Cottrell Equation: Chronoamperometry Diffusion Current
Chronoamperometry is a key electroanalytical method in CSIR NET and GATE Chemistry. In exams, scoring full marks requires understanding the Cottrell equation, which models how current decays over time at an electrode under linear diffusion control.
Physical Principle of Chronoamperometry
Chronoamperometry records current against time after applying a potential step to a stationary working electrode in an unstirred solution. An excess of supporting electrolyte is added to eliminate electrical migration, ensuring mass transport occurs only by diffusion. When the potential steps past the redox potential, analyte concentration at the electrode surface drops to zero. As the diffusion layer thickens, the concentration gradient decreases, causing the faradaic current to decay continuously.
The Cottrell Equation and Variable Units
For a planar electrode under semi-infinite linear diffusion, current is given by the Cottrell equation:
i(t) = (n × F × A × D1/2 × C) / (π1/2 × t1/2)
Variable Definitions and Units:
• i(t) = Current in Amperes (A)
• n = Electrons transferred per molecule (dimensionless)
• F = Faraday constant (≈ 96,485 C/mol, or 96,500 C/mol)
• A = Electrode area (m2 in SI, or cm2 in CGS)
• D = Diffusion coefficient (m2/s in SI, or cm2/s)
• C = Concentration (mol/m3 in SI, or mol/cm3; 1 mM = 1.0 mol/m3 = 10−6 mol/cm3)
• t = Time in seconds (s)
• π1/2 = Square root of pi (≈ 1.7725)
Plotting current i against t−1/2 yields a straight line passing through the origin. The slope of this line, termed the Cottrell slope, gives D or the electrode area A.
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: CSIR NET Chemical Sciences, GATE Chemistry (CY), IIT-JAM Chemistry
- Topic: Chronoamperometry and Cottrell Equation
- Difficulty: Advanced
Worked Examples with Step-by-Step Solutions
Question: A 1-electron reduction (n = 1) is studied at a planar electrode of area A = 0.050 cm2 in an unstirred solution of 2.00 mM analyte. The diffusion coefficient is D = 7.60 × 10−6 cm2/s. Using F = 96,485 C/mol, calculate the diffusion current at t = 2.00 s.
Solution:
Step 1: Convert parameters to SI units.
n = 1; F = 96,485 C/mol
A = 0.050 cm2 = 5.00 × 10−6 m2
C = 2.00 mM = 2.00 mol/m3
D = 7.60 × 10−6 cm2/s = 7.60 × 10−10 m2/s
t = 2.00 s
Step 2: Evaluate square roots.
D1/2 = (7.60 × 10−10)1/2 = 2.7568 × 10−5 m/s1/2
t1/2 = (2.00)1/2 = 1.4142 s1/2
π1/2 = 1.7725
Step 3: Calculate current.
Numerator = n × F × A × D1/2 × C = 1 × 96,485 × (5.00 × 10−6) × (2.7568 × 10−5) × 2.00 = 2.6599 × 10−5 C/s1/2
Denominator = π1/2 × t1/2 = 1.7725 × 1.4142 = 2.5067 s1/2
i = (2.6599 × 10−5) / 2.5067 = 1.061 × 10−5 A = 10.61 μA
Step 4: Second route check using CGS slope.
C = 2.00 × 10−6 mol/cm3; D1/2 = 2.7568 × 10−3 cm/s1/2; A = 0.050 cm2.
Slope S = (1 × 96,485 × 0.050 × 2.7568 × 10−3 × 2.00 × 10−6) / 1.7725 = 1.5007 × 10−5 A s1/2.
i = S / t1/2 = (1.5007 × 10−5) / 1.4142 = 1.061 × 10−5 A = 10.61 μA. Both routes agree.
Answer: 10.61 μA (or 1.06 × 10−5 A)
Question: For a 2-electron reduction (n = 2) at an electrode with area A = 0.200 cm2 and analyte concentration C = 1.00 mM, a plot of current i versus t−1/2 gives a slope of S = 8.00 × 10−5 A s1/2. Calculate the diffusion coefficient D in cm2/s. (Take F = 96,485 C/mol).
Solution:
Step 1: Express D1/2 from the slope formula.
S = (n × F × A × C × D1/2) / π1/2 ⇒ D1/2 = (S × π1/2) / (n × F × A × C)
Step 2: Substitute SI parameters.
S = 8.00 × 10−5 A s1/2; π1/2 = 1.7725; n = 2; F = 96,485 C/mol
A = 0.200 cm2 = 2.00 × 10−5 m2; C = 1.00 mM = 1.00 mol/m3
Step 3: Calculate D in SI units.
Numerator = (8.00 × 10−5) × 1.7725 = 1.4180 × 10−4 A s1/2
Denominator = 2 × 96,485 × (2.00 × 10−5) × 1.00 = 3.8594 C/m
D1/2 = (1.4180 × 10−4) / 3.8594 = 3.6741 × 10−5 m/s1/2
D = (3.6741 × 10−5)2 = 1.350 × 10−9 m2/s
Step 4: Convert to cm2/s.
D = (1.350 × 10−9 m2/s) × (104 cm2/m2) = 1.350 × 10−5 cm2/s
Step 5: Second route verification.
Recalculating slope: S = [2 × 96,485 × 0.200 × (1.00 × 10−6) × (1.350 × 10−5)1/2] / 1.7725 = 8.00 × 10−5 A s1/2. The result is verified.
Answer: 1.35 × 10−5 cm2/s (or 1.35 × 10−9 m2/s)
Common Mistakes in Chronoamperometry
- Concentration unit errors: Inserting molarity (mol/L) directly with cm2. In CGS units, 1 mM equals 10−6 mol/cm3, not 10−3.
- Omitting π1/2: Leaving out π1/2 (≈ 1.7725) produces an error of about 77%.
- Incorrect time power: Current is proportional to t−1/2, not t−1 or t.
- Double-layer charging current: At very short times (t < 1 ms), non-faradaic charging current distorts the signal.
- Spherical diffusion effects: At long times or with microelectrodes, convergent radial diffusion replaces linear diffusion.
Exam Relevance and Question Patterns
Chronoamperometry questions focus on calculating transport properties and interpreting linear electroanalytical plots.
| Examination | Question Types and Focus Areas | Key Skills Tested |
|---|---|---|
| CSIR NET Chemical Sciences | Electroanalytical questions on diffusion coefficients, n values, and current ratios. | Interpreting i vs t−1/2 plots and determining Cottrell slopes. |
| GATE Chemistry (CY) | Numerical Answer Type (NAT) problems determining D, A, or current. | Unit conversions between SI and CGS, applying Faraday's constant, and fractional powers. |
| IIT-JAM Chemistry & CUET-PG | Fundamental conceptual and numerical questions on diffusion-controlled currents. | Applying inverse square-root time dependence and current ratios. |
Previous-year CSIR-NET questions and timed quizzes to check what you just read.
More Concepts articles