Molar Conductivity Calculator – Conductance, Kohlrausch's Law, Ka and Ksp
An electrolyte solution conducts because its ions move. This molar conductivity calculator turns a conductivity-cell reading into Λm, builds the limiting value Λm° from ions with Kohlrausch's law, fits a dilution series, and uses conductivity to find the dissociation constant of a weak acid or the Ksp of a sparingly soluble salt. It also works forwards, giving the resistance and IR drop a solution will have in a given cell.
Conductance, conductivity and molar conductivity
A cell measures resistance R, or its inverse, the conductance G = 1/R. Conductivity κ removes the cell's geometry: κ = K_cell/R, where the cell constant K_cell = l/A is the electrode gap over the electrode area. Molar conductivity divides by concentration, Λm = κ/c. With κ in S/cm and c in mol/L, Λm = 1000·κ/c in S cm² mol⁻¹. The 1000 is a unit conversion (cm³ per litre), not a constant, and forgetting it, or mixing S cm² with S m², is the commonest error. The tool shows both units, which differ by 10⁴.
On dilution the two quantities move in opposite directions. κ falls, because each cm³ holds fewer ions, but Λm rises, because each mole of ions moves more freely.
The cell constant and KCl standards
Real electrodes don't have a clean gap and area, so cells are calibrated: fill the cell with a KCl standard of known κ, measure its resistance and multiply, K_cell = κ_std × R_std. Pick a standard close to your sample's conductivity and use the value printed on the bottle. Most meters also compensate for temperature at about 2 %/°C, which the tool can apply to an uncompensated reading.
Kohlrausch's two laws
At infinite dilution every ion migrates independently, so Λm° = ν₊λ₊° + ν₋λ₋°. For CaCl₂ that is 118.94 + 2 × 76.31 = 271.56 S cm² mol⁻¹. Tables often list λ per equivalent: λ°(½Ca²⁺) = 59.47 but λ°(Ca²⁺) = 118.94. For a strong electrolyte, Kohlrausch's square-root law, Λm = Λm° − A·√c, is a straight line in dilute solution, and extrapolating it to c = 0 gives Λm°.
A weak electrolyte such as acetic acid curves sharply upwards near c = 0, so its Λm° can't be extrapolated. Kohlrausch's first law rescues it: Λm°(CH₃COOH) = Λm°(HCl) + Λm°(CH₃COONa) − Λm°(NaCl), because Na⁺ and Cl⁻ cancel.
Strong and weak: what Λm/Λm° means
For a strong electrolyte, Λm/Λm° below 1 does not mean missing ions. Every ion is slowed by its oppositely charged ionic atmosphere, which the Debye–Hückel–Onsager limiting law describes for 1:1 salts. In 2:2 salts such as MgSO₄, ion pairing lowers Λm further. For a weak electrolyte the same ratio is roughly the fraction ionised, the Arrhenius degree of dissociation α.
Why H⁺ and OH⁻ are so fast
H⁺ and OH⁻ conduct 3–7 times better than other ions. A proton doesn't push through the water; it hops from one water molecule to the next along the hydrogen-bond network, the Grotthuss mechanism. The ion speed ladder in the Kohlrausch tab shows how far ahead they are.
Ka of acetic acid from conductivity
Measure κ of a dilute acid, compute Λm and α = Λm/Λm°, then apply Ostwald's dilution law, Ka = c·α²/(1 − α). Across a series the classical Ka drifts slightly with concentration, because it ignores activity; that is real chemistry, not noise. For activity corrections use the Ionic Strength Calculator.
Solubility of AgCl or BaSO₄ by conductivity
A saturated solution of a sparingly soluble salt is so dilute that Λm ≈ Λm°, so the solubility is s = 1000·κ_salt/Λm° and Ksp follows from the stoichiometry. The water correction is not optional: for AgCl the water's own conductivity can be nearly half the reading, and ignoring it overstates Ksp several times. For Ksp in general, see the Solubility Product Calculator.
IR drop in electrolytic cells
A cell's ohmic resistance is R = K_cell/κ, and the IR drop at current density j is j·l/κ: at a fixed current density it depends on the gap, not the electrode area. Narrow the gap to cut wasted voltage. For the resistance of a metal wire instead of a solution, use the Wire Resistance Calculator.
The formulas
G = 1/R conductance (S)
K_cell = l/A cell constant (cm⁻¹)
κ = K_cell/R = G·K_cell conductivity (S cm⁻¹)
Λm = 1000·κ/c S cm² mol⁻¹, with κ in S cm⁻¹ and c in mol L⁻¹
Λm° = ν₊λ₊° + ν₋λ₋° Kohlrausch's law of independent migration
Λm = Λm° − A·√c Kohlrausch's square-root law (strong electrolytes)
α = Λm/Λm°, K = c·α²/(1 − α) Ostwald's dilution law (weak 1:1)