Lecture 03 - Electrochemical Kinetics - Voice Over
Lecture 03 - Electrochemical Kinetics - Voice Over
Lecture 03 - Electrochemical Kinetics - Voice Over
Electrochemical Engineering
Lecture 03
Electrochemical Kinetics
Prof. Zhe Cheng, Dr. Junheng Xing
Mechanical & Materials Engineering
Florida International University
Electrochemical Kinetics
❑Definition
“…a field of electrochemistry studying the rate of electrochemical processes.”
(Wikipedia)
“The main goal of the electrochemical kinetics is to find a relationship between the
electrode overpotential and current density …” (S. N. Lvov)
❑Contents
▪ Cell potential, voltage loss, and overpotential (η)
▪ Charge transfer overpotential & Butler-Volmer equation
• Tafel relationship
• Linear approximation
▪ Mass transfer limitation
Zn Rext Cu Zn Cu
M M
e e
H2O e- Cl- Cl- H2O e- - Cl- Cl-
m Cl- e - Cl Cl m - e-
- Cl- Cl-
Cl b b
Zn2+ Zn2+
r Cu2+ Cu2+ r Cu2+ Cu2+
Zn 2+ Zn2+
a Cl- a
Cl- Zn → Zn2+ + 2e- Cl- n Cl- Cu2+ + 2e- → Cu Zn + 2e → Zn Cl- n
2+ - Cl- Cu → Cu2+ + 2e-
e e
j-V for a reversible solid oxide fuel cell (SOFC, a galvanic cell, j > 0)/solid
oxide electrolysis cell (SOEC, an electrolytic cell, j <0) in both modes
H2O/H2 = 0.9 : 1
j-V for a reversible solid oxide fuel cell (SOFC, a galvanic cell, j > 0)/solid
oxide electrolysis cell (SOEC, an electrolytic cell, j <0) in both modes
❑Definition
Difference between actual potential (e.g., measured) E and the equilibrium potential
𝑬𝒆𝒒 for an electrode (or redox or half cell) reaction:
𝜼 = 𝑬 𝑶𝒙/𝑹𝒆𝒅 − 𝑬𝒆𝒒 𝑶𝒙/𝑹𝒆𝒅
▪ For situation when i ≠ 0 or electrode (half cell) reaction is NOT at equilibrium
▪ Changes with current density & direction
❑ Significance
V
+
𝑬 𝑶𝒙/𝑹𝒆𝒅 > 𝑬𝒆𝒒 𝑶𝒙/𝑹𝒆𝒅 , 𝜼 > 𝟎
Anodic polarization (or bias) Red → Ox + ne-
η>0
E(Cu2+/Cu+) more positive than Eeq(Cu2+/Cu+)
Anodic polarization → oxidation
Cu+ (aq) → Cu2+ (aq) + e-
η<0
E(Cu2+/Cu+) more negative than Eeq(Cu2+/Cu+)
Cathodic polarization → reduction
Cu2+ (aq) + e- → Cu+ (aq)
Polarization curve of for Cu2+(aq, 0.1 mol/kg) / Cu+(aq, 0.1 mol/kg) electrode (redox or half cell) reaction at 25oC
1 atm with HCl (aq) supporting electrolyte of 8 mol/kg
S. Lvov, CRC Press (2015). ISBN: 978-1-4665-8285-9
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 10
Butler-Volmer Equation without
Mass Transport Limitation
For a one-step electrode (half cell) reaction without mass transport limitation
𝑂𝑥 + 𝑛𝑒 − ↔ 𝑅𝑒𝑑
Butler-Volmer equation describes net current density j as a function of total
overpotential for that electrode (half cell) reaction 𝜂 = 𝐸 𝑂𝑥/𝑅𝑒𝑑 − 𝐸𝑒𝑞 𝑂𝑥/𝑅𝑒𝑑
𝟏 − 𝜷 𝒏𝑭𝜼 𝜷𝒏𝑭𝜼
𝒋 = 𝒋𝐚 + 𝒋𝐜 = 𝒋𝐨 𝐞𝐱𝐩 − 𝐞𝐱𝐩 −
𝑹𝑻 𝑹𝑻
ja Anodic (oxidation) current density, in A/cm2
jc Cathodic (reduction) current density, in A/cm2
𝑗𝐨 Exchange current density, in A/cm2
β Symmetry factor, unitless
R Gas constant 8.314 J/(mol·K)
T Absolute temperature, K
Notes:
❑ Assuming uniform concentration for all active species:
• Extensively stirring and/or keep current relatively low
❑ Sign convention
• For 𝜂 & j: Anodic (oxidation) as positive (+); Cathodic (reduction) as negative (-)
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 11
Butler-Volmer Equation - Anodic Bias
β = 0.5
0.15 j (A/cm2)
Larger j0 →
T = 298.15 K j0 = 0.5 A/cm2
at the same 𝜂, n=1 0.10
much higher j j0 = 0.05 A/cm2
-0.05
-0.10
-0.15
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 16
Symmetry Factor β
1 − 𝛽 𝑛𝐹𝜂 𝛽𝑛𝐹𝜂
𝑗 = 𝑗a + 𝑗c = 𝑗o exp − exp −
𝑅𝑇 𝑅𝑇
β is an indicator of 0.20 j (A/cm2)
symmetry of the j0 = 0.05 A/cm2
polarization curve, T = 298.15 K 0.15
ranges from 0 to 1 β = 0.2
n=1 β = 0.5
0.10
▪ β = 0.5
Symmetrical 0.05
β = 0.8
η (V)
▪ β < 0.5 0.00
anodic (oxidation) -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
direction “easier” -0.05
-0.20
𝜂≫0 𝜂≪0
𝑅𝑇 𝑅𝑇 𝑅𝑇 𝑅𝑇
𝜂=− ln𝑗o + 𝑙𝑛𝑗 −𝜂 = − ln𝑗o + 𝑙𝑛|𝑗|
1 − 𝛽 𝑛𝐹 1 − 𝛽 𝑛𝐹 𝛽𝑛𝐹 𝛽𝑛𝐹
~11% error ln j
0.00
-6 -5 -4 ln𝑗o -3 -2 -1 0 1 2 3
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 22
Measurement of Overpotential for
an Electrode (half cell) Reaction
▪ Overpotential for an electrode (reaction) of
interest that is passing a current (working j Rext
electrode, WE) cannot be measured with V
respect to the other current-carrying electrode
reaction (counter electrode, CE) in an
electrochemical cell.
→A third electrode (reaction) must be - +
introduced as the reference electrode (RE)
▪ Measure potential difference for WE vs. RE
under equilibrium (j = 0) and with different CE
current density j,
WE
then the “overpotential” for WE ηRE ≈ 0
ηWE = [E (WE) – E (RE)] – [Eeq (WE) – Eeq(RE)] j
ηWE ≈ E (WE) – Eeq (WE) ηWE ηCE
could be obtained as a function of j
▪ To avoid overpotential contribution from RE, no
significant current must pass through RE, which
is achieved with a high impedance voltmeter
Data below were taken for a NiOOH electrode reaction involving a single electron. Fit
the data to obtain kinetic parameters in the Butler-Volmer equation.
Enter the data into MS Excel
2
Overpotential (V) j (A/m )
-0.10 -4.20
-0.09 -3.36
-0.08 -2.40
-0.07 -2.30 Initial guessed values for j0 and β
-0.06 -1.80
-0.05 -1.25 Calculated from B-V equation
-0.04 -1.00
-0.03 -0.80 Error = j – Calculated from B-V
-0.02 -0.50
-0.01 -0.22
0.01 0.24
0.02 0.45
0.03 0.80
0.04 1.00
0.05 1.45
0.06 1.80
0.07 2.10
0.08 2.80
0.09 3.50
0.10 4.10 = SUMSQ(J10:J29)
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 24
Example of Data Fitting to B-V Equation (2)
Use Excel Solver to fit for j0 and β values
Objective:
sum of error squared (D30) to Min
By changing variables cell:
β (cell D6) and j0 (cell D7)
Constraints:
β (cell D6) <= 1
Click “Solve”
Objective:
sum of error squared (cell
F29) to Min
Click “Solve”
Fitted values for (1‒ β) (cell E27) and j0 (cell E28) values
Sum of error squared (cell 29) greatly reduced (to minimum)
Fitted values by Tafel (0.317, 291 A/m2) different from B-V (0.336, 225 A/m2), but close – on same order!
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 30
Fitting to Tafel & Comparison w/ BV (5)
Anode 100,000 j (A/m2)
polarization
data for Cl2
production from
NaCl solution at
20oC (Kuhn and 10,000
Mortimer, J
Electrochem
Soc v120, 231
(1973))
1,000 T = 293.15 K
n=2
current density that αa, Zn 1.5 Anodic transfer coefficient for Zn electrode
corresponds to total cell αc, Zn 0.5 Cathodic transfer coefficient for Zn electrode
voltage of 1.3 V κ 60 S/m Electrolyte conductivity
𝛼𝑎 = 1 − 𝛽 𝑛
L 2 mm Electrolyte thickness
𝛼𝑐 = 𝛽𝑛
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 33
Voltage Loss in Electrochemical Cell
Example (2)
Part a) Want 𝑬𝒄𝒆𝒍𝒍 𝑮𝑪 = 𝑬𝒆𝒒 − 𝜼𝒂𝒏 + 𝜼𝒄𝒂𝒕 + |𝑰|𝑹Ω Knowing net j → back calculate η (for
Ni surface
both Zn anode and Ni cathode) based
Zn surface roughness on B-V equation
roughness factor 2 factor 100
i0 (Zn) A/m2 60 i0 (Ni) A/m2 0.61
alfa-a Zn 1.5 alfa-a Ni 0.5 Zn electrode anodic (oxidation)
alfa-c Zn 0.5 alfa-c Ni 0.5
Overpotential (V) j Zn by BV (A/m2) j Ni by BV (A/m2) current corresponding to positive
-0.300 -41194.4 -20940.3 overpotential
-0.299 -40400.5 -20536.7
-0.298 -39621.8 -20140.9 Interpolation for Zn anodic +1000 A/m2
-0.297 -38858.1 -19752.7 0.038 − 0.037
-0.296 -38109.2 -19372.0 𝜂𝑎,𝑍𝑛 = 0.037 + × 1000 − 982.4
-0.295 -37374.7 -18998.6 1046.1 − 982.4
-0.294 -36654.3 -18632.4
𝜂𝑎,𝑍𝑛 = 0.0373𝑉
0.036 922.2 92.6
0.037 982.4 95.6
0.038 1046.1 98.7 Ni electrode cathodic (reduction)
0.039 1113.6 101.8
current corresponding to negative
overpotential
-0.145 -2017.2 -1021.8
-0.144 -1978.3 -1001.9 Interpolation for Ni cathodic -1000 A/m2
-0.143 -1940.2 -982.5
𝜂𝑐,𝑁𝑖 = −0.1439𝑉
Initial guessed values for overpotential Calculated j values from B-V equation
for both Zn and Ni electrodes using the overpotential values
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 35
Voltage Loss in Electrochemical Cell
Example (2: Alternative-2)
Excel Goal Seek to provide fitted overpotential (B65, E65) for target current density (B67, E67)
𝜼𝒄𝒂𝒕
Ecell = Eeq
Ecell GC = Eeq ‒ ( 𝜼𝒂𝒏
Eeq + 𝜼𝒄𝒂𝒕 + |𝑰|𝑹Ω )
0 0 |𝜼𝒂𝒏 | x
x
|𝑰|𝑹Ω
Once electrochemical reaction starts, active species concentration often NOT uniform
𝒄𝑪𝒖𝟐+ 𝒄𝑪𝒖𝟐+
x x
When a reaction is limited by mass transfer, e.g., bringing of an active species from the
bulk to the electrolyte/electrode interface, current density often satisfy:
𝒋 ≈ 𝒏𝑭𝒌(𝒄𝐨𝒂𝒄𝒕𝒊𝒗𝒆 − 𝒄𝐬𝒂𝒄𝒕𝒊𝒗𝒆 )
When 𝒄𝐬𝒂𝒄𝒕𝒊𝒗𝒆 = 𝟎, current density reaches the max or limiting value, 𝒋𝐥𝐢𝐦 :
𝒋𝒍𝒊𝒎 ≈ 𝒏𝑭𝒌𝒄𝐨𝒂𝒄𝒕𝒊𝒗𝒆
The ratio s
𝑗 𝑐𝑎𝑐𝑡𝑖𝑣𝑒
≈1− o
𝑗lim 𝑐𝑎𝑐𝑡𝑖𝑣𝑒
Therefore, s
𝑐𝑎𝑐𝑡𝑖𝑣𝑒 𝑗
o ≈1−
𝑐𝑎𝑐𝑡𝑖𝑣𝑒 𝑗lim
s o
▪ j = 0, 𝑐𝑎𝑐𝑡𝑖𝑣𝑒 = 𝑐𝑎𝑐𝑡𝑖𝑣𝑒 , uniform concentration
s
▪ j = jlim , 𝑐𝑎𝑐𝑡𝑖𝑣𝑒 = 0, largest concentration gradient
▪ Applicable when migration mass transport is dominated by diffusion
is negligible and NOT electromigration
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 45
Mass Transfer Overpential (2)
From previous 𝑠
𝑐𝑎𝑐𝑡𝑖𝑣𝑒 𝑗
o ≈ 1 −
𝑐𝑎𝑐𝑡𝑖𝑣𝑒 𝑗lim
In this case, concentration overpotential due to mass transfer (e.g., diffusion) can be
approximated by:
𝑹𝑻 𝒄𝒔𝒂𝒄𝒕𝒊𝒗𝒆
𝜼= 𝐥𝐧 𝒐
𝒏𝑭 𝒄𝒂𝒄𝒕𝒊𝒗𝒆
𝑹𝑻 𝒋
𝜼= 𝐥𝐧 𝟏 −
𝒏𝑭 𝒋𝒍𝒊𝒎
If mass transfer limited by diffusion only, for the generalized Butler-Volmer equation
s s
𝑐𝑅𝑒𝑑 1 − 𝛽 𝑛𝐹𝜂 𝑐𝑂𝑥 𝛽𝑛𝐹𝜂
𝑗 = 𝑗o o ∙ exp − o ∙ exp −
𝑐𝑅𝑒𝑑 𝑅𝑇 𝑐𝑂𝑥 𝑅𝑇
s o s o
(𝑐𝑅𝑒𝑑 /𝑐𝑅𝑒𝑑 ) and (𝑐𝑂𝑥 /𝑐𝑂𝑥 ) are for the anodic and the cathodic processes of the same
electrode (half cell) reaction.
s s
𝑐𝑅𝑒𝑑 𝑗 𝑐𝑂𝑥 𝑗 j positive if anodic,
o ≈ 1 − o ≈ 1 + j negative if cathodic
𝑐𝑅𝑒𝑑 𝑗lim,𝑎 𝑐𝑂𝑥 𝑗lim,𝑐
Therefore,
𝑗 1 − 𝛽 𝑛𝐹𝜂 𝑗 𝛽𝑛𝐹𝜂
𝑗 = 𝑗o 1− ∙ exp − 1+ ∙ exp −
𝑗lim,a 𝑅𝑇 𝑗lim,c 𝑅𝑇
Expand, we have
From previous
1 − 𝛽 𝑛𝐹𝜂 1 − 𝛽 𝑛𝐹𝜂 𝑗0 𝛽𝑛𝐹𝜂 𝛽𝑛𝐹𝜂 𝑗0
𝑗 = exp 𝑗o − exp 𝑗 − exp − 𝑗o − exp − 𝑗
𝑅𝑇 𝑅𝑇 𝑗lim,𝑎 𝑅𝑇 𝑅𝑇 𝑗lim,c
Therefore,
1 − 𝛽 𝑛𝐹𝜂 𝛽𝑛𝐹𝜂
exp 𝑗 − exp −
𝑅𝑇 o 𝑅𝑇 𝑗o
𝑗=
1 − 𝛽 𝑛𝐹𝜂 𝑗0 𝛽𝑛𝐹𝜂 𝑗0
1 + exp + exp −
𝑅𝑇 𝑗lim,a 𝑅𝑇 𝑗lim,c
or
1 − 𝛽 𝑛𝐹𝜂 𝛽𝑛𝐹𝜂
exp − exp −
𝑅𝑇 𝑅𝑇
𝑗=
1 1 1 − 𝛽 𝑛𝐹𝜂 1 𝛽𝑛𝐹𝜂
𝑗o + 𝑗lim,a exp 𝑅𝑇 + 𝑗lim,𝑐 exp − 𝑅𝑇
η (V)
T = 298.15 K, j → limiting values
1
𝑗o = 0.001 A/cm2,
𝑗lim,a = 𝑗lim,c = 1 A/cm2 0.1
j (A/cm2)
0.01
1.20 j (A/cm2)
1.00 0.001
→ Tafel: η = a + b lg | j|
0.80
0.0001
|η| ~ 0.1-0.3 V
0.60
0.40 0.00001
-0.60
|η| very small (< ~0.015 V) 0.000
η (V)
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
-0.80 → Linear or j = η / Rct -0.001
-1.00
-0.002
-1.20
-0.003
❑Raise THREE (3) question that you don't understand for lecture videos
In case you have understood everything and don’t have that many questions, please
give corresponding number of multiple-choice problem (together with your answer)
that you feel can be used to check a student's understanding.
𝑗0 1 − 𝛽 𝑛𝐹𝑈 𝛽𝑛𝐹𝑈
≡ 𝑟𝑎 = 𝑘𝑎 𝑐𝑅𝑒𝑑 exp = 𝑘𝑐 𝑐𝑂𝑥 exp − = 𝑟𝑐
𝐹 𝑅𝑇 𝑅𝑇
Therefore,
𝑛𝐹𝑈 𝑘𝑐 𝑐𝑂𝑥
exp =
𝑅𝑇 𝑘𝑎 𝑐𝑅𝑒𝑑
𝑛𝐹𝑈 𝑘𝑐 𝑐𝑂𝑥
= ln
𝑅𝑇 𝑘𝑎 𝑐𝑅𝑒𝑑
𝑅𝑇 𝑘𝑐 𝑐𝑂𝑥
𝑈= ln
𝑛𝐹 𝑘𝑎 𝑐𝑅𝑒𝑑
which is (3.19)
𝑗0
= 𝑘𝑎 𝑐𝑅𝑒𝑑 ∙ 𝑘𝑐 𝑐𝑂𝑥 1−𝛽 ∙ 𝑘𝑎 𝑐𝑅𝑒𝑑 𝛽−1
𝐹
𝑗0
= 𝑘𝑐 𝑐𝑂𝑥 1−𝛽 ∙ 𝑘𝑎 𝑐𝑅𝑒𝑑 𝛽
𝐹
which is essentially (3.20a)
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 53
Mass Transfer on Reaction Rate
(Fuller Eq 3.33-1)
Mass transfer rate from bulk to surface:
𝑁𝑅 = 𝑘𝑚 𝑐𝑅,𝑏 − 𝑐𝑅,𝑠
At steady state, mass transfer and electrochemical reaction (charge transfer) match at
the electrode/electrode interface, where Tafel equation is assumed
𝑐𝑅,𝑠 𝛼𝑎 𝐹
𝑗 = 𝑛𝐹𝑘𝑚 𝑐𝑅,𝑏 − 𝑐𝑅,𝑠 = 𝑗0,𝑟𝑒𝑓 ∙ ∙ 𝑒𝑥𝑝 𝜂
𝑐𝑅,𝑟𝑒𝑓 𝑅𝑇 𝑠
Assuming bulk concentration equals reference/standard concentration
𝑐𝑅,𝑟𝑒𝑓 = 𝑐𝑅,𝑏
1 𝛼𝑎 𝐹
𝑛𝐹𝑘𝑚 𝑐𝑅,𝑏 = 𝑛𝐹𝑘𝑚 𝑐𝑅,𝑠 + 𝑗0, 𝑏 ∙ ∙ 𝑒𝑥𝑝 𝜂 𝑐
𝑐𝑅,𝑏 𝑅𝑇 𝑠 𝑅,𝑠
We have
1 𝛼𝑎 𝐹 𝑛𝐹𝑘𝑚 𝑐𝑅,𝑏
𝑗 = 𝑗0, 𝑏 ∙ ∙ 𝑒𝑥𝑝 𝜂 ∙
𝑐𝑅,𝑏 𝑅𝑇 𝑠 𝑛𝐹𝑘 + 𝑗 ∙ 1 ∙ 𝑒𝑥𝑝 𝛼𝑎 𝐹 𝜂
𝑚 0, 𝑏 𝑐𝑅,𝑏 𝑅𝑇 𝑠
𝛼𝑎 𝐹 𝑛𝐹𝑘𝑚
𝑗 = 𝑗0, 𝑏 ∙ 𝑒𝑥𝑝 𝜂 ∙
𝑅𝑇 𝑠 𝑛𝐹𝑘 + 𝑗 ∙ 1 ∙ 𝑒𝑥𝑝 𝛼𝑎 𝐹 𝜂
𝑚 0, 𝑏 𝑐𝑅,𝑏 𝑅𝑇 𝑠
EMA 5305 Electrochemical Engineering Zhe Cheng 3 Electrochemical Kinetics 55
Mass Transfer on Reaction Rate
(Fuller Eq 3.33-3)
From previous,
𝛼𝑎 𝐹 1
𝑗 = 𝑗0, 𝑏 ∙ 𝑒𝑥𝑝 𝜂 ∙
𝑅𝑇 𝑠 1 + 𝑗 ∙ 1 𝛼 𝐹
∙ 𝑒𝑥𝑝 𝑎 𝜂𝑠
0, 𝑏 𝑛𝐹𝑘𝑚 𝑐𝑅,𝑏 𝑅𝑇
1
𝑗=
1 1
𝛼𝑎 𝐹 +
𝑛𝐹𝑘𝑚 𝑐𝑅,𝑏
𝑗0,𝑏 ∙ 𝑒𝑥𝑝 𝜂
𝑅𝑇 𝑠
which is (3.33)
Therefore,
1 1
𝑗= =𝑐
1 1 𝑅,𝑠 1 1
𝛼𝑎 𝐹 + ∙ +
𝑛𝐹𝑘𝑚 𝑐𝑅,𝑏 𝑐𝑅,𝑏 𝑗 𝑗𝑙𝑖𝑚
𝑗0,𝑏 ∙ 𝑒𝑥𝑝 𝑅𝑇 𝜂𝑠
𝑐𝑅,𝑠 𝑗
+ =1
𝑐𝑅,𝑏 𝑗𝑙𝑖𝑚
𝑐𝑅,𝑠 𝑗
=1−
𝑐𝑅,𝑏 𝑗𝑙𝑖𝑚