Lecture #10 of 17ardo/echem/UCI-CHEM248-2019S_lecture10.pdf10-5 cm 100 nm 2.5 ms 17 ns ... A...

691

Lecture #10 of 17

692

Q: What’s in this set of lectures?

A: B&F Chapters 4 & 5 main concepts:

● Section 4.4.2: Fick’s Second Law of Diffusion

● Section 5.1: Overview of step experiments

● Section 5.2: Potential step under diffusion controlled

● Sections 5.3 & 5.9: Ultramicroelectrodes

● Sections 5.7 – 5.8: Chronoamperometry/Chronocoulometry

693

693Wightman, Anal. Chem., 1981, 53, 1125A

… scanning is “often” steady-state at a UME

… steady-state occurs when v << RTD/(nFr02)

… v (mV s-1) << 26 mV x (D/r02)… for a BASi UME with r0 = 5 𝜇m…

… (1 x 10-5 cm2 s-1) / (0.5 x 10-3 cm)2 = 26 x 40 mV s-1s

… v << 1 V s-1… Wow!

RECALL FROM LAST TIME…

694… additional/final points to address about UMEs:

1) You can buy them; how do you make them?

2) UME arrays and ensembles

3) Potential step experiments with UMEs…

4) How rapidly is steady-state attained?

695… arrays of UMEs (from B&F)…

Semi-infinite linear diffusion

Semi-infinite mixed diffusion

Semi-infinite linear diffusion, again

696

macroscopic planar electrode: 𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗

𝜋1/2𝑡1/2

… this is because a UME must behave like a macroscopic

electrode at sufficiently small times, right?... Right!

697

macroscopic planar electrode:

disk UME:

hemispherical UME:

𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗

𝜋1/2𝑡1/2


𝜋1/2𝑡1/2+ 4𝑛𝐹𝐷𝐶∗𝑟0


𝜋1/2𝑡1/2+ 2𝜋𝑛𝐹𝐷𝐶∗𝑟0



698

Cottrell eq. steady-state

current eq.

macroscopic planar electrode:

disk UME:

hemispherical UME:


𝜋1/2𝑡1/2


𝜋1/2𝑡1/2+ 4𝑛𝐹𝐷𝐶∗𝑟0





699… additional/final points to address about UMEs:

1) You can buy them; how do you make them?

2) UME arrays and ensembles… GOT IT!

3) Potential step experiments with UMEs…

4) How rapidly is steady-state attained?

700

r0 = 10 µm

5 µm

1 µm2 µm

… at early times, the current asymptotically approaches il...

C* = 1 x 10-6 moles cm-3

D = 10-5 cm2 s-1

701

r0 = 10 µm

5 µm

1 µm2 µm

C* = 1 x 10-6 moles cm-3

D = 10-5 cm2 s-1

… at early times, the current asymptotically approaches il...

702

i(t)

Time-1/2, s-1/2

note that the i(t) versus (1/t1/2) plot no longer intersects “0”…

… and D can be calculated without knowing C*… How?

2𝜋𝑛𝐹𝐷𝐶∗𝑟0



703

i(t)

Time-1/2, s-1/2

note that the i(t) versus (1/t1/2) plot no longer intersects “0”…

… and D can be calculated without knowing C*… How?

* Two equations!

… calculate D1/2C* from slope…

and DC* from y-intercept

2𝜋𝑛𝐹𝐷𝐶∗𝑟0



704… how long does it take for UME’s to attain steady-state?

r0 = 10 µm

5 µm

1 µm2 µm

Error =𝑛𝐹𝐴𝐷1/2𝐶∗


𝑛𝐹𝐴𝐷1/2𝐶∗

𝜋1/2𝑡1/2

705

r0 = 10 µm

5 µm

1 µm2 µm

25 seconds6.31.00.2

** Recall that these diffusional

processes only occur in the

absence of stirring

… how long does it take for UME’s to attain steady-state?

706

r0 r0 tsteady-state

10-3 cm 10 µm 25 s

10-4 cm 1 µm 0.25 s

10-5 cm 100 nm 2.5 ms

10-6 cm 10 nm 25 µs

707the short time limit imposed by RuCd for a macroscopic

electrode is ~100 µs…

… but UMEs are much faster!

r0 r0 tsteady-state RuCd*

10-3 cm 10 µm 25 s 1.7 µs

10-4 cm 1 µm 0.25 s 170 ns

10-5 cm 100 nm 2.5 ms 17 ns

10-6 cm 10 nm 25 µs 1.7 ns

α r02 α r0

708

r0 r0 tsteady-state RuCd*

10-3 cm 10 µm 25 s 1.7 µs

10-4 cm 1 µm 0.25 s 170 ns

10-5 cm 100 nm 2.5 ms 17 ns

10-6 cm 10 nm 25 µs 1.7 ns

… recall that Ru is nearly independent of

the WE–RE separation:

the short time limit imposed by RuCd for a macroscopic

electrode is ~100 µs…

… but UMEs are much faster!

1α r0

2 α r0 α r0-1

709

needed for il < 10 nA: a Faraday cage…

http://www.columbia.edu/cu/physics/demo-images/

… and a Keithley 428 programmable

current amplifier grounded to the cage

http://www.keithley.com/products/

… and lastly, two things you’ll really want for these experiments…

710

UMEa reference electrode

(no counter

electrode needed)

Keithley 428

Faraday cage

… there are also a few things you will not be needing…

ground the

Faraday

cage to the

Keithley

ground

711UME take-home messages:

After rapid double-layer charging…

… operate UMEs at either short times

= linear diffusion…

Can determine effects of rapid catalysis without

complications from diffuse–double-layer charging

… or long times

= steady-state radial diffusion…

Can determine D without knowing C*

712We only have time to look at techniques/publications by one

of these four UME pioneers… Guess who?

◆ Ralph “Buzz” Adams (d. Univ. Kansas)

◆ Mark Wightman (UNC Chapel Hill)

◆ Allen Bard (UT Austin)

◆ Henry White (Univ. Utah)

713Bard and scanning electrochemical microscopy (SECM)…

http://electrochem.cwru.edu/ed/encycl/art-m04-microscopy.htm

714


715


716

Fan & Bard, Science, 1995, 267, 871

717

tip

su

rfa

ce

C(x)/C*

1

0

the magic of “thin layer electrochemistry”...

distance

… Recalling Section 1.4.2 (Semi-Empirical)

𝐽𝑖 𝑥 = −𝐷𝑖𝐶𝑂∗ − 𝐶𝑂 𝑥 = 0

𝜕𝑂

718

tip

su

rfa

ce

su

rfa

ce

tip

distance

C(x)/C*

1

0

1

0

the magic of “thin layer electrochemistry”...

distance

𝐽𝑖 𝑥 = −𝐷𝑖𝐶𝑂∗ − 𝐶𝑂 𝑥 = 0

𝜕𝑂

719

su

rfa

ce

tip

distance

?

0

A feasibility assessment… 1 molecule is trapped within

a 1 µm x 1 µm x 10 nm volume between an SECM tip

and a surface. What’s the value of the limiting current?

δ0

m𝑖 = −𝑛𝐹𝐴𝐷Δ𝐶

𝛿

720

su

rfa

ce

tip

distance

?

0

δ0

m




𝑖 = −𝑛𝐹𝐴𝐷Δ𝐶

𝛿

𝐶1molecule = 1molecule1 mol

6.022 x 1023 molecules

1

10 x 10−7 cm 1 x 10−4 cm 2

𝐶1molecule = 1.66 x 10−10 mol/cm3

721

su

rfa

ce

tip

distance

?

0

δ0

m





𝛿

𝑖 = 1 eq/mol 96485 C/eq 1 x 10−4 cm 2 1 x 10−5 cm2/s1.66 x 10−10 mol/cm3

10 x 10−7 cm

= 1.6 x 10−12 A = 1.6 pA

722

su

rfa

ce

tip

distance

?

0

δ0

m

… so we’re talking about pA's. We can measure that!





𝛿

𝑖 = 1 eq/mol 96485 C/eq 1 x 10−4 cm 2 1 x 10−5 cm2/s1.66 x 10−10 mol/cm3

10 x 10−7 cm

= 1.6 x 10−12 A = 1.6 pA

723

Fan & Bard, Science, 1995, 267, 871

Reviewed in Bard & Fan, Acc. Chem. Res., 1996, 29, 572

724

Q: What was in this set of lectures?

A: B&F Chapters 4 & 5 main concepts:

● Section 4.4.2: Fick’s Second Law of Diffusion

● Section 5.1: Overview of step experiments

● Section 5.2: Potential step under diffusion controlled

● Sections 5.3 & 5.9: Ultramicroelectrodes

● Sections 5.7 – 5.8: Chronoamperometry/Chronocoulometry

725

A detailed review of Section 4.4.2 and Chapter 5

● Fick’s Second Law of Diffusion

● Linear Diffusion = time-dependent current (Cottrell Equation)

● Anson Plots for surface adsorbed species

● Radial Diffusion = time-independent current (at steady-state)

● Ultramicroelectrodes (UMEs), Scanning Electrochemical Microscopy (SECM), and single molecule electrochemistry

726

Q: What’s in this set of lectures?

A: B&F Chapter 2 main concepts:

● “Section 2.1”: Salt; Activity; Underpotential deposition

● Section 2.3: Transference numbers; Liquid-junction

potentials

● Sections 2.2 & 2.4: Donnan potential; Membrane potentials;

pH meter; Ion-selective electrodes

727Refresher… the equilibrium potential and the Nernst Equation

O + ne– ⇌ R

the standard potential

(look it up in the back of your book or in the CRC table)

activity of R

activity of O𝐸 = 𝐸0 −

𝑅𝑇

𝑛𝐹ln𝑎𝑅𝑎𝑂

728

… write the activity as the product of the activity coefficient and concentration…

Refresher… the equilibrium potential and the Nernst Equation

O + ne– ⇌ R

𝐸 = 𝐸0 −𝑅𝑇

𝑛𝐹lnγ𝑅𝐶𝑅γ𝑂𝐶𝑂


𝑛𝐹lnγ𝑅γ𝑂

−𝑅𝑇

𝑛𝐹ln𝐶𝑅𝐶𝑂


𝑛𝐹ln𝑎𝑅𝑎𝑂


E0'



−𝑅𝑇


𝐸 = 𝐸0′ −𝑅𝑇



the formal potential… this depends on the identity and

concentration of all ionizable species present in solution

E0'



−𝑅𝑇


𝐸 = 𝐸0′ −𝑅𝑇


731

Fig. 10-1, p. 268 in Skoog & West

BaSO4

CH3COOH

H2O

However, equilibrium “constants” are not… constants…

732

BaSO4(s) ⇌ SO42-(aq) + Ba2+(aq)

A (activity) = 1.0 for any pure solid compound

in its standard state at room temperature

… let’s focus on the solubility of BaSO4…

733



the activity coefficient

for SO42-

the concentration

of SO42-

734



the thermodynamic equilibrium constant

735



the concentration equilibrium constant

736

𝐼


Ionic strength, I = 0.5(zA2[A] + zB

2[B] + zC2[C] + …)

… the more charge on an ion, the greater the depression of its activity

coefficient by an inert salt…

737


𝐼


2[B] + zC2[C] + …)



… so, for NaCl, what is I?

𝐼


2[B] + zC2[C] + …)



738

Fig. 10-3, p. 272 in Skoog & West… so, for NaCl, what is I? [NaCl]!

(~10 mM) ~100 mM

739

Bockris & Reddy, Fig. 2.18


coefficient by an inert salt… but at high concentration, this trend flips!

740



… but what are these, how do we calculate them…

… and why do they depend on the concentration of salt?

741


2H2O ⇌ H3O+ + OH–

CH3COOH + H2O ⇌ H3O+ + CH3COO–

… in all three of these cases, K' > K, at not too large of an ionic strength

general observation: K' always shifts (from K) to favor

the most ionic state of the equilibrium

more ionicless ionic


𝐸 = 𝐸0′ −𝑅𝑇


Fe3+ + 1e– ⇌ Fe2+

Question: How, qualitatively, is the equilibrium potential for

Fe2+/Fe3+ affected by the addition of a supporting electrolyte, KF,

at a concentration of 0.1 M?

743

Fe3+ + 1e– ⇌ Fe2+

Refresher… the equilibrium potential and the Nernst Equation



−𝑅𝑇


𝐸 = 𝐸0′ −𝑅𝑇






… now, γFe3+ < γFe2+, agreed?



−𝑅𝑇


𝐸 = 𝐸0′ −𝑅𝑇


Fe3+ + 1e– ⇌ Fe2+




745

Table 10-2, p. 274 in Skoog & West

Debye–Hückel equation

(in water at 25 °C)

α = effective diameter of hydrated ion (nm)

− log γ𝑥 =0.51𝑧𝑥

2 𝐼

1 + 3.3α𝑥 𝐼

746Debye–Hückel equation




2 𝐼

1 + 3.3α𝑥 𝐼

… the derivation is long and the main idea is that you balance

thermal motion (Boltzmann) with electrostatics (Poisson/Gauss)

from Wiki

Physicist & P-Chemist

Peter Joseph William Debye

(1884–1966)

Physicist & P-Chemist

Erich Armand Arthur Joseph Hückel

(1896–1980)

747Debye–Hückel equation



… the “limiting law” is when I 0 (< 10 mM),

and then the D-H eq. simplifies

to just the numerator:


2 𝐼

1 + 3.3α𝑥 𝐼

Bockris & Reddy, Fig. 3.23

− log γ𝑥 = 0.51𝑧𝑥2 𝐼

748

Fe3+ + 1e– ⇌ Fe2+

… now, γFe3+ < γFe2+, agreed? So…

and

… and we conclude that in the presence of added salt…



−𝑅𝑇


lnγ𝑅γ𝑂

> 0γ𝑅γ𝑂

> 1.0




749

no added salt…



−𝑅𝑇


γ𝑂 ≈ γ𝑅 ≈ 1.0

lnγ𝑅γ𝑂

≈ 0

Fe3+ + 1e– ⇌ Fe2+




750

no added salt…



−𝑅𝑇


γ𝑂 ≈ γ𝑅 ≈ 1.0

lnγ𝑅γ𝑂

≈ 0

with added salt…

ANSWER: E0' shifts to more negative potentials

Fe3+ + 1e– ⇌ Fe2+




and lnγ𝑅γ𝑂

> 0γ𝑅γ𝑂

> 1.0

751Question: What if the redox species were negatively charged,

like [FeIII(CN)6]3–/[FeII(CN)6]

4–, and we increased the concentration

of supporting electrolyte to ~0.1 M?

[FeIII]3– + 1e– ⇌ [FeII]4–



−𝑅𝑇


with added salt…Fe(CN)6

3-/Fe(CN)64-

No added salt(note ipc and iR)

and 𝒍𝒏𝜸𝑹𝜸𝑶

< 𝟎𝜸𝑹𝜸𝑶

< 𝟏. 𝟎

ANSWER: E0' shifts to more positive potentials

increasing

supporting

electrolyte

Im-Tl+, to bring Tl+

to the electrode

Id-Tl+, to bring Tl+

to the electrode

cathodic reaction

752Question: What if the redox species were positive/neutral

charged, like Tl+/0, and we increased the concentration of

supporting electrolyte to ~0.1 M?

Tl+ + 1e– ⇌ Tl0



−𝑅𝑇


with added salt…

and 𝒍𝒏𝜸𝑹𝜸𝑶

> 𝟎𝜸𝑹𝜸𝑶

> 𝟏. 𝟎

ANSWER: E0', again, shifts to more negative potentials

753

Co0 ⇌ Co2+ + 2e–

Eeq

Underpotential deposition (UPD)… practical “activity”… even of solids!

Co2+ + 2e– ⇌ Co0

754

Co0 ⇌ Co2+ + 2e–

Eeq

Co2+ + 2e– ⇌ Co0


… but what are these

small cathodic current "bumps"

that occur at Eapp > Eeq?

755

Co0 ⇌ Co2+ + 2e–

Eeq

B&F, pg. 420

… aCo < 1… because the

activity of a solid is proportional

to its surface coverage!

Co2+ + 2e– ⇌ Co0


… but what are these

small cathodic current "bumps"

that occur at Eapp > Eeq?

756

Co0 ⇌ Co2+ + 2e–

Co2+ + 2e– ⇌ Co0

Eeq

Coδ+

gold

OPD of cobalt

gold

cobalt UPD of cobalt

Mendoza-Huizar, Robles, & Palomar-Pardavé, J. Electroanal. Chem., 2003, 545, 39


… aCo < 1… because the

activity of a solid is proportional

to its surface coverage!

B&F, pg. 420

757

an ISE (for nitrate ions)an SCE

Now on to two general liquid junctions that we care about (the most)…

758

when two ionic solutions are separated across an interface that

prevents bulk mixing of the ions, but has ionic permeability, a

potential (drop) develops called the liquid junction potential.

Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 2.3.2

same salt;

different conc.

one ion in common;

same conc.everything else.

… liquid junctions:

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