electrochemical impedance spectroscopy library
TRANSCRIPT
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Electrochemical
Impedance Spectroscopy
Library
∆O ∆R
Rt
Cdc
ZO ZR
ER@SE
December 28, 2003
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Contents
1 Reactions involving soluble species only 51.1 Redox reaction (E) . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 51.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 61.1.5 RDE (diffusion-convection) . . . . . . . . . . . . . . . . 6
1.1.6 Warburg conditions (semi-infinite linear diffusion) . . . 71.2 EE reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Kinetic equations, without
coupled homogeneous reactions . . . . . . . . . . . . . . 81.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 81.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 9
2 Reactions involving one adsorbate 112.1 Electroadsorption reaction (EAR) . . . . . . . . . . . . . . . . . . 11
2.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 112.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 122.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 12
2.2 Dissolution-passivation reaction . . . . . . . . . . . . . . . . . . . 122.2.1 Mechanism [7] . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 132.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 13
2.3 Volmer-Heyrovsky (V-H) reaction . . . . . . . . . . . . . . . . . . 142.3.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 152.3.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 15
2.4 Volmer-Tafel (V-T) reaction . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 162.4.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 17
2.5 Volmer-Heyrovsky-Tafel (V-H-T) reaction . . . . . . . . . . . . . 172.5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 17
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2.5.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 182.5.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 182.5.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 19
3 Reactions involving two adsorbates 213.1 Volmer-Heyrovsky with chemical desorption . . . . . . . . . . . . 21
3.1.1 Mechanism [6, 3, 4] . . . . . . . . . . . . . . . . . . . . . 213.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 213.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 223.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 22
3.2 Schuhmann dissolution-passivationreaction # 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Mechanism [7] . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 233.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 24
4 Reactions involving both adsorbed and soluble species 254.1 Electroadsorption reaction (EAR) with
limitation by mass transport . . . . . . . . . . . . . . . . . . . . 254.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 264.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 26
4.2 Electrosorption-desorption reaction . . . . . . . . . . . . . . . . . 264.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 274.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 274.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 28
4.3 (V-H) reaction with mass transport limitation . . . . . . . . . . . 284.3.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 284.3.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 28
4.3.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 294.3.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 30
4.4 Copper dissolution in HCl . . . . . . . . . . . . . . . . . . . . . . 314.4.1 Mechanism [5, 1, 2] . . . . . . . . . . . . . . . . . . . . . 314.4.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 314.4.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 324.4.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 32
4.5 (V-T) reaction with mass transfer limitation . . . . . . . . . . . . 334.5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 334.5.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 344.5.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 35
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Chapter 1
Reactions involving solublespecies only
1.1 Redox reaction (E)
1.1.1 Mechanism
O + eKr←→Ko
R
K r = kr exp(−αr f E ) = ko exp(−αr f (E −E ◦))
K o = ko exp(αo f E ) = ko exp(αo f (E −E ◦)) , f = F/(R T ), αo + αr = 1
1.1.2 Kinetic equations
Transformation rates
vO(t) =
−v(t), vR(t) = v(t)
Mass balance equations
Flux of soluble species : J O(0, t) = vO(t), J R(0, t) = vR(t)
Current density vs. reaction rate
if (t) = −F v(t)
Reaction rate
v(t) = −R(0, t) K o(t) + O(0, t) K r(t)
1.1.3 Steady-state conditionsSteady-state equations
Soluble species
J O(0) = − (O∗ −O(0)) mO, J R(0) = − (R∗ −R(0)) mR
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Steady-state solutions
Soluble species
R(0) =R∗ + K r (R∗/mO + O∗/mR)
1 + K o/mR + K r/mO
, O(0) =O∗ + K o (R∗/mO + O∗/mR)
1 + K o/mR + K r/mO
Current density
if =F (K o R∗ −K r O∗)
1 + K o/mR + K r/mO
1.1.4 Faradaic impedance
1.1.5 RDE (diffusion-convection)
Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z O(s) + Z R(s)
Z f (s) = 1 + K r M O(s) + K o M R(s)f F (R(0) K o αo + O(0) K r αr)
Charge transfer resistance
Rct =1
f F (R(0) K o αo + O(0) K r αr)
Concentration impedances (with ∂ Xv =∂v
∂X )
Z O = −∂ Ov M O(s)
∂ Ev= K r Rct M O(s)
Z R = ∂ Rv M R(s)∂ Ev = K o Rct M R(s)
M O(s) =1
mO
th√
τ dO s√
τ dO s, M R(s) =
1
mR
th√
τ dR s√
τ dR s
Equivalent circuit (Fig. 1.1)
∆O ∆R
Rt
Cdc
ZO ZR
Figure 1.1: Equivalent circuit for the impedance of redox reactions (RDE).
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1.1.6 Warburg conditions (semi-infinite linear diffusion)
Faradaic impedance
Z f (s) = Rct + Z O(s) + Z R(s)
only at the equilibrium potential:
E = E eq = E ◦ +1
f ln
O∗
R∗
Z f (s) =1 + K r M O(s) + K o M R(s)
f F (R∗ K o αo + O∗ K r αr)
Charge transfer resistance
Rct =1
f F (R∗ K o αo + O∗ K r αr)
Concentration impedances
Z O(s) = K r Rct M O(s), Z R(s) = K o Rct M R(s)
M O(s) =1√
s DO
, M R(s) =1√
s DR
Equivalent circuit (Fig. 1.2)
O RRt
Cdc
ZO ZR
Figure 1.2: Equivalent circuit for the impedance of redox reactions: Warburg condi-tions.
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1.2 EE reaction
1.2.1 Mechanism
RKo1←→Kr1
X + e−
XKo2←→
Kr2O + e−
K o1 = ko1 exp(αo1 f E ) = ko1 exp(αo1 f (E −E o1))
K r1 = kr1 exp(−αr1 f E ) = ko1 exp(−αr1 f (E −E o1)) , αo1 + αr1 = 1
K o2 = ko2 exp(αo2 f E ) = ko2 exp(αo2 f (E −E o2))
K r2 = kr2 exp(−αr2 f E ) = ko2 exp(−αr2 f (E −E o2)) , αo2 + αr2 = 1
1.2.2 Kinetic equations, withoutcoupled homogeneous reactions
Transformation rates
vR(t) = −v1(t), vX(t) = v1(t)− v2(t), vO(t) = v2(t)
Mass balance equations
Flux of soluble species
J R(0, t) = vR(t), J X(0, t) = vX(t), J O(0, t) = vO(t)
Current density vs. step rates
if (t) = F (v1(t) + v2(t))
Step rates
v1(t) = R(0, t) K o1(t)−X (0, t) K r1(t), v2(t) = X (0, t) K o2(t)−O(0, t) K r2(t)
1.2.3 Steady-state conditions
Steady-state equations
J R(0) = − (R∗ −R(0)) mR, J X(0) = − (X ∗ −X (0)) mX, J O(0) = − (O∗ −O(0)) mO
Steady-state solutions
Soluble species
R(0) = R∗ +R∗ K r2
mO
+X ∗ K r1
mR
+X ∗ K r1 K r2
mO mR
+
R∗ K o2mX
+R∗ K r1
mX
+R∗ K r1 K r2
mO mX
+O∗ K r1 K r2
mR mX
/
1 +K r2mO
+K o1mR
+K o1 K r2mO mR
+K o2mX
+K r1mX
+K r1 K r2mO mX
+K o1 K o2mR mX
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X (0) =
X ∗ +
X ∗ K r2mO
+X ∗ K o1
mR
+X ∗ K o1 K r2
mO mR
+
R∗ K o1mX
+O∗ K r2
mX
+R∗ K o1 K r2
mO mX
+O∗ K o1 K r2
mR mX
/
1 +K r2
mO
+K o1
mR
+K o1 K r2
mO mR
+K o2
mX
+K r1
mX
+K r1 K r2
mO mX
+K o1 K o2
mR mX
O(0) =
O∗ +
X ∗ K o2mO
+O∗ K o1
mR
+X ∗ K o1 K o2
mO mR
+
O∗ K o2mX
+O∗ K r1
mX
+R∗ K o1 K o2
mO mX
+O∗ K o1 K o2
mR mX
/
1 +K r2mO
+K o1mR
+K o1 K r2mO mR
+K o2mX
+K r1mX
+K r1 K r2mO mX
+K o1 K o2mR mX
Current density
if =
K o1 R
∗
+ K o2 X
∗
+
K o1 K o2 X ∗
mR +
K o1 K r2 R∗
mO +
2 K o1 K o2 R∗
mX −
K r1 X ∗ −K r2 O∗ − K r1 K r2 X ∗
mO
− K o1 K r2 O∗
mR
− 2 K r1 K r2 O∗
mX
/
1 +K r2mO
+K o1mR
+K o1 K r2mO mR
+K o2mX
+K r1mX
+K r1 K r2mO mX
+K o1 K o2mR mX
1.2.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z O(s) + Z R(s) + Z X(s)
Z f (s) = (1 + K o1 M O(s)) (1 + K r2 M R(s)) + (K o2 (1 + K o1 M O(s)) + K r1 (1 + K r2 M R(s))) M X(s)/
(f F (X (0) K r1 αr1 (1 + K r2 M R(s) + 2 K o2 M X(s))
+X (0) K o2 αo2 (1 + 2 K r1 M X(s)) + R(0) K r2 αr2 (1 + 2 K r1 M X(s))
+K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) + O(0) αo1 (1 + K r2 M R(s) + 2 K o2 M X(s)))))
Charge transfer resistance
Rct =1
f F (O(0) K o1 αo1 + X (0) (K o2 αo2 + K r1 αr1) + R(0) K r2 αr2)
Concentration impedances
Z O(s) = Rct K o1 M O(s) (O(0) K o1 αo1 (1 + K r2 M R(s) + K o2 M X(s))
+K r1 (X (0) αr1 (1 + K r2 M R(s)) + X (0) K o2 (αo2 + αr1) M X(s) + R(0) K r2 αr2 M X(s))) /
(X (0) K r1 αr1 (1 + K r2 M R(s) + 2 K o2 M X(s))
+X (0) K o2 αo2 (1 + 2 K r1 M X(s)) + R(0) K r2 αr2 (1 + 2 K r1 M X(s))
+K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) + O(0) αo1 (1 + K r2 M R(s) + 2 K o2 M X(s))))
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Z X(s) = Rct (K o2 −K r1) M X(s) (X (0) K o2 αo2 −X (0) K r1 αr1 (1 + K r2 M R(s))
+R(0) K r2 αr2 + K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) − αo1 O(0) (1 + K r2 M R(s)))) /
(X (0) K r1 αr1 (1 + K r2 M R(s) + 2 K o2 M X(s))
+X (0) K o2 αo2 (1 + 2 K r1 M X(s)) + R(0) K r2 αr2 (1 + 2 K r1 M X(s))
+K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) + O(0) αo1 (1 + K r2 M R(s) + 2 K o2 M X(s))))
Z R(s) = Rct K r2 M R(s) (R(0) K r2 αr2 (1 + K o1 M O(s) + K r1 M X(s))
+K o2 (X (0)αo2 + X (0)K r1 (αo2 + αr1) M X(s) + K o1 (X (0)αo2M O(s) + O(0)αo1M X(s)))) /
(X (0) K r1 αr1 (1 + K r2 M R(s) + 2 K o2 M X(s))
+X (0) K o2 αo2 (1 + 2 K r1 M X(s)) + R(0) K r2 αr2 (1 + 2 K r1 M X(s))
+K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) + O(0) αo1 (1 + K r2 M R(s) + 2 K o2 M X(s))))
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Chapter 2
Reactions involving oneadsorbate
2.1 Electroadsorption reaction (EAR)
2.1.1 Mechanism
A− + sKo←→Kr
A,s + e−
2.1.2 Kinetic equations
No mass transport limitations, Langmuir isotherm
A−(0, t) ≈ A−∗, K o = ko A−∗ exp(αo f E ) , K r = kr exp(−αr f E )
Transformation rates
vA−(t) = −v1(t), vs(t) = −v1(t), vA(t) = v1(t)
Mass balance equations
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. reaction rate
if (t) = F v(t)
Reaction rate
v(t) = θs(t) Γ K o(t)− θA(t) Γ K r(t)
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2.1.3 Steady-state conditions
Steady-state equations
Adsorbed speciesdθs/dt = 0, θA + θs = 1
Steady-state solutions
Adsorbed species
θs =K r
K o + K r, θA =
K oK o + K r
Current densityif = 0
2.1.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A(s) + Z s(s)
Z f (s) =s + K o + K r
f F Γ s (θs K o αo + θA K r αr)=
(K o + K r) (s + K o + K r)
f F s Γ K o K r
Charge transfer resistance
Rct =1
f F Γ (θs K o αo + θA K r αr)=
K o + K rf F Γ K o K r
Concentration impedancesAdsorbed species
Z A(s) =Γ K r Rct
s=
K o + K rf F s Γ K o
, Z s(s) =K o Rct
s=
K o + K rf F s Γ K r
2.2 Dissolution-passivation reaction
2.2.1 Mechanism [7]
M,sKo1←→Kr1
M2+ + s + 2 e−
A− + sKo2←→Kr2
A,s + e−
2.2.2 Kinetic equations
No mass transport limitations, Langmuir isotherm
M 2+(0, t) ≈ M 2+∗, A−(0, t) ≈ A−∗
K o1 = ko1 exp(2 αo1 f E ) , K r1 = kr1 M 2+∗ exp(−2 αr1 f E )
K o2 = ko2 A−∗ exp(αo2 f E ) , K r2 = kr2 exp(−αr2 f E )
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Transformation rates (vA stands for vA,s)
vs(t) = −v2(t), vA(t) = v2(t)
Mass balance equations
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. step rates
if (t) = F (2 v1(t) + v2(t))
Step rates
v1(t) = θs(t) Γ K o1(t)− θs(t) Γ K r1(t), v2(t) = θs(t) Γ K o2(t)− θA(t) Γ K r2(t)
2.2.3 Steady-state conditions
Steady-state equations
Adsorbed speciesdθs/dt = 0, θA + θs = 1
Steady-state solutions
Adsorbed species
θs =K o2
K o2 + K r2, θA =
K r2K o2 + K r2
Current density
if =2 F Γ (K o1 −K r1) K r2
K o2 + K r2
2.2.4 Faradaic impedance
Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A(s) + Z s(s)
Z f (s) = (s + K o2 + K r2) /
(f F Γ (θs (K o2 (s + 2 K r1) αo2 + 2 K o1 (2 (s + K o2 + K r2) αo1 −K o2 αo2) +
4 K r1 (s + K o2 + K r2) αr1) + θA (s−
2 K o1 + 2 K r1) K r2 αr2))
Z f (s) =
(K o2 + K r2) (s + K o2 + K r2)
f F Γ K r2 (4 (s + K r2) (K o1 αo1 + K r1 αr1) + K o2 (s + K o1 (−2 + 4 αo1) + K r1 (2 + 4 αr1)))
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Charge transfer resistance
Rct =1
f F Γ (4 θs K o1 αo1 + θs K o2 αo2 + 4 θs K r1 αr1 + θA K r2 αr2)
Rct =K o2 + K r2
f F Γ K r2 (K o2 + 4 K o1 αo1 + 4 K r1 αr1)Concentration impedances
Z A(s) = K r2 Rct (θs K o2 αo2 + θA K r2 αr2)/
(θs (K o2 (s + 2 K r1) αo2 + 2 K o1 (2 (s + K o2 + K r2) αo1 −K o2 αo2) +
4 K r1 (s + K o2 + K r2) αr1) + θA (s− 2 K o1 + 2 K r1) K r2 αr2)
Z A(s) =K o2 K r2 Rct
4 (s + K r2) (K o1 αo1 + K r1 αr1) + K o2 (s + K o1 (−2 + 4 αo1) + K r1 (2 + 4 αr1))
Z s(s) = −(2 K o1 + K o2 − 2 K r1) Rct (θs K o2 αo2 + θA K r2 αr2)/
(θs (−4 K o1 (s + K o2 + K r2) αo1 −K o2 (s− 2 K o1 + 2 K r1) αo2−4 K r1 (s + K o2 + K r2) αr1)− θA (s− 2 K o1 + 2 K r1) K r2 αr2)
Z s(s) =K o2 (2 K o1 + K o2 − 2 K r1) Rct
4 (s + K r2) (K o1 αo1 + K r1 αr1) + K o2 (s + K o1 (−2 + 4 αo1) + K r1 (2 + 4 αr1))
2.3 Volmer-Heyrovsky (V-H) reaction
2.3.1 Mechanism
A+ + s + e−Kr1←→Ko1
A,s
A+ + A,s + e−Kr2←→Ko2
A2 + s
2.3.2 Kinetic equations
No mass transport limitations, Langmuir isotherm
A+(0, t) ≈ A+∗, A2(0, t) ≈ A∗
2
K r1 = kr1 A+∗ exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )
K r2 = kr2 A+∗ exp(−αr2 f E ) , K o2 = ko2 A∗
2 exp(αo2 f E )
Transformation rates
vs(t) = −v1(t) + v2(t), vA(t) = v1(t)− v2(t)
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Mass balance equations
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. step rates
if (t) = −F (v1(t) + v2(t))
Step rates
v1(t) = −θA(t) Γ K o1(t) + θs(t) Γ K r1(t), v2(t) = −θs(t) Γ K o2(t) + θA(t) Γ K r2(t)
2.3.3 Steady-state conditions
Steady-state equations
Adsorbed species
dθs/dt = 0, θA + θs = 1
Steady-state solutions
Adsorbed species
θs =K o1 + K r2
K o1 + K o2 + K r1 + K r2, θA =
K o2 + K r1K o1 + K o2 + K r1 + K r2
Current density
if =2 F Γ (K o1 K o2 −K r1 K r2)
K o1 + K o2 + K r1 + K r2
2.3.4 Faradaic impedance
Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A(s) + Z s(s)
Charge transfer resistance
Rct =1
f F Γ (θA K o1 αo1 + θs K o2 αo2 + θs K r1 αr1 + θA K r2 αr2)
Concentration impedances
Z A(s) = (K o1 −K r2) Rct (θA K o1 αo1 − θs K o2 αo2 + θs K r1 αr1 − θA K r2 αr2)/
(θs K o2 (s + 2 K r1) αo2 + θs K r1 (s + 2 K o2 + 2 K r2) αr1 + θA (s + 2 K r1) K r2 αr2+
K o1 (θA (s + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))
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Z s(s) = (K o2 −K r1) Rct (− θA K o1 αo1 + θs K o2 αo2 − θs K r1 αr1 + θA K r2 αr2)/
(θs K o2 (s + 2 K r1) αo2 + θs K r1 (s + 2 K o2 + 2 K r2) αr1 + θA (s + 2 K r1) K r2 αr2+
K o1 (θA (s + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))
2.4 Volmer-Tafel (V-T) reaction
2.4.1 Mechanism
A+ + s + e−Kr1←→Ko1
A,s
2A,skd2←→kg2
A2 + 2 s
2.4.2 Kinetic equations
No mass transport limitations, Langmuir isotherm
A+(0, t) ≈ A+∗, A2(0, t) ≈ A∗
2
K r1 = kr1 A+∗ exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E ) , kg2 = kg2 A∗
2
Transformation rates
vs(t) = −v1(t) + 2 v2(t), vA(t) = v1(t)− 2 v2(t)
Mass balance equations
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. step rates
if (t) = −F v1(t)
Step rates
v1(t) = −θA(t) Γ K o1(t) + θs(t) Γ K r1(t), v2(t) = θA(t)2
Γ2 kd2 − θs(t)2
Γ2 kg2
2.4.3 Steady-state conditions
Steady-state equations
Adsorbed species
dθs/dt = 0, θA + θs = 1
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Steady-state solutions
Adsorbed species
θA =
−
4 Γ kg2 + K o1 + K r1 −
8 Γ kg2 K o1 + 8 Γ kd2 (2 Γ kg2 + K r1) + (K o1 + K r1)2
4 Γ (kd2 − kg2)
Current density
if =F
4 (kd2 − kg2)
−
4 Γ kg2 K o1 + 4 Γ kd2 K r1 + (K o1 + K r1)2
+
(K o1 + K r1)
8 Γ kg2 K o1 + 8 Γ kd2 (2 Γ kg2 + K r1) + (K o1 + K r1)
2
2.4.4 Faradaic impedance
Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A(s) + Z s(s)
Z f (s) =s + 4 θA Γ kd2 + 4 θs Γ kg2 + K o1 + K r1
F f Γ (s + 4 θA Γ kd2 + 4 θs Γ kg2) (θA K o1 αo1 + θs K r1 αr1)
Charge transfer resistance
Rct =1
f F Γ (θA K o1 αo1 + θs K r1 αr1)
Concentration impedances
Z A(s) =K o1 Rct
s + 4 θA Γ kd2 + 4 θs Γ kg2, Z s(s) =
K r1 Rct
s + 4 θA Γ kd2 + 4 θs Γ kg2
2.5 Volmer-Heyrovsky-Tafel (V-H-T) reaction
2.5.1 Mechanism
A+ + s + e−Kr1←→Ko1
A,s
A+ + A,s + e−Kr2←→Ko2
A2 + s
2A,skd3←→kg3
A2 + 2 s
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2.5.2 Kinetic equations
No mass transport limitations, Langmuir isotherm
A+(0, t) ≈ A+∗, A2(0, t) ≈ A∗
2
K r1 = kr1 A+∗ exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )
K r2 = kr2 A+∗
exp(−αr2 f E ) , K o2 = ko2 A∗
2 exp(αo2 f E ) , kg3 = k
g3 A∗
2
Transformation rates
vs(t) = −v1(t) + v2(t) + 2 v3(t), vA(t) = v1(t)− v2(t)− 2 v3(t)
Mass balance equations
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. step rates
if (t) = −F (v1(t) + v2(t))
Step rates
v1(t) = −θA(t) Γ K o1(t) + θs(t) Γ K r1(t)
v2(t) = −θs(t) Γ K o2(t) + θA(t) Γ K r2(t)
v3(t) = θA(t)2
Γ2 kd3 − θs(t)2
Γ2 kg3
2.5.3 Steady-state conditions
Steady-state equations
Adsorbed speciesdθs/dt = 0, θA + θs = 1
Steady-state solutions
Adsorbed species
θA =1
4 Γ (kg3 − kd3)(4 Γ kg3 + K o1 + K o2 + K r1 + K r2−
8 Γ (kd3 − kg3) (2 Γ kg3 + K o2 + K r1) + (4 Γ kg3 + K o1 + K o2 + K r1 + K r2)2
Current density
if = F 4 (kd3 − kg3)
4 Γ kd3 (K o2 −K r1)− (K o1 + K r1)2 + 4 Γ kg3 (−K o1 + K r2) +
(K o2 + K r2)2
+ (K o1 −K o2 + K r1 −K r2)
×
8 Γ (kd3 − kg3) (2 Γ kg3 + K o2 + K r1) + (4 Γ kg3 + K o1 + K o2 + K r1 + K r2)2
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2.5.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A(s) + Z s(s)
Z f (s) = (s + 4 θA Γ kd3 + 4 θs Γ kg3 + K o1 + K o2 + K r1 + K r2) /
(f F Γ (θs K o2 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) αo2
+θs K r1 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αr1
+θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) K r2 αr2
+K o1 (θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2))))
Charge transfer resistance
Rct =1
f F Γ (θA K o1 αo1 + θs K o2 αo2 + θs K r1 αr1 + θA K r2 αr2)
Concentration impedances
Z A(s) = (K o1 −K r2) Rct (θA K o1 αo1 − θs K o2 αo2 + θs K r1 αr1 − θA K r2 αr2)/
(θs K o2 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) αo2
+θs K r1 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αr1
+θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) K r2 αr2
+K o1 (θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))
Z s(s) = (K o2 −K r1) Rct (− θA K o1 αo1 + θs K o2 αo2 − θs K r1 αr1 + θA K r2 αr2)/
(θs K o2 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) αo2
+θs K r1 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αr1
+θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) K r2 αr2
+K o1 (θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))
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Chapter 3
Reactions involvingtwo adsorbates
3.1 Volmer-Heyrovsky with chemical desorption
3.1.1 Mechanism [6, 3, 4]
A+ + s + e−Kr1−→ A,s
A+ + A,s + e−Kr2−→ A2,s
A2,skd3−→ A2 + s
3.1.2 Kinetic equations
No mass transfer limitations, Langmuir isotherm
A+(0, t) ≈ A+∗
K r1 = kr1 A+∗ exp(−αr1 f E ) , K r2 = kr2 A+∗ exp(−αr2 f E )
Transformation rates
vs(t) = −v1(t) + v3(t), vA(t) = v1(t)− v2(t), vA2(t) = v2(t)− v3(t)
Mass balance equations
Rate of production of adsorbed species
dθs(t)dt = vs(t)Γ , dθA(t)dt = vA(t)Γ , dθA2(t)dt = vA2(t)Γ
Current density vs. step rates
if (t) = −F (v1(t) + v2(t))
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Step rates
v1(t) = θs(t) Γ K r1(t), v2(t) = θA(t) Γ K r2(t), v3(t) = θA2(t) Γ kd3
3.1.3 Steady-state conditions
Steady-state equations
Adsorbed species
dθs/dt = 0, dθA/dt = 0, θA + θA2+ θs = 1
Steady-state solutions
Adsorbed species
θA =kd3 K r1
K r1 K r2 + kd3 (K r1 + K r2), θA2
=K r1 K r2
K r1 K r2 + kd3 (K r1 + K r2)
Current density
if = −2 F Γ kd3 K r1 K r2K r1 K r2 + kd3 (K r1 + K r2)
3.1.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A(s) + Z s(s)
Charge transfer resistance
Rct =1
f F Γ (θs K r1 αr1 + θA K r2 αr2)
Concentration impedances
Z A(s) =K r2 Rct (− (θs (s + kd3) K r1 αr1) + θA (s + kd3 + K r1) K r2 αr2)
θs (s + kd3) K r1 (s + 2 K r2) αr1 + θA (s (s + kd3) + (s + 2 kd3) K r1) K r2 αr2
Z s(s) =K r1 Rct (θs K r1 (s + K r2) αr1 − kd3 (−θs K r1 αr1 + θA K r2 αr2))
θs (s + kd3) K r1 (s + 2 K r2) αr1 + θA (s (s + kd3) + (s + 2 kd3) K r1) K r2 αr2
3.2 Schuhmann dissolution-passivationreaction # 1
3.2.1 Mechanism [7]
M,sKo1←→
Kr1X,s + 2 e
X,sKo2←→Kr2
Q,s + 2 e
X,s + AKo3−→ X,s + B + 2 e
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3.2.2 Kinetic equations
No mass transfer limitations, Langmuir isotherm
A(0, t) ≈ A∗
K o1 = ko1 exp(2 αo1 f E ) , K r1 = kr1 exp(−2 αr1 f E )K o2 = ko2 exp(2 αo2 f E ) , K r2 = kr2 exp(−2 αr2 f E ) , K o3 = ko3 exp(2 αo3 f E )
Transformation rates
vs(t) = −v1(t), vX(t) = v1(t)− v2(t), vQ(t) = v2(t)
Mass balance equations
Rate of production of adsorbed species
dθs(t)
dt
=vs(t)
Γ
,dθX(t)
dt
=vX(t)
Γ
,dθQ(t)
dt
=vQ(t)
Γ
Current density vs. step rates
if (t) = 2 F (v1(t) + v2(t) + v3(t))
Step rates
v1(t) = θs(t) Γ K o1(t)− θX(t) Γ K r1(t)
v2(t) = θX(t) Γ K o2(t)− θQ(t) Γ K r2(t)
v3(t) = θX(t) Γ K o3(t)
3.2.3 Steady-state conditions
Steady-state equations
Adsorbed species
dθs/dt = 0, dθX/dt = 0, θQ + θs + θX = 1
Steady-state solutions
Adsorbed species
θQ =K o1 K o2
K r1 K r2 + K o1 (K o2 + K r2)
, θX =K o1 K r2
K r1 K r2 + K o1 (K o2 + K r2)
Current density
if =2 F Γ K o1 K o3 K r2
K r1 K r2 + K o1 (K o2 + K r2)
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3.2.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z Q(s) + Z s(s) + Z X(s)
Charge transfer resistance
Rct =1
4 f F Γ (θs K o1 αo1 + θX K o2 αo2 + θX K o3 αo3 + θX K r1 αr1 + θQ K r2 αr2)
Concentration impedances
Z Q(s) = K r2 Rct (K o2 (θX (s + K r1) αo2 + K o1 (θs αo1 + θX αo2) +
θX K r1 αr1) + θQ (s + K o1 + K r1) K r2 αr2) /θX s2 K o3 αo3 + θX s K o3 K r1 αo3 + θX s K o3 K r2 αo3 + θX K o3 K r1 K r2αo3+
θX s2 K r1 αr1 + θX s K o3 K r1 αr1 + θX s K r1 K r2 αr1 + θX K o3 K r1 K r2 αr1+
θX s K o2 ((s−K o3 + 2 K r1) αo2 + K o3 αo3 + 2 K r1 αr1) +
θQ s2 K r2 αr2 − θQ s K o3 K r2 αr2 + 2 θQ s K r1 K r2 αr2+K o1 (θs (2 s K o2 + (s + K o3) (s + K r2)) αo1 + θX s K o3 αo3+
θX K o3 K r2 αo3 + θX K o2 ((s−K o3) αo2 + K o3 αo3) + θQ s K r2 αr2 − θQ K o3 K r2 αr2))
Z s(s) = K o1 Rct (θs K o1 (s + K o2 + K r2) αo1+
K r1 (θX K o2 αo2 + θX (s + K o2 + K r2) αr1 + θQ K r2 αr2)) /θX s2 K o3 αo3 + θX s K o3 K r1 αo3 + θX s K o3 K r2 αo3 + θX K o3 K r1 K r2αo3+
θX s2 K r1 αr1 + θX s K o3 K r1 αr1 + θX s K r1 K r2 αr1 + θX K o3 K r1 K r2 αr1+
θX s K o2 ((s−K o3 + 2 K r1) αo2 + K o3 αo3 + 2 K r1 αr1) +
θQ s2 K r2 αr2
−θQ s K o3 K r2 αr2 + 2 θQ s K r1 K r2 αr2+
K o1 (θs (2 s K o2 + (s + K o3) (s + K r2)) αo1 + θX s K o3 αo3+
θX K o3 K r2 αo3 + θX K o2 ((s−K o3) αo2 + K o3 αo3) + θQ s K r2 αr2 − θQ K o3 K r2 αr2))
Z X(s) = (K o2 + K o3 −K r1) Rct (θX s K o2 αo2 − θX K r1 (s + K r2) αr1 + θQ s K r2 αr2+
K o1 (− (θs (s + K r2) αo1) + θX K o2 αo2 + θQ K r2 αr2)) /
(θX (s K o2 ((s−K o3 + 2 K r1) αo2 + K o3 αo3 + 2 K r1 αr1) +
(s + K r2) (K o3 (s + K r1) αo3 + (s + K o3) K r1 αr1)) + θQ s (s−K o3 + 2 K r1) K r2 αr2+
K o1 (θs (2 s K o2 + (s + K o3) (s + K r2)) αo1 + θX K o2 (s−K o3) αo2+
θX K o3 (s + K o2 + K r2) αo3 + θQ (s−K o3) K r2 αr2))
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Chapter 4
Reactions involvingboth adsorbedand soluble species
4.1 Electroadsorption reaction (EAR) withlimitation by mass transport
4.1.1 Mechanism
A− + sKo←→Kr
A,s + e−
4.1.2 Kinetic equations
Langmuir isotherm : K o = ko exp(αo f E ) , K r = kr exp(−αr f E )
Transformation rates
vA−(t) = −v1(t), vs(t) = −v1(t), vA(t) = v1(t)
Mass balance equations
Flux of soluble species
J A−(0, t) = vA−(t)
Rate of production of adsorbed species
dθs(t)dt = vs(t)Γ, dθA(t)dt = vA(t)Γ
Current density vs. reaction rate
if (t) = F v(t)
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Reaction rate
v(t) = A−(0, t) θs(t) Γ K o(t)− θA(t) Γ K r(t)
4.1.3 Steady-state conditions
Steady-state equations
Soluble species
J A−(0) = −
A−∗ −A−(0)
mA−
Adsorbed speciesdθs/dt = 0, θA + θs = 1
Steady-state solutions
Soluble speciesA−(0) = A−∗
Adsorbed species
θs =K r
A−∗ K o + K r, θA =
A−∗ K oA−∗ K o + K r
Current densityif = 0
4.1.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A−(s) + Z A(s) + Z s(s)
Charge transfer resistance
Rct =
1
f F Γ (A−∗ θs K o αo + θA K r αr)
Concentration impedancesSoluble species
Z A−(s) = θs Γ K o Rct M A−(s), M A−(s) =1
mA−
th√
τ A−s√
τ A−s
Adsorbed species
Z s(s) =A−∗ K o Rct
s, Z A(s) =
Γ K r Rct
s
4.2 Electrosorption-desorption reaction
4.2.1 Mechanism
A− + sKo1←→Kr1
A,s + e−
A,skd2−→ A + s
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4.2.2 Kinetic equations
Langmuir isotherm: K o1 = ko1 exp(αo1 f E ) , K r1 = kr1 exp(−αr1 f E )
Transformation rates
vA−(t) =
−v1
(t), vs(t) =
−v1
(t) + v2
(t), vA
(t) = v1
(t)−
v2
(t)
Mass balance equations
Flux of soluble species
J A−(0, t) = vA−(t)
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. step rates
if (t) = F v1(t)
Step rates
v1(t) = A−(0, t) θs(t) Γ K o1(t)− θA(t) Γ K r1(t), v2(t) = θA(t) Γ kd2
4.2.3 Steady-state conditions
Steady-state equations
Soluble speciesJ A−(0) = −
A−∗ −A−(0)
mA−
Adsorbed speciesdθs/dt = 0, θA + θs = 1
Steady-state solutions
Soluble species
A−(0) =1
2 K o1 mA−
A−∗
K o1 mA−−K r1 mA−− kd2 (Γ K o1 + mA−) + 4 Γ kd2 K o1 (kd2 + K r1) mA−+ ((A−∗ K o1 + K r1) mA−+ kd2 (−Γ K o1 + mA−))
2
Adsorbed species
θA =1
2 Γ kd2 K o1
A−∗
K o1 mA−+ K r1 mA−+ kd2 (Γ K o1 + mA−)− 4 Γ kd2 K o1 (kd2 + K r1) mA−+ ((sA−∗ K o1 + K r1) mA−+ kd2 (−Γ K o1 + mA−))
2
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Current density
if =F
2 K o1
A−∗
K o1 + K r1
mA−+ kd2 (Γ K o1 + mA−)−
4 Γ kd2 K o1 (kd2 + K r1) mA−+ A−∗
K o1 + K r1 mA−+ kd2 (−Γ K o1 + mA−)2
4.2.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A−(s) + Z A(s) + Z s(s)
Z f (s) =s + kd2 + A−(0) K o1 + K r1 + θs Γ (s + kd2) K o1 M A−(s)
f F Γ (s + kd2) (A−(0) θs K o1 αo1 + θA K r1 αr1)
M A−(s) =1
mA−
th√
τ A−s√
τ A−s
Charge transfer resistance
Rct = 1f F Γ (A−(0) θs K o1 αo1 + θA K r1 αr1)
Concentration impedancesSoluble species
Z A−(s) = θs Γ K o1 Rct M A−(s)
Adsorbed species
Z s(s) =A−(0) K o1 Rct
s + kd2, Z A(s) =
K r1 Rct
s + kd2
4.3 (V-H) reaction with mass transport limita-tion
4.3.1 Mechanism
A+ + s + e−Kr1←→Ko1
A,s
A+ + A,s + e−Kr2←→Ko2
A2 + s
4.3.2 Kinetic equations
Langmuir isotherm
K r1 = kr1 exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )
K r2 = kr2 exp(−αr2 f E ) , K o2 = ko2 exp(αo2 f E )
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Transformation rates
vA+(t) = −v1(t)− v2(t), vs(t) = −v1(t) + v2(t), vA(t) = v1(t)− v2(t)
Mass balance equations
Flux of soluble species
J A+(0, t) = vA+(t), J A2(0, t) = vA2
(t)
Rate of productions of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
Current density vs. step rate
if (t) = −F (v1(t) + v2(t))
Step rates
v1(t) =
−θA(t) Γ K o1(t) + A+(0, t) θs(t) Γ K r1(t)
v2(t) = −A2(0, t) θs(t) Γ K o2(t) + A+(0, t) θA(t) Γ K r2(t)
4.3.3 Steady-state conditions
Steady-state equations
Soluble species
J A+(0) = −
A+∗ −A+(0)
mA+, J A2(0) = − (A2
∗ −A2(0)) mA2
Adsorbed speciesdθs/dt = 0, θA + θs = 1
Steady-state solutionSoluble species (1)
A0 mA K r1 K r2 mA2A
K o1 K o2 mA2 K o2 mA A
A2 mA2
4 K o1 K r2 mA2 mA K r1 K r2 A
K o1 A2 K o2 mA2
K o1 K o22
4 K r2 mA A K o1 2 K r2 mA mA2
K o2 2 K r1 mA2A2
K r1 A
K o1 mA 2 K o1 K r2 mA2
K o2 mA2
2 K r1 K r2 mA2mA 4 K r1 K r2 mA2
A20 K r1 mA24 K r2 mA A
2 A2 mA2
mA mA A 4 A2
mA2
mA K o1 K o2 mA mA24 K r2 mA2
A2
K r2 A K o1 A2
K o2 mA
4 K o1 K r2 mA2 mA K r1 K r2 A
K o1 A2 K o2 mA2
K o1 K o22 4 K r2 mA A
K o12 K r2 mA
mA
2 K o2
2 K r1 mA
2
A2
K r1 A
K o1
mA
2 K o1 K r2 mA
2 2 mA2K o2 mA
2 2 K r1 K r2 mA2
mA 4 K r1 K r2 mA2
Adsorbed speciesCurrent density
1→ stands for =.
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Θ A 4 K o2 K r1 mA2A2
mA K o2 2 K r1 A
K o1 K r1 K r2 A K o1 A2
K o2 mA2
4 K o1 K r2 mA2 mA K r1 K r2 A
K o1 A2 K o2 mA2
K o1 K o22
4 K r2 mA A K o1 2 K r2 mA mA2
K o2 2 K r1 mA2A2
K r1 A
K o1 mA 2 K o1 K r2 mA2
4 K o1 K r2 mA2 2 K o2 2 K r1 mA2
A2
K r1 A K o1 mA
i f
F mA
4 K r1 K r2 mA2
A mA
K o1 K o2
K r1 K r2
A
K o1 A2 K o2
mA2
4 K o1 K r2 mA2 mA K r1 K r2 A
K o1 A2 K o2 mA2 K o1 K o22 4 K r2 mA A
K o1 2 K r2 mA mA2 K o2 2 K r1 mA2A2
K r1 A
K o1 mA 2 K o1 K r2 mA2
K o2 mA2
2 K r1 K r2 mA2mA 4 K r1 K r2 mA2
4.3.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A+(s) + Z A2(s) + Z A(s) + Z s(s)
Z f s s A0 K r1 K r2 K o1 2 Θ A K r2 M A s Θ s K o2 M A2s 1 K o2 A20 Θ s s A0 K r1 M A2
M A
s
Θ A
s K r2
K r1
Θ s
s 2 A
0
Θ
A Θ
s K r2
Θ s
K o2
2 A20
Θ
ss M
A2 s f F A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2
A0 Θ s K r1 Αr1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
K o1 2 A20 Θ s K o2 Αo2 A0 Θ A K r2 Αr2 Θ A Αo1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
M A+ =1
mA+
th√
τ A+ s√
τ A+ s, M A2
=1
mA2
th√
τ A2s
√τ A2
s
Transfert resistance
Rct 1
f F Θ A K o1 Αo1 A20 Θ s K o2 Αo2 A0 Θ s K r1 Αr1 A0 Θ A K r2 Αr2
Concentration impedancesSoluble species
Z A s Θ s K r1 Θ A K r2 Rct M A s
Z A2s
Θ s K o2 Rct A0 Θ s K r1 Αr1 A20 K o2 K r2 A0 Θ A s M A s Θ A K o1 Αo1 K r2 Θ A s M A s A0A20 K o2 A20 Θ s K o2 Αo2 s K o1 K r1 A0 Θ s s M A s
A0 Θ A K r2 Αr2 s K o1 K r1 A0 Θ s s M A s M A2s
A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2
A0 Θ s K r1 Αr1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
K o1 2 A20 Θ s K o2 Αo2 A0 Θ A K r2 Αr2 Θ A Αo1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
Adsorbed species
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Z As
K o1 A0 K r2 Rct A20 Θ s K o2 Αo2 2 Θ s K r1 M A s 1 A0 Θ A K r2 Αr2 2 Θ s K r1 M A s 1 Θ A K o1 Αo1
2 Θ A K r2 M A s Θ s K o2 M A2s 1 A0 Θ s K r1 Αr1 2 Θ A K r2 M A s Θ s K o2 M A2
s 1 A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2
A0 Θ s K r1 Αr1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
K o1 2 A20 Θ s K o2 Αo2 A0 Θ A K r2 Αr2 Θ A Αo1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
Z s s
A20 K o2 A0 K r1 Rct A20 Θ s K o2 Αo2 2 Θ s K r1 M A s 1 A0 Θ A K r2 Αr2 2 Θ s K r1 M A s 1 Θ A K o1Αo1 2 Θ A K r2 M A s Θ s K o2 M A2
s 1 A0 Θ s K r1 Αr1 2 Θ A K r2 M A s Θ s K o2 M A2s 1
A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2
A0 Θ s K r1 Αr1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
K o1 2 A20 Θ s K o2 Αo2 A0 Θ A K r2 Αr2 Θ A Αo1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s
4.4 Copper dissolution in HCl
4.4.1 Mechanism [5, 1, 2]
A
−
+ M,s
Ko1
←→Kr1 MA,s + e
−
A− + MA,skd2←→kg2
MA−
2 + s
4.4.2 Kinetic equations
Langmuir isotherm: K o1 = ko1 exp(αo1 f E ) , K r1 = kr1 exp(−αr1 f E )
Transformation rates
vA−(t) = −v1(t)−v2(t), vMA2−(t) = v2(t), vs(t) = −v1(t)+v2(t), vMA(t) = v1(t)−v2(t)
Mass balance equations
Flux of soluble species
J A−(0, t) = vA−(t), J MA2−(0, t) = vMA2
−(t)
Rate of production of adsorbed species
dθs(t)
dt=
vs(t)
Γ,
dθMA(t)
dt=
vM (t)
Γ
Current density vs. step rates
if (t) = F v1(t)
Step rates
v1(t) = A−(0, t) θs(t) Γ K o1(t) − θMA(t) Γ K r1(t)
v2(t) = A−(0, t) θMA(t) Γ kd2 −MA2−(0, t) θs(t) Γ kg2
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4.4.3 Steady-state conditions
Steady-state equations
Soluble species
J A−(0) =
−A−∗
−A−(0) m
A−, J
MA2−(0) =
−M A2−∗
−MA2
−(0) mMA2
−
Adsorbed species
dθs/dt = 0, θMA + θs = 1
Steady-state solutions
Soluble species
A0 mA K r1 A k d2 K o1 mMA2 k g2 mA A
K r1 MA2 mMA2
k g2 K r1 mA 4 k d2 K r1 k d2 K o1 A MA2
k g2 K r1 mA mMA2 2
4 2 k d2 K r1 k d2 A K r1 mA mMA2
k g2 2 K o1 mMA2 MA2
K o1 A
K r1 mA
2 k d2 K r1 mMA2 k g2 mA
2 2 2 k d2 K o1 k d2 K o1 mA mMA2
MA20 k g2 K r1 MA2
mMA2
mA2
k g2 K r1 mA 4 k d2 K r1 k d2 K o1 A MA2
k g2 K r1 mA mMA2 2
4 2 k d2 K r1 k d2 A K r1 mA mMA2
k g2 2 K o1 mMA2 MA2
K o1 A
K r1 mA 2 k d2 K r1 mMA2 mA
mMA2 4 2 k d2 K o1 k d2 K o1 mA mMA2
MA2
mA 4 k d2 K o1 A k d2 K o1 A
K r1 mA
4 2 k d2 K o1 k d2 K o1 mA mMA2
2 2 k g2 mA
2 mMA2
Adsorbed species
Θ MA k g2 2 K o1 A K r1 mA k g2 4 K o1 mA MA2
k d2 K o1 A
K r1 mA mMA2
k g2 K r1 mA
4 k d2 K r1
k d2 K o1
A
MA2
k g2 K r1
mA
mMA2
2
4
2 k d2 K r1
k d2 A K r1 mA mMA2
k g2 2 K o1 mMA2 MA2
K o1 A
K r1 mA 2 k d2 K r1 mMA2
4 k d2 K r1 mMA2 2 k g2 2 K o1 mMA2
MA2
K o1 A K r1 mA
Current density
i f F mA k g2 K r1 mA 4 k d2 K o1 A k d2 K o1 A
MA2 k g2 K r1 mA mMA2
k g2 K r1 mA 4 k d2 K r1 k d2 K o1 A MA2
k g2 K r1 mA mMA2
2
4 2 k d2 K r1 k d2 A K r1 mA mMA2
k g2 2 K o1 mMA2 MA2
K o1 A
K r1 mA 2 k d2 K r1 mMA2
2 k g2 mA2
4 2 k d2 K o1 k d2 K o1 mA mMA2
4.4.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A−(s) + Z MA(s) + Z s(s)
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Z f s s MA20 k 2 K 1 K 1 A0 Θ s s 2MA2
0 k 2 M A s
k 2 A0 2 A0 Θ MA Θ s K 1 Θ MA s 2 K 1 M A s
Θ s k 2 s K 1 K 1 A0 Θ s s M A s M MA2 s
f F Θ MA K 1 Α1 A0 Θ s K 1 Α1 s k 2 A0 Θ MA s M A s k 2 MA20 Θ s s M MA2
s
M A− =1
mA−
th√τ A−s√τ A−s
, M MA−2
=1
mMA−2
th
τ MA−2 s τ MA−2
s
Charge transfer resistance
Rct =1
f F Γ (A−(0) θs K o1 αo1 + θMA K r1 αr1)
Concentration impedancesSoluble species
Z A−(s) =θs Γ K o1 Rct M A−(s)
s + 2 A−(0) kd2 + kg2
2 M A2
−(0) + θs s Γ M MA2−(s)
s + kd2 (A−(0) + θMA s Γ M A−(s)) + kg2
M A2
−(0) + θs s Γ M MA2−(s)
Adsorbed species
Z MA(s) =K r1 Rct
1 + 2 θMA Γ kd2 M A−(s) + θs Γ kg2 M MA2−(s)
s + kd2 (A−(0) + θMA s Γ M A−(s)) + kg2
M A2
−(0) + θs s Γ M MA2−(s)
Z s(s) =
A−(0) K o1 Rct
1 + 2 θMA Γ kd2 M A−(s) + θs Γ kg2 M MA2
−(s)
s + kd2 (A−(0) + θMA s Γ M A−(s)) + kg2
MA2−(0) + θs s Γ M MA2−(s)
4.5 (V-T) reaction with mass transfer limitation
4.5.1 Mechanism
A+ + s + e−Kr1←→Ko1
A,s
2A,skd2←→kg2
A2 + 2 s
4.5.2 Kinetic equations
Langmuir isotherm: K r1 = kr1 exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )
Transformation rates
vA+(t) = −v1(t), vA2(t) = v2(t), vs(t) = −v1(t) + 2 v2(t), vA(t) = v1(t)− 2 v2(t)
Mass balance equations
Flux of soluble species
J A+(0, t) = vA+(t), J A2(0, t) = vA2
(t)
Rate of production of adsorbed species,
dθs(t)
dt=
vs(t)
Γ,
dθA(t)
dt=
vA(t)
Γ
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Current density vs. step rates
if (t) = −F v1(t)
Step rates
v1(t) =−
θA(t) Γ K o1(t)+A+(0, t) θs(t) Γ K r1(t), v2(t) = θA(t)2
Γ2 kd2−
A2(0, t) θs(t)2
Γ2 kg2
4.5.3 Steady-state conditions
Steady-state equations
Soluble species
J A+(0) = −
A+∗ −A+(0)
mA+, J A2(0) = (A2
∗ −A2(0)) mA2
Adsorbed speciesdθA/dt = 0, θA + θs = 1
Steady-state solutions
Soluble species: A+(0) and A2(0) are solutions of cubic equations.A+(0):
2 kd2
A+(0)Γ K r1 +
−A+∗
+ A+(0)
mA+
2mA2
=
A+∗ −A+(0)
K o1 + A+(0) K r12
mA
kg2
Γ K o1 +
A+∗ −A+(0)
mA+
2 A+∗ −A+(0)
mA+ + 2 A2
∗ mA2
A2(0):
4 (A2∗ −A2(0))
2K r1 mA2
2
K o1 + A+∗
K r1
mA+ + (A2
∗ −A2(0)) K r1 mA2
+
kd2
2 (A2
∗
−A2(0)) mA+
mA2 + Γ K r1
A
+∗
mA+
+ 2 (A2
∗
−A2(0)) mA22
=
mA+2
−
(A2∗ −A2(0))
K o1 + A+∗
K r1
2mA2
+ A2(0) kg2 (Γ K o1 + 2 (−A2
∗ + A2(0)) mA2)
Adsorbed species: θA is solution of a cubic equation
2 A2∗ Γ kg2 mA2
(−mA+ + Γ K r1 (−1 + θA)) (−1 + θA)2
+
mA+
mA2
+ Γ2 kg2 (−1 + θA)2
A+∗
K r1 (−1 + θA) + K o1 θA
+
2 Γ kd2 mA2θA
2 (mA+ + K r1 (Γ− Γ θA)) = 0
Current density: if is solution of a cubic equation
2 F if (2 Γ kd2 + K o1) K r1 mA+
if + A+∗ F mA+
mA2+
if + 2 F Γ2 kd2
K r1
2
if + A+∗
F mA+
2mA2
+
mA+2
F if
2 if kd2 + F K o12
mA2+ kg2 (if − F Γ K o1)
2(if − 2 A2
∗ F mA2)
= 0
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4.5.4 Faradaic impedance
Faradaic impedance
Z f (s) = Rct + Z A+(s) + Z A(s) + Z s(s)
Z f s s K o1 K r1 A0 Θ s s M A s 4 Θ A k d2 Θ s K r1 M A s 1
Θ s k g2 4 Θ s K r1 M A s A20 A20 Θ s s K o1 K r1 A0 Θ s s M A s M A2s
f F Θ A K o1 Αo1 A0 Θ s K r1 Αr1 s 4 Θ A k d2 Θ s k g2 4 A20 Θ s s M A2s
M A+ =1
mA+
th√
τ A+ s√
τ A+ s, M A2
=1
mA2
th√
τ A2s
√τ A2
s
Charge transfer resistance
Rct =1
f F Γ (θA K o1 αo1 + A+(0) θs K r1 αr1)
Concentration impedances Soluble species
Z A+(s) = θs Γ K r1 Rct M A+(s)
Adsorbed species
Z A(s) =K o1 Rct
1 + θs
2 Γ2 kg2 M A2(s)
s + 4 θA Γ kd2 + θs Γ kg2 (4 A2(0) + θs s Γ M A2
(s))
Z s(s) =A+(0) K r1 Rct
1 + θs
2 Γ2 kg2 M A2(s)
s + 4 θA Γ kd2 + θs Γ kg2 (4 A2(0) + θs s Γ M A2
(s))
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Copper electrodissolution in 1 M HCl at low current densities. i. Generalsteady-state study. Electrochim. Acta 43 (1998), 2469–2483.
[2] Diard, J.-P., LeCanut, J.-M., LeGorrec, B., and Montella, C.
Copper electrodissolution in 1 M HCl at low current densities. II. elec-trochemical impedance spectroscopy study. Electrochim. Acta 43 (1998),2485–2501.
[3] Diard, J.-P., LeGorrec, B., Montella, C., and Montero-Ocampo,
C. Second order electrochemical impedances and electrical resonance phe-nomenon. Electrochim. Acta 37 (1992), 177–179.
[4] Diard, J.-P., LeGorrec, B., Montella, C., and Montero-Ocampo,
C. Calculation, simulation and interpretation of electrochemical impedancediagrams. part i. second-order electrochemical impedances. J. Electroanal.
Chem. 352 (1993), 1–15.
[5] LeCanut, J.-M. Impedance faradique en presence d’un couplage elec-
trosorption-desorption, transport de matiere. C as de l’electrodissolution du
cuivre. PhD thesis, Institut National Polytechnique de Grenoble, Grenoble,1995.
[6] Montero-Ocampo, C. Impedances electrochimiques du second ordre. Ex-
emple du mecanisme de Volmer-Heyrovsky avec desorption chimique. PhDthesis, Institut National Polytechnique de Grenoble, Grenoble, 1988.
[7] Schuhmann, D. etude phenomenologique a l’aide de sch emas reaction-nels des impedances faradiques contenant des resistances negatives et desinductances. J. Electroanal. Chem. 17 (1968), 45–59.
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