Transport Numbers

Introduction

Both cations and anions move in an electrolyte, but they need not carry equal shares of the current. Transport numbers quantify those unequal shares. They matter when interpreting concentration changes near electrodes, designing salt bridges and calculating liquid-junction potentials in practical measurements.

Core explanation

The transport number of ion species i is the fraction of total ionic current attributable to that species under specified conditions: tᵢ=Iᵢ/I total in a suitable current decomposition. For a solution where one cation and one anion are the only relevant mobile charge carriers, t₊+t₋=1. Each lies between zero and one in this simple framework. Opposite physical directions of ion motion do not make one contribution negative in these positive fractions; charge sign and motion direction combine to support conventional current.

At infinite dilution for a 1:1 electrolyte, t₊°=λ₊°/(λ₊°+λ₋°) and t₋°=λ₋°/(λ₊°+λ₋°), with compatible limiting ionic conductivity definitions. If a cation has higher effective mobility, it carries a larger fraction of current. For salts with other stoichiometries, use ionic contributions including formula coefficients rather than copying the 1:1 expression unchanged.

At finite concentration, transport numbers can depend on concentration, solvent and temperature because ion interactions alter mobilities. A value measured for one solution condition is not automatically valid after dilution or replacement of a counterion. In a mixture with several ionic species, sum all species' contributions, and t values for only the two ions of one salt need not add to one if other ions carry current.

Transport numbers can be studied through concentration changes around electrodes during electrolysis, moving-boundary methods or related electrochemical measurements. Interpreting those experiments requires accounting for electrode reactions, solution volume and ion migration. A naive concentration measurement can mix diffusion and convection effects with migration, so a transport number is a model-dependent experimental inference rather than a label printed on an ion.

Transport numbers help explain junction potential. If one ion carries current or diffuses much more readily than its counterion, a concentration boundary can develop greater charge separation before the balancing electric field establishes. Salt bridges often choose ions with relatively similar mobilities to reduce this effect.

Transport number is not the fraction of total dissolved ion particles. A solution can contain equal cation and anion counts for a 1:1 salt while unequal fractions of current are carried. Their velocities and charge magnitudes matter. Nor is it the fraction of electrons passing through the external wire; electron current is linked to the total ionic current at interfaces by charge conservation.

Step-by-step reasoning

1. List all mobile ions and their charge signs. 2. Determine ionic conductivity contributions under the stated conditions. 3. Divide each contribution by the total to obtain tᵢ. 4. Check that all relevant fractions sum to one. 5. Use those fractions for transport or junction reasoning, not as ion-number fractions.

Visual explanation

Draw cations moving right and anions moving left between electrodes. Put two arrows on a conventional-current bar pointing the same way, one labeled 70% and the other 30%. In a second panel draw an unequal-mobility salt boundary producing a small charge separation.

Real-world analogy

Two teams can move supplies along one route even while walking in opposite directions if one carries positive tokens and the other carries negative tokens away. Their shares of total delivered effect depend on how fast and how much they carry, not simply how many people are on each team.

Real-world example

An electrolyte with a highly mobile cation can show t₊ well above one half even when it contains equal numbers of cations and anions. In a concentration cell, that mobility difference contributes to junction potential and must be considered in precise EMF analysis.

Why?

Why can a cation and anion each contribute positive fractions to the same conventional current while moving oppositely? Current direction includes charge sign. Positive charge moving one way and negative charge moving the other transfer electrical charge in the same conventional direction.

Common misconception

“In a 1:1 salt, t₊ must equal t₋ because there is one of each ion per formula unit.” Equal stoichiometric counts do not imply equal mobility. Their current shares differ whenever their effective ionic conductivity contributions differ.

Worked example

At infinite dilution a 1:1 electrolyte has λ₊°=50 and λ₋°=75 S cm² mol⁻¹. Total Λ°=125 S cm² mol⁻¹. Then t₊°=50/125=0.40 and t₋°=75/125=0.60. The anion carries 60% of ionic current under this limiting idealization, not 50% merely because one cation pairs with one anion.

Quick check

1. What is t₊+t₋ in a simple binary electrolyte with no other current carriers? Answer: One. 2. Is t₊ the mole fraction of cations? Answer: No; it is their fraction of ionic current.

Exam focus

Define transport number as current fraction, use correct ionic contributions and check all relevant species. Distinguish motion direction from conventional-current contribution. State concentration and temperature when quoting measured values.

Advanced insight

In concentrated electrolytes, correlated ion motion complicates the idea of assigning independent currents to each species. Transport numbers then depend on the reference frame and theoretical convention. Introductory dilute-solution formulas remain useful, but precise battery-electrolyte modeling needs more careful definitions.

Summary

Transport numbers describe how an electrolyte's ionic current is divided among species. In a simple binary dilute solution, cation and anion fractions sum to one and reflect ionic contributions rather than stoichiometric counts. They influence concentration changes and junction potentials.

Practice questions

1. If t₊=0.35 for a simple binary electrolyte, find t₋. Answer: 1−0.35=0.65. 2. Why might a transport number change after electrolyte concentration rises? Answer: Ion interactions can change the relative mobilities and conductivity contributions. 3. What additional issue arises if a third mobile ion is present? Answer: Its current contribution must be included; the selected cation and anion fractions alone may not sum to one.