Potential Sweep Methods
2-25-2014
Linear Potential Sweep Voltammetry
Ox + ne– ⇄ Red
E (V)
(-)
E(t) = Ei – vt
At Ei nonfaradaic current
COx , CRed (M)
COx*
Increasing time at |E| < |E0'|
More negative E at |E| > |E0'|
t (s)
i (A)
- Supporting electrolyte present in excess to
minimize migration.
- Solution unstirred to min. convection.
- O2 is removed by bubbling N2.
- Stationary electrodes: HDME, graphite, Pt,
etc.
Distance from Electrode (cm)
(+)
v ranges from 10 mV s-1 to ~1000 V s-1 for
conventional electrodes; up to 106 V s-1
for UMEs.
ip
ip/2
E (V)
Ep/2 Ep
(-)
Cyclic Voltammetry
COx*
E (V)
(-)
Distance from Electrode (cm)
(+)
Distance from Electrode (cm)
El
Ox + ne–  Red
ip,c
Ox – ne–  Red
ip,a
i (A)
COx , CRed (M)
COx , CRed (M)
COx*
DEp
E (V)
E0'
(-)
Ei
0
t (s)
l
- Reversal technique analogous to
double-potential step methods.
- Experiment time scale: 103 s to 10-5 s.
- For reversible electron transfer:
- DE = |Ep,a – Ep,c| = 59 mV/n
- |Ep – Ep/2| = 57 mV/n
- ip,a/ip,c = 1
El
Derivation of i-E relationship:
Description of system and conditions
O + ne– ⇄ R (Assume semi-inf linear diff & only O initially present at potential Ei where no
faradaic reaction occurs, rapid electron transfer)
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
E(t) = Ei – vt
Time dependence makes Laplace transform complicated.
Rewrite boundary condition (Nicholson & Shain):
CO(0,t) / CR(0,t) = q e –st = q S(t)
We want to determine the i-E relationship. Since E is linearly related to t, we first find
the i-t relationship. Recall that current is proportional to flux at the electrode surface.
Review of i-t derivation strategy
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
Recall Fick’s 2nd Law:
CR(x,t) / t = DR 2CR(x,t) / x2
CO(x,t) / t = DO 2CO (x,t) / x2
CO(x,0) = CO*
CR(x,0) = 0
limx  CO(x,t) = CO*
limx  CR(x,t) = 0
Recall Laplace Transform:
𝐿 𝐹(𝑡) =
∞ −𝑠𝑡
𝑒 𝐹
0
∞
𝐿 𝑑𝐹(𝑡) 𝑑𝑡 = 𝐿 𝐹′(𝑡) =
𝑒 −𝑠𝑡 𝐶𝑑𝑡 = 𝐶/𝑠
𝐿 𝐶 =
𝑡 𝑑𝑡 = 𝐹(s)
0
∞ −𝑠𝑡 ′ 𝑡
𝑒 𝐹
0
𝑑𝑡 = −𝐹 0 + 𝑠𝐹(s)
𝐿 𝑑 2 𝐹(𝑡) 𝑑𝑡 2 = −𝐹 ′ 0 − 𝑠𝐹 0 + 𝑠 2 𝐹(s)
L{CO(x,t) / t} = L{DO 2CO(x,t) / x2} 
−𝐶𝑂 (𝑥, 0) + 𝑠 𝐶𝑂 𝑥, 𝑠 = 𝐷𝑂 𝑑 2 𝐶𝑂 (𝑥, 𝑠) 𝑑𝑥 2
Solving Laplace transform differential
equation for concentration profile
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
−𝐶𝑂 ∗
𝑑2 𝐶𝑂 (𝑥, 𝑠)
𝑠
−𝐶𝑂 𝑥, 0
−
𝐶 𝑥, 𝑠 =
=
𝑑𝑥 2
𝐷𝑂 𝑂
𝐷𝑂
𝐷𝑂
−𝐶𝑂 (𝑥, 0) + 𝑠 𝐶𝑂 𝑥, 𝑠 = 𝐷𝑂 𝑑2 𝐶𝑂 (𝑥, 𝑠) 𝑑𝑥 2
To solve, consider the analogous equation:
𝑑 2 𝐶(𝑥)
− 𝑎2 𝐶 𝑥 = −𝑏
2
𝑑𝑥
2
′
2
𝑠 𝐶 𝑠 − 𝑠𝐶 0 − 𝐶 0 − 𝑎 𝐶 𝑠 = −𝑏/𝑠
−𝑏 + 𝑠 2 𝐶 0 + 𝑠𝐶′(0)
𝐴′
𝐵′
𝐷′
𝐶 𝑠 =
=
+
+
𝑠(𝑠 + 𝑎)(𝑠 − 𝑎)
𝑠+𝑎 𝑠−𝑎 𝑠
Find C(x) by inverse Laplace
Notice D’ = b/a2
𝐿 𝐴′𝑒 −𝑎𝑥 =
∞
𝑒 −𝑠𝑥 𝐴′𝑒 −𝑎𝑥 𝑑𝑥 =
0
𝐿−1
𝐶(𝑠) = 𝐶 𝑥 =
𝐿−1
∞
0
𝐴′𝑒 − 𝑠+𝑎
′
𝐴
𝑥 𝑑𝑥 =
𝑠+𝑎
𝐴′
𝐵′
𝐷′
−1
−1
+𝐿
+𝐿
= 𝐴′ 𝑒 −𝑎𝑥 + 𝐵′ 𝑒 𝑎𝑥 + 𝑏/𝑎2
𝑠+𝑎
𝑠−𝑎
𝑠
Application of boundary conditions
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
−𝐶𝑂 ∗
𝑑2 𝐶𝑂 (𝑥, 𝑠)
𝑠
−𝐶𝑂 𝑥, 0
−
𝐶 𝑥, 𝑠 =
=
𝑑𝑥 2
𝐷𝑂 𝑂
𝐷𝑂
𝐷𝑂
CO(x,t) / t = DO 2CO (x,t) / x2
𝑑 2 𝐶(𝑥)
−𝑏 + 𝑠 2 𝐶 0 + 𝑠𝐶′(0)
𝐴′
𝐵′
𝐷′
2
− 𝑎 𝐶 𝑥 = −𝑏
𝐶 𝑠 =
=
+
+
𝑠(𝑠 + 𝑎)(𝑠 − 𝑎)
𝑠+𝑎 𝑠−𝑎 𝑠
𝑑𝑥 2
𝐿−1 𝐶(𝑠) = 𝐶 𝑥 = 𝐴′ 𝑒 −𝑎𝑥 + 𝐵′ 𝑒 𝑎𝑥 + 𝑏/𝑎2
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥
𝑠/𝐷𝑂
+ 𝐵′ 𝑒 𝑥
Recall Boundary conditions:
CO(x,0) = CO*
limx  CO(x,t) = CO*
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥
lim 𝐶𝑂 𝑥, 𝑠 = 𝐶𝑂 ∗ /𝑠
𝑥→∞
𝑠/𝐷𝑂
+ 𝐶𝑂 ∗ /𝑠
𝑠/𝐷𝑂
+ 𝐶𝑂 ∗ /𝑠
CR(x,0) = 0
limx  CR(x,t) = 0
𝐶𝑅 𝑥, 𝑠 = 𝐵′ 𝑒 −𝑥
𝑠/𝐷𝑅
Determining coefficients for
concentration profile
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
Flux balance:
𝜕𝐶𝑂 (𝑥, 𝑠)
𝜕𝑥
𝐷𝑂
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥
′
−𝐴
1 1
𝐷𝑂 2 𝑠 2
−𝐵
′
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥
𝑠/𝐷𝑂
+ 𝐷𝑅
𝑥=0
+ 𝐶𝑂 ∗ /𝑠
1 1
𝐷𝑅 2 𝑠 2
𝑠/𝐷𝑂
𝜕𝐶𝑅 (𝑥, 𝑠)
𝜕𝑥
=0
𝑥=0
𝐶𝑅 𝑥, 𝑠 = 𝐵′ 𝑒 −𝑥
𝑠/𝐷𝑅
B’ = –A’x ; x = (DO/DR)(1/2)
=0
+ 𝐶𝑂 ∗ /𝑠
𝐶𝑅 𝑥, 𝑠 = 𝐴′ 𝜉𝑒 −𝑥
𝑠/𝐷𝑅
For any i-E experiment
For reversible system (potential step):
q = CO(0,t) / CR(0,t) = exp[n F (E – E0’) / (R T)]
𝐶𝑂 ∗
+ 𝐴′ = −𝜉𝜃𝐴′
𝑠
𝐶𝑂 0, 𝑠 = 𝜃𝐶𝑅 0, 𝑠
Current-time function for potential step
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥
∗
𝐶
𝑂
𝑠/𝐷𝑂
+
𝑠
𝐶𝑅 𝑥, 𝑠 = 𝐴′ 𝜉𝑒 −𝑥
For reversible system:
∗
𝐶
𝑂
𝐴′ = −
𝑠(1 + 𝜉𝜃)
𝐶𝑂 ∗
+ 𝐴′ = −𝜉𝜃𝐴′
𝑠
q = CO(0,t) / CR(0,t) = exp[n F (E – E0’) / (R T)]
𝐶𝑂 ∗
𝐶𝑂 𝑥, 𝑠 = −
𝑒 −𝑥
𝑠(1 + 𝜉𝜃)
∗
𝐶
𝑂
𝑠/𝐷𝑂 +
𝑠
Consider Flux:
𝑖 𝑠
𝜕𝐶𝑂 (𝑥, 𝑠)
−𝐽𝑂 0, 𝑠 =
= 𝐷𝑂
𝑛𝐹𝐴
𝜕𝑥
x = (DO/DR)(1/2)
𝜉𝐶𝑂 ∗
𝐶𝑅 𝑥, 𝑠 =
𝑒 −𝑥
𝑠(1 + 𝜉𝜃)
𝑖 𝑠 =
𝑥=0
𝑠/𝐷𝑅
𝑛𝐹𝐴𝐷𝑂 1/2 𝐶𝑂 ∗
1
𝑠 2 (1
+ 𝜉𝜃)
𝑖 𝑡 =
𝑠/𝐷𝑅
𝑛𝐹𝐴𝐷𝑂 1/2 𝐶𝑂 ∗
1 1
𝑡 2 𝜋 2 (1
+ 𝜉𝜃)
Concentration profile coefficients for
potential sweep
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥
∗
𝐶
𝑂
𝑠/𝐷𝑂
+
𝑠
𝐶𝑅 𝑥, 𝑠 = 𝐴′ 𝜉𝑒 −𝑥
𝑠/𝐷𝑅
For reversible system (Potential Sweep):
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
CO(0,t) / CR(0,t) = q e –st = q S(t)
s = n F v/ (R T)
q = exp[n F (E – E0’) / (R T)]
Consider Flux:
𝑖 𝑠
𝜕𝐶𝑂 (𝑥, 𝑠)
−𝐽𝑂 0, 𝑠 =
= 𝐷𝑂
𝑛𝐹𝐴
𝜕𝑥
𝑖 𝑠 =
1
1
′
−𝑛𝐹𝐴𝐷𝑂 2 𝐴 𝑠 2
𝑥=0
𝜕𝐶𝑂 (𝑥, 𝑠)
𝜕𝑥
=
𝑥=0
𝐴′ = −
−𝐴′ 𝑠1/2
𝐷𝑂 1/2
𝑖 𝑠
1 1
𝑛𝐹𝐴𝐷𝑂 2 𝑠 2
Concentration-time function at
electrode surface for potential sweep
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
For Potential Sweep:
∗
𝐶
𝑂
𝐶𝑂 𝑥, 𝑠 = 𝐴′ 𝑒 −𝑥 𝑠/𝐷𝑂 +
𝑠
𝐶𝑂 ∗
𝑖 𝑠
𝐶𝑂 0, 𝑠 = −
1 1+ 𝑠
𝑛𝐹𝐴𝐷𝑂 2 𝑠 2
𝐿−1
Convolution integral:
𝐴′
𝐶𝑂 0, 𝑠
𝐹 𝑡 − 𝜏 𝐺 𝜏 𝑑𝜏
0
𝐶𝑂 0, 𝑡 = 𝐶𝑂 −
𝐿−1
1 1
𝑛𝐹𝐴𝐷𝑂 2 𝑠 2
𝐶𝑂 ∗
𝑠
−
1
1
𝑛𝐹𝐴𝐷𝑂 2
−1
𝐿
𝑖 𝑠
𝐿−1
𝑡
𝐿 𝑓 𝑠 𝑔(𝑠) = F t ∗ G(t) =
∗
= 𝐶𝑂 0, 𝑡 =
=−
𝑖 𝑠
1 1
𝑛𝐹𝐴𝐷𝑂 2 𝑠 2
𝑡
𝐹 𝑠 𝐺(𝑠) = f t ∗ g(t) =
𝑓 𝑡 − 𝜏 𝑔 𝜏 𝑑𝜏
0
𝐿−1
𝑖 𝑠
1
𝑠2
∗
= 𝐶𝑂 −
1
𝑡
1
𝑛𝐹𝐴𝐷𝑂 2 0
1
−2
𝜋
𝑡−𝜏
1
−2
𝑖
𝜏 𝑑𝜏
Application of boundary conditions to general
concentration integral equations for potential sweep
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
For Potential Sweep:
∗
𝐶𝑂 0, 𝑡 = 𝐶𝑂 −
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
𝑡
1
1 1
𝑛𝐹𝐴𝐷𝑂 2 𝜋 2 0
−1/2
− (𝐷𝑂 𝜋)
𝑡
𝑡−𝜏
𝑡−𝜏
−
1
2𝑓
1
−
2𝑖
𝜏 𝑑𝜏
𝜏 𝑑𝜏
0
f(t) = i(t) / (n F A)
−1/2
𝐶𝑅 0, 𝑡 = (𝐷𝑅 𝜋)
𝑡
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
General equations for potential sweep (electrode kinetics/boundary conditions
not yet considered)
For reversible system (Potential Sweep):
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
q = exp[n F (E – E0’) / (R T)]
CO(0,t) / CR(0,t) = q e –st = q S(t)
s = n F v/ (R T)
Using boundary conditions to find i-t
relationship (reversible systems)
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
−1/2
− (𝐷𝑂 𝜋)
𝑡
𝑡−𝜏
−
1
2𝑓
0
𝑡
−1/2
𝜏 𝑑𝜏
𝐶𝑅 0, 𝑡 = (𝐷𝑅 𝜋)
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
f(t) = i(t) / (n F A)
For reversible system (Potential Sweep):
CO(0,t) / CR(0,t) = q e –st = q S(t)
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
s = n F v/ (R T)
q = exp[n F (E – E0’) / (R T)]
𝑡
𝑡−𝜏
0
(𝐷𝑂 𝜋)−1/2 𝐶𝑂
−
1
2𝑓
𝐶𝑅 0, 𝑡
𝜏 𝑑𝜏 =
=
(𝐷𝑅 𝜋)−1/2
0, 𝑡 = 𝐷𝑅 𝜋
1
−2
𝜃𝑆(𝑡)
𝐶𝑂 0, 𝑡
𝐷𝑅 𝜋
−
1
2 𝜃𝑆(𝑡)
=
𝐶𝑂 ∗ −𝐶𝑂 0, 𝑡
(𝐷𝑂 𝜋)−1/2
𝐶𝑂 ∗ −𝐶𝑂 0, 𝑡
𝐶𝑂 0, 𝑡 =
𝐶𝑂
𝐶𝑂
∗
𝐷𝑂 𝜋
∗
1
−2
𝐷𝑅 𝜋
1
−2
𝜃𝑆(𝑡)
+ 𝐶𝑂
∗
𝐷𝑅 𝜋
1
−2
𝜃𝑆(𝑡)
Using boundary conditions to find i-t
relationship (reversible systems)
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
For reversible system (Potential Sweep):
CO(0,t) / CR(0,t) = q e –st = q S(t)
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
𝑡
𝑡−𝜏
−
1
2𝑓
0
𝐶𝑂 0, 𝑡 =
𝐶𝑂 ∗ 𝐷𝑅 𝜋
𝐷𝑂 𝜋
1
−2
𝑡
𝑡−𝜏
0
𝐶𝑅 0, 𝑡
𝜏 𝑑𝜏 =
=
(𝐷𝑅 𝜋)−1/2
1
−2
𝜃𝑆(𝑡)
+ 𝐷𝑅 𝜋
1
−2
𝑖
1
−2
𝜃𝑆(𝑡)
𝐶𝑂 0, 𝑡
𝐷𝑅 𝜋
−
1
2 𝜃𝑆(𝑡)
𝑡
𝑡−𝜏
=
1
−2
𝑓
0
𝑛𝐹𝐴𝐶𝑂 ∗ 𝜋 1/2 𝐷𝑂 1/2
𝜏 𝑑𝜏 =
𝜉𝜃𝑆 𝑡 + 1
Solution gives i(t)  i(E). Closed-form solution cannot
be obtained, must use numerical methods.
𝐶𝑂 ∗ −𝐶𝑂 0, 𝑡
(𝐷𝑂 𝜋)−1/2
𝐶𝑂 ∗
𝜏 𝑑𝜏 =
𝐷𝑂 𝜋
1
−2
+ 𝐷𝑅 𝜋
x = (DO/DR)(1/2)
f(t) = i(t) / (n F A)
1
−2
𝜃𝑆(𝑡)
i(t) to i(E) (reversible systems)
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
For reversible system (Potential Sweep):
𝑡
𝑡−𝜏
1
−2
𝑖
0
∗
𝑛𝐹𝐴𝐶𝑂 𝜋 1/2 𝐷𝑂
𝜏 𝑑𝜏 =
𝜉𝜃𝑆 𝑡 + 1
x = (DO/DR)(1/2)
1/2
s t = n F v t/(R T) = n F (Ei – E)/(RT)
f(t) = i(t) / (n F A)
q = exp[n F (E – E0’) / (R T)]
Change from i(t)  i(E).
𝑡
𝑡−𝜏
1
−2
𝑓
𝜎𝑡
𝜏 𝑑𝜏 =
0
0
𝜎𝑡
𝜎𝑡 − 𝑧
0
1
−2 −1/2
𝜎
𝑔
𝑧
𝑡−
𝜎
1
−
2
𝑑𝑧
𝑔 𝑧
𝜎
𝐶𝑂 ∗ 𝜋 1/2 𝐷𝑂 1/2
𝑧 𝑑𝑧 =
𝜉𝜃𝑆 𝜎𝑡 + 1
f(t) = g(st); z = st
𝜎𝑡
0
𝜒(𝑧)
𝜎𝑡 − 𝑧
1 𝑑𝑧 =
2
1
𝜉𝜃𝑆 𝜎𝑡 + 1
i(E) in sweep experiments
(reversible system)
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
For reversible system (Potential Sweep):
𝜎𝑡
𝜎𝑡 − 𝑧
1
−
2 𝜎 −1/2 𝑔
0
𝐶𝑂 ∗ 𝜋 1/2 𝐷𝑂 1/2
𝑧 𝑑𝑧 =
𝜉𝜃𝑆 𝜎𝑡 + 1
q = exp[n F (E – E0’) / (R T)]
f(t) = i(t) / (n F A)
𝜎𝑡
𝜒 𝑧 =
x = (DO/DR)(1/2)
𝜒(𝑧)
1
1 𝑑𝑧 = 𝜉𝜃𝑆 𝜎𝑡 + 1
2
f(t) = g(st); z = st
0
𝜎𝑡 − 𝑧
𝐶𝑂 ∗
𝑔(𝑧)
𝑖(𝜎𝑡)
=
(𝜋𝐷𝑂 𝜎)1/2 𝑛𝐹𝐴𝐶𝑂 ∗ (𝜋𝐷𝑂 𝜎)1/2
i = n F A CO* (p DO s)1/2 c(s t)
(6.2.17)
s t = n F v t/(R T) = n F (Ei – E)/(RT)
c(st) is a pure number for any given t.
Equations for peak current
i = n F A CO* (p DO s)1/2 c(s t)
(6.2.17)
s t = n F v t/(R T) = n F (Ei – E)/(RT)
For reversible charge transfer at 25o C:
p 1/2c (st) reaches maximum of 0.4463 when n (E – E1/2) is –28.5 mV (Table 6.2.1).
Randles–Sevčik Eqn (6.2.19):
ip = (2.69 x 105) n3/2 A DO1/2 CO* v1/2
E1/2 = E0’ + (R T) ln(DR/DO)1/2/ (nF)
ip / (CO* v1/2) = constant (current function)
ip (A)
(+)
Steady-state
current
(e.g. RDV)
i (A)
v
v1/2 (V1/2 s-1/2)
E1/2
E (V)
(-)
Depletion of
O near
electrode
More on peak shape (reversible systems)
Depletion of
O near
electrode
v
i (A)
i (A)
(+)
Equal O, R: i ~0.852ip
Recall Cottrell Eqn for E-step (5.2.11):
i(t) = n F A DO1/2 CO* / (p t)1/2
E1/2
E (V)
t (s)
(-)
(+)
Ep = E1/2 – 28.5/n mV
Ep/2 = E1/2 + 28.0/n mV
i (A)
For reversible charge transfer at 25o C:
ip
ip/2
E (V)
Ep/2 Ep
(-)
LSV with spherical electrodes and UMEs
i = n F A CO* (p DO s)1/2 c(s t) + n F A DO CO* f(s t) / r0
i (spherical correction)
i (plane)
s t = n F v t/(R T) = n F (Ei – E)/(RT)
i (A)
(+)
Spherical term dominates if v << R T D / (n F r02)
If 0.5 mm radius, D = 10-5 cm2/s, T = 298 K,
steady state up to 10 V/s.
v
E (V)
(-)
Double-layer C and uncompensated R
During potential step experiment (stationary, constant A electrode), charging current
disappears after few RuCd (time constant). In potential sweep, ich always flows.
𝑖𝑐ℎ = 𝑣𝐶𝑑 +
−𝑡
𝐸𝑖
𝑅
− 𝑣𝐶𝑑 𝑒 𝑠 𝐶𝑑
𝑅𝑠
Section 1.2.4. in B&F
Recall Randles–Sevčik Eqn (6.2.19):
ip = (2.69 x 105) n3/2 A DO1/2 CO* v1/2
|ich| = A Cd v
Limit for max useful scan rate and
min concentration
Also: EWE = Eapp + iR
ich more important at high v.
v = 100a
i (A)
i (A)
ip
ich
E (V)
(-)
ich
E (V)
v = 900a
ip
ip
i (A)
v=a
(+) x 40
(+) x 20
(+) x 1
|ich| / ip = Cd v1/2 (10-5) / (2.69 n3/2 DO1/2 CO*)
(-)
ich
E (V)
(-)
Irreversible e– transfer (boundary condition)
O + ne– ⇄ R (Assume semi-inf linear diff, only O initially present, and rapid charge-transfer
kinetics)
Recall boundary condition for reversible system (Potential Sweep):
CO(0,t) / CR(0,t) = q e –st = q S(t)
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
And the general concentration equations:
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
−1/2
− (𝐷𝑂 𝜋)
𝑡
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
−1/2
𝐶𝑅 0, 𝑡 = (𝐷𝑅 𝜋)
𝑡
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
f(t) = i(t) / (n F A)
Led to:
i = n F A CO* (p DO s)1/2 c(s t)
(6.2.17)
kf
O +  R (Assume semi-inf linear diff, only O initially present, and totally irreversible,
one-electron, one-step) The corresponding boundary condition is:
e–
𝑖
𝜕𝐶𝑂 (𝑥, 𝑡)
−𝐽𝑂 0, 𝑡 =
= 𝐷𝑂
𝐹𝐴
𝜕𝑥
𝑥=0
= 𝑘𝑓 (𝑡)𝐶𝑂 (0, 𝑡)
Sec. 5.5 in B&F
𝑘𝑓 𝑡 = 𝑘 0 𝑒 −𝛼𝑓
𝐸 𝑡 −𝐸0′
Irreversible e– transfer i(E)
kf
O +  R (Assume semi-inf linear diff, only O initially present, and totally irreversible,
one-electron, one-step) The corresponding boundary condition is:
e–
𝑖
𝜕𝐶𝑂 (𝑥, 𝑡)
−𝐽𝑂 0, 𝑡 =
= 𝐷𝑂
𝐹𝐴
𝜕𝑥
= 𝑘𝑓 (𝑡)𝐶𝑂 (0, 𝑡)
𝑘𝑓 𝑡 = 𝑘 0 𝑒 −𝛼𝑓
𝑥=0
f = F / (R T)
And the general concentration equations:
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
−1/2
− (𝐷𝑂 𝜋)
𝑡
𝑡−𝜏
0
−
1
2𝑓
𝜏 𝑑𝜏
𝐸 𝑡 −𝐸0′
−1/2
𝐶𝑅 0, 𝑡 = (𝐷𝑅 𝜋)
𝑡
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
f(t) = i(t) / (n F A)
Leads to:
i = F A CO* (p DO v)1/2 [(a F) / (R T)]1/2 c(bt)
Compare to Reversible:
i = n F A CO* (p DO s)1/2 c(s t)
(6.3.6)
(6.2.17)
b=afv
a transfer
coefficient
Equations for peak current (irreversible)
i = F A CO* (p DO v)1/2 [(a F) / (R T)]1/2 c(bt)
One-step, one-electron
(6.3.6)
For irreversible charge transfer at 25o C:
p 1/2c (bt) reaches maximum of 0.4958 (Table 6.3.1).
Peak current (6.3.8):
ip = (2.99 x 105) n (ana)1/2 A DO1/2 CO* v1/2
Compare to (6.2.19):
ip = (2.69 x 105) n3/2 A DO1/2 CO* v1/2
0<a<1
(+)
Since ana always < n, ip, irrev smaller than ip, rev.
Don’t forget: EWE = Eapp + iR
Reversible reaction can appear irreversible if iR
drop not compensated! Min by using 3electrode cell, positive feedback compensation
at v > 1 V/s, consider UME
i (A)
na is the number of electrons involved
in the rate-limiting step.
E (V)
E0'
(-)
Peak shape for irreversible system
i = n F A CO* (p DO v)1/2 [(a F) / (R T)]1/2 c(bt)
(6.3.6)
Peak current (6.3.8):
ip = (2.99 x 105) n (ana)1/2 A DO1/2 CO* v1/2
Compare to (6.2.19):
ip = (2.69 x 105) n3/2 A DO1/2 CO* v1/2
|Ep – Ep/2| = 48/(a na ) mV
0<a<1
|Ep – Ep/2| = 57/n mV
Irreversible peak is broader and smaller
than reversible peak. As a decreases,
peak widens and decreases in magnitude.
1/2
𝑅𝑇
𝐷
𝛼𝐹𝑣
𝑂
0′
𝐸𝑝 = 𝐸 −
0.78 + ln 0 + ln
𝛼𝐹
𝑘
𝑅𝑇
1/2
i (A)
(+)
Compare to reversible:
Unlike reversible, Ep is a function of scan rate for irreversible.
Ep shifts more negative for reduction by 30/(a na) m V at
25oC for each 10-fold increase in scan rate.
E (V)
E0'
(-)
Quasireversible e– transfer (boundary condition)
Recall boundary condition and i(E) relationship for reversible system (Potential Sweep):
CO(0,t) / CR(0,t) = q e –st = q S(t)
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
i = n F A CO* (p DO s)1/2 c(s t)
(6.2.17)
And the boundary condition and i(E) relationship for the totally irreversible system:
−𝐽𝑂 0, 𝑡 =
𝑖
𝜕𝐶𝑂 (𝑥, 𝑡)
= 𝐷𝑂
𝐹𝐴
𝜕𝑥
= 𝑘𝑓 (𝑡)𝐶𝑂 (0, 𝑡)
𝑥=0
i = F A CO* (p DO v)1/2 [(a F) / (R T)]1/2 c(bt)
𝑘𝑓 𝑡 = 𝑘 0 𝑒 −𝛼𝑓
𝐸 𝑡 −𝐸 0′
f = F / (R T)
(6.3.6)
Now consider reactions where electron transfer limitations make consideration of the
reverse reaction necessary (quasireversible – Matsuda and Ayabe):
kf
–
O + e ⇄ R (Assume semi-inf linear diff, only O initially present, one-electron, one-step)
The corresponding boundary condition is:
kb
𝑖
𝜕𝐶𝑂 (𝑥, 𝑡)
−𝐽𝑂 0, 𝑡 =
= 𝐷𝑂
𝐹𝐴
𝜕𝑥
= 𝑘 0 𝑒 −𝛼𝑓 𝐸
𝑥=0
𝑡 −𝐸 0′
𝐶𝑂 0, 𝑡 − 𝐶𝑅 (0, 𝑡)𝑒 𝑓 𝐸
𝑡 −𝐸 0′
Quasireversible e– transfer equations
Now consider reactions where electron transfer limitations make consideration of the
reverse reaction necessary (quasireversible – Matsuda and Ayabe):
kf
–
O + e ⇄ R (Assume semi-inf linear diff, only O initially present, one-electron, one-step)
kb
The corresponding boundary condition is:
𝑖
𝜕𝐶𝑂 (𝑥, 𝑡)
−𝐽𝑂 0, 𝑡 =
= 𝐷𝑂
𝐹𝐴
𝜕𝑥
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
−1/2
− (𝐷𝑂 𝜋)
= 𝑘 0 𝑒 −𝛼𝑓 𝐸
𝐶𝑂 0, 𝑡 − 𝐶𝑅 (0, 𝑡)𝑒 𝑓 𝐸
𝑡 −𝐸 0′
𝑥=0
𝑡
𝑡−𝜏
0
𝑡 −𝐸 0′
−
1
2𝑓
𝜏 𝑑𝜏
−1/2
𝐶𝑅 0, 𝑡 = (𝐷𝑅 𝜋)
𝑡
𝑡−𝜏
−
1
2𝑓
0
Matsuda and Ayabe showed peak shape and characteristics to be function of a and
parameter L:
Λ=
i = F A CO* (DO v)1/2 [F/ (R T)]1/2 Y(E)
𝑘0
𝐷𝑂 1−𝛼 𝐷𝑅 𝛼 𝑓𝑣
1/2
(6.4.5)
𝜏 𝑑𝜏
Quasireversible, reversible, irreversible
Now consider reactions where electron transfer limitations make consideration of the
reverse reaction necessary (quasireversible – Matsuda and Ayabe):
Matsuda and Ayabe showed peak shape and characteristics to be
function of a and parameter L:
𝑘0
Λ=
1/2
𝐷𝑂 1−𝛼 𝐷𝑅 𝛼 𝑓𝑣
i = F A CO* (DO v)1/2 [F/ (R T)]1/2 Y(E)
(6.4.5)
kf
O + e– ⇄ R
kb
i = n F A CO* (p DO s)1/2 c(s t)
(6.2.17)
i = F A CO* (p DO v)1/2 [(a F) / (R T)]1/2 c(bt)
(6.3.6)
Y(E) is shown in Fig. 6.4.1. L > 10, behavior of Y(E) approaches that of reversible system.
System
|Ep – Ep/2|at 25oC (mV)
Reversible
57/n
Irreversible
48/(a na )
Quasireversible
26D(L, a)
L ≥ 10, D(L, a) ≈ 2.2 |Ep – Ep/2| ≈ 57
L < 10-2 & a = 0.5, D(L, a) ≈ 3.7 
|Ep – Ep/2| ≈ 96
Scan rate and kinetics
System
|Ep – Ep/2|at 25oC (mV)
Reversible
57/n
Irreversible
48/(a na )
Quasireversible
26D(L, a)
L ≥ 10, D(L, a) ≈ 2.2 |Ep – Ep/2| ≈ 57
L < 10-2 & a = 0.5, D(L, a) ≈ 3.7 
|Ep – Ep/2| ≈ 96
Depending on L (v), a system may show rev, quasirev or irrev behavior. Appearance of
kinetic effects depends on time window of experiment.
At small v (long t)  rev, while at large v (short t)  irrev.
For n = 1, a = 0.5, T = 25oC, and D = 10-5 cm2/s with L ≈ k0/(39 D v)1/2
Matsuda and Ayabe suggested
Reversible: L ≥ 15 ; k0 ≥ 0.3v1/2 cm/s
Quasireversible: 15 ≥ L ≥ 10 –2(1+a) ; 0.3v1/2 cm/s ≥ k0 ≥ 2 x 10-5 v1/2 cm/s
Totally irreversible: L ≤ 10 –2(1+a) ; k0 ≤ 2 x 10-5 v1/2 cm/s
Increased v makes rxn more difficult (broader wave, shifted negatively for reduction)
Also need fast recorder < 0.5 V s-1 ; computer or oscilloscope > 0.5 V s-1
iR compensation as iR drop distorts linear input ramp of potential sweep
EWE = Eapp + iR
Cyclic Voltammetry
COx*
E (V)
(-)
Distance from Electrode (cm)
(+)
Distance from Electrode (cm)
El
Ox + ne–  Red
ip,c
Ox – ne–  Red
ip,a
i (A)
COx , CRed (M)
COx , CRed (M)
COx*
DEp
E (V)
E0'
(-)
Ei
0
t (s)
l
- Reversal technique analogous to
double-potential step methods.
- Experiment time scale: 103 s to 10-5 s.
- For reversible electron transfer:
- DE = |Ep,a – Ep,c| = 59 mV/n
- |Ep – Ep/2| = 57 mV/n
- ip,a/ip,c = 1
El
(-)
Cyclic Voltammetry
(boundary value problem)
El
E (V)
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
− (𝐷𝑂 𝜋)
Ei
0
t (s)
−1/2
𝐶𝑅 0, 𝑡 = (𝐷𝑅 𝜋)
l
𝑡
−1/2
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
𝑡
𝑡−𝜏
−
1
2𝑓
𝜏 𝑑𝜏
0
f(t) = i(t) / (n F A)
(0 < t ≤ l) E = Ei – vt
(t > l) E = Ei – 2vl + vt
(0 < t ≤ l)
CO(0,t) / CR(0,t) = exp[n F (Ei – vt – E0’) / (R T)]
q = exp[n F (E – E0’) / (R T)]
CO(0,t) / CR(0,t) = q e –st = q S(t)
s = n F v/ (R T)
(t > l)
CO(0,t) / CR(0,t) = exp[n F (Ei – 2vl + vt – E0’) / (R T)]
CO(0,t) / CR(0,t) = q e st – 2sl = q S(t)
Shape of curve on reversal segment dependent on El!
(+)
ip,c El1
CV curve shape
El2
Shape of curve on reversal segment dependent on El!
(isp)0
ip,a
i (A)
(ip,a)0 El
E (V)
𝑖𝑝,𝑎
𝑖𝑝,𝑎
=
𝑖𝑝,𝑐
𝑖𝑝,𝑐
0
+
- For reversible electron transfer: ip,a/ip,c = 1
- regardless of v and El if El > 35/n mV past Ep,c and D,
but must measure ip,a from decaying cathodic current.
(-)
0.485 𝑖𝑠𝑝
𝑖𝑝,𝑐
If El is > 35/n mV beyond Ep,c, peak shape generally
the same, like forward segment but plotted in
opposite direction.
0
+ 0.086
Suggestion by Nicholson if actual baseline
for measuring ip,a cannot be determined.
Also consider: allowing current to decay to nearly zero then performing reverse sweep.
Don’t forget charging current: peak must be measured from proper baseline.
|ich| = A Cd v
Real experiments: ip measurement imprecise due to uncertainty in correction for charging
current, definition of baseline. CV not ideal for quantitative properties reliant on ip
(concentration, rate constants), but offers ease of interpreting qual and semi-quan behavior.
CV curve shape
(+)
ip,c El1
ip,a
El2
(isp)0
i (A)
(ip,a)0 El
E (V)
(-)
Shape of curve on reversal segment
dependent on El!
If El is > 35/n mV beyond Ep,c, peak shape
generally the same, like forward segment
but plotted in opposite direction.
DEp is a slight function of El, but always close to
59/n mV (25oC).
Repeated cycling  steady state (decreased ip,c and increased ip,a) DEp = 58/n mV (25oC).
CV curve shape (quasirev)
Recall for quasirev:
𝛬=
𝑘0
𝐷𝑂 1−𝛼 𝐷𝑅 𝛼 𝑓𝑣
𝜓 = 𝛬𝜋 −1/2
1/2
𝐷𝑂 𝛼/2 0
𝑘
𝐷𝑅
=
𝜋𝐷𝑂 𝑓𝑣 1/2
y (at 25oC)
DEp (mV)
20
61
5
65
2
72
0.5
105
0.25
141
DEp is a function of v, k0, a, and El.
If El is 90/n mV beyond peak, El effect is small.
0.3 < a < 0.7, DEp nearly ind of a, only depends on y
(Nicholson method) Variation of DEp with v y estimate k0
Beware of uncomp R: ipRu must be small. Effect of Ru most important when currents large and
k0  rev limit (DEp only slightly diff from rev value)
Multicomponent or multistep charge transfer
(-)
(+)
Other strategies:
O' + n'e-
(-)
Notice i’p must be measured from decaying
current of first wave.
i (A)
Assume current decays as t-1/2 to get baseline.
Recall Cottrell Eqn for E-step (5.2.11):
i(t) = n F A DO1/2 CO* / (p t)1/2
t (s)
(-)
E (V)
R'
E (V)
i'p
E (V)
R
i (A)
O + ne-
Reestablish init. cond. then go
through with full scan
Hold E ~60/n mV beyond Ep to
get more accurate baseline
t (s)
Stop scan for 20-50 seconds, allow
current to decay. Requires
convection-free conditions.
t (s)
Consider difference between CV and steady-state voltammetry,
pulse voltammetry methods.
Multistep charge transfer and wave order problem
(+)
Mulitstep (O + n1e  R1, then R1 + n2e  R2)
similar to two component, but shape depends
on DE0 = E20 – E10, reversibility of each step, n1
and n2.
R
i (A)
O + ne-
i'p
O' + n'e-
E (V)
R'
(-)
-100 mV < DE0 < 0  single broad wave Ep ind of scan rate
DE0 = 0  single peak with ip between ip(1e), ip(2e) and Ep – Ep/2 = 21 mV
DE0 > 180 mV, 2nd easier than 1st  single wave characteristic of 2e reduction
DE0 = –(2RT/F)ln2 = –35.6 mV (25oC), no interaction between R and O  observed
wave has shape of one-e transfer (consider redox polymer with k noninteracting
centers: –(2RT/F)ln k)
DE0 more positive than –(2RT/F)ln2  2nd e transfer assisted by first
(+)
Coupled chemical reactions and CV shape
O + ne ⇄ R
k
i (A)
R + Z  R-Z (not electroactive)
E (V)
(+)
(-)
O + ne ⇄ R
k
i (A)
R+ZO+P
E (V)
(-)
Cases involving adsorption
See Wopschall & Shain Anal. Chem. 1967, 39: 1514-1542
O+eR
Osoln
I
Rsoln
Rads
I
Oads
G
Rxn coord
O weakly
adsorbed
V
Post-wave
DG = -nFE
R weakly
adsorbed
Adsorption stabilizes species
(lower G), otherwise wouldn’t
occur spontaneously
V
Pre-wave
Osoln
G
O strongly
adsorbed
Oads
Rsoln
Rads
Rxn coord
R strongly
adsorbed
Variation of CV shape w/ scan rate (V) (single charge-transfer events)
I.
II.
Reversible charge transfer (er): O + ne- ⇄ R
𝑘
−
Irreversible charge transfer (ei): 𝑂 + 𝑛𝑒 → 𝑅
𝑘𝑓
III. Rev Chem rxn preceding rev ct (crer): 𝑍 ⇄ 𝑂; 𝑂 + 𝑛𝑒 − ⇄𝑅
𝑘𝑏
𝑘𝑓
𝑘
−
IV. Rev Chem Rxn preceding irrev ct (crei): 𝑍 ⇄ 𝑂; 𝑂 + 𝑛𝑒 → 𝑅
𝑘𝑏
𝑘𝑓
V. Rev ct followed by rev chem rxn (ercr): 𝑂 + 𝑛𝑒 − ⇄𝑅; 𝑅 ⇄ 𝑍
𝑘𝑏
𝑘𝑓
−
VI. Rev ct followed by irrev chem rxn (erci): 𝑂 + 𝑛𝑒 ⇄𝑅; 𝑅 → 𝑍
𝑘𝑓 ′
VII. Catalytic rxn w/ rev ct (erci’): 𝑂 + 𝑛𝑒 ⇄𝑅; 𝑅 + 𝑍 → 𝑂
−
𝑘
𝑘𝑓 ′
VIII. Catalytic rxn w/ irrev ct (eici’): 𝑂 + 𝑛𝑒 → 𝑅; 𝑅 + 𝑍 → 𝑂
−
Reprinted with permission from Nicholson & Shain, Anal. Chem. 1964, 36: 706-723. Copyright 1964 American Chemical Society.
Linear sweep methods
• Signal shape can depend on electrode size, v, k0, a, El
• Good for qualitative or semi-quantitative description
of system based on ip, DEp
• Experimental data compared to mathematical models
• ip measurement complicated by baseline correction
– ich, iR drop, reversal baseline
– mathematical corrections, additional experiments
• High scan rates: quasireversible behavior, large iR
effect
• Detection limit ~10-6 or 10-7 M at best