linear sweep voltammetry and cyclic voltammetry (lsv & cv) katalin neuróhr research institute...

LINEAR SWEEP VOLTAMMETRY LINEAR SWEEP VOLTAMMETRY AND CYCLIC VOLTAMMETRYAND CYCLIC VOLTAMMETRY

(LSV & CV)(LSV & CV)

Katalin NeuróhrKatalin NeuróhrResearch Institute for Solid State Physics and

Optics (RISSPO), Hungarian Academy of Sciences

221. 11. 20111. 11. 2011

22

IntroductionIntroduction LSVLSV

voltammetric methodevoltammetric methode the current at the working electrode is measured while the the current at the working electrode is measured while the

potential between the working electrode and the reference potential between the working electrode and the reference electrode is swept linearly in timeelectrode is swept linearly in time

ends when it reaches a set potentialends when it reaches a set potential

CVCV a type of potentiodynamic electrochemical measurementa type of potentiodynamic electrochemical measurement the working electrode potential is ramped linearly versus time the working electrode potential is ramped linearly versus time

like linear sweep voltammetrylike linear sweep voltammetry when cyclic voltammetry reaches a set potential, the potentialwhen cyclic voltammetry reaches a set potential, the potential

of the working electrodeof the working electrode ramp is inverted ramp is inverted the current at the working electrode is plotted versus the the current at the working electrode is plotted versus the

applied voltage to give the cyclic voltammogram traceapplied voltage to give the cyclic voltammogram trace

33

LSV – Nernstian LSV – Nernstian (Reversible) System(Reversible) System

Fundamentals Consider the reactionConsider the reaction We assume here th We assume here the e Erdey-Grúz–Erdey-Grúz–

VolmerVolmer equation equation

andand(1)(1)

The summary of the kinetics parameters:The summary of the kinetics parameters:

where where is a dimensionless parameter, is a dimensionless parameter,

if if > 10, the voltammogram is reversible> 10, the voltammogram is reversible

if if 0.1 0.1, the voltammogram is irreversible, the voltammogram is irreversible

if if 0.1 0.1 > 10, the voltammogram is quasireversible> 10, the voltammogram is quasireversible

RneO

2/1/ DFvRTks

RT

zFkzFcj

αexpaRa

RTεzF

kzFcjα1

expkOk

44

LSV – NerLSV – Nernnstian stian (Reversible) System(Reversible) System

Solution of the Boundary Value ProblemSolution of the Boundary Value Problem The equations governing this case are The equations governing this case are

(2)(2)

(3)(3)

(4)(4)

The The initial condition, (3), merely express the initial condition, (3), merely express the homogeneity of the homogeneity of the ssolution beforeolution before the experiment starts at the experiment starts at t t = 0= 0, , and the semi-infinite and the semi-infinite condition,condition, (4), is an assertion(4), is an assertion that regions distant from thethat regions distant from the electrode are unperturbed by the experiment. The third condition,electrode are unperturbed by the experiment. The third condition, (3),(3), expresses the condition at the electrode surface after theexpresses the condition at the electrode surface after the potential transition,potential transition, and it embodies the particular experiment we and it embodies the particular experiment we have at hand.have at hand.

2

2 ),(),(

x

txCD

t

txC OO

O

2

2 ),(),(

x

txCD

t

txC RR

R

OO CxC )0,( 0)0,( xCR

OO

xCtxC ),(lim 0),(lim

txCR

x

00),0( tfortCO

55


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem and the fluxe balance isand the fluxe balance is

(5)(5)

It is It is convenient convenient to rewrite (1)to rewrite (1)

(6)(6)

Consider Consider reaction, linear diffusion, only species O, with reaction, linear diffusion, only species O, with the electrode held initially at the electrode held initially at EEii potential potential

(7)(7)

the potential is swept linearly at the potential is swept linearly at vv If the rate of electron transfer is rapid at the electrode surfaceIf the rate of electron transfer is rapid at the electrode surface

(8)(8)

0),(),(

00

x

RR

x

OO x

txCD

x

txCD

'0exp

,0

,0EE

RT

nF

tC

tC

R

O

RneO

vtEtE i

'0exp,0

,0EvtE

RT

nFtf

tC

tCi

R

Ο

66


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem after Laplace transformation of (after Laplace transformation of (22), the application), the application of conditionsof conditions

(3) and (4) yields(3) and (4) yields

(9)(9)

(10)(10)

xDsOO

OesAs

CsxC /,

xDsR

ResBsxC /,

77


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem 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

(11)

Where and and . Laplace transformation of the diffusion equations and application of the initial and semi-infinite conditions leads to [see (9)]:

(12)

tSetC

tC t

R

Ο

,0

,0

tetS '0/exp EERTnF i vRTnF /

xD

ssA

s

CsxC

O

OO

2/1

exp,

88

LSV – NerLSV – Nernnstian stian (Reversible) System(Reversible) SystemSolution of the Boundary Value ProblemSolution of the Boundary Value Problem

After tAfter the transform of the currenthe transform of the current is given by is given by

((1313))

Laplace transformationLaplace transformation (14)(14)

(15)(15)

An initial conditionsAn initial conditions ( (tt = 0) and two boundar = 0) and two boundaryy conditions in x. Typically conditions in x. Typically one takes for the initial state:one takes for the initial state:

(16)(16)

and one uses the semi-infinite limit:and one uses the semi-infinite limit:

(17)(17)

0

,

x

OO x

sxCnFADsi

2

2 ),(),(

x

txCD

t

txC

CxC )0,(

CtxCx

),(lim

0

dttfetfLsF st

99


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem Generally:Generally:

((1818))

For variable For variable tt, we obtain, we obtain

(19)(19)

(20)(20)

(21)(21)

0...0'0 121 nnnnn FFsFssfsFL

2

2 ,,

dx

sxCdDCsxCs

D

CsxC

D

s

dx

sxCd

,,2

2

xDssBxDssAsC

sxC 2/12/1 /exp'/exp',

1010


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem The semi-infinite limit can be transformed toThe semi-infinite limit can be transformed to

((2222))

hence, B’(s) must be zero for the conditions at hand. Therefore,hence, B’(s) must be zero for the conditions at hand. Therefore,

(23)(23)

andand

(24)(24)

Final evFinal evoolluution of tion of and C(x,t) depends on the and C(x,t) depends on the boundarboundaryy condition. condition.

xDssAs

CsxC 2/1/exp',

s

CtxC

x

),(lim

xDssALCtxC 2/11 /exp',

sxC ,

1111


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem Combining Combining (13)(13) with ( with (1212) and inverting) and inverting, we obtain, we obtain

((2525))

By lettingBy letting

(26)(26)

((2525) can be written ) can be written

(27)(27)

dtiDnFACtCt

OOO2/1

0

12/1*,0

nFA

if

dtfDCtCt

OOO2/1

0

2/1*,0

1212


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem AnAn expression for C expression for CRR(0,t) can be obtained (assuming R is initially (0,t) can be obtained (assuming R is initially

absent):absent):

(28)(28)

The derivation of (The derivation of (2727) and () and (2828) employed only the linear diffusion ) employed only the linear diffusion equations,equations, initial conditions, semi-infinite conditions, and the flux initial conditions, semi-infinite conditions, and the flux balancebalance.. No assumption related to electrode kinetics or technique No assumption related to electrode kinetics or technique was made; hence (27) and (28) are general. was made; hence (27) and (28) are general. From these equations From these equations and the boundary condition for LSV (and the boundary condition for LSV (1111), we obtain), we obtain

(29)(29)

(30)(30)

dtfDtCt

RR2/1

0

2/1,0

2/12/1

*2/1

0 )(

OR

Ot

DDtS

Cdtf

1)(

*2/12/12/1

0

tS

CDnFAdti OO

t

wwhere, as beforehere, as before,,2/1

R

O

D

D

1313


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem

The solution of this last integral equation would be the function The solution of this last integral equation would be the function i i ((tt), embodying the desired current/time curve, or since potential is ), embodying the desired current/time curve, or since potential is linearly related to time, the current/potential equation. A closed-linearly related to time, the current/potential equation. A closed-form solution of form solution of (30)(30) cannot be obtained, and a numerical method cannot be obtained, and a numerical method must be employed.must be employed.

Before solving (30) numerically, it is convenient (a) to change from Before solving (30) numerically, it is convenient (a) to change from i(t) i(t) to to i(E), i(E), since that is the way in which the data are usually since that is the way in which the data are usually considered, and (b) to put the equation in a dimensionless form so considered, and (b) to put the equation in a dimensionless form so that a single numerical solution will give results that will be useful that a single numerical solution will give results that will be useful under any experimental conditions. This is accomplished by using under any experimental conditions. This is accomplished by using the following substitution:the following substitution:

(31)(31)

EERT

nFvt

RT

nFt i

1414


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem Let Let . With . With , so that, so that ,, , , atat

andand atat , we obtain , we obtain

(32)(32)

so that (so that (3030) can be written) can be written

((3333))

Of finally, dividing byOf finally, dividing by , we obtain, we obtain

((3434)) wherewhere

((3535))

gf z /z /dzd 0z 0tz t

dzz

tzgdtftt 2/1

0

2/1

0

tS

DCdzztzg OO

t

1

2/1*2/12/1

0

2/1*OO DC

tSzt

dzzt

1

1

02/1

2/12/1

OOOO DnFAC

tit

DC

zgz

1515


Solution of the Boundary Value ProblemSolution of the Boundary Value Problem Note that (Note that (3434) is the desired equation in terms of the ) is the desired equation in terms of the

dimensionless variablesdimensionless variables , , , ,, , and . Thus at any value and . Thus at any value ofof , which is a function of , which is a function of EE,, can be obtained by solution of can be obtained by solution of ((3434) and, form it, the current can be obtained by rearrangement of ) and, form it, the current can be obtained by rearrangement of ((3535):):

((3636))

tDnFACi OO 2/1

z tS t tS t

tAt any given point,At any given point, is a pure number,is a pure number, soso

that (that (3636) gives the functional relationship) gives the functional relationship

betweenbetween the current at any point on the LSVthe current at any point on the LSV

curve and the variables.curve and the variables. Specifically,Specifically,ii is is

proportional toproportional to and and vv1/21/2.The solution of (.The solution of (3434))

has been carried out numericallyhas been carried out numerically

(Nicholson and Shain), by a series(Nicholson and Shain), by a series solutionsolution

(Sevcik, Reinmuth), analitically in items of(Sevcik, Reinmuth), analitically in items of

an integral thatan integral that must be evaluted numerically. must be evaluted numerically. Fig.1. Linear potential sweep voltammogram in terms of dimensionless current function. Values on the potential axis are for 13oC

OC

1616


Peak CurrentPeak Current and Potentialand Potential The function The function , and hence the current, reaches a maximum where , and hence the current, reaches a maximum where

. . From (From (3636) the peak current, ) the peak current, iipp, is, is((3737))

At At 2525 ooC for in cmC for in cm22, , DD00** in cm in cm22/s, /s, CCOO

** mol/cm mol/cm33, and , and vv in V/s, in V/s, iipp is is(38)(38)

The peak potential, The peak potential, EEpp is ismV at mV at 2525 o oCC (39)(39)

Difficult to determineDifficult to determine TThe half-peak potential, he half-peak potential, EEp/2p/2, which is, which is

mV at mV at 2525 ooCC (40)(40)

t,2/1 4463.0,2/1 t

2/12/12/3

2/13

4463.0 vCADnRT

Fi OOp

2/12/12/351069.2 vCADni OOp

nnF

RTEEp /5.28109.12/1

nEnF

RTEEp /0.28109.1 2/12/12/

1717


Peak Current and PotentialPeak Current and Potential

Note that Note that EE1/1/22 is located just about midway between is located just about midway between EEpp and and EEpp/2/2, and , and that a convenient diagnosticthat a convenient diagnostic for a nernstian wave isfor a nernstian wave is

(41)(41)

Thuse for a reversible wave, Thuse for a reversible wave, EEpp is independent of scan rate and is independent of scan rate and iip p is is proportional to proportional to vv1/21/2. The latter property indicates diffusion control . The latter property indicates diffusion control and is analogous to the variation of and is analogous to the variation of iidd in chronoamperometry. A in chronoamperometry. A convenient in LSV is convenient in LSV is , which deends on , which deends on nn3/23/2 and and DDOO

1/2. 1/2. This This constant can be used to estimate constant can be used to estimate nn for an electrode reaction, if a for an electrode reaction, if a value of value of DDOO can be estimated. can be estimated.

nnF

RTEE pp /5.5620.22/

Op Cvi 2/1/

1818

LSV – Totally Irreversible LSV – Totally Irreversible SystemsSystems

The Boundary Value ProblemThe Boundary Value Problem For a totally irreversible one-step, one-electron reactionFor a totally irreversible one-step, one-electron reaction

(( ) the nerstian ) the nerstian boundary condition, (boundary condition, (88), is replaced by), is replaced by

(42)(42)

Where Where

(43)(43)

Introducing Introducing E(t)E(t) from ( from (77) into () into (43)43) yields yields

(44)(44)

where where b = afvb = afv and and

(45)(45)

ReO fk tCtkx

txCD

FA

iOf

x

OO ,0

,

0

'00 exp EtEfktk f

btOfiOf etCktCtk ,0,0

'00 exp EEfkk ifi

1919


The Boundary Value ProblemThe Boundary Value Problem The solution follows in an analogous manner to that described in The solution follows in an analogous manner to that described in

Totally Reverse Systems and again requires a numerical solution of Totally Reverse Systems and again requires a numerical solution of an integral equation. The current is given byan integral equation. The current is given by

(46)(46)

(47)(47)

wherewhere is a function tabulated from a table. is a function tabulated from a table. ii at any at any point on the point on the wave varies with wave varies with vvl/2l/2 andand DD00

**.. For spherical electrodes, values of the spherical correction factor, For spherical electrodes, values of the spherical correction factor,

employed in the equation employed in the equation

(48)(48)

btbDFACi OO 2/1

btRT

FvDFACi OO 2/1

2/12/12/1

bt

0r

btCFADplaneii OO

tb,

2020


Peak Current and PotentialPeak Current and Potential The function The function goes throug a maximum at goes throug a maximum at . .

Introduction of this value into (36) yields te following for the peak Introduction of this value into (36) yields te following for the peak current:current: (49)(49)

Where the units are as for (49). From a table, the peak potential is Where the units are as for (49). From a table, the peak potential is given bygiven by

mV at 25 mV at 25 ooCC (50) (50)

oror

(51)(51)

(52)(52)

bt 4958.0,2/1 tb

2/12/12/151099.2 vDACi OOp

34.521.0ln

0

2/10'

F

RT

k

bD

F

RTEE O

p

2/1

0

2/10 lnln780.0'

RT

Fv

k

D

F

RTEE O

p

mVF

RTEE pp

7.47857.12/

2121

LSV – Quasireversible LSV – Quasireversible SystemsSystems

Matsuda and Ayabe coined the term Matsuda and Ayabe coined the term quasireveriblequasireverible for raections that for raections that show electron transfer kinetic limitations where the reverse reaction show electron transfer kinetic limitations where the reverse reaction has to be considered, and they provided the fisrt treatment of such has to be considered, and they provided the fisrt treatment of such systems. systems. For the one-step, one-electron case,For the one-step, one-electron case, the the corresponding boundary condition iscorresponding boundary condition is

((5353))

The shape of the peak and the various peak parameters were shown The shape of the peak and the various peak parameters were shown to be functions of ato be functions of a and a parameter and a parameter , defined as, defined as

(54)(54) ((55)55)

The current is given byThe current is given by ((5656))

ReOb

f

k

k

'0'0

,0,0,

0

EtEf

RO

EtEf

x

OO etCtCek

x

txCD

2/11

0

fvDD

k

RO

DDD RO or for 2/1

0

Dfv

k

EvfCFADi OO 2/12/12/1

2222

LSV – Quasireversible LSV – Quasireversible SystemsSystems

when when > 10, the behavior approachesthat of a reversible system. > 10, the behavior approachesthat of a reversible system. The The values of values of iipp, , EEрр, and , and EEp/2p/2 depend on depend on and and . The peak . The peak ccurrent is given urrent is given byby

((5757)) where where iipp(rev) is the reversible (rev) is the reversible iipp value ( value (3737)). Note. Note that for a that for a

quasireversible reaction, quasireversible reaction, iipp is not proportional to is not proportional to vv11/2/2.. The peak potential isThe peak potential is

((5858))

For the half-peak potential, we haveFor the half-peak potential, we have

((5959))

, Krevii pp

mVKF

RTKEEp ,26,2/1

mVF

RTEE pp ,26,2/

2323

CV CV (Experimental (Experimental basis)basis)

The basic scheme involves application of a potential sweep to theThe basic scheme involves application of a potential sweep to the wworking electrode. The various parameters of interest are shown in orking electrode. The various parameters of interest are shown in

Fig.Fig. 2. 2. In linear sweep voltammetry the potential scan is done in only one, In linear sweep voltammetry the potential scan is done in only one,

stopping at a chosen value, stopping at a chosen value, EEff for example atfor example at t = t t = t11 in Fig. in Fig. 22.. The The scan direction can be positive or negative and, in principle, thescan direction can be positive or negative and, in principle, the sweep rate can have any value. sweep rate can have any value.

Fig.2. Variation of applied potential with time in cyclic voltammetry, showing the initial potential, Ei9 the final potential, f, maximum, Emax, and minimum, Emin, potentials. The sweep rate \dE/dt\ = v. For linear sweep voltammetry consider only one segment. The fact that the initial sweep is positive is purely illustrative.

In cyclic voltammetry, on reaching t = t1

the sweep direction is invertedas shown in Fig. 2 and swept until Emin, then inverted and swept to Emax, etc.

2424

CV CV (Experimental (Experimental basis)basis)

The important parameters areThe important parameters are

• • the initial potential, the initial potential, EEii

• • the initial sweep directionthe initial sweep direction

• • the sweep rate, the sweep rate, vv

• • the maximum potential, the maximum potential, EEmaxmax

• • the minimum potential, the minimum potential, EEminmin

• • the final potential, the final potential, EEf f

It is not common, but can sometimes be convenient, to change the It is not common, but can sometimes be convenient, to change the valuesvalues

of of EEmaxmax and E and Eminmin between successive cycles. between successive cycles.

The toThe total current istal current is

((6060))fdfdfC IvCI

dt

dECIII

2525

CV CV (at planar (at planar electrode)electrode)

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 areThe boundary conditions are

(63 a)(63 a)

(63 b)(63 b)

(63 c)(63 c)

(63 d)(63 d)

RneO

2

2

x

OD

t

OO

2

2

x

RD

t

RR

0t 0x OO 0R

0t 0x OO 0R

0t 0x

0

OR

OO x

RD

x

OD

t0 vtEE i

t tvvEE i

2626

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)

CV CV (at planar electrode)(at planar electrode) – Reversible Systems– Reversible Systems

'0exp EERT

nF

R

O

tDOnFAI O 2/1

vRT

nF

EE

RT

nFt i

Thus the current is dependent on the square root of the sweep rate.

2727

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

(68)

(69)Secondly, the peak current in amperes is

(70)


t 2/1

nD

D

nF

RTEE

R

Ocp /0285.0ln

2/1

'0,

nEE rcp /0285.02/1,

2/12/12/351069.2 vOADnI Op

2828

With A measured in cm2, Do in cm2s-1, [O]oo 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

(71)

If the scan direction is inverted after passing the peak for a reduction reaction, then a cyclic voltammogram, as shown schematically in Fig. 3, 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)


mVnnF

RTEE cpcp

6.562.2,2/,

n

xnEE r

ap /0285.02/1,

1/ ,, cpap II

2929

The shape of the anodic curve is always the same, 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.

We can summarize all the information in a diagnostic for linear sweep and cyclic voltammograms of reversible reactions:


• Ip v1/2

• Ep independent of v

•Ep - Ep/2= 56.6/n mV

and for cyclic voltammetry alone•Ep,a – Ep,c = 57.0/n mV

•Ip,a/Ip,c = 1

apcp EorEEE ,,

Fig.3. Cyclic voltammogram for a reversible system

3030

As is clear from (65), Ip T-1/2: so, if experiments are conducted at temperatures other than 298 K, the correction in Ip is easy to do. Sometimes, and this is one of the disadvantages of conventional analysis of cyclic voltammograms, it is not possible to measure the baseline with sufficient precision in order to measure Ipa. However, it is a good approximation to apply the following expression in terms of the peak current measured from the current axis (Ipa)0 and the current at the inversion potential (I)0 (see Fig. 1)

(74)


086.0

485.0

,

0

,

,

,

, cpcp

Oap

cp

ap

I

I

I

I

I

I

3131

The capacitive contribution to the total current as given in (60) should also be taken into account. Writing If = Ipc we have, from (60) and

(75)

Substituting typical values (Cd = 20 Fcm-2, Do=10-5cm2s-1 and [О]∞ = 10-7 molcm-3 (10-4 M), and n = 1) we obtain

(76)


ODn

vC

I

I

O

d

cp

C2/12/3

52/1

, 69.2

10

2/1

,

24.0 vI

I

cp

C

This ratio is 0.1 for v = 0.18Vs; if [О]∞ is an order of magnitude higher, i.e. 10-3 M, then the ratio is only 0.01. This shows the advantage of using concentrations as high as possible, millimolar concentrations representing the upper limit.

3232

In the case of an irreversible reaction of the type , linear sweep and cyclic voltammetry lead to the same voltammetric profile, since no inverse peak appears on inversing the scan direction. To solve (61) and (62), a fifth boundary condition to add to boundary conditions (63) is

(77) for a reduction, where

(78) and

(79)

CV CV (at planar electrode)(at planar electrode) – – IrrIrreversible Systemseversible Systems

ObtkOkx

OD ccO exp'

'0'0

' /exp EERTFnkk icc

RTFvnb c /'

n' being the number of electrons transferred in the rate-determining step.

RneO

3333

As for the reversible case, the mathematical solution in the Laplace domain cannot be inverted analytically. Numerical inversion leads to

(80)

and the values of are tabulated, having a maximum of 0.4958 for E = Ep, see a Table. The voltammetric curve is shown in

Fig. 4. The peak current in amperes is

(81)


btRT

FnvDOnFAI c

Oc 2/1

2/1'2/112

t 2/1

2/12/12/1'5, 1099.2 vDOAnnI Occp

Fig.4. Linear sweep voltammogram for an irreversible system.In cyclic voltammetry, on inverting the sweep direction,One obtains only the continuation of current decay

3434

with the units the same as in (70). The peak potential is given by

(82)

an alternative expression for Ip is obtained from combining (81) and (82), leading to

(83) From data such as those in a Table it can be deduced that

mV and that mV.


b

k

D

Fn

RTEE O

ccp ln

2

1ln780.0

0

2/1

''0

,

'0

,0,

'exp227.0 EE

RT

FnkOnFAI cp

ccp

'/7.472/ nEE pp '6.29lg/ nvddEp

With respect to reversible systems the waves are shifted to morenegative potentials (reduction), Ep depending on sweep rate. The peaksare broader and lower.

3535

For quasi-reversible systems the kinetics of the oxidation and reduction reactions have to be considered simultaneously. The mathematical solution is, therefore, more complex, but there are numerical theoretical solutions.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.

CV CV (at planar electrode)(at planar electrode) – Quasi-– Quasi-rreversible eversible SystemsSystems

Fig.5. The effect of increasing irreversibility on the shape of cyclic voltammograms.

36

Peak shape and associated parameters are conveniently expressed by a parameter, , which is a quantitative measure of reversibility, being effectively the ratio kinetics/transport,

(84)

When DR = DO = D

(85) showing that small corresponds to large v (i.e. large ). The following ranges were suggested for the different types of

system at stationary planar electrodes:

2/1)1(0

2/1

0 // ccccRORO DDkDDk

2/12/10 Dk


37

The transition between these zones is shown schematically in Fig.6. In the case of cyclic voltammograms and for 0.3 < < 0.7, Ep is almost exclusively dependent on and hardly varies with . This can be useful for the calculation of k0 using a table, or by interpolation from a working curve drawn using these data.

Fig.6. Transition from a reversible to an irreversible system on increasing sweep rate.


3838

CV – Adsorbed speciesCV – 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. There are two types of situation:• the rate of reaction of adsorbed species is much greater than ofspecies 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

(87)

2

,

/1

/

RO

ROiOc

bb

bbAnFI

'0exp EERT

nF

3939

CV – Adsorbed speciesCV – Adsorbed species The peak current for reduction, Ip,c, is obtained when (bo/bR) = 1,

that is

(88)

having the same magnitude for an oxidation. The peak potential is then

(89)

RT

vAFnI iO

cp 4,

22

,

R

Op b

b

nF

RTEE ln'0

Fig.7. Cyclic voltammogram for a reversible system of species adsorbed on the electrode.

The value of Ep is the same for oxidation and for reduction.If the adsorption isotherm is of Langmuir type and (bO/bR) = 1, then the voltammetric profile is described by the function (1 + )-

2. From this it can be calculated that the peak width at half height is 90.6/n mV.This is all shown schematically in Fig.7.

4040

CV – Spherical CV – Spherical electrodeselectrodes

Considering first potential sweep for a reversible system one obtains

(90) For a reduction, in which r0 is the electrode radius and t a

currentfunction different from (t). The peak current is

(91) where ro is in cm and the other units are as in (70) for a planar

electrode. Since the spherical correction does not depend on scan rate we can consider spherical electrodes as if they were planar, plus a spherical correction.

0r

tOnFADII Oplanar

0

5,, 10725.0

r

OnADII O

planarpcp

4141

CV – Spherical CV – Spherical electrodeselectrodes

The current function for the spherical correction, t), is shown in a table; as can be seen it follows a sigmoidal I-Е profile. For irreversible systems the corresponding expressions are

(92)

(93)

0

,

,

r

tbOnFADII O

planarpp

0

5,, 10670.0

r

OnADII O

planarpcp

where r0 is in cm and the other units are as in (81) for a planarelectrode, the spherical correction being, once more, independent of scanrate. Values of the current function for the spherical correction, (bt)are given in a table. As for reversible systems, the spherical correctionby itself corresponds to a sigmoidal I-Е profile, though of lower slope.

Thank you for your Thank you for your attention!attention!

linear sweep voltammetry and cyclic voltammetry (lsv & cv) katalin neuróhr research institute...

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