Debye-Hückel-Onsager Equation: Strong Electrolytes
The Debye-Hückel-Onsager equation describes conductance in dilute solutions of strong electrolytes. In competitive exams, questions examine why strong electrolytes exhibit lower molar conductivity at higher concentrations. While weak electrolytes show reduced dissociation, strong electrolytes remain 100% ionised. Their conductivity decrease arises entirely from interionic electrostatic forces. Kohlrausch discovered empirically that molar conductivity varies linearly with the square root of concentration. Debye, Hückel, and Onsager provided the theoretical derivation by analyzing the ionic atmosphere.
Physical Origin of Conductance Decrease
At infinite dilution, ions migrate independently, reaching limiting molar conductivity (Λm0). At finite concentration, an oppositely charged ionic atmosphere forms around each ion, creating retarding forces via the relaxation effect and electrophoretic effect.
The Two Retarding Effects
1. Relaxation Effect (Asymmetry Effect): At rest, the ionic atmosphere is symmetric. Under an electric field, rebuilding the atmosphere ahead and relaxing it behind takes a finite relaxation time (τ). This lag creates excess opposite charge behind the moving ion, exerting backward electrostatic drag proportional to √c.
2. Electrophoretic Effect: Oppositely charged counter-ions migrate counter to the central ion, dragging associated solvent shells along. The central ion moves against this solvent counter-flow. This viscous drag retards ion motion, governed by solvent viscosity (η) and permittivity.
The Debye-Hückel-Onsager Equation
Onsager combined Brownian motion with interionic electrostatic forces to derive the molar conductivity equation:
Debye-Hückel-Onsager Equation:
Λm = Λm0 - (A + B Λm0) √c
Or expressed concisely as: Λm = Λm0 - S √c
Where:
Λm = molar conductivity at concentration c (S cm2 mol-1 or S m2 mol-1; note that 1 S m2 mol-1 = 104 S cm2 mol-1)
Λm0 = limiting molar conductivity at infinite dilution (same units as Λm)
c = electrolyte concentration in molarity (mol dm-3 or mol L-1)
S = Onsager coefficient (theoretical slope), given by S = A + B Λm0
A = electrophoretic coefficient; for a 1:1 electrolyte in water at 25 °C, A = 60.20 S cm2 dm1.5 mol-1.5
B = relaxation coefficient; for a 1:1 electrolyte in water at 25 °C, B = 0.229 dm1.5 mol-0.5
In water at 25 °C (298.15 K) for 1:1 electrolytes: Λm = Λm0 - (60.20 + 0.229 Λm0) √c.
Worked Numerical Examples
Question: The limiting molar conductivity of KCl in water at 25 °C is Λm0 = 149.86 S cm2 mol-1. Using the Debye-Hückel-Onsager coefficients A = 60.20 S cm2 dm1.5 mol-1.5 and B = 0.229 dm1.5 mol-0.5, calculate the molar conductivity Λm of an aqueous KCl solution at concentration c = 0.0040 mol dm-3.
Step 1: Calculate square root of concentration: √c = √(0.0040) = 0.063246 mol0.5 dm-1.5.
Step 2: Relaxation coefficient term: B Λm0 = 0.229 × 149.86 = 34.318 S cm2 dm1.5 mol-1.5.
Step 3: Calculate Onsager slope: S = A + B Λm0 = 60.20 + 34.318 = 94.518 S cm2 dm1.5 mol-1.5.
Step 4: Conductivity reduction: S √c = 94.518 × 0.063246 = 5.978 S cm2 mol-1.
Step 5: Compute molar conductivity: Λm = 149.86 - 5.978 = 143.882 ≈ 143.88 S cm2 mol-1.
Double-check by factoring: Λm = 149.86(1 - 0.229 × 0.063246) - 60.20(0.063246) = 147.690 - 3.807 = 143.88 S cm2 mol-1.
Answer: 143.88 S cm2 mol-1
Question: The molar conductivities of a 1:1 strong electrolyte in water at 25 °C are measured at two concentrations: at c1 = 0.0016 mol dm-3, Λm1 = 126.38 S cm2 mol-1; at c2 = 0.0049 mol dm-3, Λm2 = 123.50 S cm2 mol-1. Assuming the electrolyte obeys Λm = Λm0 - S √c, determine the experimental slope S and limiting molar conductivity Λm0.
Step 1: Square root of concentrations: √c1 = √(0.0016) = 0.040 mol0.5 dm-1.5; √c2 = √(0.0049) = 0.070 mol0.5 dm-1.5.
Step 2: Differences: Δ√c = 0.070 - 0.040 = 0.030 mol0.5 dm-1.5; ΔΛm = 123.50 - 126.38 = -2.88 S cm2 mol-1.
Step 3: Calculate slope: Slope = -S = ΔΛm / Δ√c = -2.88 / 0.030 = -96.00 S cm2 dm1.5 mol-1.5. Thus S = 96.00 S cm2 dm1.5 mol-1.5.
Step 4: Compute Λm0 from first point: Λm0 = 126.38 + (96.00 × 0.040) = 126.38 + 3.84 = 130.22 S cm2 mol-1.
Double-check with second point: Λm0 = 123.50 + (96.00 × 0.070) = 130.22 S cm2 mol-1, matching the first point.
Answer: S = 96.00 S cm2 dm1.5 mol-1.5, Λm0 = 130.22 S cm2 mol-1
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: CSIR NET, GATE, IIT JAM, CUET PG
- Topic: Debye-Hückel-Onsager Conductance Equation
- Difficulty: Advanced
Common Mistakes Students Make
1. Attributing decrease in conductance to incomplete dissociation: Strong electrolytes are 100% ionised. The decrease in molar conductivity with concentration is caused solely by interionic retarding forces (relaxation and electrophoretic effects), not incomplete ionisation.
2. Using concentration directly instead of square root: Students often write Λm = Λm0 - S c. The Kohlrausch and Onsager relationship depends strictly on √c. Plotting against c yields curvature rather than a straight line.
3. Applying the limiting law at high concentrations: The Debye-Hückel-Onsager equation is valid only in dilute solutions (c ≤ 0.01 mol dm-3). At higher concentrations, ion association and higher-order effects cause positive deviations from the Onsager line.
4. Confusing unit conversions between cm2 and m2: Remember that 1 S m2 mol-1 = 104 S cm2 mol-1. Forgetting this conversion leads to errors of 104 in numerical calculations.
Exam Relevance and Problem Types
| Examination | Question Format | Key Skills Tested |
|---|---|---|
| CSIR NET | Theoretical MCQs and graphical analysis | Differentiating electrophoretic vs relaxation effects, solvent viscosity dependence, and high-concentration deviations. |
| GATE | Numerical answer type and concept MCQs | Computing theoretical Onsager slope S, finding Λm at given concentration, and determining Λm0 from intercepts. |
| IIT JAM | Multiple-choice and numerical questions | Applying Kohlrausch's law alongside Onsager theory, calculating molar conductivity from specific conductance, and slope analysis. |
| CUET PG | Objective conceptual questions | Definitions of relaxation and electrophoretic drag, dependence on electrolyte valence, and validity limits. |
Turn this concept into marks: solve previous-year CSIR-NET questions and take a timed quiz.
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