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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
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2+ 4𝑛𝐹𝐷𝐶∗𝑟0
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2+ 2𝜋𝑛𝐹𝐷𝐶∗𝑟0
… this is because a UME must behave like a macroscopic
electrode at sufficiently small times, right?... Right!
698
Cottrell eq. steady-state
current eq.
macroscopic planar electrode:
disk UME:
hemispherical UME:
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2+ 4𝑛𝐹𝐷𝐶∗𝑟0
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2+ 2𝜋𝑛𝐹𝐷𝐶∗𝑟0
… this is because a UME must behave like a macroscopic
electrode at sufficiently small times, right?... Right!
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
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2+ 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
𝑖 𝑡 =𝑛𝐹𝐴𝐷1/2𝐶∗
𝜋1/2𝑡1/2+ 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∗
𝜋1/2𝑡1/2+ 2𝜋𝑛𝐹𝐷𝐶∗𝑟0
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
http://electrochem.cwru.edu/ed/encycl/art-m04-microscopy.htm
715
http://electrochem.cwru.edu/ed/encycl/art-m04-microscopy.htm
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
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?
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
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?
𝑖 = −𝑛𝐹𝐴𝐷Δ𝐶
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!
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?
𝑖 = −𝑛𝐹𝐴𝐷Δ𝐶
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
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹ln𝑎𝑅𝑎𝑂
729Refresher… the equilibrium potential and the Nernst Equation
E0'
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
𝐸 = 𝐸0′ −𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
730Refresher… the equilibrium potential and the Nernst Equation
the formal potential… this depends on the identity and
concentration of all ionizable species present in solution
E0'
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
𝐸 = 𝐸0′ −𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
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
BaSO4(s) ⇌ SO42-(aq) + Ba2+(aq)
… let’s focus on the solubility of BaSO4…
the activity coefficient
for SO42-
the concentration
of SO42-
734
BaSO4(s) ⇌ SO42-(aq) + Ba2+(aq)
… let’s focus on the solubility of BaSO4…
the thermodynamic equilibrium constant
735
BaSO4(s) ⇌ SO42-(aq) + Ba2+(aq)
… let’s focus on the solubility of BaSO4…
the concentration equilibrium constant
736
𝐼
Fig. 10-3, p. 272 in Skoog & West
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
Fig. 10-3, p. 272 in Skoog & West
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…
… so, for NaCl, what is I?
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…
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
… the more charge on an ion, the greater the depression of its activity
coefficient by an inert salt… but at high concentration, this trend flips!
740
BaSO4(s) ⇌ SO42-(aq) + Ba2+(aq)
… let’s focus on the solubility of BaSO4…
… but what are these, how do we calculate them…
… and why do they depend on the concentration of salt?
741
BaSO4(s) ⇌ SO42-(aq) + Ba2+(aq)
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
742Refresher… the equilibrium potential and the Nernst Equation
𝐸 = 𝐸0′ −𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
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
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
𝐸 = 𝐸0′ −𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
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?
744Refresher… the equilibrium potential and the Nernst Equation
… now, γFe3+ < γFe2+, agreed?
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
𝐸 = 𝐸0′ −𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
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?
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
(in water at 25 °C)
α = effective diameter of hydrated ion (nm)
− log γ𝑥 =0.51𝑧𝑥
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
(in water at 25 °C)
α = effective diameter of hydrated ion (nm)
… the “limiting law” is when I 0 (< 10 mM),
and then the D-H eq. simplifies
to just the numerator:
− log γ𝑥 =0.51𝑧𝑥
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…
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
lnγ𝑅γ𝑂
> 0γ𝑅γ𝑂
> 1.0
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?
749
no added salt…
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
γ𝑂 ≈ γ𝑅 ≈ 1.0
lnγ𝑅γ𝑂
≈ 0
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?
750
no added salt…
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
γ𝑂 ≈ γ𝑅 ≈ 1.0
lnγ𝑅γ𝑂
≈ 0
with added salt…
ANSWER: E0' shifts to more negative potentials
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?
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–
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
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
𝐸 = 𝐸0 −𝑅𝑇
𝑛𝐹lnγ𝑅γ𝑂
−𝑅𝑇
𝑛𝐹ln𝐶𝑅𝐶𝑂
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
Underpotential deposition (UPD)… practical “activity”… even of solids!
… 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
Underpotential deposition (UPD)… practical “activity”… even of solids!
… 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
Underpotential deposition (UPD)… practical “activity”… even of solids!
… 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: