Transport Number: Hittorf Method and Ionic Mobility Numericals
When current flows through an electrolyte, both the cations and the anions carry it, but not equally. The fraction of the total current carried by one ion is its transport number (or transference number). Questions on Hittorf's method, ionic mobility and the link between t and Kohlrausch's ionic conductivities appear regularly in the electrochemistry sections of IIT-JAM, GATE, CSIR-NET and CUET-PG.
Definitions and key relations
t+ + t− = 1
Ionic mobility: u = λ / F (cm2 V−1 s−1 when λ is in S cm2 mol−1)
- t+, t− are dimensionless fractions between 0 and 1.
- u is the mobility: the drift speed of the ion per unit electric field (cm/s per V/cm).
- λ+, λ− are the molar ionic conductivities and Λ = λ+ + λ− for a 1:1 electrolyte at the limit of infinite dilution.
- F = 96485 C mol−1.
Hittorf's method
Electrolyse a solution between two electrodes. Chemical change at the electrodes depends only on the total charge, but the change in the amount of electrolyte near each electrode depends on how fast the ions move. Divide the cell into anode, middle and cathode compartments. Analyse the anode and cathode compartments before and after, and measure the total charge with a coulometer in series.
For a silver nitrate solution with silver electrodes, with n moles of electrons passed:
- Anode compartment: n mol of Ag dissolves, adding n mol Ag+. Meanwhile t+n mol Ag+ migrates out and t−n mol NO3− migrates in. Net gain of AgNO3 = n − t+n = t−n.
- Cathode compartment: n mol Ag+ is deposited, t+n mol Ag+ migrates in. Net loss of AgNO3 = n − t+n = t−n.
So with Ag electrodes, the anode compartment gains and the cathode compartment loses the same amount of AgNO3, equal to t− times the total moles of electrons. (With inert electrodes the accounting changes, so always write down what happens at each electrode first.)
Example 1: t+ from the anode compartment. A silver coulometer in series with a Hittorf cell deposits 0.3200 g of Ag. Analysis shows that the AgNO3 in the anode compartment (silver electrodes) has increased by 0.2671 g. Find t+ and t− for Ag+ and NO3−. Use M(Ag) = 107.87 and M(AgNO3) = 169.87 g mol−1.
Moles of electrons n = 0.3200 / 107.87 = 2.9665 × 10−3 mol (1 mol Ag = 1 mol e−).
Gain of AgNO3 = 0.2671 / 169.87 = 1.5724 × 10−3 mol.
t− = gain / n = 1.5724 / 2.9665 = 0.530, so t+ = 1 − 0.530 = 0.470.
Check: n(1 − t+) = 2.9665 × 0.5300 = 1.572 mmol, which matches the measured gain.
Example 2: t− from the cathode compartment. In a second experiment on another AgNO3 solution, 0.4500 g of Ag is deposited in the coulometer. The cathode compartment loses 2.210 mmol of AgNO3. Find t− and t+.
n = 0.4500 / 107.87 = 4.1717 × 10−3 mol = 4.1717 mmol.
Loss at the cathode = t−n, so t− = 2.210 / 4.1717 = 0.530.
t+ = 1 − 0.530 = 0.470.
The two compartments agree, which is the standard check that the experiment was done correctly.
Example 3: mobility from ionic conductivity. For KCl at infinite dilution take λ(K+) = 73.5 and λ(Cl−) = 76.3 S cm2 mol−1 (typical textbook values). Find t+, t− and the mobilities.
Λ = 73.5 + 76.3 = 149.8 S cm2 mol−1.
t+ = 73.5 / 149.8 = 0.491; t− = 76.3 / 149.8 = 0.509; the sum is 1.000.
u(K+) = 73.5 / 96485 = 7.62 × 10−4 cm2 V−1 s−1.
u(Cl−) = 76.3 / 96485 = 7.91 × 10−4 cm2 V−1 s−1.
Drift speed of K+ in a field of 10 V cm−1 = 7.62 × 10−4 × 10 = 7.62 × 10−3 cm s−1. The ions drift slowly, less than a hundredth of a centimetre per second.
Example 4: finding ionic conductivities from t and Λ. For NaCl, Λ = 126.5 S cm2 mol−1 at infinite dilution and the question states t(Na+) = 0.396. Find λ(Na+) and λ(Cl−).
λ(Na+) = t+ × Λ = 0.396 × 126.5 = 50.1 S cm2 mol−1.
λ(Cl−) = 126.5 − 50.1 = 76.4 S cm2 mol−1 (t− = 0.604, and 0.604 × 126.5 = 76.4, which agrees).
Common mistakes
- Saying the anode compartment of AgNO3 loses electrolyte. With Ag electrodes it gains, because Ag dissolves there.
- Using the compartment change as t+ itself. With Ag electrodes it equals t− times n.
- Forgetting that mobility has units of cm2 V−1 s−1, not cm/s.
- Using the Hittorf formula for inert electrodes without rewriting the electrode reactions.
- Applying t = λ/Λ to a 2:1 electrolyte without including the charges and stoichiometry.
- Thinking that the ion with the larger t is always the larger ion. A larger t means higher mobility, which also depends on hydration.
Exam-relevance table
| Item | Formula | Typical question |
|---|---|---|
| Transport number | t+ = u+/(u++u−) | Find t from mobilities |
| Hittorf method | Change in compartment = t × n | Find t from analysis data |
| Mobility | u = λ/F | Convert λ to u, find drift speed |
| Kohlrausch link | λ+ = t+Λ | Split Λ into ionic parts |
Exact data tables and constants differ from paper to paper, so use the values given in your question. Check the current official notification for the syllabus of your exam.
Keep F, atomic masses and other constants at full precision while you work through these numericals.
Open the Scientific Constants Tool →Previous-year CSIR-NET questions and timed quizzes to check what you just read.
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