Lecture #23 of 26 - chem.uci.edu

825

Lecture #23 of 26

826

Interfacial Charge-Transfer Kinetics

Chapter 3

827

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, Transition-StateTheory (Activated Complex Theory),Marcus Theory)

● Sections 3.2, 3.3, 3.4 & 3.6: Heterogeneous ET (Butler–Volmer Equation, Tafel Equation,Volcano Plot, Gerischer Theory,Quantum Mechanical Tunneling)

● Section 3.5: Multistep ET Mechanisms

828… and where α (or β) introduces asymmetry into this J–E curve

the Butler–Volmer Equation

----

++++

–

Note: These quadrants are flipped… but at least they are (–, –) and (+, +) now

829note that for a one-electron-transfer reaction and α (or β) < 1/2,oxidation is preferentially accelerated at any η value

----

++++


830

----

++++


note that for a one-electron-transfer reaction and α (or β) > 1/2,reduction is preferentially accelerated at any η value

831

Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 3.3.3

… now, more specifically, α (actually β) 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 triangles on the right…

tan θ = αFE/xtan ϕ = (1 – α)FE/x

832


… if the barrier is symmetrical…

this means that the cathodic and anodic barriers are affected equallyby the change in potential.

833

this means that a change in the electrode potential affects the anodicbarrier more than the cathodicbarrier.


… if the R side is steeper than the O side…

Note that in the limit of a vertical potential-energy “curve” for R at the crossingpoint, α = 0 and 100% of the potential change accelerates oxidation.

834


… 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 anodicbarrier.

Note that in the limit of a vertical potential-energy “curve” for O at the crossingpoint, α = 1 and 100% of the potential change accelerates reduction.

835

so small means η < 30 mV (αfη = (0.5)(1 / 26 mV)(30 mV) = 0.58)

B–V: exp(0) = 1 T/M: 1 + 0 = 1 (error = 0%)B–V: exp(1) = 2.7 T/M: 1 + 1 = 2 (error = -26%)

B–V: exp(0.58) = 1.78 T/M: 1 + 0.58 = 1.5 (error = -11%)

What’s small for one term?

… two limiting cases for the Butler–Volmer Equation 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

836

Note: no α!… and ohmic


… 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

837

… if, instead, η is large, then either ic or ia can be neglected…… and we obtain the famous Tafel Equation 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


–

𝑖 = −𝑖0 exp −α𝑓𝜂 ...

𝑖 = +𝑖0 exp − 1 − α 𝑓𝜂 …

838

Slope-1 = Tafel Slope(“mV per decade”)

Note: The x axis is flipped… sorry

… “η is large” means > 60 mV or so…

… both β and i0 can be obtained from a J–E curve in one direction…

839

Quick Quiz: 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: The x axis is flipped… sorry

… “η is large” means > 60 mV or so…

… Take-home point: Always report over-potentials at a specific current density

840What does real data look like, so that I can indicate the overpotential?

… wait, where is E0(O2,H+/H2O)?

… 1.23 V vs. SHE… but howcan we tell where that occursgiven this SCE scale and nomention of the pH value?

… Oh… this is in base?Gotcha! Now I see whyplotting on an RHEscale is nice forH+-dependentreactions

841

Yup! … this is a consequence of achange in the mechanism of the reaction, resulting from a change in the chemical state of the catalyst, for example…

… but even in this case, overpotentials can and should be reported for a given current density

… wait, the Tafel Slope (in units of mV/decade) changes?

842

O + ne– ⇋ Rn– (insoluble)

CR* = 0

Recall, and for clarity, that we have already encountered anoverpotential… and seen a case where it is important…

CO* = the bulk concentration of O

e.g. Ag+ + e– ⇋ Ag0

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

Eeq

843


CR* = 0


CO* = the bulk concentration of O

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

Eeq

… Interpretation: An extra potential, beyond Eeq, is required to drive mass

transfer of species to the electrode surface…

844

𝜼𝐜𝐨𝐧𝐜 =𝑹𝑻

𝒏𝑭ln

𝒊𝒍 − 𝒊

𝒊𝒍


… Interpretation: An extra potential, beyond Eeq, is required to drive mass

transfer of species to the electrode surface…

What’s happening here (not electrocatalysis)? 845


846

An overpotential that is derived from rate-limiting mass transferalone is called a concentration overpotential, ηconc…

… it is also called a concentration polarization.

Kinetic overpotential is often just called overpotential, but can also be called activation overpotential…

… okay, that summarizes this topic nicely… there is lots more to learn that your book lays out in great detail related to complex reactions and mechanisms, but this suffices for our course

847

Q: What was in this set of lectures?A: B&F Chapter 3 main concepts:

● Sections 3.1 & 3.6: Homogeneous Electron-Transfer (ET)(Arrhenius, Eyring, Transition-StateTheory (Activated Complex Theory),Marcus Theory)

● Sections 3.2, 3.3, 3.4 & 3.6: Heterogeneous ET (Butler–Volmer Equation, Tafel Equation,Volcano Plot, Gerischer Theory,Quantum Mechanical Tunneling)

● Section 3.5: Multistep ET Mechanisms

848Let’s summarize the steady-state behavior from the entire course…

… by examining I–E data when stirred away from electrode (non-hysteretic)…

… and for each applied potential, E, let’s think about what resistance (not impedance, as no AC contribution) is limiting the observed behavior…

Incorrect circuit representation of an electrochemical cell(but good enough at this stage for illustration purposes)

RuRCT RMTRDL




… so, we want to know what dictates at each E…

Fe3+/2+ and H+

electrocatalysis

H+/H2electrocatalysis

𝑅 =𝜕𝐼

𝜕𝐸

−1

850RECALL… Let’s compare total capacitance (C) and differential capacitance (Cd) as follows:

Ez = pzc




… so, we want to know what dictates at each E…

… but this will be difficult because we have several convoluting factors…

… what are the limiting behaviors of each major resistance and can we begin to piece out which resistance is responsible for observed steady-state I–Ebehavior at various E, while recalling that Eapp(I) = ECT(I) + EMT(I) + Eu(I) + …? …

𝑅 =𝜕𝐼

𝜕𝐸

−1

Correct circuit representation of an electrochemical cell

RCT ZMT

Ru

CDL

when this is fully charged,it has infinite resistance…… equals an open circuit

852… what are the limiting behaviors of each major resistance and can we piece out which is responsible for observed steady-state I–E behavior? …

electro-catalysis

masstransfer

(solution)ohmic drop

853Let’s try some examples… EXAMPLE #1

electro-catalysis

masstransfer



electro-catalysis

masstransfer


862Let’s try some examples… EXAMPLE #5… You get the idea! …… what do ELJ, EDonnan, and/or activity coeffs. do to these plots? Shift them left/right!

electro-catalysis

masstransfer


I (A)

E (V)

863Let’s try some examples… EXAMPLE #5… You get the idea! …… what do ELJ, EDonnan, and/or activity coeffs. do to these plots? Shift them left/right!

electro-catalysis

masstransfer

(solution)ohmic drop… while a little difficult to look at initially…

… ultimately, this may be clearer, because for each I, add the series E values…… in fact many fuel cell J–E data are plotted this way (J is fine when same areas)

E (V)

I (A)

864But how do we learn even more about our complex systems?

Steady-state reactions and processes can be amazingly complex (e.g. see everything we have covered thus far in the course)… ideally, we need to piece out each mechanistic component from interrelated processes… we do this by performing studies over various time regimes… thus, we need to change the temporal response of our measurements!

• R(R)DE: stirring removes mass-transport limits, which is nice… rotating the electrode does the same thing… so precisely change the rotation rate… we can also surround the disk/button by a second ring electrode to observe products of redox reactions

• EIS: sweep/scan potentials over very small range… as CVs… but then change the region (DC)… and also change the sweep/scan rate (AC)… model using ZFit/ZSim

• CV: change the scan rate… mechanisms by Saveant’s “Foot of the Wave” analysis (e.g. ECE, etc.)… model using BASi DigiSim, EC Lab, etc.

• UME: sweep/scan forward and backward very quickly

Costentin, Drouet, Robert & Savéant, J. Am. Chem. Soc., 2012, 134, 11235

Costentin, Nocera & Brodsky, PNAS, 2017, 114, 11303

865

Q: What’s in this final set of lectures?A: B&F Chapters 9, 10, and 6 main concepts:

● Sections 9.1 – 9.4: Rotating (Ring-)Disk Electrochemistry

● Sections 10.1 – 10.4: Electrochemical Impedance Spectroscopy

● Sections 6.1 – 6.6, 11.7, 14.3: Linear Sweep Voltammetry (LSV), Cyclic Voltammetry (CV), Thin-Layer Electrochemistry, Molecular Electrocatalysis

… to learn even more about your experimental systems…… go beyond steady-state conditions and modulate things!

Lecture #23 of 26 - chem.uci.edu

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