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Theoretical concepts and application of a Rotating disk electrode (RDE) and Rotating
Ring disk electrode
Concentrated summer course in electrochemistry, September, 2014
Bar-Ilan University
Outline
• Introduction & Determination of general equations for hydrodynamic systems
• RDE equation and application
• RRDE equation and application
Not steady state system (not stirred)
*
𝑶+ 𝒏𝒆 ↔ 𝑹
* o
Mass Transport Mechanisms
Diffusion
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- -
- -
-
-
+
+
+
+
+
+
+
+
Migration
Convection
𝑱𝒋(𝒙) = −𝑫𝒋𝝏𝑪𝒋(𝒙)
𝝏𝒙−𝒛𝒊𝑭
𝑹𝑻𝑫𝒊𝑪𝒊
𝝏𝝋(𝒙)
𝝏𝒙
𝑰𝒋 = 𝒏𝑭𝑨𝑫𝒋𝝏𝑪𝒋(𝒙)
𝝏𝒙
𝑱𝒋 = −𝑫𝒋𝜵𝑪𝒋 −𝒛𝒋𝑭
𝑹𝑻𝑫𝒋𝑪𝒋𝜵𝝋 + 𝑪𝒋𝒗
Nernst-Plank equation
𝑱𝒋(𝒙) =𝑰𝒋
𝒛𝒋𝑭𝒋𝑨
Steady State system
𝑪𝑶 Bulk solution
(stirred)
Stagnant 𝑪𝑶 *
𝑶 + 𝒏𝒆 ↔ 𝑹
Convective diffusion equation
𝑱𝒊 = −𝑫𝒊𝜵𝑪𝒊 −𝒛𝒊𝑭
𝑹𝑻𝑫𝒊𝑪𝒊𝜵𝝋 + 𝑪𝒊𝒗
𝝏𝑪𝒊𝝏𝒕= −𝜵𝑱𝒊
𝝏𝑪𝒊𝝏𝒕= 𝑫𝒊𝜵
𝟐𝑪𝒊 − 𝒗𝜵𝑪𝒊
𝑰𝒍 = 𝒏𝑭𝑨𝒎𝟎𝑪𝟎∗
𝒎𝟎 =𝑫𝟎𝜹𝟎
Hydrodynamic electrode
Rotating disc electrode (RDE) Wall-jet electrode (WJE)
The tubular and channel electrode
• Well-defined hydrodynamics flow to electrode
surface.
• The mathematical equations are available for the calculation of different parameters.
• Rate of material transport depends in a well defined manner on the rotation speed of the electrode.
• Systems with RDE are relatively simple for fabrication and operation.
RDE- specialized hydrodynamic electrode used in the study of the
kinetics and mechanism of electrode reaction
Convective diffusion equation: RDE
𝒗𝒚𝝏𝑪𝟎𝝏𝒚
= 𝑫𝟎𝝏𝟐𝑪𝟎𝝏𝒚𝟐
𝝏𝑪𝒊𝝏𝒕= 𝟎
𝒗𝒓𝝏𝑪𝟎𝝏𝒓
+𝒗𝝋
𝒓
𝝏𝑪𝟎𝝏𝝋
+ 𝒗𝒚𝝏𝑪𝟎𝝏𝒚
= 𝑫𝟎𝝏𝟐𝑪𝟎𝝏𝒚𝟐
+𝝏𝟐𝑪𝟎𝝏𝒓𝟐
+𝟏
𝒓
𝝏𝑪𝟎𝝏𝒓
+𝟏
𝒓𝟐𝝏𝟐𝑪𝟎𝝏𝝋𝟐
=0
=0
=0
=0
=0
𝝏𝑪𝒊𝝏𝒕= 𝑫𝒊𝜵
𝟐𝑪𝒊 − 𝒗𝜵𝑪𝒊
Convective diffusion equation expressed in cylindrical polar-co-ordinates (r, z, φ)
Convective diffusion equation: RDE
𝜕2𝐶𝑂𝜕𝑦2
= −𝑦2
𝐵
𝜕𝐶𝑂𝜕𝑦
𝑣𝑦 = 0.51𝜔3/2𝑣−1/2𝑦2
𝐴 = 𝜋𝑟2
𝐵 = 𝐷𝑂 𝜔3/2𝑣−1/2/0.51
Allen J. Bard and Larry R. Faulkner, ELECTROCHEMICAL METHODS, 2nd edition, 2001
𝒊 = 𝒏𝑭𝑨𝑫𝟎𝝏𝑪𝟎(𝒙)
𝝏𝒚𝒚=𝟎
Levich equation Diffusion limited Current
𝑪𝟎 =𝝏𝑪𝟎𝝏𝒚
𝒚=𝟎
𝟎. 𝟖𝟗𝟑𝟒𝟑𝑫𝟎𝝎
−𝟑/𝟐𝜸𝟏/𝟐
𝟎. 𝟓𝟏
𝟏/𝟑
*
𝒊𝑳 = 𝟎. 𝟔𝟐𝒏𝑭𝑨𝑫𝟐/𝟑𝒗−𝟏/𝟔𝑪𝟎𝝎
𝟏/𝟐 *
𝟏
𝒊=𝟏
𝒊𝑲+𝟏
𝒊𝒍,𝒄=𝟏
𝒊𝑲+
𝟏
𝟎. 𝟔𝟐𝒏𝑭𝑨𝑫𝟐/𝟑𝒗−𝟏/𝟔𝑪𝟎𝝎𝟏/𝟐
Levich
Koutecky-Levich
Levich study (For a simple electrochemical system where the
rate of the half reaction is governed only by mass transport to the electrode surface)
The limiting current increases linearly with the square root of the rotation rate and the line intercepts the vertical axis at zero.
2/1
R
6/13/2
RLACnFAD620.0i
𝑹 ↔ 𝑶+ 𝒏𝒆
Measuring Limiting Currents
Koutecky-Levich Analysis - the rate of a half reaction
occurring at an electrode surface is limited by a combination of mass transport and sluggish kinetics
2/16/13/2 )620.0/1(/1/1 CnFADii k
ki/1
Determination of the activation controlled current density
𝑖𝑘 = 𝐹𝐴𝑘𝑓(𝐸)𝐶0 *
Application of RDE
0
6/12/13/262.0 CnFDil 0
2/1 Cil
i-E curves as a function of rotation rate
23 1 FeeFe
eFeFe 132
Fast electron transfer
k3Fe(CN)6 10mM & K4Fe(CN)6 20mM
i-E curves as a function of K4Fe(CN)6 concentration. Rotation rate: 2000 rpm
eFeFe 132
Increase in RDE rotation speed and electroactive species concentration cause an increase in the limiting current density.
23 1 FeeFe
0
6/12/13/262.0 CnFDil
Plots iL vs. ω0.5 , working solution: K3Fe(CN)6 (10mM)+ K4Fe(CN)6 (20mM)in Na2SO4(0.1M)
Plots iL vs. C. working solution: K3Fe(CN)6 (10mM)+ K4Fe(CN)6 (20-60mM)in Na2SO4(0.1M).
Rotation speed 2000 rpm.
Calculation of diffusion coefficient
0
6/13/262.0 CnFD 1/2ωliIL vs. ω0.5
0
6/12/13/262.0 CnFDil IL vs. C
20.66 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒2+2/3 × (1.1 × 10−6)1/6× 20
𝐷𝐹𝑒2+ = 2.5 × 10−9𝑚2𝑠−1
10.44 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒3+2/3 × (1.1 × 10−6)1/6× 10
𝐷𝐹𝑒3+ = 2.6 × 10−9𝑚2𝑠−1
15.59 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒2+2/3 × (1.1 × 10−6)1/6× 14.47
𝐷𝐹𝑒2+ = 2.7 × 10−9𝑚2𝑠−1
RRDE
Teflon Ring
Disk
RRDE
𝑶 + 𝒏𝒆 ↔ 𝑹 Disk- Ed, id
𝑹 ↔ 𝑶+ 𝒏𝒆 Ring-Er, ir
𝒗𝒓𝝏𝑪𝒐𝝏𝒓
− 𝒗𝒚𝝏𝑪𝑹𝝏𝒚
=𝑫𝑹𝑩′
𝟏
𝒚
𝝏𝟐𝑪𝑹𝝏𝒚𝟐
Steady-state ring convective-diffusion equation
𝒊𝑹 = 𝒏𝑭𝑫𝑹𝟐𝝅 𝝏𝑪𝑹𝝏𝒚
𝒓𝟑
𝒓𝟐
𝒓𝒅𝒓
𝑵𝒆𝒎𝒑𝒊𝒓𝒊𝒄𝒂𝒍 = −𝒊𝑳,𝒓𝒊𝒏𝒈/𝒊𝑳,𝒅𝒊𝒔𝒌
𝑩′ = 𝟎. 𝟓𝟏𝝎𝟑/𝟐𝜸−𝟏/𝟐
lim𝑦→∞
𝐶𝑅 = 0 R initially absent in solution
Bulk concentration of O is CO*
𝐹𝑒(𝐶𝑁)6 + 𝒆 → 𝐹𝑒(𝐶𝑁)6 3- 4-
Reduction of ferricyanide to ferrocyanide at disk
𝐹𝑒(𝐶𝑁)6 → 𝐹𝑒(𝐶𝑁)6 + 𝒆 4- 3-
Oxidation of ferrocyanide to ferricyanide at ring
K3Fe(CN)6 10mM in 1M KNO3
RRDE experiment
𝑵𝒆𝒎𝒑𝒊𝒓𝒊𝒄𝒂𝒍 = −𝒊𝑳,𝒓𝒊𝒏𝒈/𝒊𝑳,𝒅𝒊𝒔𝒌