Ionic Conductance, Kohlrausch’s Law and Transport Numbers

The quantities

Conductivity is conductance per unit dimension of the cell. Dividing by concentration gives molar conductivity, which measures the conducting ability per mole of electrolyte and is therefore the quantity to compare between solutions.

Molar conductivity increases on dilution for every electrolyte, but the shape of the increase differs sharply between strong and weak electrolytes — and that difference is the topic's central observation.

Strong versus weak electrolytes

Kohlrausch's law of independent migration

At infinite dilution, ions move independently and each contributes a characteristic amount:

Because the contributions are additive and independent of the counter-ion, the limiting conductivity of an electrolyte that cannot be measured directly can be assembled from ones that can. Combining measured values for three strong electrolytes to obtain the value for a weak acid is the standard numerical.

What it lets you calculate

Degree of dissociation

Measure molar conductivity at a given concentration, divide by the limiting value from Kohlrausch's law, and the ratio is the fraction dissociated.

Dissociation constant

Substituting that degree of dissociation into the equilibrium expression gives the acid dissociation constant. This is a genuinely useful method, since it requires only conductance measurements.

Solubility of a sparingly soluble salt

For a salt too insoluble to weigh out usefully, measuring the conductivity of its saturated solution and dividing by the limiting molar conductivity gives the concentration — hence the solubility and the solubility product. Correcting for the conductivity of the water itself is essential here, and forgetting to do so is the usual error.

Transport numbers

The transport number of an ion is the fraction of the total current it carries:

Ions do not carry current equally, because they differ in mobility. Transport numbers are measured by the Hittorf method, which follows concentration changes around the electrodes, or by the moving boundary method, which tracks a visible boundary between two solutions.

Why hydrogen and hydroxide ions are exceptional

Both have anomalously high mobilities, far above what their size would suggest. The explanation is the Grotthuss mechanism: rather than the ion physically migrating the whole distance, a proton is passed along a chain of hydrogen-bonded water molecules, so charge moves much faster than matter.

This is a favourite question, and the answer must describe the hopping mechanism rather than simply noting that the ions are small.

Frequently asked questions

Why does conductivity itself fall on dilution while molar conductivity rises?

Because conductivity depends on the number of ions per unit volume, which falls as the solution is diluted. Molar conductivity divides by concentration, so it isolates the per-mole ability to conduct, which improves as interionic interference decreases.

Why do strong electrolytes plot linearly against the square root of concentration?

Because interionic effects, described by Debye–Hückel–Onsager theory, vary with the square root of ionic strength. That is the same square-root dependence seen in activity coefficients.

Why must water conductivity be subtracted in solubility measurements?

Because a sparingly soluble salt contributes very little conductivity, comparable to that of the water itself. Ignoring the solvent contribution can produce an error of tens of percent.

Can transport numbers exceed one?

No. They are fractions of the total current and must sum to one. A value outside zero to one indicates an experimental or arithmetic error.