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American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

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Kinetic and Diffusion Control with Rotating Electrodes (RDE and RRDE)

Achim Habekost

Former: University of education Ludwigsburg, Ludwigsburg, Germany

Abstract: This article provides a structured, didactic introduction to hydrodynamic electrochemistry using rotating disk (RDE) and ring-disk (RRDE) electrodes. The Fe²⁺/Fe³⁺ redox couple is used as a model system to illustrate the experimental separation of kinetic- and diffusion-controlled current contributions. In diffusion- and kinetic-controlled reactions, the reaction velocity step is diffusion- or kinetic-controlled, respectively. Two additional systems supplement this one: the copper system and the organic hydroquinone/quinone (HQ/Q) system. At first, Cu²⁺ is reduced to Cu⁺, followed by subsequent reduction to Cu. This is an electrochemical three-component system. The RDE results of the HQ/Q system are compared by fitting the cyclic voltammogram. Then, the calculated diffusion coefficient and velocity constant are compared to the RDE measurement results. The quantitative evaluation using Levich analysis is illustrated step by step. We also introduce the RRDE technique as a mechanistic tool that enables direct correlation between charge transfer and product formation. The Koutecky–Levich formalism provides a more rigorous treatment of mixed kinetic and mass-transport control. In this approach, the experimentally measured reciprocal total current is expressed as the sum of the kinetic- and diffusion-limited currents according to the serial resistance of the electrode and solution. From a didactic perspective, Koutecky–Levich analysis is powerful because it enables students to visualize the transition from kinetic to diffusion control using experimental data. All calculations can be found in the Supporting Information. This manuscript is intended for advanced undergraduate and graduate education in electrochemistry.

Keywords: Rotating disk electrode, rotating ring–disk electrode, Levich equation, Koutecky–Levich analysis

Received January 24, 2026; Revised February 26, 2026; Accepted March 05, 2026

Citation: Achim Habekost, Kinetic and Diffusion Control with Rotating Electrodes (RDE and RRDE). World Journal of Chemical Education, 14 (1). 1-10. January 2026. doi:10.12691/wjce-14-1-1

1. Introduction

Electrochemical reactions are governed by the interplay between interfacial charge transfer kinetics and the mass transport of reactive species to and from the electrode. In conventional voltammetry, it is often difficult to distinguish between these contributions. Hydrodynamic techniques, most notably the rotating disk electrode (RDE), provide a powerful experimental approach that imposes well-defined transport conditions [1,2,3,4]. This enables a quantitative separation of kinetic and diffusion effects. The rotating ring–disk electrode (RRDE) further builds on this concept, enabling the direct detection of reaction products and intermediates.

Under hydrodynamic conditions at an RDE, the steady-state current is determined by the balance of convective transport, diffusion, and interfacial charge transfer. In the limit of fast kinetics, the current becomes purely mass-transport limited and follows the Levich equation:

iL = 0.62 · n · F · A · D²/₃ · ν⁻¹/₆ · ω¹/₂ · C*

where:

iL is the diffusion-limited (Levich) current (A)

n is the number of electrons transferred per molecule or ion

F is the Faraday constant (96,485 C·mol⁻¹)

A is the geometric area of the disk electrode (m²)

D is the diffusion coefficient of the electroactive species (m²·s⁻¹)

ν is the kinematic viscosity of the solution (m²·s⁻¹); typically, ≈1.0×10⁻⁶ m²·s⁻¹ for dilute aqueous solutions

ω is the angular rotation rate (rad·s⁻¹); the relation to the rotation rate N (rpm) is: ω = 2πN/60

C*: Bulk concentration of the electroactive species [mol·m⁻³]

The rotating disk electrode generates a well-defined hydrodynamic flow field characterized by axial convection of the electrolyte toward the disk surface and radial flow away from the electrode (Figure 1). This flow pattern results in a stable diffusion layer of thickness δ ∼ ω⁻¹ᐟ², which forms the physical basis of the Levich equation and ensures the reproducibility of RDE measurements.

Figure 1. Flow field near the electrode.

When electron-transfer kinetics are finite, the measured current reflects contributions from both kinetics and mass transport. This situation is described by the Koutecky–Levich equation, which separates both contributions through graphical analysis. This means that the concentration at the electrode is equal to the bulk concentration, C*, i.e., the reactants are replenished faster than they are converted. The Koutecky–Levich equation applies here:

1/i = 1/ik + 1/(0.62·n·F·A·D²/₃·ν−1/6·ω¹/₂·C*)

Beyond classical kinetic studies, RDE and RRDE techniques are widely regarded as gold-standard tools in modern electrochemical energy research. Prominent applications include oxygen reduction reaction (ORR) studies in fuel cell catalyst development and the investigation of hydrogen and oxygen evolution reactions [5,6], where controlled hydrodynamics facilitate the removal of gas bubbles from the electrode surface.

Didactic literature offers suggestions on how to integrate hydrodynamic electrochemistry into basic electrochemistry practicals. For example, this can be achieved by using a low-cost RDE setup (e.g. motors from the maker sector), in catalyst research and in STEM education [7,8,9,10,11,12,13].

Learning Objectives:

• Understand the principles of hydrodynamic control in electrochemistry using rotating electrodes.

• Distinguish experimentally between kinetic and diffusion-limited current regimes.

• Evaluate collection efficiency and mass balance in coupled electrochemical reactions.

• Apply Levich and Koutecky-Levich analysis to quantitatively interpret electrochemical data.

• Interpret RRDE measurements in terms of reaction mechanisms and product stability.

• It is important to apply the appropriate potential limits in order to prevent gas bubble.

2. Case Study I: Fe²⁺/Fe³⁺

All experiments were performed using a Metrohm rotating electrode system (Autolab RRDE 608 and Autolab Motor Controller, both Metrohm) equipped with a bipotentiostat (Autolab PGSTAT 204, Metrohm). A platinum disk electrode served as working electrode for RDE experiments, while a combined Pt disk / glassy carbon ring electrode (RRDE Pt-GC, 3.109.4080, Metrohm) was used for RRDE measurements. An Ag/AgCl (saturated KCl) electrode was employed as reference electrode and a platinum wire as counter electrode. The electrolyte consisted of an aqueous solution of 5 mmol L⁻¹ K₃[Fe(CN)₆] in 20 mmol L⁻¹ Na₂SO₄. Linear sweep voltammetry was performed at a scan rate of 0.05 V s⁻¹.

Figure 2 shows the experimental setup.

Figure 2. Top: Measurement setup with motor control (Metrohm), bipotentiostat (Metrohm), and measuring cell containing a reference electrode (Ag/AgCl, left); a rotating electrode connected to the rotary motor (center); and a counter electrode (Pt, right). Bottom: RRDE (Pt disk and GC ring, Metrohm)

RDE measurements yielded sigmoidal current–potential curves, and the limiting currents increased systematically with the rotation rate (D in Figure 3). Levich analysis of the diffusion-limited region (Figure 4, top) produced a diffusion coefficient of D(Fe³⁺) ≈ 4.1 × 10⁻⁹ m²/s, which is in good agreement with values reported in the literature [5]. Koutecky-Levich analysis 3 (Figure 4, bottom) of the kinetic region K in Figure 3 provided access to the kinetic current density and yielded a heterogeneous rate constant of approximately 2.0 × 10⁻⁴ m/s and an exchange current density of about 0.48*10⁻³ A/cm², indicating fast electron-transfer kinetics on platinum. For calculations, see the Supporting Information.

Figure 3. Current-voltage curve for various rotation speeds (500 to 3,000 rpm) of the Pt-RDE in an aqueous solution of K₃[Fe(III)(CN)₆]/K₄[Fe(II)(CN)₆] (5*10-3 mol each) in 20 mmol of Na₂SO₄. D indicates diffusion limitation and K indicates kinetic limitation. The arrow indicates the scan direction

Figure 4. Top: Levich plot at 1.1 V, bottom: Koutecki-Levich plot at 0.5 V

3. Case Study II: Copper System in Chloride Electrolyte (Cu²⁺/Cu⁺/Cu⁰)

This case study illustrates a more complex electrochemical system involving copper in chloridic media. The data originate from cyclic voltammetry (CV) and rotating ring–disk electrode (RRDE) measurements of 5 mM CuSO₄ in 0.5 M KCl. The measurements were taken using the same platinum (Pt) disk and glassy carbon (GC) ring electrode as in Case Study I.

Figure 5 shows the cyclic voltammogram.

At high chloride activity, Cu(II) does not solely exist as the aquo complex, but rather forms an equilibrium mixture of chloro complexes (e.g., [CuCl₄]²⁻ and mixed species) [6]. Chloride also stabilizes Cu(I) in solution (CuCl(aq), CuCl₂⁻, and CuCl₃²⁻), favoring stepwise reduction:

Cu(II) + e⁻ ⇌ Cu(I) (couple 0.05/0.15 V) and Cu(I) + e⁻ ⇌ Cu(0) (metal deposition; couple -0.37/-0.13 V).

Figure 5. Cyclic voltammogram of Cu²⁺/Cu⁺/Cu in a solution of 5 mM CuSO₄ in 0.5 M KCl

Copper deposition and stripping on platinum are often accompanied by nucleation phenomena, underpotential deposition effects, and the possible formation of Cu–Cl surface films. The RRDE configuration is therefore valuable because it detects soluble intermediates, such as Cu(I), at the ring electrode.

The CV shows the following: (i) pronounced cathodic peaks around 0.05 V and −0.37 V, which is consistent with the onset of Cu⁺ formation and copper deposition; (ii) a sharp anodic stripping peak near −0.13 V during the reverse scan; and (iii) a broader anodic wave at 0.14 V.

The peak separation, ΔEp (1) = 0.233 V, and the pronounced asymmetry between the forward and reverse scans are characteristic of metal deposition and stripping processes influenced by nucleation, surface restructuring, and complexation effects. The peak separation of the Cu²⁺/Cu⁺ redox couple is about 90 mV, the redox reaction is quasi-reversible.

In the RRDE experiments (Figure 6), the disk current increases with the rotation rate and exhibits a distinct, diffusion-limited plateau consistent with Levich behavior. When the disk is cathodically polarized, the ring current rises concurrently, indicating that a reduced copper species (Cu⁺) is generated at the disk (beginning at approximately −0.2 V), transported to the ring, and reoxidized there (ring potential: −0.2 V). At approximately −0.4 V, Cu⁺ is reduced to copper (Cu), and the ring current diminishes.

Figure 6. Dotted lines: Linear sweep voltammetry (LSV) of Cu²⁺. Colored solid lines: Ring current at 0.05 V Rotation rates: 500 (top lines) to 3,500 (bottom lines). The arrow indicates the scan direction

This behavior provides strong experimental evidence for a stepwise Cu(II) → Cu(I) → Cu(0) mechanism.

RRDE-Based Collection Efficiency: An average collection efficiency of N ≈ 37% was obtained from the RRDE dataset by evaluating the ratio |I_(ring)/I_(disk)| = 3×10⁻⁵ / 8×10⁻⁴ in the high-current regime. This value is consistent with the presence of a soluble intermediate and confirms that a significant fraction of the species generated at the disk is transported to and detected at the ring.

Figure 7 shows the Levich and Koutecki-Levich plots at different potentials: The diffusion region is shown at -0.8 V and -0.2 V, and the kinetic region is shown at 0.05 V and -0.45 V.

Calculating the diffusion coefficients of Cu⁺ and Cu²⁺ (see Supporting Information) yields the following results: D(blue, Cu⁺, −0.8 V) = 3.05 × 10⁻⁹ m²/s, and D(red, Cu²⁺, +0.2 V) = 1.2 × 10⁻⁸ m²/s.

Figure 7. Levich plot (top) and the Koutecki-Levich plot (bottom) of the copper system. The top blue line is at -0.8 V and the top red line is at -0.2 V. The bottom blue line is at 0.05 V and the bottom red line is at -0.45 V

Cu²⁺ is a classic hard cation with predominantly electrostatic hydration. Despite its greater hydration enthalpy, the hydrate shell is more dynamic and compact. Therefore, the effective hydrodynamic radius can be smaller than that of Cu⁺ in a KCl-containing solution.

Cu⁺ is a soft cation with high polarizability and a strong tendency to form covalent interactions with ligands. In an aqueous solution, Cu⁺ forms relatively stable, partially structured hydrate or chloro complexes (e.g., CuCl₂⁻, CuCl₃²⁻). This complexation significantly increases the effective hydrodynamic radius.

The experimental finding, D(Cu²⁺) > D(Cu⁺), reflects the effective size and solvation dynamics of the real diffusing species rather than the charge.

Cu²⁺ is predominantly aquated, resulting in a more compact solvate structure and larger D, which is consistent with known ion mobilities and electrochemical observations. In chloride-containing media, Cu(I) species often exhibit slower mass transport kinetics and complex chemistry, whereas Cu(II) behaves more like an ideal aqua cation.

The calculated reaction rates are ik = −9.44109 × 10⁻⁴ A for the blue line (Cu⁺, −0.45 V) and ik = −1.10993 × 10⁻² A for the red line (Cu²⁺, 0.05 V).

Interpreting the reaction constants is not straightforward because the Cu²⁺ ion undergoes labile complexation, resulting in faster kinetics. Cu⁺ is predominantly chlorocomplexed and less prevalent in the bulk solution. Consequently, i_k is significantly smaller. Note that the concentrations of Cu²⁺ and Cu⁺ are quite different.

4. Case Study III: Hydroquinone/Quinone (HQ/Q) Redox System

The HQ/Q redox couple is a classic example of an organic electron-transfer system.

Q + 2H⁺ + 2e⁻ ⇌ H₂Q.

Unlike the copper-ion examples discussed above, this system enables students to compare hydrodynamic experiment data with CV data fitting. The CV data exhibit quasi-reversible behavior (Figure 8), and the RRDE measurements (Figure 9) confirm that the disk current corresponds to the formation of quinone, which can be reduced back to hydroquinone at the ring under appropriate collection potentials.

Figure 8. shows the CV of hydroquinone (5 mmol) in a 20 mmol KCl solution. The solid black line represents the experiment, and the dotted red line represents the data fitting curve. Arrow indicates scan direction. Scan rate: 0.05 V/s

The potential difference between the anodic and cathodic peaks is approximately 160 millivolts (mV), which is about 130 mV higher than that of a reversible two-electron redox couple. The redox reaction is quasi-reversible. Fitting the CV data with the DigiElch program [14,15,16] yields the following values: Standard potential E0 = 0.32 V, ks = 2 × 10⁻⁶ m/s, and diffusion coefficient (hydroquinone) = diffusion coefficient (benzoquinone) ≈ 7.2 × 10⁻¹⁰ m²/s.

Figure 9 shows the disk and ring currents at different rotation rates (500 to 3,000 rpm) and disk potentials (from 0.2 to 0.6 V) at a ring potential of 0.2 V.

Starting at approximately 0.3 V, HQ is oxidized to Q, resulting in an increase in disk current (dotted lines). At the same potential, the ring current becomes negative because the formed HQ is reduced to Q.

Figure 9. Dotted lines: LSV of HQ (0.2 V to 0.6 V); colored solid lines: ring current at 0.2 V. Rotation rates: 500 (bottom dotted and top solid lines) to 3,000 (top dotted and bottom solid lines). The arrow indicates the scan direction

Figure 10 shows Levich and Koutecki-Levich plots at different potentials. The diffusion region at 0.6 V and the kinetic region at 0.425 V are shown.

The Levich plot yields a diffusion coefficient of 7.7×10⁻¹⁰ m²/s (see Supporting Information), a value similar to that obtained by fitting the CV.

The Koutecki-Levich plot results in an exchange current of k ≈ 7.1×10⁻⁶ m/s, which is three times higher than the CV fitting value.

Figure 10. Top: Levich plot at 0.6 V. Bottom: Koutecki-Levich plot at 0.425 V.

For the HQ/Q system, it is emphasized that the rate constant obtained from Koutecky–Levich analysis corresponds to an apparent, overpotential-dependent heterogeneous rate constant ks. This value reflects the applied potential during the KL analysis and therefore cannot be directly compared with the standard rate constant ks derived from numerical cyclic voltammetry fitting (which is related to the standard potential). Using Butler–Volmer kinetics, k can be related to ks through an exponential potential correction, which substantially reduces the apparent discrepancy between both methods (see Supporting Information).

Even small differences in D result in significant deviations in ks.

5. Conclusions

Combining RDE and RRDE techniques provides a powerful experimental framework for distinguishing between kinetic and mass-transport effects in electrochemistry. Beyond their analytical value, these methods are ideal for teaching purposes because they enable students to connect theory, experimentation, and quantitative data analysis directly. The system is an ideal model for introducing hydrodynamic electrochemistry in advanced laboratory courses.

Experimental limitations and error sources (RDE/RRDE): Common experimental limitations in rotating electrode experiments include electrode eccentricity or misalignment, deviations from ideal laminar flow at high rotation rates, incomplete deaeration, oxygen ingress, gas bubble adhesion during ORR-relevant potentials, and inaccuracies in rotation rate calibration. Such effects primarily influence the diffusion-limited current and may introduce systematic errors in Levich and Koutecky–Levich analyses.

Teaching notes and student exercises

Suggested questions for students:

Why does the limiting current increase with the square root of the rotation rate?

How can one distinguish experimentally between kinetic and diffusion control using RDE data?

What assumptions are required for the validity of the Levich equation?

Why is the RRDE technique particularly useful for detecting reaction intermediates?

How would side reactions affect the collection efficiency?

Please note the main differences between RDE versus static cyclic voltammetry (From a didactic perspective, rotating disk electrode experiments offer steady-state currents, well-defined mass transport, and direct access to diffusion coefficients and kinetic parameters. Static cyclic voltammetry, by contrast, is better suited for qualitative mechanistic screening, assessment of reversibility, and surface-sensitive phenomena. The complementary use of both techniques provides a comprehensive teaching framework).

Supporting Information

A). Fe²⁺/Fe³⁺: Levich and Koutecky–Levich Theory

Levich Equation (Rotating Disk Electrode, RDE)

The Levich equation describes the diffusion-limited limiting current of an electrochemical reaction at a rotating disk electrode under laminar flow conditions.

Diffusion-limited current:

i_L = 0.62 · n · F · A · D^(2/3) · ν^(−1/6) · ω^(1/2) · C* (1)

Parameters:

i_L: diffusion-limited (Levich) current [A]

n: number of electrons transferred per molecule/ion

F: Faraday constant (96485 C·mol⁻¹)

A: geometric area of the disk electrode [m²]

D: diffusion coefficient of the electroactive species [m²·s⁻¹]

ν: kinematic viscosity of the solution (ν = η/ρ) [m²·s⁻¹], typically ≈ 1.0 × 10⁻⁶ m²·s⁻¹ for dilute aqueous solutions

ω: angular rotation rate [rad·s⁻¹]; relation to rotation rate N (rpm): ω = 2πN/60

C*: bulk concentration of the electroactive species [mol·m⁻³]

Validity and assumptions:

• Laminar flow above the rotating disk (typical for moderate rotation rates and common electrolyte viscosities)

• Steady-state conditions and pure convection–diffusion control; electron-transfer kinetics are sufficiently fast

• Dilute solution with constant transport properties (D and ν)

• Under diffusion limitation, the surface concentration approaches zero (c(0) ≈ 0)

The goal of Levich analysis is to determine the electron number n and/or the diffusion coefficient D from experimentally measured limiting currents.

Graphical evaluation (Levich plot)

Plotting i_L versus ω^(1/2) yields a straight line under diffusion control.

From the slope m of the Levich plot:

n = m / (0.62 · F · A · D^(2/3) · ν^(−1/6) · C*)

D = [ m / (0.62 · n · F · A · ν^(−1/6) · C*) ]^(3/2)

Koutecky–Levich (KL) Equation

The Koutecky–Levich equation describes the relationship between the measured current, the kinetically controlled current, and the mass-transport-limited current at a rotating disk electrode. It allows separation of electrode kinetics and mass transport.

General form:

1 / i = 1 / i_k + 1 / i_L

With substitution of the Levich equation:

1 / i = 1 / i_k + 1 / (0.62 · n · F · A · D^(2/3) · ν^(−1/6) · ω^(1/2) · C*)

Parameters:

i: measured steady-state current [A]

i_k: kinetically controlled current [A]

i_L: diffusion-limited current [A]

n: number of electrons transferred

F: Faraday constant (96485 C·mol⁻¹)

A: electrode area [m²]

D: diffusion coefficient [m²·s⁻¹]

ν: kinematic viscosity [m²·s⁻¹]

ω: angular rotation rate [rad·s⁻¹]

C*: bulk concentration [mol·m⁻³]

The Koutecky–Levich equation can be derived by combining the Levich expression with the Butler–Volmer equation:

i_k = i_0 [ exp(α n F η / RT) − exp(−(1−α) n F η / RT) ]

where η = E − E_eq is the overpotential, i_0 is the exchange current, T is the absolute temperature, and α is the charge-transfer coefficient (typically 0.3–0.7; α = 0.5 corresponds to a symmetric energy barrier).

At sufficiently large anodic or cathodic overpotentials, one exponential term dominates (Tafel approximation), and the kinetic current becomes independent of mass transport.

Physical meaning:

The measured current reflects a series combination of kinetic resistance and mass-transport resistance.

If i ≪ i_k (kinetic region), i ≈ i_k.

If i ≈ i_L (diffusion-controlled region), mass transport dominates.

Plotting 1/i versus ω^(−1/2) yields a straight line (Koutecky–Levich plot). The intercept corresponds to 1/i_k, providing direct access to the kinetic current.

The kinetic current is related to the heterogeneous rate constant k by:

k = i_k / (n · F · A · C*)

Example Calculations

A) Diffusion coefficient from Levich analysis

Given:

A = 19.63 mm² = 1.963 × 10⁻⁵ m²

n = 1

Slope m = 3.0 × 10⁻⁵ A·s^(1/2)

F = 96485 C·mol⁻¹

C* = 1 mol·m⁻³

ν = 1.0 × 10⁻⁶ m²·s⁻¹ (water, 25 °C)

ν^(−1/6) = 10

Denominator:

0.62 · n · F · A · ν^(−1/6) · C* ≈ 11.74

m / denominator ≈ 2.56 × 10⁻⁶

D = (2.56 × 10⁻⁶)^(3/2) ≈ 4.1 × 10⁻⁹ m²·s⁻¹

= 4.1 × 10⁻⁵ cm²·s⁻¹

Koutecky–Levich analysis: kinetic current

Given intercept of KL plot:

1 / i_k = 2660.7 A⁻¹

i_k = 1 / 2660.7 A = 3.76 × 10⁻⁴ A

Electrode area: A = 19.63 mm² = 0.1963 cm²

Kinetic current density:

j_k = i_k / A = 1.915 mA·cm⁻²

C) Heterogeneous rate constant

i_k = n · F · A · k · C*

k = i_k / (n · F · A · C*)

With the values above:

k ≈ 2.0 × 10⁻⁴ m·s⁻¹ = 2.0 × 10⁻² cm·s⁻¹

Note: The unit m·s⁻¹ is characteristic of heterogeneous interfacial reactions and differs from the unit s⁻¹ used for homogeneous first-order reactions.

Butler–Volmer and exchange current density (Fe³⁺/Fe²⁺ example)

Given:

1 / i_k = 2660.7 A⁻¹ → i_k = 3.76 × 10⁻⁴ A

A = 0.1963 cm² → j_k = 1.915 mA·cm⁻²

Formal potential:

E°(Fe³⁺/Fe²⁺) = +0.771 V vs SHE

E(Ag/AgCl, sat. KCl) = +0.197 V vs SHE

→ E_eq ≈ 0.574 V vs Ag/AgCl

Measurement at E = 0.500 V:

η = E − E_eq = −0.074 V

Butler–Volmer (α = 0.5):

j = 2 j_0 sinh(Fη / 2RT)

From j = j_k:

j_0 ≈ 0.48 mA·cm⁻²

B). Copper system

Calculation of Diffusion Coefficients (Levich Analysis)

Given parameters:

Electrode area A = 1.963 × 10⁻⁵ m²

Number of electrons n = 1

Faraday constant F = 96485 C mol⁻¹

Bulk concentration C* = 5.0 mol m⁻³

Kinematic viscosity ν = 1.0 × 10⁻⁶ m² s⁻¹

Levich equation:

|i_L| = 0.62 n F A D^(2/3) ν^(−1/6) C* ω^(1/2)

The slopes were determined versus √(rpm) and converted to √(rad s⁻¹):

ω = 2π·rpm / 60 → √ω = √(2π/60) · √(rpm)

Conversion factor: √(2π/60) = 0.32360

Slopes:

Blue: m = 4.00 × 10⁻⁵ A s^(1/2) rpm^(−1/2) → 1.24 × 10⁻⁴ A s^(1/2) (rad s⁻¹)^(−1/2)

Orange: m = 1.00 × 10⁻⁴ A s^(1/2) rpm^(−1/2) → 3.09 × 10⁻⁴ A s^(1/2) (rad s⁻¹)^(−1/2)

Calculated diffusion coefficients:

D (blue, −0.8 V, Cu⁺) = 3.055 × 10⁻⁹ m² s⁻¹ = 3.05 × 10⁻⁵ cm² s⁻¹

D (orange, +0.2 V, Cu²⁺) = 1.207 × 10⁻⁸ m² s⁻¹ = 1.2 × 10⁻⁴ cm² s⁻¹

Koutecký–Levich Analysis

1/i = 1/ik + 1/(B ω^(1/2))

y = c + a x with c = 1/i_k and a = 1/B

Levich constant:

B = 0.62 n F A D^(2/3) ν^(−1/6) C*

D = (|B| / (0.62 n F A C* ν^(−1/6)))^(3/2)

Input parameters:

A = 1.963 × 10⁻⁵ m²

n = 1

F = 96485 C mol⁻¹

C* = 5 mol m⁻³

ν = 1.0 × 10⁻⁶ m² s⁻¹

0.62 n F A C* ν^(−1/6) = 58.714

Blue fit:

y = −12555 x − 90.1

ik = −0.0110993 A

B = −7.96495 × 10⁻⁵ A s^(1/2)

D = 1.580 × 10⁻⁹ m² s⁻¹

Red fit (0.02 V):

y = −30328 x − 1.06 × 10³

ik = −9.44109 × 10⁻⁴ A

B = −3.29728 × 10⁻⁵ A s^(1/2)

D = 4.208 × 10⁻¹⁰ m² s⁻¹

C). HQ/Q system

Results from data fitting with DigiElch:

Calculation of Diffusion Coefficients (Levich Analysis)

Levich (disk current):

IL = 0.62 n F A D^(2/3) ν^(-1/6) C ω^(1/2)

Slope of the plot I versus ω^(1/2):

m = 0.62 n F A D^(2/3) ν^(-1/6) C

D = (m / (0.62 n F A C) · ν^(1/6) )^(3/2)

• from the fit: m = 1.0 × 10^(-4) A (rad s^(-1))^(-1/2)

• n = 2

• A = 0.195 cm²

• C = 5 × 10^(-3) mol L^(-1) = 5 × 10^(-6) mol cm^(-3)

• water at 25 °C: ν ≈ 8.93 × 10^(-3) cm² s^(-1)

• F = 96485 C mol^(-1)

Result

D ≈ 7.72 × 10^(-6) cm² s^(-1)

Koutecki-Levich

From the Koutecký–Levich equation:

1 / I = 1 / Ik + 1 / L and I_L = B ω^(1/2)

Plotting 1 / I versus 1 / ω^(1/2) yields:

• Intercept b = 1 / I_k

• Slope m_KL = 1 / B

From your linear fit:

y = 5468 x + 757.6

Therefore:

1 / Ik = 757.6 A⁻¹ ⇒ Ik = 1 / 757.6 = 1.320 × 10⁻³ A

Ik ≈ 1.32 mA

Under the common assumption that the kinetic current can be described by a pseudo-first-order reaction with respect to the dissolved species:

Ik = n F A k C

with:

• n = 2

• F = 96485 C mol⁻¹

• A = 0.195 cm²

• C = 5 × 10⁻³ mol L⁻¹ = 5 × 10⁻⁶ mol cm⁻³

This gives:

k = Ik / (n F A C) = (1.320 × 10⁻³) / (2 · 96485 · 0.195 · 5 × 10⁻⁶) ≈ 7.02 × 10⁻³ cm s⁻¹

k ≈ 7.0 × 10⁻³ cm s⁻¹ = 7.0 × 10⁻⁵ m s⁻¹

This is the apparent heterogeneous rate constant in the sense of Ik = n F A k C. For a full Butler–Volmer analysis, the potential (overpotential), possibly α, and the distinction between C* oxidized / reduced would additionally be required.

Potential-Corrected Rate Constant from Koutecký–Levich Analysis

1) The K–L value at 0.42 V is an overpotential-dependent apparent rate, not the standard rate constant k⁰.

For Butler–Volmer kinetics with only hydroquinone (HQ) present (so the backward term is negligible), the anodic kinetic current is approximately:

ik ≈ n F A k⁰ C_HQ exp((1 − α) f η)

If this is forced into the simplified form i_k = n F A k_app C, one obtains:

k_app(η) ≈ k⁰ exp((1 − α) f η)

At 25 °C: f = F / RT ≈ 38.94 V⁻¹

With α = 0.415 → (1 − α) = 0.585, and η = 0.10 V:

exp((1 − α) f η) = exp(0.585 × 38.94 × 0.10) ≈ exp(2.28) ≈ 9.8

Thus, the potential-corrected estimate of the standard rate constant is:

k⁰ ≈ (7.0 × 10⁻³ cm s⁻¹) / 9.8 ≈ 7.1 × 10⁻⁴ cm s⁻¹

This correction alone reduces the discrepancy between the K–L-derived value and the DigiElch-fitted value by approximately one order of magnitude.

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