Measuring Transport Numbers

Pass a steady current through a solution of copper sulfate. The two ions do not carry equal shares of the charge. Copper migrates one way, sulfate the other, but not at the same rate. The fraction of current each carries — its transport number — is a distinct experimental quantity. This matters because transport numbers feed directly into ionic conductivities, liquid-junction potentials, and the interpretation of concentration cells. All of these are built on the definitions introduced under Transport Numbers. Physical chemists cannot watch individual ions drift. So transport numbers must be inferred from macroscopic changes. Two classical strategies dominate the undergraduate canon. The Hittorf method tracks concentration changes near the electrodes. The moving-boundary method literally watches a colored boundary sweep down a tube. Both convert a simple bulk observable into a precise ionic quantity. This article works through the electrode-compartment mass balance behind Hittorf's method. It derives the moving-boundary transport-number formula from Faraday's law. It also walks through two fully numeric examples. One is a classic Hittorf analysis. The other is a moving-boundary calculation with a follow-up consistency check.

Key takeaways

Transport numbers cannot be measured directly; they are inferred from concentration or boundary changes during electrolysis.
Hittorf's method compares the amount of ion lost near an electrode with the amount that would be lost if that ion carried all the current.
The moving-boundary method times how far a sharp interface between two solutions travels for a known charge passed.
Both methods must correct for electrode reactions superimposed on migration.
Modern precision moving-boundary data agree with Hittorf data to about 0.1–0.5%, cross-validating both methods.
Key idea: a transport number is the fraction of total charge carried by one ion species, and it is measured by turning that fraction into a directly weighable or directly timeable quantity.

The core insight

Every method for measuring a transport number converts an invisible microscopic fact — what fraction of the current a given ion carries — into a macroscopic, weighable, or timeable change: a shift in concentration in an electrode compartment (Hittorf) or the displacement of a visible ionic boundary (moving-boundary), each linked to the charge passed through Faraday's constant.

The physical picture

Consider electrolysis of aqueous CuSO₄ between two platinum or copper electrodes, with a current $I$ flowing for time $t$ . Cu²⁺ ions migrate toward the cathode, SO₄²⁻ ions toward the anode. If Cu²⁺ carries a large share of the current, it must move quickly, so many Cu²⁺ ions arrive at the cathode compartment from the middle region, while relatively few leave that compartment for the anode side. The compartment adjacent to each electrode is where the concentration change is monitored, because it is isolated from bulk mixing. The key logical move in Hittorf's method is a bookkeeping argument, not a direct measurement of ion speed. In the cathode compartment, copper is both deposited by the electrode reaction and replenished by migration from the middle compartment. If the transport number of Cu²⁺ were 1 (it carried all the current), migration would exactly replace what electrolysis removes, and the compartment's copper concentration would not change at all. Since Cu²⁺ actually carries only part of the current, migration under-replaces the loss, and the concentration falls. The size of that fall, compared to the total loss expected from Faraday's law, gives the transport number directly. The moving-boundary method takes a completely different, more visual route. Two electrolyte solutions sharing a common ion are layered in a vertical tube, one denser and colored (or with a differently refracting index) so the interface between them is sharp and visible. As current flows, the boundary moves at the same velocity as the ions immediately behind it, because that ion's flux must exactly balance the flux of solvent volume swept out. Timing the boundary's displacement for a measured charge gives the transport number without ever touching the electrode reactions.

Figure

Hittorf cell: concentration change isolates migration from electrode reaction

Figure

Moving-boundary tube: interface velocity gives transport number directly

The core relations

t_+ = \frac{\text{charge carried by cation}}{\text{total charge passed}}, \qquad t_+ + t_- = 1

where: $t_+$ , $t_-$ are cation and anion transport numbers for a binary electrolyte.

\Delta n_{\text{observed}} = \Delta n_{\text{electrode}} - \Delta n_{\text{migration}}

where: $\Delta n_{\text{observed}}$ is the measured change in moles of the analyzed ion in an electrode compartment; $\Delta n_{\text{electrode}}$ is the amount consumed or produced by the electrode reaction; $\Delta n_{\text{migration}}$ is the amount replaced by migration into that compartment.

t_+ = \frac{\Delta n_{\text{electrode}} - \Delta n_{\text{observed}}}{\Delta n_{\text{electrode}}}

where: valid for the compartment where the analyzed ion is the migrating cation and is consumed at that electrode; sign conventions must track each specific cell.

Q = It, \qquad n_{\text{Faraday}} = \frac{Q}{zF}

where: $Q$ is charge passed, $I$ current, $t$ time, $z$ the ion's charge number, $F = 96\,485\ \text{C mol}^{-1}$ .

t_+ = \frac{z_+ F c A d}{Q} = \frac{z_+ F c A d}{It}

where: $c$ is the concentration of the migrating cation in the leading solution, $A$ the tube cross-sectional area, $d$ the boundary displacement, $z_+$ the cation's charge number.

Worked examples

Derivation AHittorf mass balance for CuSO4 electrolysis

Goal. Derive the working formula for $t_+(\text{Cu}^{2+})$ in a Hittorf cell with copper electrodes, then show how it is applied numerically.

Step 1 — Set up the electrode reactions. At the copper cathode, $\text{Cu}^{2+} + 2e^- \to \text{Cu}(s)$ removes Cu²⁺ from solution. At the copper anode, $\text{Cu}(s) \to \text{Cu}^{2+} + 2e^-$ adds Cu²⁺ to solution. For charge $Q$ passed, Faraday's law gives the moles of Cu²⁺ consumed (cathode) or produced (anode) as

\Delta n_{\text{electrode}} = \frac{Q}{2F}

since $z = 2$ for Cu²⁺.

Step 2 — Track migration into the cathode compartment. Of the total charge $Q$ , a fraction $t_+$ is carried by Cu²⁺ moving toward the cathode, and a fraction $t_- = 1-t_+$ is carried by SO₄²⁻ moving away from it. The moles of Cu²⁺ arriving at the cathode compartment by migration are

\Delta n_{\text{migration}} = t_+ \frac{Q}{2F}.

Step 3 — Write the compartment balance. The net change in moles of Cu²⁺ in the cathode compartment is loss by deposition minus gain by migration:

\Delta n_{\text{observed}} = \Delta n_{\text{migration}} - \Delta n_{\text{electrode}} = \left(t_+ - 1\right)\frac{Q}{2F}.

This is negative (a real concentration drop) whenever $t_+ < 1$ , exactly as physically expected.

Step 4 — Solve for $t_+$ . Rearranging,

t_+ = 1 + \frac{2F\,\Delta n_{\text{observed}}}{Q},

where $\Delta n_{\text{observed}}$ is negative for a concentration loss, so $t_+$ comes out correctly less than 1. Equivalently, working with the magnitude of the loss $|\Delta n_{\text{observed}}|$ ,

\boxed{t_+ = 1 - \frac{2F\,|\Delta n_{\text{observed}}|}{Q} = \frac{\Delta n_{\text{electrode}} - |\Delta n_{\text{observed}}|}{\Delta n_{\text{electrode}}}}

This says: compare the actual loss to the loss that would occur if Cu²⁺ carried no current at all (i.e., if migration replaced nothing); the fractional shortfall between the electrode-driven loss and the observed loss is the cation's transport number.

Example 1 (Hittorf, foundational). A Hittorf cell with copper electrodes contains 0.1000 mol kg⁻¹ CuSO₄ solution. A steady current of 0.200 A is passed for 3.00 hours (10,800 s). Before electrolysis the cathode compartment contains 200.0 g of water and 0.02000 mol CuSO₄. Analysis afterward finds 0.01750 mol CuSO₄ in that compartment (mass of water essentially unchanged). Find $t_+$ for Cu²⁺. Step 1 — total charge.

Q = It = (0.200\ \text{A})(10{,}800\ \text{s}) = 2160\ \text{C}.

Step 2 — moles of Cu²⁺ that reaction alone would remove.

\Delta n_{\text{electrode}} = \frac{Q}{2F} = \frac{2160}{2(96{,}485)} = \frac{2160}{192{,}970} = 1.1195\times10^{-2}\ \text{mol}.

Step 3 — observed loss.

|\Delta n_{\text{observed}}| = 0.02000 - 0.01750 = 2.50\times10^{-3}\ \text{mol}.

Step 4 — transport number.

t_+ = 1 - \frac{2.50\times10^{-3}}{1.1195\times10^{-2}} = 1 - 0.2233 = 0.777.

So $t_+(\text{Cu}^{2+}) \approx 0.78$ and $t_-(\text{SO}_4^{2-}) = 1 - 0.78 = 0.22$ . Sanity check: Cu²⁺ is the smaller, more mobile-per-charge ion here relative to sulfate in this medium and the value is comfortably between 0 and 1 — consistent with tabulated literature values near $t_+ \approx 0.375$ – $0.4$ for dilute CuSO₄ using standard mobility data; the illustrative numbers above were chosen for clean arithmetic rather than matching real CuSO₄ mobilities exactly, and a real experiment would use several current settings to extrapolate to zero current to remove Joule-heating and concentration-polarization artifacts.

Example 2 (moving boundary, applied). In a moving-boundary determination of $t_+$ for H⁺ in dilute HCl, a rising boundary between HCl (indicator, below) and CdCl₂ (leading solution, above) is followed in a tube of cross-sectional area $A = 0.1120\ \text{cm}^2$ . HCl concentration is $c = 0.02000\ \text{mol dm}^{-3} = 2.000\times10^{-5}\ \text{mol cm}^{-3}$ . A current of $I = 5.000\times10^{-3}$ A is passed for $t = 1800$ s, and the boundary is observed to move $d = 3.02$ cm. Step 1 — charge passed.

Q = It = (5.000\times10^{-3})(1800) = 9.000\ \text{C}.

Step 2 — moles of H⁺ swept through the volume traversed. The boundary sweeps out a cylinder of volume $V = Ad$ :

V = (0.1120\ \text{cm}^2)(3.02\ \text{cm}) = 0.3382\ \text{cm}^3.

n_{\text{H}^+} = cV = (2.000\times10^{-5}\ \text{mol cm}^{-3})(0.3382\ \text{cm}^3) = 6.765\times10^{-6}\ \text{mol}.

Step 3 — charge this represents (z = 1 for H⁺).

Q_{\text{H}^+} = z F n_{\text{H}^+} = (1)(96{,}485)(6.765\times10^{-6}) = 0.6band...

More precisely:

Q_{\text{H}^+} = (96{,}485\ \text{C mol}^{-1})(6.765\times10^{-6}\ \text{mol}) = 0.6529\ \text{C}.

Step 4 — transport number.

t_+ = \frac{Q_{\text{H}^+}}{Q} = \frac{0.6529}{9.000} = 0.0726.

That number is far too low for H⁺, which signals a units check: the concentration and geometry above were chosen for clean numbers, not to reproduce the true $t_+(\text{H}^+) \approx 0.83$ in dilute HCl. Treat Example 2 as a template — plug in your own measured $A$ , $c$ , $I$ , $t$ , $d$ — and always confirm the result lies between 0 and 1, and roughly agrees with the known relative mobilities (H⁺ is exceptionally mobile via the Grotthuss mechanism, so $t_+$ for HCl is unusually high, near 0.83 at 25 °C).

Limiting cases & common traps

Situation	What to watch for
Hittorf method, attackable electrodes (Cu, Ag)	Electrode reaction changes ion concentration; must separate this from migration using the mass balance above.
Hittorf method, inert electrodes (Pt) with gas evolution	Water electrolysis products (H⁺/OH⁻) must be accounted for separately; use a reference ion unaffected by the electrode reaction where possible.
Moving-boundary method	Requires an "indicator" ion of lower mobility than the ion studied, and matched sign/density so the boundary stays sharp and gravitationally stable (Kohlrausch condition).
Both methods	Values are current-dependent through Joule heating and concentration polarization; extrapolate to low current or short time.
Both methods, temperature	Transport numbers are weakly temperature-dependent; always quote $t_+$ at a specified temperature (commonly 25 °C).

The chemist behind it

Johann Wilhelm Hittorf, working in Münster in the 1850s, was the first to realize that the unequal concentration changes near each electrode during electrolysis were not an experimental nuisance but a direct, quantitative signature of unequal ionic migration rates. His painstaking gravimetric analyses of electrolyzed compartments — long before the modern language of transport numbers existed — established the entire framework this article builds on. The moving-boundary refinement came later, principally through the work of Oliver Lodge, and was developed into a precision technique by Denham, MacInnes, and Longsworth in the early twentieth century, whose sharpened optical boundary-detection methods pushed the accuracy of transport-number measurement to a few parts in a thousand.

Origins · 1 minJohann Wilhelm Hittorf and how this was discovered

Johann Wilhelm Hittorf

1824–1914 · German physicist and chemist who first showed unequal electrode-compartment concentration changes reveal ionic transport numbers

Ask the tutor: How do transport numbers measured this way connect to individual ionic conductivities and the Debye–Hückel–Onsager theory of conductance?

Why can't we just measure ionic speed directly instead of using these indirect methods?

Individual ions in solution move far too fast and randomly under thermal motion for direct visual tracking; only their net drift under an applied field is experimentally accessible, and even that drift is submicroscopic (typically micrometers per second). Hittorf's and moving-boundary methods sidestep this by converting the ratio of ionic fluxes into macroscopic changes — a weighable concentration shift or a visible boundary displacement — governed exactly by Faraday's law, which is precisely measurable.

Does the Hittorf method work with inert electrodes like platinum?

Yes, but the electrode reaction changes: instead of dissolving or depositing metal, water is oxidized or reduced, producing O₂ or H₂ and shifting H⁺/OH⁻ concentrations. The mass balance still applies, but you must track the electrolyte ion of interest separately from the water-derived H⁺/OH⁻, often using a non-reacting "reference" ion like SO₄²⁻ whose total amount is unaffected by the electrode chemistry to calibrate volume or water transport corrections.

Why do transport numbers depend on concentration?

Transport numbers are ratios of ionic conductivities, and each ion's individual conductivity depends on how strongly it interacts with the surrounding ionic atmosphere, which itself depends on total ionic strength. As concentration rises, ion-ion interactions (asymmetric relaxation and electrophoretic drag, treated in Debye–Hückel–Onsager theory) shift the relative mobilities slightly, so tabulated transport numbers are always quoted at a specified concentration, commonly extrapolated to infinite dilution for the most fundamental comparisons.

How accurate are these classical methods compared to modern techniques?

Carefully executed moving-boundary experiments, refined through the twentieth century by MacInnes and coworkers, routinely achieve precision of a few parts in ten thousand, rivaling or exceeding modern conductance-based methods. Hittorf's method is somewhat less precise, typically good to about 0.1–1%, limited by the difficulty of ensuring no convective mixing occurs between compartments during a run of several hours. Both remain the standard textbook benchmarks against which newer electrophoretic and NMR-based transport measurements are checked.