Transport Numbers

Dissolve a pinch of copper sulfate at one end of a tube, and pass a steady current through the solution. Copper ions drift toward the cathode, sulfate ions drift toward the anode, but they do not move at the same speed. This asymmetry means the two ions do not share the current equally. Transport numbers quantify exactly this imbalance. They tell us what fraction of the total current each ion species carries in an electrolyte solution. The idea matters because conductivity alone only gives the sum of ionic contributions. It cannot separate a cation's role from an anion's role. This article defines the transport number rigorously, connects it to ionic mobility, and derives the working equations. You will see the formal definition first. Then you will work through two numeric examples. Finally you will meet the classic Hittorf method that measures these numbers experimentally.

Key takeaways

The transport number $t_i$ is the fraction of total current carried by ion species $i$ in solution.
Transport numbers satisfy $t_+ + t_- = 1$ for a simple binary electrolyte.
$t_i$ depends on both the ionic mobility $u_i$ and the ion's charge number, weighted by concentration.
Transport numbers are not fixed properties of an ion alone; they depend on the identity of the partner ion.
Hittorf and moving-boundary experiments measure transport numbers directly from concentration or boundary changes.
Key idea: each ion's share of the current is proportional to its contribution to the solution's conductivity, not to its concentration alone.

The core insight

A transport number is the fraction of the total ionic current that a given ion carries, and because faster or more highly charged ions sweep more charge past any cross-section per unit time, this fraction is set by each ion's molar conductivity relative to the total molar conductivity of the electrolyte.

The physical picture

Picture a solution of electrolyte between two flat electrodes, with a uniform electric field $E$ pointing from anode to cathode. Every ion feels a force $zeE$ and drifts with a terminal velocity set by its ionic mobility, $u_i$ , defined so that the drift speed is $s_i = u_i E$ . Cations drift one way, anions drift the other way, and both motions push conventional current in the same direction, because a positive charge moving right and a negative charge moving left both correspond to current flowing right. The total current density is the sum of contributions from every ionic species present. But the ions do not usually move at the same speed. A small, highly mobile ion like H⁺ (mobility about $3.63\times10^{-7}\ \text{m}^2\text{s}^{-1}\text{V}^{-1}$ ) vastly outpaces a bulky ion like acetate. When mobilities differ, the faster ion carries a disproportionate share of the current, even though both species carry exactly the same total charge through electroneutrality in a simple 1:1 salt. This is precisely what the transport number captures: not how much charge an ion carries in total (that is fixed by stoichiometry), but how much of the current, i.e. charge per unit time crossing a reference plane, it is responsible for. A transport number close to 1 means that ion is the dominant charge-shuttle; a transport number close to 0 means the ion barely contributes, even though it is present in full concentration. It helps to visualize the drift picture directly, and then to compare relative contributions with a real numerical example, KCl(aq), where the two ions happen to have unusually similar mobilities.

Figure

Migration of cations and anions between electrodes carrying current I

The core relations

t_i = \frac{I_i}{I_{\text{total}}}

where $I_i$ is the current carried by ion $i$ and $I_{\text{total}}$ the total ionic current.

t_+ + t_- = 1

where this holds for any solution containing only one cation and one anion type.

t_i = \frac{z_i u_i c_i}{\sum_j z_j u_j c_j}

where $z_i$ is the (unsigned) charge number, $u_i$ the ionic mobility, $c_i$ the concentration of ion $i$ .

t_i = \frac{\lambda_i}{\Lambda_m}

where $\lambda_i$ is the molar ionic conductivity of ion $i$ and $\Lambda_m = \lambda_+ + \lambda_-$ is the molar conductivity of the electrolyte.

t_+ = \frac{u_+}{u_+ + u_-}, \qquad t_- = \frac{u_-}{u_+ + u_-}

where this simplified form applies to a 1:1 electrolyte with equal stoichiometric numbers of each ion.

Derivation AFrom ionic drift velocities to the transport number

Goal. Show that the transport number of an ion equals its fractional contribution to the total ionic current, and reduce this to a ratio of mobilities for a 1:1 electrolyte.

Step 1. Current from a single ionic species. Consider ions of species $i$ , charge number $z_i$ , number density $n_i$ (ions per unit volume), drifting with speed $s_i$ through a tube of cross-sectional area $A$ . In time $\Delta t$ , every ion within a distance $s_i \Delta t$ of a reference plane crosses it. The volume swept is $A s_i \Delta t$ , so the number of ions crossing is $n_i A s_i \Delta t$ , each carrying charge $z_i e$ . The current contributed is

I_i = \frac{z_i e \, n_i A s_i \Delta t}{\Delta t} = z_i e n_i A s_i .

Step 2. Introduce mobility. By definition the ionic mobility relates drift speed to field strength, $s_i = u_i E$ . Substituting,

I_i = z_i e n_i A u_i E .

This shows the current from species $i$ scales with its charge, concentration, mobility, and the applied field — exactly the quantities that distinguish one ion from another.

Step 3. Total current and the transport number definition. For a solution containing cations and anions only, the total current is the sum over both species,

I_{\text{total}} = I_+ + I_- = eAE\left(z_+ n_+ u_+ + z_- n_- u_-\right).

Both terms are positive contributions to current in the same direction (cations and anions drift oppositely but push current the same way). The transport number is the ratio

t_i = \frac{I_i}{I_{\text{total}}} = \frac{z_i n_i u_i}{z_+ n_+ u_+ + z_- n_- u_-} .

Converting number density to molar concentration ( $n_i = c_i N_A$ ) leaves the ratio unchanged, giving the concentration-mobility form quoted above.

Step 4. Reduce to the 1:1 case. For a 1:1 electrolyte such as KCl, electroneutrality demands $z_+ n_+ = z_- n_-$ (equal numbers of unit positive and unit negative charges). The common factor $z n$ cancels top and bottom, leaving

t_+ = \frac{u_+}{u_+ + u_-}, \qquad t_- = \frac{u_-}{u_+ + u_-}.

Step 5. Connect to molar conductivity. Since the molar ionic conductivity is defined as $\lambda_i = z_i u_i F$ (with $F$ the Faraday constant), dividing $I_i$ by $I_{\text{total}}$ is equivalent to dividing $\lambda_i$ by $\Lambda_m = \lambda_+ + \lambda_-$ . This gives the practically useful form

\boxed{t_i = \dfrac{\lambda_i}{\Lambda_m}}

which lets transport numbers be computed directly from tabulated limiting molar conductivities.

Worked examples

Example 1 — KCl(aq), foundational. At 298 K the limiting molar ionic conductivities are $\lambda(\text{K}^+) = 7.35\ \text{mS m}^2\text{mol}^{-1}$ and $\lambda(\text{Cl}^-) = 7.63\ \text{mS m}^2\text{mol}^{-1}$ (equivalently $73.5$ and $76.3\ \text{S cm}^2\text{mol}^{-1}$ , standard tabulated values). Step 1: total molar conductivity.

\Lambda_m = \lambda_+ + \lambda_- = 73.5 + 76.3 = 149.8\ \text{S cm}^2\text{mol}^{-1}.

Step 2: transport numbers.

t_+ = \frac{73.5}{149.8} = 0.4907, \qquad t_- = \frac{76.3}{149.8} = 0.5093.

Check: $t_+ + t_- = 0.4907 + 0.5093 = 1.0000$ . ✓ This matches the well-known experimental result that K⁺ and Cl⁻ are almost equally mobile, so KCl is the classic choice for salt bridges — neither ion dominates charge transport, minimizing junction potentials.

Example 2 — CaCl₂(aq), applied, unequal stoichiometry. Now consider calcium chloride, where the cation is doubly charged and there are two chloride ions per formula unit. At 298 K the limiting molar ionic conductivities (per mole of charge, i.e. "equivalent" conductivities) are

\lambda\!\left(\tfrac12\text{Ca}^{2+}\right) = 5.95\ \text{mS m}^2\text{mol}^{-1}, \qquad \lambda(\text{Cl}^-) = 7.63\ \text{mS m}^2\text{mol}^{-1}

(i.e., $59.5$ and $76.3\ \text{S cm}^2\text{mol}^{-1}$ on a per-equivalent basis). Working per mole of charge keeps the stoichiometric factors consistent, since transport numbers must be computed from equivalent conductivities when ion charges differ. Step 1: Total equivalent conductivity. Because each mole of CaCl₂ supplies 2 equivalents of Ca²⁺ and 2 equivalents of Cl⁻, we sum the two ionic equivalent conductivities directly:

\Lambda_{\text{eq}} = 59.5 + 76.3 = 135.8\ \text{S cm}^2\text{mol}^{-1}_{\text{eq}}.

Step 2: Transport number of Ca²⁺.

t(\text{Ca}^{2+}) = \frac{59.5}{135.8} = 0.438.

Step 3: Transport number of Cl⁻.

t(\text{Cl}^-) = \frac{76.3}{135.8} = 0.562.

Check: $0.438 + 0.562 = 1.000$ . ✓ Even though there are twice as many Cl⁻ ions as Ca²⁺ ions in solution, this is already built into the equivalent-conductivity bookkeeping (each Cl⁻ carries one unit charge and there are two per formula unit, matching the two positive charges on one Ca²⁺). The result shows chloride still carries the majority of the current, but less dominantly than the raw ion count might suggest, because Ca²⁺, despite being doubly charged, is a fairly mobile small ion once hydration effects are included. Sanity check on magnitude: both transport numbers lie between 0 and 1, and their sum is exactly 1, confirming internal consistency with the defining relation $t_+ + t_- = 1$ .

Limiting cases & common traps

Situation	What happens	Common error
Equal mobilities ( $u_+ = u_-$ )	$t_+ = t_- = 0.5$ exactly	Assuming this holds for all salts (it is special to ions like K⁺/Cl⁻)
One ion much faster (e.g. H⁺ in HCl)	$t_+ \to 1$ , $t_- \to 0$	Forgetting that H⁺ transport is anomalously fast due to the Grotthuss mechanism
Multivalent ions	Must use equivalent conductivities, not molar	Mixing raw molar conductivity with unit charge formulas, breaking $t_+ + t_- = 1$
Mixed electrolytes (two salts sharing an ion)	Each ion's $t_i$ depends on the full ionic composition	Treating $t_i$ as an intrinsic constant of the ion alone
Very dilute vs. concentrated solution	$t_i$ drifts slightly with concentration due to ionic interactions	Assuming transport numbers are strictly constant across all concentrations

The chemist behind it

Transport numbers were first measured systematically by Johann Wilhelm Hittorf in the 1850s. Hittorf passed a steady current through an electrolyte cell divided into anode, middle, and cathode compartments, and analyzed how the concentration of electrolyte changed near each electrode after electrolysis. Because the change in concentration near an electrode depends on how much of the current that region's ions carried away versus how much arrived, careful mass-balance bookkeeping yields the transport number directly. This "Hittorf method" remains the conceptual ancestor of modern transport-number measurements, including the moving-boundary technique, which tracks a visible boundary between two electrolytes as current passes and computes $t_i$ from the boundary's displacement.

Origins · 1 minJohann Wilhelm Hittorf and how this was discovered

Johann Wilhelm Hittorf

1824–1914 · German chemist and physicist who first measured transport numbers via electrolysis mass-balance, laying groundwork for quantitative electrochemistry

Ask the tutor: How does the Hittorf method turn a simple concentration change into a precise transport number?

Do transport numbers always add up to exactly 1?

Yes, for any solution containing exactly one cation species and one anion species, $t_+ + t_- = 1$ by definition, since together they must account for the entire current. In solutions with more than two ionic species — for example, a mixture of two salts sharing a common ion — the transport numbers of all species present still sum to 1, but each individual $t_i$ becomes smaller and depends on the full ionic composition, not just one partner ion.

How is a transport number different from an ionic mobility?

Ionic mobility, $u_i$ , is an intrinsic property of an ion in a given solvent at a given temperature. It measures drift speed per unit field, independent of what other ions are present. A transport number, by contrast, is a relative quantity: it is the ion's mobility contribution divided by the total mobility-weighted contribution from all ions in that specific solution. Changing the counter-ion changes $t_i$ even though $u_i$ itself may stay nearly the same.

Why is the transport number of H⁺ or OH⁻ unusually large?

H⁺ and OH⁻ move through water via the Grotthuss mechanism, a rapid hopping of protons along hydrogen-bonded water chains, rather than simple physical diffusion of the ion itself. This gives them mobilities several times larger than typical ions like Na⁺ or Cl⁻. In HCl solution, for instance, $t(\text{H}^+)$ is around 0.82–0.83 at moderate concentrations, meaning hydrogen ions carry the large majority of the current despite being present in equal molar amount to Cl⁻.

How are transport numbers actually measured in the laboratory?

The two classic techniques are the Hittorf method and the moving-boundary method. The Hittorf method electrolyzes a three-compartment cell and measures concentration changes near each electrode after a known charge has passed, then back-calculates $t_i$ from mass balance. The moving-boundary method directly tracks the physical displacement of a visible boundary between two electrolyte solutions as current flows, giving $t_i$ from the volume swept per unit charge. Both are explored in detail in the discussion of measuring transport numbers.