Basic potential step and sweep methods

Contents
1. BASIC POTENTIAL STEP METHODS
 Potential Step Under Diffusion Control
 Sampled-current Voltammetry For Reversible Electrode Reactions
 Sampled-current Voltammetry For Quasireversible and irreversible
Electrode Reactions
 Chronoamperometric Reversal Techniques
2. POTENTIAL SWEEP METHODS
LINEAR SWEEP VOLTAMMETRY (LSV)
 LSV – Nernstian (Reversible) System
 LSV – Totally Irreversible Systems
 LSV – Quasireversible Systems
CYCLIC VOLTAMMETRY (CV)
 CV – Nernstian (Reversible) System
 CV – Totally Irreversible Systems
 CV – Quasireversible Systems

Potentiostat has control of the voltage across
the working electrode-counter electrode pair,
and it adjusts this voltage to maintain the
potential difference between the working and
reference electrodes
Excitation
(perturbation)
System Detection
observation of response
BASIC POTENTIAL STEP METHODS

Time
E
E1
E2
Ox + ne-  Red
Distance (x)
C
o
Co(t=0)
t1 t2
t3
(a) Waveform for a step experiment in which species О is electroinactive at E1, but is reduced at a diffusion-
limited rate at E2- (b) Concentration profiles for various times into the experiment, (c) Current flow vs. time.
Consider the following waveform applied in a basic potential step experiment. Let us consider
its effect on the interface between a solid electrode and an unstirred solution containing an
electroactive species. We know that there is a potential region where faradaic processes do not
occur; let E1 be in this region.
we can also find a more negative potential at which the kinetics for reduction of species are
become so rapid that no oxidized species can coexist with the electrode, and its surface
concentration goes nearly to zero. Consider E2 to be in this "mass-transfer-limited" region.
What is the response of the system to the step perturbation?
t
i
0
Potential Step Under Diffusion Control

E𝐸0′
1
2
di
i
Ox + ne-  Red
There is a limiting current
during this condition
Now if at t=0 lets step
the potential to
maximum
E𝐸0′
1
2
di
i 400mv
During this time the limited
current will be changed and
current starts to flow do to the
applied high potential. We
expect non equilibrium current
flow, and Lets see how this non
equilibrium current can be
derived.𝑡 = 0
CO
CR
𝑡 = 𝑡1 At any distance between the electrode and solution
Apply the Fick's diffusion equation and solve for
Concentration using Laplace transformation.
The solve for the current ….
Cont..
c
t

Formulation of general equation for electrode dynamics
2
2
),(),(
x
txC
D
t
txC O
O
O





2
2
),(),(
x
txC
D
t
txC R
R
R





The initial condition, (a), merely express the homogeneity of the solution before the
experiment starts at t = 0, and the semi-infinite condition, (b), is an assertion that
regions distant from the electrode are unperturbed by the experiment.
Solving this two equation with the following initial condition will
help to calculate the Concentration of species as function of
potential or current …

 OO CxC )0,(


 OO
x
CtxC ),(lim
0)0,( xCR
0),(lim 

txCR
x
At t=0
semi-infinite condition At t= infinity
(a),
(b),
Using mass conservation properties , when we oxidized n mole of reactant n mole of
product will be created So that the net flux must be Zero
𝐷0
𝜕𝐶0(𝑥, 𝑡)
𝜕𝑥 𝑥=0
+ 𝐷 𝑅
𝜕𝐶 𝑅(𝑥, 𝑡)
𝜕𝑥 𝑥=0
= 0
Semi infinite
linear diffusion
A planar electrode (e.g., a platinum disk) and an unstirred solution
(no convection)

−𝐽0 0, 𝑡 =
𝑖(𝑡)
𝑛𝐹𝐴
= 𝐷0
𝜕𝐶0(𝑥, 𝑡)
𝜕𝑥 𝑥=0
There are four cases
1. CO(0,t) = 0 , call Cottrell Equation
2. Reversible case
3. Irreversible cases
Case 1 Cottrell Equation
1D planer diffusion with no kinetic flow, no migration
and no convection And the only flow is diffusional
current
1/21/2
1/2
onFAD
)()(
t
C
titi o
d



Cottrell Equation
t
i
t
𝒊𝒕 𝟏/𝟐
2/1
1/2
onFAD


oC
Laplacian transformation
The calculation of the diffusion-limited current, id and the concentration profile, C0(X,
t), involves the solution of the linear diffusion equation:
Cont..
0
),(),(
0
2
2
0
















 x
R
R
x
O
O
x
txC
D
t
txC
D

Oads
Rads
CO(x,0) = CO*
CO(0,t) = 0
LimCO(x,t) = CO*
x∞
1/21/2
1/2
onFAD
)()(
t
C
titi o
d



Cottrell Equation





 

)(
)0(
2
D *
1/2
1/2
o
t
xcc
tnFA
i oo

buildup of the diffusion layer
tDt oo  )(
Co*
DOt =
30 m
1 m
30 nm
at t =
1 s
1 ms
1 s
Cont..
 unstirred solution where the diffusion layer continues to grow with time.

The Cottrell equation works only if we step the potential from zero to the plateau region.
the others step doesn’t fulfill the boundary condition of Cottrell equation
𝐸0′
1/21/2
1/2
onFAD
)(
t
C
ti o



1/21/2
1/2
RnFAD
)(
t
C
ti R



The Cottrell equation for
both R and O can be written
at diffusion limited region
1/21/2
1/2
0nFAD
)(
t
C
ti o



t
i
The validity Cottrell equation was verified in detail by the classic
experiments of Kolthoff and Laitinen,
In practical measurements of the i-t behavior under "Cottrell
conditions have some practical limitation
• Potentiostatic limitations.
• Limitations in the recording device.
• Limitations due to convection.
• Limitations imposed by charging current (ic)
(due to non faradic current)
𝒊𝒄 = 𝑬/𝑹𝒔 ∙ exp(−𝒕/(𝑹𝒔𝑪 𝒅)) 𝒊 𝒕 𝑻 = 𝒊(𝒕) 𝑪 + 𝒊(𝒕) 𝑭
𝒊(𝒕) 𝑪
𝒊(𝒕) 𝑭
Cont..

SAMPLED-CURRENT VOLTAMMETRY FOR REVERSIBLE ELECTRODE REACTIONS
Linear diffusion at a planar electrode, Reversible electrode reaction Stepped to an arbitrary potential
)
)0(
)0(
ln'
R
O0



xc
xc
nF
RT
EE 






 )(exp)
)0(
)0( 0
R
O
EE
RT
F
n
xc
xc

CO(x,0) = CO*
LimCO(x,t) = CO*
x∞
CR(x,0) = 0
LimCR(x,t) = 0
x∞
𝜕𝐶0(𝑥,𝑡)
𝜕𝑡
= 𝐷0
𝜕2 𝐶0(𝑥,𝑡)
𝜕2 𝑥
for 1D
)1(
nFAD
)( 1/21/2
1/2
o
 


t
C
ti o
),0(
),0(C
and 00
tC
t
D
D
RR
 
Consider again the reaction О + 𝑛𝑒 ⇔ 𝑅 in a Cottrell-like experiment at an electrode where
semi-infinite linear diffusion applies, but this time Let us treat potential steps of at any
magnitude. We begin each experiment at a potential at which no current flows; and at t = 0, we
change E instantaneously to a value anywhere on the reduction curve. We assume here that
charge-transfer kinetics are very rapid, so that
This relation is the general response function for a step experiment in a reversible system. The Cottrell
equation, is a special case for the diffusion-limited region, which requires a very negative E - E° so that 𝜃 →0
𝜕𝐶 𝑅(𝑥,𝑡)
𝜕𝑡
= 𝐷 𝑅
𝜕2 𝐶 𝑅(𝑥,𝑡)
𝜕2 𝑥
for 1D

At very negative potentials,   0, and
i(t)  id which is express the Cottrell
plateau , means when the potential
steps to the plateau It is convenient to
represent the Cottrell current as id(t) )1(
)(
)(


ti
ti d
Now we see that for a reversible couple, every current-time curve has the same shape; but
its magnitude is scaled by 1/(1 + 𝜉𝜃) according to the potential to which the step is made.
Shape of I-E Curve
In sampled-current voltammetry, our goal is to obtain an 𝑖(𝜏) − 𝐸 curve by
(a) performing several step experiments with different final potentials E,
(b) sampling the current response at a fixed time 𝜏 after the step, and
(c) plotting i(𝜏) 𝑣𝑠. E.
Here we consider the shape of this curve for a reversible couple and the kinds of
information one can obtain from it.
which can be rewritten as
𝜉𝜃=0
𝜉𝜃=1
𝜉𝜃=2
𝑖
𝑡
CO(0,t) = 0 , call Cottrell Equation Then the above equation
becomes 𝒊 𝒅(𝒕) = 𝒊(𝒕)
𝜉𝜃=0
𝜉𝜃=1
𝜉𝜃=2
𝑖
𝐸
Cont..
)1(
nFAD
)( 1/21/2
1/2
o
 


t
C
ti o

)(
)()(
lnln 2/1
0
2/1
'0


i
ii
nF
RT
D
D
nF
RT
EE dR 

When 𝑖(𝜏) = id(T)/2, the current ratio becomes unity so that the
third term vanishes. The potential for which this is so is E1/2,
the half-wave potential:
2/1
0
2/1
'0
2/1 ln
D
D
nF
RT
EE R

)1(
)(
)(,fixedfor




 di
i
)(
)()(



i
iid 

Here we monitoring the current as function other parameter 𝜏 rather
than measuring the whole current, and drawing the potential for
different values of 𝜏
𝜉𝜃=0
𝜉𝜃=1
𝜉𝜃=2
𝑖
𝑡
𝜏 𝜏′ 𝜏′′
Theses are potential that we have drawn from the current sampled at 𝜏 points , and we call this
method sampled-current voltamogram for reversible process.
𝜏
τ′
τ′′
𝑖
𝐸𝐸1/2
id(T)
Cont..

is often written as
)(
)()(
ln2/1


i
ii
nF
RT
EE d 

We can also check the reversibility by drawing
These equations describe the voltammogram for a
reversible system in sampled-current voltammetry
as long as semi-infinite linear diffusion holds.
)(
)()(
lnln 2/1
0
2/1
'0


i
ii
nF
RT
D
D
nF
RT
EE dR 

)(
)()(
logvs


i
ii
E d 
If we get a slope 2.303
𝑅𝑇
𝑛𝐹
𝑜𝑟
59.1
𝑛
𝑚𝑉 𝑎𝑡 250
𝐶
𝑡ℎ𝑒𝑛 𝑅𝑒𝑣𝑒𝑟𝑠𝑖𝑏𝑖𝑙𝑒
)1( 
 di
i
1/21/2
1/2
0nFAD
t
C
i o
d



 For Cottrell current  For reversible system
Cont..
 Concentrat
ion profile






 
)(
)(
1Ct)(0, 00
ti
ti
C
d
)(
)(
Ct)(0, 0
ti
ti
C
d
R

 

In contrast with the reversible cases just examined, the interfacial electron-transfer kinetics in the
systems considered here are not so fast. Thus kinetic parameters such as 𝑘 𝑓, 𝑘 𝑏, 𝑘0
and a influence
the responses to potential steps.
Sampled-current Voltammetry For Quasireversible and irreversible Electrode Reactions
𝑂 + 𝑛𝑒 ⇋ 𝑅
𝑘 𝑏
𝑘 𝑓
)](exp[ '00
EE
RT
nF
kk f  
)]()1exp[( '00
EE
RT
nF
kkb  
),0(),0(
0
0
0 tCktCk
x
C
D
nFA
i
Rbof
x










With
)()exp( 2/12
0 HterfctHCnFAki f


After Laplace transformation
the current will be
For the case when R is initially present
)()exp()( 2/12
0 HterfctHCkCknFAi Rbf


2/12/1
0 R
bf
D
k
D
k
H Where
At a given step potential, 𝑘 𝑓, 𝑘 𝑏, and
H are constants. The product
exp(𝑥2
)erfc(𝑥) is unity for x = 0, but
falls monotonically toward zero as x
becomes large; thus the current-time
curve has the shape shown in Figure.

0CFAk f
t
i
At t=0 and CR is zero

 0)0( CnFAkti f
Current decay after the application of a step to a potential
where species О is reduced with quasireversible kinetics.

)1(Htand,)()exp()(Where),(
)1( 2/1
0
2
1
2
1
22
1
11 




D
tK
erfcFF
i
i
fd
Since semi-infinite linear diffusion applies, the diffusion-limited current is the
Cottrell current, which is easily recognized in the factor preceding the brackets
It is a very compact representation of the way in which the current in a step experiment depends
on potential and time, and it holds for all kinetic regimes: reversible, quasireversible, and
totally irreversible.
The function F1 𝜆 manifests the kinetic effects on the current in terms of the dimensionless
parameter 𝜆.
 Small value of 𝜆 implies a strong kinetic influence on the current,
 Large value of 𝜆 corresponds to a situation where the kinetics are facile and the response is
controlled by diffusion.
The function F1 𝜆 rises monotonically from a value of zero at λ= 0 toward an asymptote of
unity as λ becomes large
• simply by recognizing that with reversible kinetics,λ is very large, so that F1 𝜆 is always unity.
Cont..

Totally Irreversible Reactions
The backward component of the electrode reaction becomes progressively less important at
potentials further to the negative side of E° . If k° is very small, a sizable activation of Kf is
required for all points where appreciable current flows, and Kъ is suppressed consistently to a
negligible level. The irreversible regime is defined by the condition that
𝒌𝒃
𝒌𝒇
~ 0 (i.e., 𝜃≪ 0) over
the whole of the voltammetric wave. becomes
)()exp( 2/12
0 HterfctHCnFAki f


)()exp( 2/1
0
2/12
0
2
0
D
tK
erfc
D
tK
CnFAki
ff
f


2/1
0
2
1
22
1
1 where),()exp()(
D
tK
erfcF
i
i f
d
 
)(
)1(
1 

F
i
i d

And becomes
The half-wave potential for an irreversible wave occurs where 𝐅𝟏 𝝀 = 0.5, which is where 𝝀 =
0.433. If kf follows the usual exponential form and 𝑡 = 𝜏, then
  433.0)(exp '0
2/12/1
0
2/10
 EEf
D
k

 By taking logarithms and rearranging, one obtains






 2/1
0
2/10
'0
2/1
31.2
ln
D
k
F
RT
EE


where the second term is the displacement
required to activate the kinetics. Obviously
provides a simple way to evaluate k° if a is
known

If the electrode in the step experiment is spherical rather than planar (e.g., a hanging mercury
drop), one must consider a spherical diffusion field, and Fick's second law becomes,
𝜕𝐶0(𝑟, 𝑡)
𝜕𝑡
= 𝐷0
𝜕2
𝐶0(𝑟, 𝑡)
𝜕𝑟2
+
2 𝜕𝐶0(𝑟, 𝑡)
𝑟 𝜕𝑟
where r is the radial distance from the electrode
center. The boundary conditions are then
Semi-Infinite Spherical Diffusion
  







 
0
2/1
0
0o
11
nFAD)(
rtD
Ctid

𝑖𝑑 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 = 𝑖𝑑 𝑙𝑖𝑛𝑒𝑎𝑟 +
𝑛𝐹𝐴𝐷0 𝐶0
∗
𝑟0
Thus the diffusion current for the spherical case is just that for the
linear situation plus a constant term. For a planar electrode
For a planar electrode, lim
𝑡→∞
𝑖 𝑑 = 0
but in the spherical case,
lim
𝑡→∞
𝑖 𝑑 =
𝑛𝐹𝐴𝐷0 𝐶0
∗
𝑟0
  0
0o
2/1
0
0o nFADnFAD
)(
r
C
tD
C
tid



Long times
The spherical character of the electrode becomes
important, and the mass transport process is
dominated by radial or spherical diffusion.
where r0 is the radius of the electrode.
Cr(r,0) = CO*
LimCr(r,t) = CO*
r∞
Cr(r0,t) = 0

Chronoamperometry
1/21/2
1/2
0nFAD
)(
t
C
ti o



Chronoamperometry/chronocoulometry
Chronoamperometry (CA) is used to study diffusion-
controlled electrochemical reactions and complex
electrochemical mechanisms. It is performed by
applying an initial potential at which no faradaic
reaction is occurring, then stepping the potential to a
value at which the electrochemical reaction of interest
takes place. The solution is generally unstirred and the
current is measured throughout the experiment.
– Measurement of surface area
– Measurement of diffusion co-efficient
– Determination of heterogeneous rate constant
– Determination of diffusion layer thickness
E2
E1
0
ic
Cottrell current
time
t
t
t
I
Q

  d t
it  O O
 1 / 2
t1 / 2
nF AD 1 / 2
C *
Ef
Ei
0
t
i
t
Q
t
 dt
Chronocoulometry
Chronocoulometry (CC) is chronoamperometry in which the cell current is integrated to calculate
charge. The presentation of the charge allows more convenient access to certain information in
the experiment. Chronocoulometry is particularly useful for studying adsorption processes and
other surface-constrained reactions, such as modified electrodes.
Least distorted by
Q
t
Q = nFN
potential rise
2/1
2/11/2
0AD2
)(

tCnF
tQ o


Chronocoulometry
Q t 
2nFAD C tO O
1/2  1/2
1/2
Qc
ic
Cottrell current

Charge due to cottrell current
Qt
2nFAD C tO O
1/2  1/2
1/2 Qc
Interfacial capacitance charge
1
What if redox species (O) is adsorbed on electrode surface?
Q vs. t1/2
O
Q t  
2nFAD C tO O
1/2
1/2  1/2
Q Qc ads
Extra charge produced by the adsorbed
reactant
Qc
Q
t1/2
2/1
1/2
0AD2


oCnF
𝑄𝑐 + 𝑄𝐹
𝑄𝐹
If there is offset of the charge at t=0 it is due to
chagrining current and most of the time it
subtracted From other charge.
Q
t1/2
2/1
1/2
0AD2


oCnF
𝑄𝑐 + 𝑄𝐹 + 𝑄𝐴𝑑𝑠
𝑄𝐹
Qads
Qads  nFAO the quantity of adsorbed reactants
𝑄𝐴𝑑𝑠
𝑄𝐶
Cont..

Double potential step
chronoamperometry.
(a) Typical waveform.
(b) Current response
Now consider the effect of the potential program displayed in
The forward step, that is, the transition from 𝐸1 to E2 at t = 0, is exactly the chronoamperometric
Experiment. For a period 𝜏, it causes a buildup of the reduction product in the region near the
electrode. However, in the second phase of the experiment, after t = 𝜏, the potential returns to 𝐸1
where only the oxidized form is stable at the electrode. The anion radical cannot coexist
there; hence a large anodic current flows as it begins to reoxidize, then the current declines
in magnitude (Figure b) as the depletion effect sets in. This experiment, called double potential
step chronoamperometry. It is example of a reversal technique.
Potential Reversal Techniques
double potential step chronoamperometry

Since the experiment for 0 < 𝑡 ≤ 𝜏 is
identical to that treated before
1/21/2
1/2
onFAD
)(
t
C
ti o
f



The current during the reversal step
turns out to be 









2/12/11/2
1/2
0 1
)1(
1nFAD
)(
t
C
ti o
R

(the forward current )
If 𝑡𝑓 𝑎𝑛𝑑𝑡 𝑅 values are selected in pairs so that
𝑡 𝑅 − 𝜏 = 𝑡𝑓 always, then
2/1
11
)(








Rf
R
ti
ti 
𝑖𝑓 𝑡 𝑓 = 𝜏 and 𝑡 𝑅 = 2𝜏
𝑡 𝑅 − 𝜏 = 𝑡𝑓
Under this condition
293.0
)(


f
R
i
ti
𝐸
𝑡𝑓
𝜏 2𝜏
𝑡 𝑅
Cont..
 If 𝑡𝑓 𝑎𝑛𝑑𝑡 𝑅are the times at which the current measurements are made, then for the purely
diffusion-limited case described by the above equation
FORWARD STEP
REVERSE STEP
2
1
2
1
)(
)(















R
f
R
f
f
R
t
t
t
t
i
ti

𝑡𝑓
𝑡 𝑅

The usual observables in controlled-potential experiments are currents as functions of
time or potential. In some experiments, it is useful to record the integral of the current
versus time. Since the integral is the amount of charge passed, these methods are
coulometric approaches. The most prominent examples are chronocoulometry and
double potential step chronocoulometry, which are the integral analogs of the
corresponding chronoamperometric approaches.
Step waveform and response curve for double potential step chronocoulometry.
Charge that is injected by reduction in the
forward step is withdrawn by oxidation in the
reversal.
Forward Reversed
Cont..

The complete electrochemical behavior of a system can be obtained through a series of steps to
different potentials with recording of the current-time curves, as described before which yields a
three-dimensional i-t-E surface . However, the accumulation and analysis of these data can be
tedious especially when a stationary electrode is used. Also, it is not easy to recognize the
presence of different species (i.e., to observe waves) from the recorded i-t curves alone, and
potential steps that are very closely spaced (e.g., 1mV apart) are needed for the derivation of
well-resolved i-E curves.
More information can be gained in a single experiment by sweeping the potential with time and
recording the i-E curve directly. Usually the potential is varied linearly with time (i.e., the
applied signal is a voltage ramp) with sweep rates v ranging from 10 mV/s (1 V traversed in 100
s) to about 1000 V/s with conventional electrodes and up to 106
𝑉/𝑠 with UMEs.
If we record the current as a function of potential, it is also equivalent to recording current
versus time. The formal name for the method is linear potential sweep chronoamperometry,
but most workers refer to it as linear sweep voltammetry (LSV).
LINEAR SWEEP VOLTAMMETRY AND
CYCLIC VOLTAMMETRY
(LSV & CV)

If the scan is begun at a potential well positive of E 0' for the reduction, only nonfaradaic
currents flow for a while. When the electrode potential reaches the vicinity of E0' the
reduction begins and current starts to flow.
t
Ei
E
E2
I
EE0'Ei E(or t)
i
(a) Linear potential sweep or ramp starting at E;. (b) Resulting i-E curve. (c) Concentration
profiles of A and A-; for potentials beyond the peak.
(a)
(b)
(c)
The concentration profiles near the electrode are like those shown in Figure c. Let us
consider what happens if we reverse the potential scan
Linear Sweep Method (LSV)

Linear Sweep Method (LSV)25
Reduction begins and
current starts to flow
A +e- →A‧
Mass transfer of A reaches
maximum rate
Nonfaradaic current flow
Co approaches to zero and
diffuse layer grows
Cont..

LSV – Nernstian (Reversible) System
We consider again the reaction 𝑶 + 𝒏𝒆 ⇋ 𝑹 , assuming semi-infinite linear diffusion and a
solution initially containing only species 0, with the electrode held initially at a potential 𝐸𝑖, where
no electrode reaction occurs. These initial conditions are identical to those we used previously.
The equations governing this case are
2
2
),(),(
x
txC
D
t
txC O
O
O





2
2
),(),(
x
txC
D
t
txC R
R
R





(2)

 OO CxC )0,( 0)0,( xCR00),0(  tfortCO (3)


 OO
x
CtxC ),(lim 0),(lim 

txCR
x
(4)
The initial condition, (3), merely express the homogeneity of the solution before the experiment
starts at t = 0, and the semi-infinite condition, (4), is an assertion that regions distant from the
electrode are unperturbed by the experiment. The third condition, (3), expresses the condition at
the electrode surface after the potential transition, and it embodies the particular experiment we
have at hand.
and the fluxe balance is
It is convenient to rewrite
0
),(),(
00
















 x
R
R
x
O
O
x
txC
D
x
txC
D (5)
 
 
 



 '0
exp
,0
,0
EE
RT
nF
tC
tC
R
O
 (6)

Cont..
  vtEtE i  (7)
the potential is swept linearly at v
If the rate of electron transfer is rapid at the electrode surface
   














'0
exp
,0
,0
EvtE
RT
nF
tf
tC
tC
i
R
Ο
(8)
    xDsO
O
O
esA
s
C
sxC
/
,



    xDs
R
R
esBsxC
/
,



(9)
(10)
after Laplace transformation of (2), the
application of conditions (3) and (4)
yields
The time dependence is significant because the Laplace transformation of (8) cannot be
obtained. The problem was first considered by Randles and Sevcik; the tratement and notation
here follow the later work of Nicholson and Shain.The boundary condition
  t
etS 
where    '0
/expand EERTnF i 
 vRTnF /and 
Laplace transformation of the diffusion
equations and application of the initial and
semi-infinite conditions leads to [see (9)]:
 tSe
tC
tC t
R
Ο
 
 










,0
,0
(11)
   
















x
D
s
sA
s
C
sxC
O
O
O
2/1
exp, (12)

Solution of the Boundary Value Problem
After the transform of the current is given by  
 
0
,










x
O
O
x
sxC
nFADsi
(13)
Laplace transformation       




0
dttfetfLsF st
(14)
2
2
),(),(
x
txC
D
t
txC





(15)
An initial conditions (t = 0) and two boundary conditions
in x. Typically one takes for the initial state
and one uses the semi-infinite limit


 CtxC
x
),(lim (17)

 CxC )0,( (16)
Generally
For variable t, we obtain
 
         0...0'0 121 
 nnnnn
FFsFssfsFL (18)
   
2
2
,
,
dx
sxCd
DCsxCs   (19)
   
D
C
sxC
D
s
dx
sxCd 
 ,
,
2
2
(20)
           xDssBxDssA
s
C
sxC
2/12/1
/exp'/exp', 

(21)

 sxC ,
The semi-infinite limit can be transformed to
s
C
txC
x


),(lim (22)
      xDssA
s
C
sxC
2/1
/exp', 

       xDssALCtxC
2/11
/exp',  
(23) (24)and
hence, B’(s) must be zero for the conditions at hand. Therefore,
Final evolution of and C(x,t) depends on the boundary condition.
 Combining (13) with (12) and inverting, we obtain
(25)
 By letting
(26)
 (25) can be written
(27)
         dtiDnFACtC
t
OOO
2/1
0
12/1*
,0

 
   
nFA
i
f

 
        dtfDCtC
t
OOO
2/1
0
2/1*
,0

 

 An expression for CR(0,t) can be obtained (assuming R is initially absent):
(28)
 The derivation of (27) and (28) employed only the linear diffusion equations, initial
conditions, semi-infinite conditions, and the flux balance. No assumption related to
electrode kinetics or technique was made; hence (27) and (28) are general. From
these equations and the boundary condition for LSV (11), we obtain
(29)
(30)
        dtfDtC
t
RR
2/1
0
2/1
,0

 
  
    2/12/1
*
2/1
0 )(




OR
O
t
DDtS
C
dtf


  
 1)(
*2/12/1
2/1
0



 


tS
CDnFA
dti OO
t
where, as before,
2/1







R
O
D
D

The solution of this last integral equation would be the function i (t), embodying the desired
current/time curve, or since potential is linearly related to time, the current/potential equation. A
closed-form solution of (30) cannot be obtained, and a numerical method must be employed.
Before solving (30) numerically, it is convenient (a) to change from i(t) to i(E), since that is the
way in which the data are usually considered, and (b) to put the equation in a dimensionless form
so that a single numerical solution will give results that will be useful under any experimental
conditions. This is accomplished by using the following substitution

3131
 Let . With , so that , , at
and at , we obtain
(32)
 so that (30) can be written
(33)
 Of finally, dividing by , we obtain
(34)
 where
(35)
    gf  z  /z  /dzd  0z 0
tz  t
    



dzz
tzgdtf
tt 2/1
0
2/1
0


 






  
 
 tS
DC
dzztzg OO
t





 
 1
2/1*
2/12/1
0
  2/1*
OO DC 
 
   tSzt
dzz
t





 1
1
0
2/1
   
 
 
  2/12/1




OOOO DnFAC
tit
DC
zg
z 

 EE
RT
nF
vt
RT
nF
t i 





 (31)

32
 t
Note that (34) is the desired equation in terms of the dimensionless variables
𝜒 𝑧 , 𝜉, 𝜃, 𝑠 𝜎𝑡 and 𝜎𝑡 . Thus at any value of 𝑠 𝜎𝑡 , which is a function of E, 𝜒 𝜎𝑡 , can be
obtained by solution of (34) and, form it, the current can be obtained by rearrangement of (35):
   tDnFACi OO  2/1
 (36)
At any given point,𝝌(𝝈𝒕)is a pure number, so that (36) gives the functional relationship
between the current at any point on the LSV curve and the variables.
Specifically, I is proportional to 𝐶0
⋆
and v1/2. The solution of (34) has been carried ou
numerically (Nicholson and Shain), by a series solution (Sevcik, Reinmuth), analitically in items
of an integral that must be evaluted numerically.
The general result of solving equation 34 is a set of values of 𝜒 𝜎𝑡 , (see following Table and
Figure ) as a function of 𝜎𝑡 or n(E - E1/2).
   tDnFACi OO  2/1

 EE
RT
nF
vt
RT
nF
t i 






𝜒 𝜎𝑡 ,
   
 
 
  2/12/1




OOOO DnFAC
tit
DC
zg
z 


 
   tSzt
dzz
t





 1
1
0
2/1
The general result of solving equation 34 is a set of values of 𝜒 𝜎𝑡 , (see following Table and
Figure ) as a function of 𝜎𝑡 or n(E - E1/2).
   tDnFACi OO

2/1

 EE
RT
nF
vt
RT
nF
t i 





   
 
 
  2/12/1




OOOO DnFAC
tit
DC
zg
z 

At any given point,𝝌(𝝈𝒕)is a pure number, so that (36) gives the functional relationship
between the current at any point on the LSV curve and the variables.
Specifically, I is proportional to 𝐶0
⋆
and v1/2. The solution of (34) has been carried out
numerically (Nicholson and Shain), by a series solution (Sevcik, Reinmuth), analytically in items
of an integral that must be evaluated numerically
Note that (34) is the desired equation in terms of the dimensionless variables
𝜒 𝑧 , 𝜉, 𝜃, 𝑠 𝜎𝑡 and 𝜎𝑡 . Thus at any value of 𝑠 𝜎𝑡 , which is a function of E, 𝜒 𝜎𝑡 , can be
obtained by solution of (34) and, form it, the current can be obtained by rearrangement of (35):
34

Cont..
34
   tDnFACi OO  2/1
   
 
 
  2/12/1




OOOO DnFAC
tit
DC
zg
z 

Table :Current Functions for reversible charge transfer
  
 1)(
*2/12/1
2/1
0



 


tS
CDnFA
dti OO
t
 
   tSzt
dzz
t





 1
1
0
2/1
 EE
RT
nF
vt
RT
nF
t i 






2/1







R
O
D
D

   '0
/exp
,0
,0
EERTnF
tC
tC
i
R
Ο












  vtEtE i 

𝒏 𝑬 − 𝑬 𝟏
𝟐
=
𝑹𝑻
𝑭
𝒍𝒏𝝃 + 𝒏 𝑬𝒊 − 𝑬 𝟎′
−
𝑹𝑻
𝑭
𝝈𝒕ln
𝛑𝟏/𝟐
𝛘(𝛔𝐭)=
𝐢
𝐧𝐅𝐀𝐂𝟎
∗
𝐃𝟎
∗(
𝐧𝐅
𝐑𝐓
)𝟏/𝟐
𝐯𝟏/𝟐
𝑛𝑜𝑡 𝑡ℎ𝑎𝑡 𝒍𝒏 𝝃𝜽𝑺 𝝈𝒕 = 𝒏𝒇 𝑬 − 𝑬 𝟏
𝟐
, 𝒘𝒉𝒆𝒓𝒆
2/1
0
2/1
0'0
2/1 ln
D
D
nF
RT
EE 
Cont..

 t,2/1

  4463.0,2/1
tThe function , and hence the current, reaches a maximum where .
From (36) the peak current, ip, is
Peak Current and Potential
At 25 oC for in cm2, D0
* in cm2/s, CO
*
mol/cm3, and v in V/s, ip is
 The peak potential, Ep is
 The half-peak
potential, Ep/2,
which is
2/12/12/3
2/13
4463.0 vCADn
RT
F
i OOp







 (37)
  2/12/12/35
1069.2 vCADni OOp

 (38)
n
nF
RT
EEp /5.28109.12/1  mV at 25 oC (39)
nE
nF
RT
EEp /0.28109.1 2/12/12/  (40)mV at 25 oC
For a Nernstian wave is
 Thuse for a reversible wave, Ep is independent of scan
rate and ip is proportional to v1/2. The latter property
indicates diffusion control and is analogous to the
variation of id with 𝑡
−1
2 in chronoamperometry.
mV
5.56
20.22/
nnF
RT
EE pp  (41)
ip
A
A cm2
Co* mol/cm3
Do
cm2/sec
ν V/sec

𝑖 = 𝑖 𝑝𝑙𝑎𝑛𝑒 + 𝑖 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
     t
r
DnFACtDnFACi OOO  





 
0
0
2/1 1
For LSV with a spherical electrode (e.g., a hanging mercury drop), a similar treatment can
be presented , and the resulting current is
where r0 is the radius of the electrode and 𝜙 𝜎𝑡 is a tabulated function (see Table above ). For
large values of v and with electrodes of conventional size the i (plane) term is much larger than
the spherical correction term, and the electrode can be considered planar under these conditions.
Basically, the same considerations also applies to hemispherical and ultra microelectrodes at fast
scan rates. However, for a UME, where r0 is small, the second term will dominate at sufficiently
small scan rates. One can show from above that this is true when 𝜐 ≪ 𝑅𝑇
𝑅𝑇𝐷
𝑛𝐹𝑟0
2
Cont..

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| = A Cd v
Recall Randles–Sevčik Eqn
ich more important at high v.
i(A) E (V)(+) (-)
ich
ip
i(A)
E (V)
(+)
(-)
ich
ip
i(A)
E (V)
(+)
(-)
ich
ip
x1
x20
x40
v = a
v = 100a
v = 900a
𝑖 𝑐ℎ = 𝑣𝐶 𝑑 +
𝐸𝑖
𝑅 𝑠
− 𝑣𝐶 𝑑 𝑒
−𝑡
𝑅 𝑠 𝐶 𝑑
ip = (2.69 x 105) n3/2 A DO
1/2 CO* v1/2

3939
LSV – Totally Irreversible Systems
ReO fk
For a totally irreversible one-step, one-electron reaction ( ) the nerstian
boundary condition, (8), is replaced by
Where
Introducing E(t) from (7) into (43) yields
 where 𝑏 = 𝛼𝑓𝑣 and
 
   tCtk
x
txC
D
FA
i
Of
x
O
O ,0
,
0










(42)
     '
00
exp EtEfktk f   (43)
      bt
OfiOf etCktCtk ,0,0  (44)
  '
00
exp EEfkk ifi   (45)
The solution follows in an analogous manner to that described in Reversible Systems and again
requires a numerical solution of an integral equation. The current is given by
where is a function tabulated from a table. i at any point on the wave varies with v l/2
and D0
*. For spherical electrodes, values of the spherical correction factor , employed
in the equation
 bt
 tb,
   btbDFACi OO  2/1
  bt
RT
F
vDFACi OO 
 2/1
2/1
2/12/1






 
(46) (47)
 
 
0r
btCFAD
planeii OO 
 (48)

Table 2 . Current Functions for Irreversible Charge Transfer
Cont..

Peak Current and Potential
 The function goes through a maximum at .
Introduction of this value into (36) yields te following for the peak current:
(49)
 From above table, the peak potential is given by
at 25 oC (50)
or
(51)
(52)
 bt   4958.0,2/1
tb
  2/12/12/15
1099.2 vDACi OOp

 
   
mV
F
RT
k
bD
F
RT
EE O
p 34.521.0ln 0
2/1
0'

































2/1
0
2/1
0
lnln780.0
'
RT
Fv
k
D
F
RT
EE O
p


mV
F
RT
EE pp

7.47857.1
2/ 
where Ep/2 is the potential where the current is at half the peak value. For a totally irreversible
wave, Ep is a function of scan rate, shifting (for a reduction) in a negative direction by an amount
1.15𝑅𝑇
𝛼𝐹
(𝑜𝑟 30/𝛼 𝑚𝑉 𝑎𝑡 25°𝐶) for each tenfold increase in v.

Peak shape for Reversible and irreversible system
i(A)
E (V)
(+)
(-)
E0'
(+)
(-)
E0'
(+)
(-)
E0'
i(A)
E (V)
(+)
(-)
E0'
(+)
(-)
E0'
(+)
(-)
E0'
Irreversible peak is broader and
smaller than reversible peak. As 
decreases, peak widens and
decreases in magnitude.
Unlike reversible, Ep is a function of scan rate
for irreversible. Ep shifts more negative for
reduction by 30/() m V at 25oC for each 10-
fold increase in scan rate.
𝐸 𝑝 = 𝐸0′
−
𝑅𝑇
𝛼𝐹
0.78 + ln
𝐷 𝑂
1/2
𝑘0
+ ln
𝛼𝐹𝑣
𝑅𝑇
1/2
  2/12/12/15
1099.2 vDACi OOp

 
  2/12/12/35
1069.2 vCADni OOp



43
LSV – Quasireversible Systems
ReO
b
f
k
k


   
   
 















 


 


 

'0'0
,0,0
, 0
0
EtEf
RO
EtEf
x
O
O etCtCek
x
txC
D

(54)  2/11
0
fvDD
k
RO


DDD RO or for
  2/1
0
Dfv
k
 (55)
 EvfCFADi OO   2/12/12/1
(56)
quasireverible for reactions that show electron transfer kinetic limitations where the reverse
reaction has to be considered, and they provided the fisrt treatment of such systems. For the one-
step, one-electron case, the corresponding boundary condition is
The shape of the peak and the various peak parameters were shown to be functions of 𝛼 and a
parameter , defined as
The current is given by
(53)
when  > 10, the behavior approaches that of a reversible system. The values of ip, Eр, and
Ep/2 depend on  and 𝛼. The peak current is given by
where ip(rev) is the reversible ip value (37).
N.B for a quasireversible reaction, ip is not
proportional to v1/2.
For the half-peak potential, we have
The peak potential is
   , Krevii pp (57)
   mVK
F
RT
KEEp  ,26,2/1 





 (58)
    mV
F
RT
EE pp  ,26,2/ 






(59)
quasireversible current function,

(a) Cyclic potential sweep. (b) Resulting cyclic voltarnmogram
cyclic voltammetry, in which the direction of the potential is reversed at the end of the
first scan. Thus, the waveform is usually of the form of an isosceles triangle.
The Advantage is that, the product of the electron transfer reaction that occurred in the
forward scan can be probed again in the reverse scan.
Powerful tool for the determination of formal redox potentials, detection of chemical
reactions that precede or follow the electrochemical reaction and evaluation of electron
transfer kinetics.
2. Cyclic Voltammetry

E(V)
t (s)
(-)
E
Ei
0 
- Reversal technique analogous to double-
potential step methods.
- Experiment time scale: 103 s to 10-5 s.
The reversal experiment in linear scan voltammetry is carried out by switching the direction of
the scan at a certain time,𝑡 = 𝜆(or at the switching potential, E 𝜆). Thus the potential
is given at any time by
( 𝟎 < 𝒕 ≤ 𝝀 ) , 𝑬 = 𝑬𝒊 − 𝒗𝒕
𝒕 > 𝝀 , 𝑬 = 𝑬{− 𝟐𝒗𝝀 + 𝒗𝒕
In linear sweep voltammetry the potential scan
is done in only one, stopping at a chosen value
E
I
Some important parametres
The initial potential, Ei , the initial sweep direction, the
sweep rate, v, the maximum potential, Emax , the minimum
potential, Emin, the final potential, Ef
fdfdfC IvCI
dt
dE
CIII 
Cont..

4646
 For simple electron transfer , with only O initially present in
solution. The initial sweep direction is therefore negative. The observed faradaic
current depends on the kinetics and transport by diffusion of the electroactive
species. It is thus necessary to solve the equations
(61) and (62)
 The boundary conditions are
(63 a)
(63 b)
(63 c)
(63 d)
RneO 
0t 0x
0t 0x
0t 0x
t0 vtEE i 
t    tvvEE i
   
2
2
x
R
D
t
R
R





   
2
2
x
O
D
t
O
O





     OO   0R
     OO   0R
    0















O
R
O
O
x
R
D
x
O
D
CV (at planar electrode) – Reversible Systems

 The final boundary condition for a reversible system is the Nernst equation
(64)
 Solution of the diffusion equations leads to a result in the Laplace domain that
cannot be inverted analytically, numerical inversion being necessary. The final
result, after inversion, can be expressed in the form
(65)
where and (66, 67)
 
 
 





 '0
exp EE
RT
nF
R
O
     tDOnFAI O  2/1

v
RT
nF






  EE
RT
nF
t i 
 Thus the current is dependent on the square root of the sweep rate.
Indicate some quantitative parameters in the curve, which can be deduced from data in table.
First, the current function, , passes through a maximum value of 0.4463 at a
reduction peak potential Ep,c of
 t 2/1
n
D
D
nF
RT
EE
R
O
cp /0285.0ln
2/1
'0
, 






nEE r
cp /0285.02/1, 
(68)
(69)
  2/12/12/35
1069.2 vOADnI Op  (70)
Cont..

With A measured in cm2, Do in cm2s-1, in mol cm-3 and v in Vs-1, substituting T =
164 К in (65) and (66)—an equation first obtained by Randles and Sevcik. Thirdly, the
difference in potential between the potential at half height of the peak, Ep/2,c (I = IP,c/2),
and Epc is given by
If the scan direction is inverted after passing the peak for a reduction reaction, then a
cyclic voltammogram, as shown schematically in Fig. below, is obtained. It has been
shown that, if the inversion potential, E, is at least 35/n mV after Epc, then
(72)
In which x = 0 for E «Epc and is 3 mV for |Epc - E | = 80/n mV.
In this case
(73)
mV
nnF
RT
EE cpcp
6.56
2.2,2/, 
n
x
nEE r
ap  /0285.02/1,
1/ ,, cpap II
(71)
 O

 The shape of the anodic curve is independent of E, but the value of E alters the position of
the anodic curve in relation to the current axis. For this reason Ip,a should be measured from a
baseline that is a continuation of the cathodic curve, as shown in Fig. 3.
apcp EorEEE ,,  Fig.3. Cyclic voltammogram
for a reversible system
• Ip  v1/2
• Ep independent of v
•Ep - Ep/2= 56.6/n mV
and for CV
•Ep,a – Ep,c = 57.0/n mV
•Ip,a/Ip,c = 1
for LSV

• In cyclic voltammetry, on inverting the sweep direction, One obtains only the continuation of
current decay
• With respect to reversible systems the waves are shifted to more negative potentials (reduction), Ep
depending on sweep rate. The peaks are broader and lower
(82)
The peak current in amperes is
CV (at planar electrode) – Irreversible Systems
    2/12/12/1'5
, 1099.2 vDOAnnI Occp  
Fig.4. Linear sweep voltammogram for an irreversible system.






 b
k
D
Fn
RT
EE O
c
cp ln
2
1
ln780.0
0
2/1
'
'0
,

n' being the number of electrons transferred in
the rate-determining step.
From data such as those in a Table it can be
deduced that
 '/7.472/ nEE pp  mV

CV (at planar electrode) – Quasi-reversible Systems
Fig.5. The effect of increasing
irreversibility on the shape of cyclic
voltammograms.
Peak shape and associated parameters are
conveniently expressed by a parameter, , which
is a quantitative measure of reversibility, being
effectively the ratio kinetics/transport,
 When DR = DO = D
    2/1)1(
0
2/1
0 //   cccc
RORO DDkDDk 
 (84)
2/12/1
0 
 Dk (85)
As a general conclusion, the extent of irreversibility
increases with increase in sweep rate, while at the same time
there is a decrease in the peak current relative to the
reversible case and an increasing separation between anodic
and cathodic peaks, shown in Fig.5.
Matsuda and Ayabe suggested
Reversible:  ≥ 15 ; k0 ≥ 0.3v1/2 cm/s
Quasireversible: 15 ≥  ≥ 10 –2(1+) ; 0.3v1/2 cm/s ≥ k0 ≥ 2 x 10-5 v1/2 cm/s
Totally irreversible:  ≤ 10 –2(1+) ; k0 ≤ 2 x 10-5 v1/2 cm/s
System |Ep – Ep/2|at
25oC (mV)
Reversible 57/n
Irreversible 48/( )
Quasireversible 26(, )

5252
CV – Adsorbed species
 If the reagent or product of an electrode reaction is adsorbed strongly or weakly on
the electrode, the form of the voltammetric wave is modified.
 The rate of reaction of adsorbed species is much greater than of species in solution
 It is necessary to consider the reactions of both adsorbed species and of those in
solution.
 For a reversible reaction in which only the adsorbed species О and R contribute to
the total current, the current-potential curve for О initially adsorbed and for fast
electrode kinetics is given by (86)
 where o,i is the surface concentration of adsorbed O, before the experiment begins,
on an electrode of area А,  = (nF/RT)v, bo and bR express the adsorption energy
of О and R respectively, and
 
   2
,
/1
/


RO
ROiO
c
bb
bbAnF
I



 



 '0
exp EE
RT
nF
 (87)
 The peak current for reduction, Ip,c, is obtained when (bo/bR)  = 1, that is
RT
vAFn
I
iO
cp
4
,
22
,


having the same magnitude for an
oxidation. The peak potential is then 






R
O
p
b
b
nF
RT
EE ln'0

Cases involving adsorption
R weakly
adsorbed
R strongly
adsorbed
V
Pre-wave
O weakly
adsorbed
O strongly
adsorbed
Post-waveV
Wopschall & Shain Anal. Chem. 39: 1514-1542

Basic potential step and sweep methods

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