Liquid Junction Potential: Henderson Equation Numericals
In advanced physical chemistry, liquid junction potential is an essential topic in potentiometry and electrochemical thermodynamics. When two electrolyte solutions of different concentrations or ionic compositions meet, an electrical potential difference develops across the liquid-liquid boundary. In competitive examinations such as CSIR NET, GATE, and IIT-JAM, numerical problems regularly test the Henderson equation, ionic transport numbers, and the quantitative link between concentration cells with transference and cells without transference. Mastering these core formulas allows students to solve multi-step problems accurately while avoiding common sign errors and conversion pitfalls.
Origin of Liquid Junction Potential
When two solutions of unequal concentrations (such as 0.10 M HCl and 0.01 M HCl) are brought into contact, ions diffuse from the concentrated side toward the dilute side. Because hydronium ions possess much higher mobility than chloride ions due to the Grotthuss proton-hopping mechanism in water, hydronium ions diffuse faster into the dilute compartment. This unequal migration creates a microscopic charge separation: the dilute side acquires a net positive charge, while the concentrated side retains a net negative charge. This potential difference is the liquid junction potential, denoted E_j. The resulting electric field retards faster cations and accelerates slower anions until equal steady-state velocities are attained.
The Henderson Equation and Key Mathematical Formulas
For two solutions of the same uni-univalent (1:1) electrolyte at concentrations c1 and c2 (where c1 > c2), the Henderson equation expresses the liquid junction potential directly through ionic transport numbers.
Henderson Equation for 1:1 Electrolyte Junctions:
E_j = (t_+ - t_-) * (RT / F) * ln(c1 / c2)
At 298.15 K (25 °C) in base-10 logarithmic form:
E_j = (t_+ - t_-) * 0.05916 V * log10(c1 / c2) = (2 * t_+ - 1) * 0.05916 V * log10(c1 / c2)
Where:
- E_j = liquid junction potential (volts, V)
- t_+ = cation transport number (dimensionless, where t_+ + t_- = 1)
- t_- = anion transport number (dimensionless)
- R = universal gas constant = 8.31446 J mol−1 K−1
- T = absolute temperature = 298.15 K
- F = Faraday constant = 96485 C mol−1
- c1, c2 = molar concentrations of concentrated and dilute solutions (mol L−1)
- 0.05916 V = 2.3026 * RT / F at 298.15 K (often written as 0.0592 V)
When two different 1:1 electrolytes share a common ion at equal concentration c, the Lewis-Sargent relation applies: E_j = 0.05916 V * log10(Λ°_1 / Λ°_2), where Λ°_1 and Λ°_2 are limiting molar conductivities in S cm2 mol−1.
Minimizing Liquid Junction Potential with a Salt Bridge
In laboratory cells, a salt bridge containing concentrated KCl, KNO3, or NH4NO3 connects the half-cells. Because K+ and Cl− have almost equal ionic mobilities (t_+ ≈ 0.49 and t_- ≈ 0.51), the factor (t_+ − t_-) is nearly zero. This reduces E_j to approximately 1 to 2 mV, preventing junction interference in potentiometric measurements.
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: CSIR NET Chemical Sciences, GATE Chemistry, IIT-JAM Chemistry, CUET-PG
- Topic: Liquid Junction Potential and Henderson Equation Numericals
- Difficulty: Advanced
Worked Numerical Examples
Question: At 298.15 K, a liquid boundary is formed between 0.100 M HCl and 0.010 M HCl. The transport number of H+ is 0.830. Calculate the liquid junction potential E_j across this junction in millivolts using 2.3026 RT / F = 0.05916 V.
Step-by-step solution:
Step 1: Identify given concentrations and transport number.
c1 = 0.100 M, c2 = 0.010 M, t_+ = 0.830.
Step 2: Determine the anion transport number.
t_- = 1.000 − 0.830 = 0.170.
Step 3: Compute the transport number difference.
t_+ − t_- = 0.830 − 0.170 = +0.660.
Step 4: Compute the concentration ratio.
c1 / c2 = 0.100 / 0.010 = 10.0, so log10(10.0) = 1.000.
Step 5: Calculate E_j using Route 1.
E_j = (t_+ − t_-) * 0.05916 V * log10(c1 / c2) = 0.660 * 0.05916 V * 1.000 = +0.03905 V = +39.05 mV.
Step 6: Verify via Route 2 using the single transport number formula.
E_j = (2 * t_+ − 1) * 0.05916 V * log10(c1 / c2) = (2 * 0.830 − 1) * 0.05916 V * 1.000 = 0.660 * 0.05916 V = +39.05 mV.
Both independent routes confirm an identical value of +39.05 mV.
Answer: +39.05 mV
Question: At 298.15 K, a concentration cell with transference Pt | H2(1 bar) | HCl(0.020 M) :: HCl(0.200 M) | H2(1 bar) | Pt has a measured cell potential E_wt = 0.0280 V. The corresponding cell without transference has E_wot = 0.05916 V. Calculate the liquid junction potential E_j and find the transport numbers t_+ and t_-.
Step-by-step solution:
Step 1: Relate cell potentials to find E_j (Route 1).
For a cell reversible to cations: E_wt = E_wot + E_j.
E_j = E_wt − E_wot = 0.0280 V − 0.05916 V = −0.03116 V = −31.16 mV.
Step 2: Determine the anion transport number.
E_wt = 2 * t_- * E_wot, which gives t_- = 0.0280 V / (2 * 0.05916 V) = 0.0280 / 0.11832 = 0.2366.
Step 3: Determine the cation transport number.
t_+ = 1.0000 − 0.2366 = 0.7634.
Step 4: Verify E_j using the Henderson relation (Route 2).
E_j = (t_+ − t_-) * 0.05916 V * log10(0.020 / 0.200) = (0.7634 − 0.2366) * 0.05916 V * (−1.000) = 0.5268 * 0.05916 V * (−1.000) = −0.03117 V = −31.17 mV.
Both mathematical paths match closely at −31.16 mV.
Answer: E_j = −31.16 mV; t_+ = 0.7634; t_- = 0.2366
Common Mistakes to Avoid
Watch out for these common errors in competitive electrochemistry exams:
- Inverting the sign of E_j: Always verify which compartment is positive. Because faster cations diffuse into the dilute side, that compartment becomes positive relative to the concentrated side.
- Confusing E_wt with E_j: The potential with transference E_wt includes electrode reactions. For cation-reversible cells, E_wt = 2 * t_- * E_wot, whereas E_j = E_wt − E_wot.
- Mixing natural log and base-10 log: The 0.05916 V factor contains 2.3026. Do not multiply by 2.3026 again when using log10.
- Substituting t_+ instead of t_-: For cation-reversible electrodes, cell EMF with transference depends on the anion transport number t_-, not t_+.
- Assuming salt bridges eliminate E_j completely: A salt bridge suppresses E_j to 1 to 2 mV, but cannot reduce it to absolute zero.
Exam Relevance and Question Patterns
| Examination | Question Format | Key Focus Areas |
|---|---|---|
| CSIR NET Chemical Sciences | Multiple-choice and numerical questions | Henderson equation derivations, transport numbers, and cell transference relationships |
| GATE Chemistry | Numerical Answer Type (NAT) problems | Direct calculations of E_j and determining ionic mobilities or transport numbers |
| IIT-JAM Chemistry | Multiple-choice and numerical questions | Concentration cells with and without transference and role of salt bridges |
| CUET-PG Chemistry | Objective conceptual questions | Conditions for zero junction potential and basic formula applications |
Previous-year CSIR-NET questions and timed quizzes to check what you just read.
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