁻¹·s⁻¹.

If electrodes are 20 cm apart with 100 V, potential gradient = 5 V/cm, drift velocity v = u × gradient = 2.60×10^-3 cm·s⁻¹.

3️⃣ Solubility of Sparingly Soluble Salts

For a saturated solution of a salt like AgCl, the specific conductance κ (after subtracting water conductance) is related to solubility (c in mol/L) by: c = κ / Λ_m^∞. Λ_m^∞ obtained via Kohlrausch sum.

AgCl : Λ_m^∞ = λ(Ag⁺) + λ(Cl^–) = 61.92 + 76.34 = 138.26 S·cm²·mol⁻¹
Measured κ (sat. AgCl) = 1.89×10^-6 S·cm⁻¹ ⇒ c = (1.89×10^-6)/138.26 ≈ 1.367×10^-5 mol·cm⁻³ → 1.367×10^-2 mol/L.

4️⃣ Degree of Dissociation (α) & Dissociation Constant (K_a)

For weak electrolyte at concentration c, measured molar conductivity Λ_m^c gives α = Λ_m^c / Λ_m^∞. Then K_a = c α²/(1-α).

For 0.001 M CH₃COOH: Λ_m^c = 48.15 , Λ_m^∞ = 390.7 ⇒ α = 48.15/390.7 ≈ 0.1233
K_a = (0.001 × 0.1233²)/(1-0.1233) = 1.73×10^-5 (literature 1.75×10^-5)

5️⃣ Transport Numbers (t₊, t_–) at Infinite Dilution

t₊ = ν₊λ₊^∞ / Λ_m^∞ , t_– = ν_–λ_–^∞ / Λ_m^∞

Example: HCl → t_H+ = 349.8/(349.8+76.34)=0.821 ; t_Cl-=0.179.

⚡ DIAGRAM 2: Conductivity Cell for Saturated Solution Measurement

[Space for schematic: platinum electrodes immersed in saturated sparingly soluble salt solution, connected to conductance bridge. Used to measure κ and compute solubility via Kohlrausch sum.]

[ ▪▪ Electrodes ▪▪ ] —— Salt solution ——

🧮 Interactive Calculators

➕ Compute Λ_m^∞ for any electrolyte

Select Cation

Select Anion

👉 Result will appear here

* For 1:1 electrolytes ν₊=ν_–=1; for CaSO₄ stoichiometry considered in calculation.

💧 Solubility of AgCl (sparingly soluble)

Enter measured specific conductance κ (S·cm⁻¹) of saturated AgCl solution (corrected for water). The known Λ_m^∞ (AgCl) = 138.26 S·cm²·mol⁻¹ is used.

κ (S·cm⁻¹)

Solubility = 1.367×10⁻² mol/L (using default κ)

🎬 Interactive Video Lecture Series

Download Complete Notes for Kohlrausch’s Law Below

download PDF

(4) Calculation of the Degree of Dissociation or Conductance Ratio

An electrolyte’s apparent degree of dissociation, α, at dilution V can be found using the formula, α * λv/λ_∞ where λv is the electrolyte’s equivalent conductance at dilution V, and λ_∞is its equivalent conductance at infinite dilution.
Kohlrausch’s Law states that this is equal to the sum λ _anion and λ _cation.

(5) Calculation of the Ionic product for water

The observed specific conductance of the purest water at 25oC is 5.54 × 10^-8mho cm^-1 . The conductance of one litre of water containing 1 gram eqvt of it would be:

When the temperature is the same, the conductance of H⁺ions andOH^– ions is:

λ_H+= 349.8 mho cm^-1

λ _OH– = 198.5 mho cm^-1

According to Kohlrausch’s Law

λ_H₂_O= λ_H++ λ _OH–

349.8 + 198.5 = 548.3 mho cm^-1

Since one water molecule gives one H⁺ ion and one OH^– ion

H₂O = H⁺+ OH^–

Assuming that conductance and ionic concentration are proportionate, we have

[H⁺] = [OH^–] = (is 5.54 × 10^-8) / 548.3= 1.01 × 10^-7g ion litre^-1

The ionic product of water is then

For most purposes, the value of Kw is taken to be 10^-14

⚡ Kohlrausch’s Law of Independent Migration of Ions

Statement & Fundamental Equation

📊 Experimental verification (NaCl at 25°C)

📋 Table of Limiting Ionic Conductivities (25°C, aqueous)

Applications of Kohlrausch’s Law