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![Page 1: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/1.jpg)
1183
Electrode kinetics, … finally!
Q: What’s in this set of lectures?
A: B&F Chapter 3 main concepts:
● Sections 3.1 & 3.6: Homogeneous Electron-Transfer
(ET) (Arrhenius, Eyring, TST (ACT),
Marcus Theory)
● Sections 3.2, 3.3, 3.4 & 3.6: Heterogeneous ET (Butler–Volmer
Eq, Tafel Eq, Volcano Plot,
Gerischer Theory, Quantum
Mechanical Tunneling)
● Section 3.5: Multistep ET Mechanisms
![Page 2: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/2.jpg)
… Marcus Theory… the idea…
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1992/marcus-lecture.pdf
–ΔG0 < λ –ΔG0 > λ
–ΔG0 = λ
The nuclear reorganization energy, λ, is the
free energy required to reorganize the solvent
(outer) and bonds (inner) when the electron
moves from the reactant to product energy
potential energy well… but at the nuclear
arrangement of the reactant and for ΔG0 = 0
• Minor assumptions to go from internal (potential) energy to free energy (ΔG = ΔH – TΔS)
• Three regions of electron transfer• (I) Normal, (II) Barrierless, (III) Inverted
1184
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1185here’s a thought experiment that gets us an expression for kf:
What happens to ΔGcǂ and ΔGa
ǂ when the potential is changed by E?
1) “O” is stabilized (i.e. lowered) by F(E – E0’)…
2) … and the barrier height decreased by (1 – α)F(E – E0’)…
3) … the net change in the cathodic barrier is the difference:
F(E – E0’) – (1 – α)F(E – E0’) = αF(E – E0’)
NOTE: It’s positive; the cathodic barrier became larger.
4) … and the anodic barrier just decreased by (1 – α)F(E – E0’)…
* Linearized Marcus Theory
* assume Eeq = E0’
![Page 4: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/4.jpg)
1186
Butler–Volmer Equation:
eliminate effects due to mass transfer…
… stir well… or pass a small current…
or use surface-
adsorbed species!–
… now plug these into our expression for the current:
replace (E – E0') with η = (E – Eeq)…
… and i0 (B&F, pp. 98–99)
–
–
–
![Page 5: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/5.jpg)
1187What do these equations predict?
the Current–Overpotential Equation
–
(-)
(+)
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)...
now that I edited them
![Page 6: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/6.jpg)
1188What do these equations predict?
the Current–Overpotential Equation
the exponential increase of ic
… the exponential increase of ia
–
(-)
(+)
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)...
now that I edited them
![Page 7: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/7.jpg)
1189
i0
i0
What do these equations predict?
the Current–Overpotential Equation
–
(-)
(+)
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)...
now that I edited them
![Page 8: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/8.jpg)
1190if mass transport effects can be neglected (by rapidly stirring the solution, for example), then the Butler-Volmer Eq. is valid:
–
(-)
(+)
* Note: These quadrants are
flipped but at least they are
(-, -) and (+, +) now
![Page 9: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/9.jpg)
1191if mass transport effects can be neglected (by rapidly stirring the solution, for example), then the Butler-Volmer Eq. is valid:
i0 (j0) is called the
exchange current
(density) and is the
current that is equal
and opposite at an
electrode at
equilibrium (think
microscopic
reversibility)…
… it is the most
convenient indicator
of the kinetic facility
of an electrochemical
reaction
–
(-)
(+)
* Note: These quadrants are
flipped but at least they are
(-, -) and (+, +) now
![Page 10: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/10.jpg)
1192… a challenge in all types of kinetic analyses is making the mass-transfer-limited current, il, high enough so that a kinetically-controlled reaction rate is observed…
(-)
(+)
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)
![Page 11: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/11.jpg)
1193
http://www.scielo.br/img/revistas/qn/v28n6/26839t1.gif
j0 can vary over twenty orders of magnitude! Consider just one reaction: proton reduction (H2 evolution)…
1010!
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1194
http://www.scielo.br/img/revistas/qn/v28n6/26839t1.gif
j0 can vary over twenty orders of magnitude! Consider just one reaction: proton reduction (H2 evolution)…
1010!
To test the
materials
below Pt,
don’t use a
CE made
of Pt as in
acid it
dissolves!
![Page 13: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/13.jpg)
1195Sabatier Principle and Volcano plots for, for example, proton reduction (H2 evolution)…
Parsons, Trans. Faraday Soc., 1958, 54, 1053
Trasatti, Electroanal. Chem. Interfac. Electrochem., 1972, 39, 163
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1196Sabatier Principle and Volcano plots for, for example, proton reduction (H2 evolution)…
Parsons, Trans. Faraday Soc., 1958, 54, 1053
Trasatti, Electroanal. Chem. Interfac. Electrochem., 1972, 39, 163
Ni Mo
HHHHH
HHH+
e–
“Spillover”
(synergism)
![Page 15: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/15.jpg)
1197Simple, multistep electron-transfer mechanisms…
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1198Simple, multistep electron-transfer mechanisms…
![Page 17: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/17.jpg)
1199Simple, multistep electron-transfer mechanisms…
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1200Simple, multistep electron-transfer mechanisms…
… Tafel slope depends on rate-determining step… fun kinetics!
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1201Simple, multistep electron-transfer mechanisms…
… are not so simple… imagine CO2 + 8e– + 8H+ CH4 + 2H2O
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1202… and where α introduces asymmetry into this J–E curve
the Butler–Volmer Equation
-
-
-
-
+
+
+
+
–
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)
![Page 21: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/21.jpg)
1203note that when α < 1/2, at any η value,oxidation is preferentially accelerated
-
-
-
-
+
+
+
+
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)
![Page 22: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/22.jpg)
1204note that when α > 1/2, at any η value,reduction is preferentially accelerated
-
-
-
-
+
+
+
+
* Note: These quadrants are flipped…but at least they are (-, -) and (+, +)
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1205
Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 3.3.3
… now, more specifically, α is related to the symmetry of the
barrier in the vicinity of the crossing point…
tan = opposite/adjacent
derive this by
applying “TOA” to
the two right
triangles on the
right…
tan θ = αFE/x
tan ϕ = (1 –α)FE/x
![Page 24: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/24.jpg)
1206
Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 3.3.4
… if the barrier is symmetrical…
this means that the cathodic and anodic barriers are affected equally by the change in potential.
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1207
this means that a change in the electrode potential affects the anodic barrier more than the cathodic barrier.
Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 3.3.4
… if the R side is steeper than the O side…
Note that in the limit of a vertical potential-energy “curve” for R at thecrossing point, α = 0 and 100% of the potential change accelerates oxidation.
![Page 26: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/26.jpg)
1208
Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 3.3.4
… if the R side is more shallow than the O side…
this means that a change in the electrode potential affects the cathodic barrier more than the anodic barrier.
Note that in the limit of a vertical potential-energy “curve” for O at thecrossing point, α = 1 and 100% of the potentialchange accelerates reduction.
![Page 27: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/27.jpg)
1209two limiting cases for the B-V Eq. are important…
… first, if η is small, then exp(x) can be approximated by a Taylor/Maclaurin series expansion as 1 + x…
–
𝑖 = −𝑖0 1 + −α𝑓𝜂 − 1 + 1 − α 𝑓𝜂 = +𝑖0𝑓𝜂
![Page 28: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/28.jpg)
1210
so small means η < 30 mV (αfη = (0.5)(1 / 26 mV)(30 mV) = 0.58)
exp(0) = 1 1 + 0 = 1 (error = 0%)exp(1) = 2.7 1 + 1 = 2 (error = -26%)
(error = -11%)exp(0.58) = 1.78 1 + 0.58 = 1.58
What’s small?
two limiting cases for the B-V Eq. are important…
… first, if η is small, then exp(x) can be approximated by a Taylor/Maclaurin series expansion as 1 + x…
–
𝑖 = −𝑖0 1 + −α𝑓𝜂 − 1 + 1 − α 𝑓𝜂 = +𝑖0𝑓𝜂
![Page 29: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/29.jpg)
1211
Note: no α!
… and ohmic
two limiting cases for the B-V Eq. are important…
… first, if η is small, then exp(x) can be approximated by a Taylor/Maclaurin series expansion as 1 + x…
–
𝑖 = −𝑖0 1 + −α𝑓𝜂 − 1 + 1 − α 𝑓𝜂 = +𝑖0𝑓𝜂
![Page 30: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/30.jpg)
1212
… if, instead, η is large, then either ic or ia can be neglected…… and we obtain the famous Tafel Eq. which has two versions:
for η << 0: (current negative, or reducing/cathodic)
for η >> 0: (current positive, or oxidizing/anodic)
ln 𝑖 = ln 𝑖0 + 1 − α 𝑓η
ln 𝑖 = ln 𝑖0 − α𝑓η
two limiting cases for the B-V Eq. are important…
–
𝑖 = −𝑖0 exp −α𝑓𝜂 ...
𝑖 = +𝑖0 exp − 1 − α 𝑓𝜂 …
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1213
both α and i0 (and thus k0) and are obtained from a J–E curve in one direction…
Slope-1 = Tafel Slope
(“mV per decade”)* Note: The x axis is flipped; sorry.
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1214
… note: “η is large” means > 120 mV or so
Slope-1 = Tafel Slope
(“mV per decade”)* Note: The x axis is flipped; sorry.
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1215
… note: “η is large” means > 120 mV or so
* Note: The x axis is flipped; sorry.
Which catalyst is best?
(A) jo = 10-4 A cm-2 and 120 mV decade-1
(B) jo = 10-7 A cm-2 and 60 mV decade-1
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1216
… note: “η is large” means > 120 mV or so
* Note: The x axis is flipped; sorry.
Which catalyst is best?
(A) jo = 10-4 A cm-2 and 120 mV decade-1
(B) jo = 10-7 A cm-2 and 60 mV decade-1
It depends on the desired j…
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1217
… note: “η is large” means > 120 mV or so
* Note: The x axis is flipped; sorry.
Which catalyst is best?
(A) jo = 10-4 A cm-2 and 120 mV decade-1
(B) jo = 10-7 A cm-2 and 60 mV decade-1
It depends on the desired j…
For 1 mA cm-2, (A) is best… but…
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1218
Which catalyst is best?
(A) jo = 10-4 A cm-2 and 120 mV decade-1
(B) jo = 10-7 A cm-2 and 60 mV decade-1
It depends on the desired j…
For 1 mA cm-2, (A) is best… but…
… for 1 A cm-2, (B) is best…
… where catalyst (A) requires η = 480 mV,
while catalyst (B) requires η = 420 mV!
… note: “η is large” means > 120 mV or so
* Note: The x axis is flipped; sorry.
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• Laboratory #7 Wrap-Up will be held right now• Discuss the activity
• Provide feedback
• Recommend other activities Fe3+/2+ and H+
electrocatalysis
(care of Sunny,
Mackenzie, & Kyle)
y-axis: current
y-axis: log |current|
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• Laboratory #7 Wrap-Up will be held right now• Discuss the activity
• Provide feedback
• Recommend other activities
Trotochaud, Young, Ranney, and Boettcher, JACS, 2014, 136, 6744
OH–/H2O electrocatalysis
(care of Simon, Peter, & Hong)
RHE + 1.229 V
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1221What does real data look like?
… wait, what is E0(O2,H+/H2O)?
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1222What does real data look like?
… wait, what is E0(O2,H+/H2O)?
… 1.23 V vs. SHE… Huh?
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1223What does real data look like?
… wait, what is E0(O2,H+/H2O)?
… 1.23 V vs. SHE… Huh?
Oh, this is in base? Gotcha!
Now I see why RHE is a nice
RE for these systems
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1224
Yup! … this is a consequence of a change in the mechanism of the reaction, resulting from a change in the chemical state of the catalyst, for example.
… wait, the Tafel Slope (in units of mV/decade) changes?
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1225Recall (and for clarity) that we have already encountered the overpotential, and seen a case where it is important…
O + ne- ⇋ Rn- (insoluble)
![Page 44: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/44.jpg)
1226
CO* = the bulk concentration of O
e.g. Ag+ + e- ⇋ Ag0
Recall (and for clarity) that we have already encountered the overpotential, and seen a case where it is important…
O + ne- ⇋ Rn- (insoluble)
CR* = 0
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1227
O + ne- ⇋ Rn- (insoluble)
CO* = the bulk concentration of O
CR* = 0
e.g. Ag+ + e- ⇋ Ag0
Recall (and for clarity) that we have already encountered the overpotential, and seen a case where it is important…
Repeating a derivation akin to one we did in Chapter 1…
𝑬 = 𝑬𝟎′ +𝑹𝑻
𝒏𝑭ln 𝐶𝑂
∗ +𝑹𝑻
𝒏𝑭ln
𝒊𝒍 − 𝒊
𝒊𝒍
![Page 46: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/46.jpg)
1228
Eeq
Recall (and for clarity) that we have already encountered the overpotential, and seen a case where it is important…
O + ne- ⇋ Rn- (insoluble)
Repeating a derivation akin to one we did in Chapter 1…
𝑬 = 𝑬𝟎′ +𝑹𝑻
𝒏𝑭ln 𝐶𝑂
∗ +𝑹𝑻
𝒏𝑭ln
𝒊𝒍 − 𝒊
𝒊𝒍
𝑬 − 𝑬𝐞𝐪 = 𝜼𝐜𝐨𝐧𝐜 =𝑹𝑻
𝒏𝑭ln
𝒊𝒍 − 𝒊
𝒊𝒍
![Page 47: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/47.jpg)
1229Recall (and for clarity) that we have already encountered the overpotential, and seen a case where it is important…
O + ne- ⇋ Rn- (insoluble)
Interpretation: An extra voltage, beyond the Eeq, is required todrive mass transport of species to the electrode surface.
Repeating a derivation akin to one we did in Chapter 1…
Eeq
𝑬 = 𝑬𝟎′ +𝑹𝑻
𝒏𝑭ln 𝐶𝑂
∗ +𝑹𝑻
𝒏𝑭ln
𝒊𝒍 − 𝒊
𝒊𝒍
𝑬 − 𝑬𝐞𝐪 = 𝜼𝐜𝐨𝐧𝐜 =𝑹𝑻
𝒏𝑭ln
𝒊𝒍 − 𝒊
𝒊𝒍
![Page 48: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/48.jpg)
1230Recall (and for clarity) that we have already encountered the overpotential, and seen a case where it is important…
Interpretation: An extra voltage, beyond the Eeq, is required todrive mass transport of species to the electrode surface.
𝜼𝐜𝐨𝐧𝐜 =𝑹𝑻
𝒏𝑭ln
𝒊𝒍 − 𝒊
𝒊𝒍
![Page 49: Electrode kinetics, … finally! - UCI Department of …ardo/echem/UCI-CHEM248-2017W_lecture16.pdfElectrode kinetics, … finally! Q: ... Chem. Interfac. Electrochem., 1972, 39, 163.](https://reader036.fdocuments.us/reader036/viewer/2022082214/5aa0f00b7f8b9a71178ee879/html5/thumbnails/49.jpg)
O + ne- ⇋ Rn- (insoluble)
What’s happening here (not electrocatalysis)? 1231
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1232
An overpotential that is derived from rate-limiting mass transport alone is called a concentration overpotential, ηconc.
It’s also called: concentration polarization.
Kinetic overpotential is often just called overpotential, but can also be called activation overpotential.
… we are almost finished this topic…