Cyclic Voltammetry Simulator

Cyclic Voltammetry Simulator - Basics of Cyclic Voltammetry

Theoretical • Semi-infinite planar diffusion • Nicholson–Shain convolution • Butler–Volmer kinetics • IUPAC sign convention

Dr. M Kanagasabapathy
Associate Professor of Chemistry
Rajapalayam Rajus' College
Madurai Kamaraj University
Rajapalayam, (TN) 626117, India

CV Simulation

Scan Rate Overlay

Randles–Ševčík (ip vs √ν)

Concentration Plot (ip vs C)

Theory & Equations

Input Parameters

Formal Reduction Potential E°' (V vs ref) Centre of CV  |  Fe³⁺/Fe²⁺ ≈ 0.77 V  |  Ferrocene ≈ 0.40 V

Concentration C (mol/L) 1 mM = 0.001  |  10 mM = 0.01  |  ip ∝ C (linear)

Electrons Transferred (n) Integer 1–8  |  ΔEp = 59.16/n mV at 298 K

Diffusion Coefficient D (cm²/s) 1×10⁻⁵ — typical small molecule (aq) 5×10⁻⁶ — moderate 1×10⁻⁶ — large molecule / viscous 5×10⁻⁷ — polymer / gel 1×10⁻⁷ — very slow Custom →

Custom D (cm²/s) e.g. 2.4e-5 for ferrocene in MeCN

Scan Rate ν (mV/s) 5 mV/s 10 mV/s 25 mV/s 50 mV/s 100 mV/s 200 mV/s 500 mV/s 1000 mV/s

Electrode Kinetics k° (cm/s) Reversible — Nernstian (k° → ∞) Quasi-rev fast — k° = 0.1 Quasi-rev moderate — k° = 0.01 Quasi-rev slow — k° = 10⁻³ Irreversible — k° = 10⁻⁵ Custom k° →

Custom k° (cm/s) Smaller → more irreversible behavior

Transfer Coefficient α 0.5 = symmetric barrier  |  affects peak asymmetry

Potential Window ± from E°' (V) Scan covers E°' ± this value. Min 0.15 V recommended.

T = 298 K (25°C) A = 1 cm² F/RT = 38.924 V⁻¹ 59.16 mV thermal D_O = D_R (ideal) E_start = E°'+window (auto) E_switch = E°'−window (auto) IUPAC: cathodic i < 0

Sign Convention (IUPAC): Reduction O + ne⁻ → R = cathodic = negative current. Oxidation R → O + ne⁻ = anodic = positive current. Some older textbooks (American convention) show cathodic peaks pointing upward — the opposite.

Cathodic peak i_pc

—

μA (negative = reduction)

Anodic peak i_pa

—

μA (positive = oxidation)

Cathodic peak E_pc

—

V vs ref

Anodic peak E_pa

—

V vs ref

Peak Separation ΔE_p

—

mV  (theory: 59.16/n)

Half-wave E_1/2 ≈ E°'

—

V vs ref

|i_pa/i_pc| ratio

—

1.000 = ideal reversible

Randles–Ševčík i_p

—

μA (theoretical)

Nicholson ψ

—

ψ >7 rev | 0.001–7 quasi

Regime

—

 

Cyclic Voltammogram

Multi Scan Rate Overlay

Runs 6 scan rates simultaneously using all parameters from Tab 1. Peak current i_p grows as √ν. Shapes remain identical for a reversible system — only amplitude changes.

CV Overlay — Scan Rates: 5, 10, 25, 50, 100, 200 mV/s

Randles–Ševčík Diagnostic: i_p vs √ν

For diffusion-controlled systems, |i_pc| and i_pa are linearly proportional to √ν. A straight line through the origin is consistent with diffusion control under this ideal model. The slope gives D if n, A, C are known.

i_p vs √ν — Randles–Ševčík linearity

Concentration Diagnostic: i_p vs C

Randles–Ševčík predicts i_p is linearly proportional to C (concentration). This is consistent with an ideal diffusion-controlled system and is the basis of analytical calibration curves in electrochemistry.

i_p vs C — Linearity (analytical calibration)

Core Equations

Redox reaction: O + n e⁻ ⇌ R Nernst equation (equilibrium surface condition): E = E°' + (RT/nF)·ln(C_O/C_R) = E°' + (0.02569/n)·ln(C_O/C_R) Overpotential: η = E − E°' (negative η drives reduction; positive η drives oxidation) Butler–Volmer kinetics — IUPAC sign (anodic = positive i): i = nFA·[ k°·C_R(0,t)·exp((1−α)nFη/RT) − k°·C_O(0,t)·exp(−αnFη/RT) ] where kf = k°·exp(−αnFη/RT) [reduction], kb = k°·exp((1−α)nFη/RT) [oxidation] Jreduction = kf·C_O − kb·C_R (reduction flux, +ve cathodic) i (IUPAC) = −nFA·Jreduction = nFA·[kb·C_R − kf·C_O] Randles–Ševčík peak current — REVERSIBLE, 298 K: ip = 0.4463·n·F·A·C·√(nFDν/RT) ip (A) = 2.687×10⁵ · n^(3/2) · D^(1/2) · C (mol/cm³) · ν^(1/2) · A (cm²) Randles–Ševčík peak current — IRREVERSIBLE, 298 K: ip = 0.4958·n·F·A·C·√(αnFDν/RT) ip (A) = 2.990×10⁵ · (αn)^(1/2) · n · D^(1/2) · C (mol/cm³) · ν^(1/2) · A (cm²) Peak separation — reversible, 298 K: ΔEp = |Epa − Epc| = 59.16/n mV (independent of ν, D, C) Nicholson reversibility parameter ψ: ψ = k° / √(D·nFν/RT) [ψ > 7: reversible | 0.001–7: quasi | < 0.001: irreversible] Irreversible cathodic peak potential (shifts with ν): Epc = E°' − (RT/αnF)·[ 0.780 + 0.5·ln(αnFDν/RT) − ln(k°) ] Epc shifts −(30/αn) mV per decade of ν (Tafel-like shift, Bard & Faulkner eq 6.2.32) Diffusion layer thickness (grows during scan): δ (cm) ≈ √(πDt) → δ (μm) ≈ 10⁴·√(πDt) Example: D=10⁻⁵ cm²/s, ν=10 mV/s, window=350 mV → t=35 s → δ ≈ 332 μm Cottrell equation (current after potential step / after peak): i(t) = nFAC·√(D/πt) → i ∝ t^(−½) Half-wave potential: E½ = (Epa + Epc)/2 ≈ E°' (exact for reversible; approximate for quasi-rev)

What is cyclic voltammetry?

The working electrode potential is linearly swept from E_start → E_switch (forward sweep) then back to E_start (reverse sweep), while current is recorded. The resulting i–E curve — the cyclic voltammogram — encodes thermodynamics (E°'), kinetics (k°, α), and transport (D, C).

Why does a peak appear?

As E passes E°', O reduces rapidly → current surges. But O near the electrode depletes faster than diffusion replenishes it. The diffusion layer grows as √(Dt) — supply falls — current drops after the peak. This competition between electrochemical reaction and mass transport creates the characteristic peak shape.

Cathodic sweep (forward)

• E sweeps from +window → −window (more reducing)
• At E ≈ E°': O + ne⁻ → R begins
• IUPAC: cathodic current = negative; peaks at E_pc
• Post-peak: diffusion limitation → Cottrell decay
• R builds up near electrode surface

Anodic sweep (return)

• E sweeps back toward +window
• R near electrode is re-oxidized: R → O + ne⁻
• Anodic (positive) current peak at E_pa
• For reversible: |i_pa| = |i_pc| → |i_pa/i_pc| = 1 (note: with IUPAC sign, i_pc < 0 and i_pa > 0)
• Two peaks together form the "duck" or "butterfly" shape

Reversibility — what ψ means

• Reversible (ψ > 7): k° ≫ mass transport rate. Nernst equation obeyed at all times. ΔEp = 59.16/n mV. |i_pa/i_pc| = 1. ip ∝ √ν.
• Quasi-rev (0.001 < ψ < 7): Both peaks visible. ΔEp > 59/n mV and grows with ν. Peaks broaden.
• Irreversible (ψ < 0.001): Return peak absent. Epc shifts −30/(αn) mV per decade of ν.

Effect of each parameter

• C ↑: ip ∝ C — taller peaks, unchanged shape (calibration basis)
• D ↑: ip ∝ √D — taller peaks
• ν ↑: ip ∝ √ν — taller peaks; ΔEp unchanged (reversible)
• n ↑: ip ∝ n^(3/2) (rev); ΔEp = 59.16/n mV narrower
• k° ↓: ΔEp increases, peaks shift apart, anodic peak shrinks
• α ≠ 0.5: only affects quasi-rev/irreversible systems — shifts peak position and creates asymmetric peak widths; has NO effect in the fully reversible (Nernstian) limit

Diagnostic checklist (real experiments)

• ip vs √ν = straight line through origin → consistent with diffusion control (ideal model)
• ip vs C = straight line through origin → analytical linearity ✓ (ideal model; real CVs may show offset due to background current or adsorption)
• ΔEp = 59.16/n mV (independent of ν) → reversible ✓
• |i_pa/i_pc| = 1 → stable product R ✓
• E½ = E°' (independent of ν) → thermodynamic stability ✓
• ΔEp increases with ν → quasi-reversible kinetics
• No return peak → irreversible or coupled chemical reaction

Model assumptions & limitations

• Semi-infinite planar diffusion (no convection, no stirring)
• D_O = D_R (equal diffusion coefficients)
• Electrode is macroscopic: radius ≫ diffusion layer √(Dt)
• No double-layer charging current (purely Faradaic)
• No adsorption, no coupled chemical reactions (EC, CE, etc.)
• Bulk concentration C* uniform and constant far from electrode

Numerical method: Nicholson–Shain convolution integral with 1 mV potential steps. Reversible case: Nernst equation enforced directly at each step (exact, no Butler-Volmer needed). Quasi/irreversible: Butler–Volmer solved implicitly (single linear equation per step — numerically stable for the parameter ranges applicable to this model). Convolution kernel: g[m] = √(Δt)·(√m − √(m−1)), which is the exact semi-infinite diffusion Green's function.

Refer: Cyclic Voltammetry Basics

 

Book on Simulated Cyclic Voltammograms: Basics of Electrochemical Kinetics