🧲 Hittorf’s Rule: Ion Transport & Transference Numbers
Hittorf’s rule (also known as Hittorf’s law of ion migration) describes how the concentration of ions changes around electrodes during electrolysis. It states that the loss of concentration around any electrode is directly proportional to the speed of the ion moving away from that electrode. This fundamental principle led to the concept of transport numbers (transference numbers), which quantify the fraction of electric current carried by each ion species in an electrolyte.
⚡ The Core Idea: Ion Speeds & Concentration Changes
When an electric current passes through an electrolytic cell, cations move toward the cathode and anions move toward the anode. However, cations and anions generally move at different speeds. Hittorf discovered that the decrease in concentration around an electrode is proportional to the velocity of the ion that migrates away from it. For a binary electrolyte:
where \(v_+\) is the cation speed and \(v_-\) the anion speed (cm/s under unit potential gradient).
🧪 Hittorf’s Experimental Cell & The Three Compartments
Hittorf used a three‑compartment cell separated by two porous diaphragms (imaginary planes AA′ and BB′). The middle compartment remains unaffected by electrode reactions, while the outer compartments (anode and cathode compartments) experience concentration changes. The classic illustration with 13 ion‑pairs (4 in each outer compartment, 5 in the middle) before electrolysis demonstrates the effect of unequal ion speeds.
🔬 DIAGRAM 1: Hittorf’s Three‑Compartment Cell
[Anode compartment | AA′ | Middle compartment | BB′ | Cathode compartment]
Ions migrate according to their speeds – concentration changes only in outer compartments.
📖 Three Illustrative Cases (from Hittorf’s original scheme)
- Case (i) – Only anions move: Anions travel toward the anode; cations remain stationary. Concentration decreases only in the cathode compartment.
- Case (ii) – Equal speeds (\(v_+ = v_-\)): Equal number of cations and anions leave the outer compartments. Concentration decreases equally in both anode and cathode compartments.
- Case (iii) – Cations move twice as fast (\(v_+ = 2v_-\)): Anode compartment loses twice as many ion‑pairs as the cathode compartment. Concentration loss ratio = \(v_+/v_- = 2\).
These observations prove that the amount of electrolyte lost from an electrode compartment is directly proportional to the speed of the ion moving away from that electrode. The ions are always discharged in equivalent amounts at both electrodes, but the concentration changes differ due to unequal ionic velocities.
📊 DIAGRAM 2: Ion movement at different speed ratios
[Schematic: Position I (before electrolysis) → Position II, III, IV after electrolysis showing concentration shifts]
Cations (blue +) , Anions (red −) — arrows indicate direction and relative speed.
🔢 Transport (Transference) Numbers: Definition & Relation to Speeds
The transport number \(t_i\) of an ion is the fraction of total current carried by that ion. Since current is proportional to ion velocity and charge, we have:
Clearly, \(t_+ + t_- = 1\). Hittorf’s concentration loss ratio directly gives the ratio of transport numbers:
📊 Transport Numbers from Ionic Conductivities (Modern Method)
At infinite dilution, the transport numbers can be calculated using limiting ionic conductivities \(\lambda_+^\infty\) and \(\lambda_-^\infty\):
Where \(\nu_+\) and \(\nu_-\) are the number of cations and anions per formula unit. The table below lists limiting ionic conductivities at 25 °C for selected ions.
| Cation | \(\lambda_+^\infty\) (S·cm²·mol⁻¹) | Anion | \(\lambda_-^\infty\) (S·cm²·mol⁻¹) |
|---|---|---|---|
| H⁺ | 349.8 | OH⁻ | 198.6 |
| Na⁺ | 50.11 | Cl⁻ | 76.34 |
| K⁺ | 73.52 | CH₃COO⁻ | 40.9 |
| Ag⁺ | 61.92 | NO₃⁻ | 71.44 |
| Ca²⁺ | 119.0 | SO₄²⁻ | 160.0 |
⚙️ Transport Number Calculator (from Limiting Ionic Conductivities)
✅ For NaCl (\(\nu_+=\nu_-=1\)): \(t_{\text{Na}^+} = 50.11/(50.11+76.34)=0.396\), \(t_{\text{Cl}^-}=0.604\).
🏃♂️ Transport Numbers from Ionic Speeds (Hittorf’s Direct Principle)
Speeds are related to conductivity: \(\lambda_i = z_i F u_i\). Typical values shown for Na⁺ and Cl⁻.
🧪 Practical Importance of Transport Numbers
Transport numbers are essential in:
- Designing efficient electrochemical cells and batteries.
- Determining ionic mobilities from conductivity measurements.
- Calculating liquid junction potentials and diffusion coefficients.
- Understanding membrane selectivity in electrodialysis and fuel cells.
📌 Important Assumption & Electrode Reactions
Hittorf’s analysis assumes that discharged ions do not react with the electrode material (inert electrodes). When ions react with electrodes (e.g., Ag⁺ depositing on silver), the concentration changes may differ, but the transport numbers can still be obtained by correcting for the electrode reaction. The fundamental relation \(t_+/t_- = v_+/v_-\) remains valid.
📚 Summary of Key Equations (Hittorf’s Rule & Transport Numbers)
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