Ebook Electrochemical methods Fundamentals and applications (2nd edition) Part 2

CHAPTER

12
ELECTRODE REACTIONS
WITH COUPLED
HOMOGENEOUS
CHEMICAL REACTIONS
12.1 CLASSIFICATION OF REACTIONS
The previous chapters dealt with a number of electrochemical techniques and the responses obtained when the electroactive species (O) is converted in a heterogeneous electron-transfer reaction to the product (R). This reaction is often a simple one-electron
transfer, such as an outer-sphere reaction where no chemical bonds in species О are broken and no new bonds are formed. Typical reactions of this type are
Fe(CN)^" + e <=± Fe(CN)£~

Ar + e ^ ArT
where Ar is an aromatic species and ArT is a radical anion. In many cases the electrontransfer reaction is coupled to homogeneous reactions that involve species О or R. For example, О may not be present initially at an appreciable concentration, but may be
produced during the electrode reaction from another, nonelectroactive species. More frequently, R is not stable and reacts (e.g., with solvent or supporting electrolyte). Sometimes a substance that reacts with product R is intentionally added so that the rate of the
reaction can be determined by an electrochemical technique or a new product can be produced. In this chapter, we will survey the general classes of coupled homogeneous chemical reactions and discuss how electrochemical methods can be used to elucidate the
mechanisms of these reactions.
Electrochemical methods are widely applied to the study of reactions of organic
and inorganic species, since they can be used to obtain both thermodynamic and
kinetic information and are applicable in many solvents. Moreover, as described
below, reactions can be examined over a wide time window by electrochemical techniques (submicroseconds to hours). Finally, these methods have the special feature
that the species of interest (e.g., R) can be synthesized in the vicinity of the electrode by the electron-transfer reaction and then be immediately detected and analyzed
electrochemically.
The initial investigations of coupled chemical reactions were carried out by
Brdicka, Wiesner, and others of the Czechoslovakian polarographic school in the
1940s; since that time countless papers dealing with the theory and application of dif471


472

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions

ferent electroanalytical techniques to the study of coupled reactions have appeared. It
is beyond the scope of this textbook to attempt to treat this area exhaustively. The
reader is instead referred to monographs and review articles dealing with different aspects of it (1-9).
Before discussing the electrochemical techniques themselves, let us consider some
general pathways that typify the overall electrochemical reactions of many soluble organic and inorganic species. We represent our general stable reactant as RX and consider what reactions can occur following an initial one-electron oxidation or reduction
(Figure 12.1.1). For example, if RX is an organic species, R can be a hydrocarbon moiety (alkyl, aryl) and X can represent a substituent (e.g., H, OH, Cl, Br, NH 2 , NO 2 , CN,
CO2", . . .). In some cases, the product of the one-electron reaction is stable and leads to
production of a radical ion (path 1). Often the addition of an electron to an antibonding
orbital or the removal of an electron from a bonding orbital will weaken a chemical
bond. This can lead to a rearrangement of the molecule (path 3) or, if X is a good "leaving group," reaction paths 6 and 7 can occur. Sometimes, for example, with an olefinic

Ox

©RX :

-RH©

(RX-)'
(3)
®

^RXE-T

(5)

(a) General reduction paths

-Red

©RX 2 +

RXNu+ — ^

RXNu2

©

(b) General oxidation paths

Figure 12.1.1 Schematic representation of possible reaction paths following reduction and
oxidation of species БОС. (a) Reduction paths leading to (1) a stable reduced species, such as a
radical anion; (2) uptake of a second electron (ЕЕ); (3) rearrangement (EC); (4) dimerization (EC2);
(5) reaction with an electrophile, E€ + , to produce a radical followed by an additional electron
transfer and further reaction (ECEC); (6) loss of X" followed by dimerization (ECC2); (7) loss of
X~ followed by a second electron transfer and protonation (ECEC); (8) reaction with an oxidized
species, Ox, in solution (EC), (b) Oxidation paths leading to (1) a stable oxidized species, such as a
radical cation; (2) loss of a second electron (ЕЕ); (3) rearrangement (EC); (4) dimerization (EC2);
(5) reaction with a nucleophile, Nu~, followed by an additional electron transfer and further
reaction (ECEC); (6) loss of X + followed by dimerization (ECC2); (7) loss of X + followed by a
second electron transfer and reaction with OH~ (ECEC); (8) reaction with a reduced species, Red,
in solution (EC). Note that charges shown on products, reactants, and intermediates are arbitrary.
For example, the initial species could be RX~, the attacking electrophile could be uncharged, etc.


12.1 Classification of Reactions i 473
reactant, dimerization takes place (path 4) (with the possibility of further oligimerization and polymerization reactions). Finally, reactions of intermediates with solution
components are possible. These include the reaction of RXT with an electrophile, E€ +
(i.e., a Lewis acid like H + , CO2, SO2) or of RX+* with a nucleophile, Nu~ (i.e., a Lewis
base like OH~, CN~, NH3) (path 5). An electron-transfer reaction with a nonelectroactive species present in solution (Ox or Red) can also occur (path 8). In general, the addition of an electron produces a species that is more basic than the parent so that
protonation can occur (i.e., RXT in path 5 with E€ + being H + ). Likewise, removal of

an electron from a molecule produces a species that is more acidic than the parent, so
that loss of a proton can occur (i.e., RX^ in path 7 with X + being H + ). Similar pathways take place following an initial electron-transfer reaction with an organometallic
species or coordination compound. For example, oxidation or reduction can be followed
with loss of a ligand or rearrangement.
It is convenient to classify the different possible reaction schemes by using letters to
signify the nature of the steps. "E" represents an electron transfer at the electrode surface, and "C" represents a homogeneous chemical reaction (10). Thus a reaction mechanism in which the sequence involves a chemical reaction of the product after the electron
transfer would be designated an EC reaction. In the equations that follow, substances
designated X, Y, and Z are assumed to be not electroactive in the potential range of interest. It is also convenient to subdivide the different types of reactions into (1) those that
involve only a single electron-transfer reaction at the electrode and (2) those that involve
two or more E-steps.

12.1.1

Reactions with One E Step
(a) CE Reaction (Preceding Reaction)
Y<=±0
O+

rce^R

(12.1.1)
(12.1.2)

Here the electroactive species, O, is generated by a reaction that precedes the electron
transfer at the electrode. An example of the CE scheme is the reduction of formaldehyde
at mercury in aqueous solutions. Formaldehyde exists as a nonreducible hydrated form,
H2C(OH)2, in equilibrium with the reducible form, H 2 C=O:
он
H2C

^ H?C = О + H2O
Ч

он

<

12Л

3

- >

The equilibrium constant of (12.1.3) favors the hydrated form. Thus the forward reaction
in (12.1.3) precedes the reduction of H 2 C=O, and under some conditions the current will
be governed by the kinetics of this reaction (yielding a so-called kinetic current). Other
examples of this case involve reduction of some weak acids and the conjugate base anions, the reduction of aldoses, and the reduction of metal complexes.
(b) EC Reaction (Following Reaction)
O + ne±±R
R<±X

(12.1.4)
(12.1.5)

In this case the product of the electrode reaction, R, reacts (e.g., with solvent) to produce a species that is not electroactive at potentials where the reduction of О is occurring.


474 i Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
An example of this scheme is the oxidation of p-aminophenol (PAP) at a platinum electrode in aqueous acidic solutions:
\ = ! М Н + 2Н+ + 2г

NH2 ^ = ^ O=/
(PAP)

(12.1.6)

(Ql)
+ NH3

(12.1.7)

(BQ)

where the quinone imine (QI) formed in the initial electron-transfer reaction undergoes a hydrolysis reaction to form benzoquinone (BQ), which is neither oxidized nor
reduced at these potentials. This type of reaction sequence occurs quite frequently,
since the electrochemical oxidation or reduction of a substance often produces a reactive species. For example, the one-electron reductions and oxidations that are characteristic of organic compounds in aprotic solvents [e.g., in acetonitrile (CH 3 CN) or
MAf-dimethylformamide (Me 2 NHC=O)] produce radicals or radical ions that tend to
dimerize:
R + e^±RT
T

2R ->R^~

(12.1.8)
(12.1.9)

e.g., where R is an activated olefin, such as diethyl fumarate (see Figure 12.1.1, path 4). In
this example, the reaction that follows the electron transfer is a second-order reaction, and
this case is sometimes designated as an EC 2 reaction. Sometimes, yet another chemical
reaction follows the first; for example, in the dimerization of olefins, there is a concluding

(two-step) protonation process:
R\~ + 2 H + - * R 2 H 2

(12.1.10)

This sequence is an ECC (or EC2C) reaction. The products of one-electron transfers can
also rearrange (see Figure 12.1.1, path 3), because a bond is weakened. For similar reasons, electron transfers can also lead to loss of ligands, substitution, or isomerization in
coordination compounds. Examples include
+

[Cp*Re(CO) 2 (p-N 2 C 6 H 4 OMe)] + e -> [Cp*Re(CO)2(p-N2C6H4OMe)] -^
Cp*Re(CO) 2 N 2 + C 6 H 4 OMe
in

n

Co Br 2 en 2 + 6H 2 O + e -> Co (H 2 O) 6 + 2Br~ + 2en

(12.1.11a)
(12.1.11b)

5

(where Cp* = 77 -СзМе5 and en = ethylenediamine). In many cases, the product formed
in the following reaction can undergo an additional electron-transfer reaction, leading to
an ECE sequence, discussed in Section 12.1.2(b).
(c) Catalytic (EC) Reaction
O + ne ±± R

t

(12.1.12)

,
I

R + Z -> O + Y
(12.1.13)
A special type of EC process involves reaction of R with a nonelectroactive species,
Z, in solution to regenerate О (Figure 12.1.1, path 8). If species Z is present in large excess compared to O, then (12.1.13) is a pseudo-first-order reaction. An example of this


12.1 Classification of Reactions

475

scheme is the reduction of Ti(IV) in the presence of a substance that can oxidize Ti(III),
suchasNH 2 OHorClO^:
Ti(IV) + e --> Ti(III)

t

I

СЮ3-, NH 2 OH

(12.1.14)

Since hydroxylamine and chlorate ion can be reduced by Ti(III), they should be reducible
directly at the mercury electrode at the potentials needed to generate Ti(III); however, the

direct reductions do not occur because the rates at the electrode are very small. Other examples of E C reactions are the reduction of Fe(III) in the presence of H 2 O 2 and the oxidation
of I~ in the presence of oxalate. An important E C reaction involves reductions at mercury
where the product can reduce protons or solvent (a so-called "catalytic" hydrogen reaction).

12.1.2.

Reactions with Two or More E Steps
(а) ЕЕ Reaction
A + £?*=> В
В +
£?
E§

(12.1.15)
(12.1.16)

The product of the first electron-transfer reaction may undergo a second electrontransfer step at potentials either more or less negative than that for the first step (Figure
12.1.1, path 2). Of particular interest is the case where the second electron transfer is thermodynamically easier than the first. In this situation, a multielectron overall response
arises. In general, the addition of an electron to a molecule or atom results in a species that
is more difficult to reduce, considering only the electrostatics; that is, R~ is more difficult
to reduce than R. Similarly, R + is more difficult to oxidize than R. In the gas phase, the
ionization potential (IP) for R + is almost always much higher, by 5 eV or more, than that
for R (e.g., Zn, IPj = 9.4 eV and IP 2 = 18 eV). Thus one would generally expect a species
to undergo step wise one-electron reduction or oxidation reactions. However, if one or
more electron-transfer steps involve significant structural change such as a rearrangement
or a large change in solvation, then the standard potentials of the electron-transfer reactions
can shift to promote the second electron transfer and produce an apparent multielectron
wave. Thus one can argue that the oxidation of Zn proceeds in an apparent two-electron
reaction to Zn 2 + , because this species is much more highly solvated and stabilized than

Zn + . Apparent multielectron-transfer reactions are also observed when there are several
identical groups on a molecule that do not interact with one another, such as,
R-(CH2)6-R + 2e ?± [ ^R-(CH2)6-R^

(12.1.17)

where R = 9-anthryl or 4-nitrophenyl. This same principle holds in the reduction or
oxidation of many polymers, such as (CH2-CHR')X> where R' is an electroactive group
like ferrocene. The electrochemical response appears as a single wave, representing an
x-electron EEE . . . (or xE) reaction. This result contrasts sharply with the multistep
electron-transfer behavior found with fullerene (C 6 0 ), which shows six resolved, oneelectron cathodic waves (an overall 6E sequence), where each step is thermodynamically
more difficult than the preceding one (11).
Whenever more than one electron-transfer reaction occurs in the overall sequence,
such as in an ЕЕ reaction sequence, one must consider the possibility of solution-phase
electron-transfer reactions, such as for (12.1.15) and (12.1.16), the disproportionation of B:
2B<=±A + C
or the reverse reaction (the comproportionation of A and C).

(12.1.18)


476

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
(h) ECE Reaction

Ox + n ^ ^ R ! • £?
(
Rl

O2 +

(12Л.20)

i?O2

tt2e^R2

(12.1.19)

° E\

(12.1.21)

When the product of the following chemical reaction is electroactive at potentials of the
Oi/Ri electron-transfer reaction, a second electron-transfer reaction can take place (Figure
12.1.1, paths 5 and 7). An example of this scheme is the reduction of a halonitroaromatic
compound in an aprotic medium (e.g., in liquid ammonia or A/,Af-dimethylformamide),
where the reaction proceeds as follows (X = Cl, Br, I):
XC 6 H 4 NO 2 + e <± XC 6 H 4 NO 2 T
XC 6 H 4 NO 2 T -» X " + -C 6 H 4 NO 2
•C6H4NO2 + e ^± TC 6 H 4 NO 2
TC 6 H 4 NO 2 + H + -> C 6 H 5 NO 2

(12.1.22)
(12.1.23)
(12.1.24)
(12.1.25)

Since protonation follows the second electron-transfer step, this is actually an ECEC reaction sequence. The assignment of such a sequence is not as straightforward as it might

first appear, however. Because species O 2 is more easily reduced than Oi (i.e., E\ < £ 2 ),
species R\ diffusing away from the electrode is capable of reducing O2. Thus, for the example mentioned above, the following reaction can occur:
XC6H4NO2T + -C6H4NO2 ^± XC6H4NO2 + TC6H4NO2

(12.1.26)

It is not simple to distinguish between this case, where the second electron transfer occurs
in bulk solution [sometimes called the DISP mechanism], and the true ECE case where
the second electron transfer occurs at the electrode surface (12).
Another variety of this type of reaction scheme, which we will designate ECE', occurs when the reduction of O 2 takes place at more negative potentials than O\ (i.e.,
E\ > £ 2 ). In this case the reaction observed at the first reduction wave is an EC process;
however, the second reduction wave will be characteristic of an ECE reaction.

(c) ECE Reaction
This case occurs when the product of a chemical reaction following the reduction of A at
the electrode is oxidized at potentials where A is reduced (hence the backward arrow on
the second E) (13):
A + e^±A~
A"->B~
B--ez±B

(12.1.27)
(12.1.28)
(12.1.29)

Charges are explicitly indicated here only to emphasize the different directions of the two
E steps. As with ЕЕ and ECE reactions, one needs to include the possibility of a solution
electron-transfer reaction also taking place:
A~ + B ^ ± B ~ + A

(12.1.30)

An example of this case is the reduction of Сг(СН)^~ in 2 M NaOH (in the absence of dissolved CN"). In this case, reduction of the kinetically inert Cr(CN)|" (A) to the labile
Cr(CN)£~ (A") causes rapid loss of CN~ to form Cr(OH)n(H2O)^Ij; (B~) which is immediately oxidized to Сг(ОН)п(Н2О)б1„ (В). Additional reactions of this type include isomerizations and other structural changes that occur on electron transfer.


12.1 Classification of Reactions

477

Note that the overall reaction for this scheme is simply A —> B, with no net transfer
of electrons. Thus, at a suitable potential, the electrode accelerates a reaction that presumably would proceed slowly without the electrode. An interesting extension of this mechanism is the electron-transfer-catalyzed substitution reaction (equivalent to the organic
chemist's S R N 1 mechanism) (7, 14):
RX + e«±RX~

(12.1.31)

R X " ^ R + X~
R + Nu~ -> RNu"
RNu~ - e+± RNu

(12.1.32)
(12.1.33)
(12.1.34)

along with the occurrence of the solution phase reaction
RX + RNu" -» RX" + RNu

(12.1.35)

Again the overall reaction does not involve any net transfer of electrons and is equivalent
to the simple substitution reaction
RX + Nu" -» RNu + X "

(12.1.36)

(d) Square Schemes
Two electron-transfer reactions can be coupled to two chemical reactions in a cyclic pattern called a "square scheme" (15):
A ' + e

IT

0

<=> ° A

It

(12.1.37)

В ' + e ° +± ° В
This mechanism often occurs when there is a structural change on reduction, such as a
cis-trans isomerization. An example of this scheme for an oxidation reaction is found in
the
electrochemistry
of
ds-W(CO) 2 (DPE) 2
[where
DPE
=

1,2Z?/5'(diphenylphosphino)ethane], where the cis-form (C) on oxidation yields C + , which
isomerizes to the trans species, T + . More complex reaction mechanisms result from coupling several square schemes together to form meshes (e.g., ladders ox fences) (8).
(e) Other Reaction Patterns
Under the subheadings above, we have considered some of the more important general
electrode reactions involving coupled homogeneous and heterogeneous steps. A great variety of other reaction schemes is possible. Many can be treated as combinations or variants of the general cases that we delineated above. In all schemes, the observed behavior
depends on the reversibility or irreversibility of the electron transfer and the homogeneous reactions (i.e., the importance of the back reactions). For example, subclasses of EC
reactions can be distinguished depending on whether the reactions are reversible (r), quasireversible (q), or irreversible (i); thus we can differentiate E r C r , E r Q, E q Q, etc. There
has been much interest and success since the 1960s in the elucidation of complex reaction
schemes by application of electrochemical methods, along with identification of intermediates by spectroscopic techniques (see Chapter 17) and judicious variation of solvent and
reaction conditions. A complex example is the reduction of nitrobenzene (PhNO2) to
phenylhydroxylamine in liquid ammonia in the presence of proton donor (ROH), which
has been analyzed as an EECCEEC process (16):
PhNO 2 + e <=* PhNO 2 T
PhNO 2 T + e<± PhNO^"

(12.1.38)
(12.1.39)


478

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
PhNO^" + ROH <=t PhNOH" + RO"
О
PhNOH~ -» PhNO (nitrosobenzene) + OH~
PhNO + e i=± PhNOT
PhNO" + e <=t PhNO 2 "
H
PhNO 2 " + 2ROH -> PhNOH + 2RO"

12.1.3

(12.1.40)
(12.1.41)
(12.1.42)
(12.1.43)
(12.1.44)

Effects of Coupled Reactions on Measurements
In general, a perturbing chemical reaction can affect the primary measured parameter of
the forward reaction (e.g., the limiting or peak current in voltammetry), the forward reaction's characteristic potentials (e.g., Ey2 or £ p ), and the reversal parameters (e.g., ipjipc).
A qualitative understanding of how different types of reactions affect the different parameters of a given technique is useful in choosing reaction schemes as candidates for more
detailed analysis in a given situation. We assume here that the characteristics of the unperturbed electrode reaction (O + ne ^ R) have already been determined, so we focus
now on how the perturbing coupled reaction affects these characteristics.
(a) Effect on Primary Forward Parameters (i, Q, т,.,.)
The extent to which the limiting current for the forward reaction (O + ne —> R) is affected
by the coupled reaction depends on the reaction scheme. For an EC reaction, the flux of О
is not changed very much, so that any index of that flux, such as the limiting current (or
Qf or Tf), is only slightly perturbed. On the other hand, the limiting current for a catalytic
reaction (EC) will be increased, because О is continuously replenished by the reaction.
The extent of this increase will depend on the duration (or characteristic time) of the experiment. For very short-duration experiments, this limiting current will be near that for
the unperturbed reaction, since the regenerating reaction will not have sufficient time to
regenerate О in appreciable amounts. For longer-duration experiments, the limiting current will be larger than in the unperturbed case. Similar considerations apply to the ECE
mechanism, except that for longer-duration experiments an upper bound for the limiting
current is reached.
(b) Effect on Characteristic Potentials (Ет,
Ep,...)
The manner in which the potential of the forward reaction is affected depends not only on
the type of coupled reaction and experimental duration, but also on the reversibility of
electron transfer. Consider the E r Q case; that is, a reversible (nernstian) electrode reaction

followed by an irreversible chemical reaction:
O +

rce^R^X

(12.1.45)

The potential of the electrode during the experiment is given by the Nernst equation:
^

<12Л-46>

= 0)
where CQ(X = 0)/CR(X = 0) is determined by the experimental conditions. The effect
of the following reaction is to decrease CR(X = 0) and hence to increase CQ(X = 0)/
CR(X = 0)- Thus the potential will be more positive at any current level than in the
absence of the perturbation, and the wave will shift toward positive potentials. (This case
was considered with steady-state approximations in Section 1.5.2.) For an EC reaction


12.1 Classification of Reactions

479

where the electron transfer is totally irreversible, the following reaction causes no change in
characteristic potential, because the i-E characteristic contains no term involving CR(x = 0).
(c) Effect on Reversal Parameters (граЛ'рс, т г / т ^ . . . )
Reversal results are usually very sensitive to perturbing chemical reactions, For example,
in the E r Q case for cyclic voltammetry, /pa//pc would be 1 in the absence of the perturbation (or in chronopotentiometry rr/Tf would be 1/3). In the presence of the following reaction, /pa/zpc < 1 (or Tr/rf < 1/3) because R is removed from near the electrode surface
by reaction, as well as by diffusion. A similar effect will be found for a catalytic (EC)

reaction, where not only is the reverse contribution decreased, but the forward parameter
is increased.

12.1.4 Time Windows and Accessible Rate Constants
The previous discussion makes it generally clear that the effect of a perturbing reaction on
the measured parameters of an electrode process depends on the extent to which that reaction proceeds during the course of the electrochemical experiment. Consequently, it is
valuable to be able to compare characteristic time for reaction with a characteristic time
for observation. The characteristic lifetime of a chemical reaction with rate constant к can
be taken as t\ = Ilk for a first-order reaction or t'2 = 1/fcQ for a second-order (e.g., dimerization) reaction, where Q is the initial concentration of reactant. One can easily show
that t\ is the time required for the reactant concentration to drop to 37% of its initial value
in a first-order process, and that t2 is the time required for the concentration to drop to
one-half of Q in a second-order process. Each electrochemical method is also described
by a characteristic time, r, which is a measure of the period during which a stable electroactive species can communicate with the electrode. If this characteristic time is small
compared to t\ or t'2, then the experimental response will be largely unperturbed by the
coupled chemistry and will reflect only the heterogeneous electron transfer. If t' << r,
the perturbing reaction will have a large effect.
For a given method with a particular apparatus, a certain range of т (a time window)
exists. The shortest useful r is frequently determined by double-layer charging and instrumental response (which can be governed by the excitation apparatus, the measuring
apparatus, or the cell design). The longest available т is often governed by the onset of
natural convection or changes in the electrode surface. The achievable time window is
different for the different electrochemical techniques (Table 12.1.1). To study a coupled
reaction, one must be able to find conditions that place the reaction's characteristic lifetime within the time window of the chosen technique. Potential step and voltammetric
methods are applicable to reactions that are fast enough to occur within the diffusion
layer near the electrode surface. Thus these methods would be useful for studying firstorder reactions with rate constants of about 0.02 to 107 s" 1 . To reach the upper limit, a
UME would have to be employed, where the characteristic time is governed by the electrode radius, r 0 , and is —Го/D. Rapid reactions can also be studied by ac methods and
with the SECM (where the characteristic time depends on the spacing between the tip
and substrate, d, and is ~d2ID). Coulometric methods are applicable to slower reactions
that take place outside of the diffusion layer. The main strategy adopted in studying a reaction is to systematically change the experimental variable controlling the characteristic
time of the technique (e.g., sweep rate, rotation rate, or applied current) and then to determine how the forward parameters (e.g., iplvmC, irmIC, or ц1а)112С), the characteristic
potentials (e.g., Ep and £1/2), and the reversal parameters (ipjipc> h^b QJQd respond.

The directions and extents of variation of these provide diagnostic criteria for establish-


480

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions

TABLE 12.1.1

Approximate Time Windows for Different Electrochemical Techniques
Usual range of
parameter"

Technique

Time parameter

ас Impedance

1/(0 = (27Г/Г (S)
(/=freq. inHz)

Rotating disk electrode
voltammetry
Scanning electrochemical
microscopy
Ultramicroelectrode at steady state
Chronopotentiometry
Chronoamperometry
Chronocoulometry

Linear scan voltammetry
Cyclic voltammetry
dc Polarography
Coulometry
Macroscale electrolysis

l/a> = (2wf)~ (s)
(/ = rotation rate, in r/s)
SID

2

1

l

Time window
b
(s)

(o= 10" - l O ^ "
c

rllD
t(s)
т (Forward phase duration, s)
т (Forward phase duration, s)
RTIFv (s)
RT/Fv (s)
*max (drop time, s)

t (electrolysis duration, s)
t (electrolysis duration, s)

0)

= 30-1000 s"

1

1

5

10" -100
3

10" -0.03
7

d = 10 nm-10 fim

10" -0.1

r0 = 0.1-25 jLtm
6
10~ -50 s
7
10~ -10 s
7
10" -10 s

6
v = 0.02-10 V/s
6
v = 0.02-10 V/s
1-5 s
100-3000 s
100-3000 s

10~ -l
6
10" -50
7
10~ -10
7
10~ -10
7
10~ -l
7
10~ -l
1-5
100-3000
100-3000

5

°This represents a readily available range; these limits can often be extended to shorter times under favorable conditions.
For example, potential and current steps in the nanosecond range and potential sweeps above 10 6 V/s have been reported.
*This time window should be considered only approximate. A better description of the conditions under which a chemical
reaction will cause a perturbation of the electrochemical response can be given in terms of the dimensionless rate
parameter, Л, discussed in Section 12.3.

This is sometimes also given in a term that includes the kinematic viscosity, v, and diffusion coefficient, D, (both with
units of cm2/s), such as, (1.61)V/3/(w£>1/3).

ing the type of mechanism involved, and the measurements themselves provide data for
evaluation of the magnitudes of the rate constants of the coupled reactions.

12.2 FUNDAMENTALS OF THEORY FOR VOLTAMMETRIC
AND CHRONOPOTENTIOMETRIC METHODS
12.2.1 Basic Principles
The theoretical treatments for the different voltammetric methods (e.g., polarography, linear sweep voltammetry, and chronopotentiometry) and the various kinetic cases generally
follow the procedures described previously. The appropriate partial differential equations
(usually the diffusion equations modified to take account of the coupled reactions producing or consuming the species of interest) are solved with the requisite initial and boundary
conditions. For example, consider the E r Q reaction scheme:
О + ne ^ R (at electrode)
k

R —> Y (in solution)

(12.2.1)
(12.2.2)

For species O, the unmodified diffusion equation still applies, since О is not involved directly in reaction (12.2.2); thus

°L дхг J

(12.2.3)


12.2 Fundamentals of Theory for Voltammetric and Chronopotentiometric Methods 4 481
For the species R, however, Fick's law must be modified because, at a given location in

solution, R is removed not only by diffusion but also by the first-order chemical reaction.
Since the rate of change of the concentration of R caused by the chemical reaction is

[dCR(x, Ol
Мй~П

=-*CR(*.O

(12.2.4)

Ol

L
Jchem.rxn.
the appropriate equation for species R is
dt
The initial conditions, assuming only О is initially present, are, as usual,
Co(x, 0) = Cg

CR(x, 0) = 0

(12.2.6)

The usual boundary conditions for the flux at the electrode surface

D
and as x - ^ °°,

[dCo(x, Ol

=

r
°HH-O "Ч~^~1-о

lim cg(jc, 0 = Q) ° °

x—» 0 0

l i m

x—>°°

CR(x, 0 = 0

(12 2 7)

--

(12.2.8)

also apply. The sixth needed boundary condition depends on the particular technique and
the reversibility of the electron-transfer reaction (12.2.1), just as described in Chapters
5-10. For example, for a potential step experiment to the limiting cathodic current region,
Q)(0, 0 = 0- F ° r a s ^ p to an arbitrary potential, assuming (12.2.1) is reversible, the requisite condition is [see (5.4.6)]
_2—!— = 0 = exp —(E

— E?)

(12.2.9)

and, for chronopotentiometry,
(12.2.10)
Note that equations need not be written for species Y, since its concentration does not
affect the current or the potential. If reaction (12.2.2) were reversible, however, the concentration of species Y would appear in the equation for dCR(x, t)/dt, and an equation for
dCy(;t, t)/dt and initial and boundary conditions for Y would have to be supplied (see
entry 3 in Table 12.2.1). Generally, then, the equations for the theoretical treatment are
deduced in a straightforward manner from the diffusion equation and the appropriate homogeneous reaction rate equations. In Table 12.2.1, equations for several different reaction schemes and the appropriate boundary conditions for potential-step, potential-sweep,
and current-step techniques are given.
Solutions of the equations appropriate for a given reaction scheme are obtained by (a)
approximation methods, (b) Laplace transform or related techniques to yield closed form
solutions, (c) digital simulation methods, and (d) other numerical methods. Approximation methods, such as those based on the reaction layer concept as described in Section
1.5.2, are sometimes useful in showing the dependence of measured variables on various
parameters and in yielding rough values of rate constants. With the availability of digital
simulation methods, they are now rarely used. Laplace transform techniques can sometimes be employed with first-order coupled chemical reactions, often with judicious substitutions and combinations of the equations. Only rarely can closed-form solutions be
obtained, such as in Section 12.2.2. For most reaction schemes, direct numerical solution


5.

ErC2[

4. ErC{

3. ETCT

2. CTE{

1. CrEr

Case

2R-Xx

dt

•=

DR

(as above)

dC0
d2C0
-r=
D
—
—
o
dt
dx2

О + ne ^± R

C R = CY = 0

(equation for CY not required)

kbCY

(as above)

(Note 1

R->Y

dx1

2

^? С С

^ Со

С о + C Y = С*

с о /с у - /^

О + ne ^± R

(as above)

dt

dC-v
d Cv
• = DY — j - + kfCR - kbCY
(Note 1)
Ol

dx
(as above, with k b = 0)
(as above)

kb

О + яе ^± R

О + лг -> R

О + ne ^± R

Reactions

Diffusion equations
(all x and t)

General initial and
semi-infinite
boundary conditions
(t = 0 andx-> oo)

TABLE 12.2.1 Modified Diffusion Equations and Boundary Conditions for
Several Different Coupled Homogeneous Chemical Reactions in Voltammetry

o(*7T)

'o

k C

(as CrEr above)

(as CrEr above)

(as CrEr above)

(Note 3)

D

=

• = es(t)

(Note 2)

Cn

еЫ

Potential step and
sweep boundary
conditions (at x = 0)

a b o v e

)

nFADo

(as above)

(as above)

(as above)

(as

dx

Current step
boundary conditions (at x = 0)


2

дСт
fa
U

~ R\

д Ст
2

2

^f^Rl

дС0
д С0
— — = Do — — + kfCR
d t
2
dx
^CR
дCR

C

~ V2^\4

(Note 5)

~Q

СО2

(as CTEr above)

For potential step to potential E,

в = exp jjf(E ~ E0') \ ' ' S(f) = 1

J

/vi

J l ( ^ - £ 0 ' ) ° * ^> = ^— v

|_ /vi

[nF
1
For potential step, в} = exp ~^(E ~ Ef)

There are two flux balance equations analogous to that in Note \{a), one written for each of the redox couples.
VnF
,1
(Note 5) For potential sweep, 0j = exp -^f{E-x ~ Ef) ' ' ^'pertains to Oj + ще ^± Rj

(Note 4)

u

R1 ~ O2 ~ <-R2 ~~

(Note 4)

C

[Note l(a)]

CQ = Co

For potential sweep, в = expl -=^=,(E-. — E°) I * * .У(/) = expf — r = yf I ' ° E: = initial potential
L
J
\
/

у = scan rate

(дС0\
(дСк\
(dCY\
(a)Do[-^)
- -DR I—^
• "(/,)Z) Y —^
=0
\ dx Jx=0
\ dx Jx=0
\ dx Jx=0

*f

О + пе^К

(Note 3) For sweep from E\ at scan rate v or for step to E\ with у = 0, к' = ко exp

(Note 2)

(Notel)

6. EYC[

/дСп

(as above)

484 • Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
of the differential equations or digital simulation, especially when higher-order reactions
are involved, is the method of choice. Commercial computer programs, such as, DigiSim
(17), ELSIM (18), and CVSIM (19), are available for some methods. A brief discussion
of digital simulation with coupled homogeneous reactions is given in Section B.3.
For rotating disk electrode studies, the appropriate kinetic terms are added to the convective-diffusion equations. For ac techniques, the equations in Table 12.2.1 are solved
for CQ(0, t) and CR(0, i) in a form obtained by convolution [equivalent to (10.2.14) and
(10.2.15) for the appropriate case]. Substitution of the current expression, (10.2.3), then
yields the final relationships.

12.2.2

Solution of the ErCi Scheme in Current
Step (Chronopotentiometric) Methods
To illustrate the analytical approach to solving problems involving coupled chemical reactions and the treatment of the theoretical results, we consider the E r Q scheme for a constant-current excitation. Although chronopotentiometric methods are now rarely used in
practice to study such reactions, this is a good technique for illustrating the Laplace transform method, the nature of the changes caused by the coupled reaction, and the "zone diagram" approach for visualizing the effects of changes in time scale and rate constant.
Analogous principles apply for cyclic voltammetry, where only numerical solutions are
available. The equations governing the E r Q case are given as entry 4 in Table 12.2.1 and
were discussed in Section 12.2.1.
(a) Forward Reaction
The equation for Co(x, t) is the same as that in the absence of the following reaction, that
is, (8.2.13):

Thus, the forward transition time, r f [when CQ(0, t) = 0], is unperturbed, and /Tf1/2/Co is
a constant given by (8.2.14). However, CR(x, t) is affected by the following reaction, and
this causes the E— t curve to be different. The Laplace transform of (12.2.5) with initial
condition (12.2.6) yields
sCR(x, s) = DR\

^ — ^ - kCR(x, s)

(12.2.12)

Solution of this equation with the boundary condition lim CR(x, s) = 0 gives

CR(x,s) = CR(0, s) exp - f e ^ )

x\

(12.2.14)

With the boundary condition

this finally yields
1/2

^/2

^

(12.2.16)


12.2 Fundamentals of Theory for Voltammetric and Chronopotentiometric Methods

485

For the forward step at constant current,
Ks) = |

(12.2.17)

and, from the inverse transform,

(b) Potential-Time Behavior
From (8.2.14) and (12.2.11),

(11220)

i ^

For a reversible electron-transfer reaction, the Nernst equation applies, that is,

The E-t curve is obtained from (12.2.19) to (12.2.21):

ШМ

(T

]/2

1/

\

O

)

nF

)

(12.2.22)

± ^ )

(12.2.23)

/2

rf[(fe) V

ъЩр0)
112

'

e
m

[тг zrf[(kt) }\

n F

This can be written
(12.2.24a)

(12.2.24b)
The term (RT/nF) In E represents the perturbation caused by the chemical reaction.

It is instructive to examine the limiting behavior of S as a function of the dimensionless
m
1/2
l/2
m
product let. For (kt) < 0.1, erf[(fo) ] « 2(kt) /ir (see Section A.3), or E = 1, and the
second term of (12.2.24a) is zero. In other words, the following reaction will have no effect
for sufficiently small к or short times. This condition can be considered to define the pure difm
fusion-controlled zone. As (ki) increases, E becomes smaller, so that the E-t curve is
m
1/2
shifted toward more positive potentials. For example, when (kt) = 1, erf[(A/) ] = 0.84,
S = 0.75 and the wave is shifted 7 mV on the potential axis in a positive direction. When
m
m
112
(kt) > 2, erf [(kt) ] approaches the asymptote of 1, so that E = VKirlkt) . This represents
the limiting region for large к or t, and leads to the E-t equation for the pure kinetic zone:

E = Em

+

( ff)
) iin(^)
n ( ^ ) ++ ((f
f ) ln ( ^ - ,-)

(12.2.25)

Note that this equation is very similar in form to that for a totally irreversible electrontransfer reaction with no coupled chemical reaction, (8.3.6), and predicts a linear variation
1/2
m
of E with ln(r - t ) in this zone. This equation can also be written as

f) i£)+(Б


486

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
or, at t = т/4, Е = £ T /4, where
ET/4 =

%)и4и) + 1£)ыь)

Ey2

(12.2.27)

A plot of ET/4 vs. log (Jet) is shown in Figure 12.2.1. Note that the limiting diffusion
1
and kinetic zones are described by the solid lines, and the dashed curve represents the
exact equation, (12.2.24). Of course, the boundaries of these zones depend on the approximation employed, and the applicability of the limiting equations depends on the accuracy
of the electrochemical measurements. For example, if potential measurements are made to
the nearest 1 mV, the pure kinetic zone will be reached (for n — \ and 25°C) when 25.7
1/2
m
In [erf(fo) ] ^ 1 mV or when (kt) > 1.5.
(c) Current Reversal

The treatment involving current reversal employs the same equations and utilizes the
zero-shift-theorem method, as in Section 8.4.2. Thus, for reversal of current at time t\
(where t\ ^ т\),
i(t) = i ~ Sh(t)(2i)

(12.2.28)

where St (t) is the step function, equal to 0 (t < t\) and 1 (t > ti). Then
(12.2.29)

2 е-'*
and from (12.2.16),

c R (a> ) W—
s

i Г

in
nFADlAs(s
+ k)m

s(s + k)

(12.2.30)
m

\

Г)

0

и
-

30

2 -

- 60

£ т / 4 -£ 1 / 2 3 -

- 90

1

=Y

RT/nF 4
5 ~~

KP
Pure
kinetic
zone

KD

DP
Pure
diffusion
zone

- 120

\
\.
\.

- 150
—
\ .

6

180

7

210
I

I

I

-2

I

I

I

I

I

I

I

I

I

I

2
log (to)

Figure 12.2.1 Variation of £ т/4 with log(At) for chronopotentiometry with the E r Q reaction
scheme. Zone KD is a transition region between the pure diffusion and pure kinetic situations.
'The concept of using zone diagrams to describe behavior within a given mechanistic framework was developed
extensively by Saveant and coworkers. Many examples are covered below. The labels given to the zones (e.g.,
DP, KD, and KP in Figure 12.2.1) are typically derived from their work and are based on their French-language
abbreviations.

12.3 Theory for Transient Voltammetry and Chronopotentiometry i 487

Figure 12.2.2 Variation of T2lt\ with kt\ for
chronopotentiometry with the E r Q reaction scheme.

The inverse transform yields

Jl

kuz

l

kiU

(12.2.31)

At the reverse transition time, t — t\ + т2, CR(0, i) — 0, so that
-|l/2i

=

Let us again examine the limiting behavior. When kt\ is small (diffusion zone),
erf[(&T2)1/2] approaches 2(&T2)1/2/TT-1/2 and erf{[&(^ + r 2 )] 1 / 2 } approaches 2[k(ti +
тг)\т1тгт. Under these conditions (12.2.32) becomes identical to the equation for unperturbed reversal chronopotentiometry and т 2 = ^/3 (see equation 8.4.9). When kt\ is large
(kinetic zone), r 2 approaches 0. The variation of т2/^ with kt\ is shown in Figure 12.2.2
(20-22). Note that kinetic information can be obtained from reversal measurements only
in the intermediate zone (0.1 ^ kt\ ^ 5). The actual value of к is obtained by determining
T2lt\ for different values of t\ and fitting the data to the working curve shown in Figure

12.2.2 (23). Kinetic information can also be obtained in the kinetic zone from the shift of
potential with T\\ however, EP must be known for the electron-transfer step before an actual value of к can be determined.
The treatment given here is typical of those required for other reaction schemes and
techniques. These treatments result in the establishment of (a) diagnostic criteria for distinguishing one mechanistic scheme from another and (b) working curves or tables that
can be used to evaluate rate constants. A survey of results is given in Section 12.3.

12.3 THEORY FOR TRANSIENT VOLTAMMETRY
AND CHRONOPOTENTIOMETRY
We examine here the theoretical treatments for cyclic voltammetry and other transient
techniques (chronoamperometry, chronopotentiometry) for a broad set of reaction
schemes, all of which are introduced in Section 12.2. When one wants to investigate an
electrochemical reaction scheme, one almost always turns first to CV. Although all transient methods can, in principle, explore the same i-E-t space to obtain the needed data,
cyclic voltammetry allows one to see easily the effects of E and t on the current in a single
experiment (Figure 6.1.1). Moreover, correction of the faradaic current for capacitive effects and adsorption is relatively straightforward with CV (compared, for example, to
chronopotentiometry). If capacitive effects are so large that good CV behavior is not obtained, methods such as square wave or pulse voltammetry might be preferable. On the
other hand, CV suffers from the fact that heterogeneous kinetics can affect the observed
response and can complicate the extraction of accurate rate constants for homogeneous re-


488 • Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
actions. Measurements in which the potential is stepped to values where the heterogeneous reaction is mass-transfer controlled, like potential step and rotating disk electrode
methods, do not have this problem. Thus, after a reaction mechanism has been elucidated
and semiquantitative results have been obtained by CV, one often turns to other methods,
like chronocoulometry or RDE methods, to obtain better values of kinetic parameters.
In the sections that follow, we first examine typical cyclic voltammetric responses for
the different reaction mechanisms and then show how consideration of zone diagrams and
theoretical responses can be used to recognize the reaction scheme and extract kinetic parameters. After the discussion of CV, other transient techniques for the same reaction
scheme are discussed. We will not describe results in detail, but rather attempt to show
important limiting cases and equations that are useful for recognizing a given reaction sequence and estimating rate constants.

12.3.1

Preceding Reaction—C r E r

Y^±O

(12.3.1)

O + ne*±R

(12.3.2)

К = к{/къ = CO(JC, O)/CY0t, 0)

(12.3.3)

The behavior of this system depends on the magnitudes of both first-order rate constants, к{ and къ (s" 1 ), and the equilibrium constant, K. It is convenient to describe the reactions in terms of dimensionless parameters related to the rate constants of the reactions
(or the characteristic reaction lifetimes) and the duration of the experiment. For the C r E r
case in the context of a potential step experiment of duration t, these are conveniently expressed by К and A = (fcf + kb)t. For different methods and mechanisms, A is defined in
particular ways, as given in Table 12.3.1.
It is instructive to think about the behavior according to the zone diagram (24) in Figure
12.3.1, which defines how the electrochemical parameters are affected by A and K, and when
the limiting behavior will be observed within a given accuracy. When К is large (e.g., К ^
20), the equilibrium in (12.3.1) lies so far to the right that most of the material exists in the
electroactive form, O. The preceding reaction then has little effect on the electrochemical response, which is essentially the unperturbed nernstian behavior. Similarly, when kf and къ are
small compared to the experimental time scale (e.g., A < 0.1), the preceding reaction cannot
occur appreciably on the experimental time scale. Thus it again has little effect and a nernstian response results, but with the effective initial concentration of O, Co(x, 0), being given by
*, 0) = ^ ^ y

(12.3.4)

TABLE 12.3.1 Dimensionless Parameters for Various Methods
Technique

Time
parameter(s)

Chronoamperometry and polarography

t

(kf

+ kb)t

Linear sweep and cyclic voltammetry

l/v

(*f

+ *b) (RT]

Chronopotentiometry

r

(kf

Rotating disk electrode

a

Or 8/fji=

1.61

I/to

Dimensionless kinetic parameter, A, for
CrEr
ErC[
EXC{

v

kt

[nFj

(kf + kb)/co

a

К

k Cjt
RT\
nF)

*'Cz (RT\

v {nF)

kr

к'ф

k/co

к'С*ш


12.3 Theory for Transient Voltammetry and Chronopotentiometry • 489
2 --

DP

1 ^*v.

DM
4

Kl

^ ^ \ . (?)

S -1
DP
-2
-3
-4

-2

_© ®
1
-1

1

^ ^ « s ^

KP

1
1

1

1

2

3

1
4

rV 1
5

6

log A.

Figure 12.3.1 C r E r reaction diagram with zones for different types of electrochemical behavior as
a function of К and Л (defined in Table 12.3.1). The zones are DP, pure diffusion; DM, diffusion
modified by equilibrium constant of preceding reaction; KP, pure kinetics; and KI, intermediate kinetics. The circled numbers correspond to the boundaries calculated in Section 12.3.l(c). [Adapted
with permission from J.-M. Saveant and E. Vianello, Electrochim. Ada, 8, 905 (1963). Copyright
1963, Pergamon Press PLC]

where
C o (*, 0) + C Y fc 0) = C*

(12.3.5)

When Л is very large, (12.3.1) is so mobile that it can always be considered at equilibrium. In this case, the behavior is yet again nernstian, but the wave is shifted along the potential axis from its unperturbed position by an extent that depends on the magnitude of K,
as discussed in Sections 1.5.1 and 5.4.4. This shift is a thermodynamic effect reflecting
the energy by which species О is stabilized by the equilibrium. The extent of this zone, in
the upper-right portion of Figure 12.3.1, depends on К and Л. When К is small and Л is
large, the reaction is so fast that the reactants can be considered to be at steady-state values within the reaction layer near the electrode surface, and the differential equations governing the system can be solved by setting the derivatives with respect to time equal to
zero (the "reaction-layer treatment"). This is the pure kinetic zone. A more quantitative
description of how the limits of these zones are chosen is given in Section 12.3. l(c).
(a) Linear Sweep and Cyclic Voltammetric Methods
The shape of the i-E curve depends on the values of К and A; that is, on the region of interest in Figure 12.3.1 (24, 25). Curves for this scheme with К = 10~3, kf = 1СГ2 s" 1 ,
kb = 10 s" 1 , and scan rates, v, of 0.01 to 10 V/s (A of 26 to 0.026) are shown in Figure
12.3.2. It is instructive to correlate these curves to the appropriate points in the zone
diagram (Figure 12.3.1). In all cases, log К = - 3 . At the high scan rate (v = 10 V/s,
log A = —1.6), the operating point is in the DP region, and a diffusion-controlled voltammogram with little contribution from the preceding reaction is observed. The behavior appears essentially as an unperturbed reversible reaction with an initial concentration of О
determined by the small equilibrium constant of reaction (12.3.1). As v decreases, one
proceeds horizontally at the log К = —3 level across the zone diagram toward larger log

A values. At v = 1 V/s (log A = —0.6), the operating point is in the KI region. We enter
the KP region at still smaller scan rates, such as v = 0.01 V/s (log A = 1.6). In this region, the response is totally governed by the rate at which О is supplied by the forward re-


490

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
3r-

-400

Figure 12.3.2 Cyclic voltammograms for the C r E r case. A ^ В; В + e ^± C, where EQ/C = 0 V
C* = 1 mM, A = 1 cm 2 , DA = DB = Dc = 10" 5 cm2/s, К = 1СГ3, kf = 10"- 22 - 1
kb = 10 s" 1 , Г = 25°C, and scan rates, v of (1) 10; (2) 1; (3) 0.1; (4) 0.01 V/s.

action, rather than by diffusion. The current attains a steady-state value, indicated by the
cathodic plateau independent of the scan rate.
Since the observed i-E response depends upon K, &f, къ, and v, in addition to Д C,
and n, a full representation of the CV behavior in terms of these parameters would involve
a large number of plots. The results can be given more economically by plotting in terms
of the dimensionless parameters К and Л and by normalizing the current, as shown in Figure 12.3.3.
As discussed in the introduction to this section, the behavior is diffusion-controlled in
regions DP and DM.
In the pure kinetic region (KP), the i-E curve takes on an S-shape (rather than the
usual peak-shape) and the current attains a steady-state value, Z'L, independent of v, given
by
iL = nFADmC*K(kf

+ къ)

т

(12.3.6)

In this region, the half-peak (i.e., half-plateau) potential Ep/2 is given by
Evll = E0' - Q211RTInF - (RTIlnF) In Л

(12.3.7)

The shift of Ер/2 with v is
dEp/2/d In v = RTHnF

(12.3.8)

thus at 25°C a tenfold increase in v causes the reduction peak to shift by 29/n mV in the
positive direction. As v increases (so that Л decreases) and the system enters the zone of
intermediate kinetics (zone KI, Figure 12.3.1), the shift of Ep/2 with scan rate becomes
smaller and finally is independent of v in the diffusion zone (DP). The shift of Ep/2 with
the dimensionless parameter K\ is shown in Figure 12.3.4 (25). A working curve showing
the ratio of the kinetic peak current, /k, to the diffusion-controlled current, /d (attained at
very slow scan rates), has been proposed (25) (Figure 12.3.5) and has been shown to fit
the empirical equation

*d

1.02 + ОЛЛ/КЛ/Т

(12.3.9)

12.3 Theory for Transient Voltammetry and Chronopotentiometry *4 491
4-10- 2

^=10"2

1.5
KXV

If 5-Jjn 4 ^

-

•

—

— - — —

—

0.5

^

—

.

,

.—

a

^

20

10

30

2

2-Ю-

А

I 13-Ю-2
2

.ж

-2

Ю-2

к.2"10"1

й

у-1

— — - — ,

ю6
ю7

д о ^
10

* — — • —

20

30

Figure 12.3.3 Curves of current [plotted
as 7rmx((Tt)/KXm, where x(p-t) is defined
as in (6.2.16)] vs. potential at К = 10~2
(upper) and К = 10~4 (lower), at different
values of Л = (RT/nF) [(kf + kb)/v] shown
on each curve for the CrEr reaction scheme.
[Reprinted with permission from J.-M.
Saveant and E. Vianello, Electrochim.
Acta, 8, 905 (1963). Copyright 1963, Pergamon Press PLC]

In cyclic voltammetry, the anodic portion on the reverse scan is not affected as much
as the forward response by the coupled reaction (Figure 12.3.2). The ratio of /pa//pC (with
/pa measured from the extension of the cathodic curve as described in Section 6.5) increases with increasing scan rate as shown in the working curve in Figure 12.3.6 (25). The

actual i-E curves can be drawn using series solutions or a table given by Nicholson and
Shain (25) or by digital simulation.
(b) Polarographic and Chronoamperometric Methods
The current of interest is that at the limiting current plateau, that is, for CQ(0, i) = 0. For
a planar electrode, assuming equal diffusion coefficients for all species (DQ — Dy = D)


492

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions

0.0

+1.0

-1.0

-2.0

log (KXV2)

Figure 12.3.4 Variation of £ p / 2 with KXm for
the CrEr reaction scheme. Potential axis is
n(Ep/2 ~ Em) ~ (RT/F) ln[K/(l + K)]. u-> shows
direction of increasing scan rate. [Reprinted with
permission from R. S. Nicholson and I. Shain,
Anal. Chem., 36, 706 (1964). Copyright 1964,
American Chemical Society.]

and the chemical equilibrium favoring Y [K «

by (26)
/ = nFAC*Dmk\l2Km
Letting (kfKt)m

1, CY(x, 0) ~ C*], the current is given

exp(kfKt) evfc[(k{Kt)m]

(12.3.10)

= Z, this can be written
/ = nFADmC*rmZexp(Z2)

erfc(Z)

(12.3.11)

Note that this is of the same form as the current for a totally irreversible wave [see
(5.5.27)]. For large values of kb the function Z exp(Z2) erfc(Z) approaches u~m and the
current becomes the diffusion-controlled value, /in the diffusion zone on the right side in Figure 12.3.1. Equation 12.3.11 can be
written
4-=
7T 1/2 Zexp(Z 2 )erfc(Z)
l

(12.3.12)

[Compare with (5.5.28) and Figure 5.5.2.] For small values of the argument Z, exp(Z2)
erfc(Z) « Z, and (12.3.12) yields the same current given in equation 12.3.6 with К «

1,
that is,
i - idirm(kfKf)in

=

nFADmC*(k{K)m

(12.3.13)

which is independent of t and governed by the rate of conversion of Y to O.
These equations hold for polarography as well (within the expanding plane approximation) with t = tm2iX (the drop time) and the area A given by (7.1.3). The approach of
Section 7.2.2 applies. Treatments taking account of spherical diffusion and unequal diffusion coefficients have also been presented (27, 28). Note how the limiting current in po-

2

4

6

1/2 1

(/a r

8

Figure 12.3.5 Working curve of /k//d vs.
(KXm)~l for the C r E r reaction scheme. [Reprinted
with permission from R. S. Nicholson and I. Shain,
Anal Chem., 36, 706 (1964). Copyright 1964,

American Chemical Society.]


12.3 Theory for Transient Voltammetry and Chronopotentiometry

493

Figure 12.3.6 Ratio of anodic to cathodic peak
currents as a function of the kinetic parameters for
the C r E r reaction scheme. [Reprinted with permission from R. S. Nicholson and I. Shain, Anal.
Chem., 36, 706 (1964). Copyright 1964, American
Chemical Society.]
larography varies with r m a x or the height of the mercury column, hC0YY. For large kf (in the
diffusion region), / varies as t]^ or as hlj2n. For small &f, where (12.3.13) applies, / is independent of both tmax and hcon
(c) Chronopotentiometric Methods
The i-r behavior is governed by the following equation (29-31)
\l/2
nFAC*(rrDy
ir 1 / 2 =

(12.3.14)

The first term on the right is the value for the diffusion-controlled reaction, ir\/2.
Using the definition of A (Table 12.3.1), this equation can then be written
irm =

2K\ 1/2

(12.3.15)

or
(12.3.16)

The variation of irm with A 1/2 for several different values of К is shown in Figure
12.3.7. This equation is also useful in examining the limiting behavior of ir112 and defin1/2
ing the different zones of interest as in Figure 12.3.1. Consider A
< 0.4, where
1/2
1/2
1/2
erf(A )/A approaches the limiting value of 2/тг (within ca. 5%) and (12.3.16) yields
l/2
/2
(ir /ir\ ) ~ K/(l + K), which is the diffusion-controlled response corrected by calculat-

Figure 12.3.7 Variation of irm/iTlJ2 with A for various values of К (indicated on curve) for
chronopotentiometric study of the C r E r reaction scheme.


494

Chapter 12. Electrode Reactions with Coupled Homogeneous Chemical Reactions
ing CQ(X, 0) from C*. Thus this condition, or log Л < —0.8, defines the left boundary
ш
1/2
(line 1). For large Л (e.g., Л > 1.4), erf(A ) « 1 and (12.3.16) yields

^

/1 + <шбГ

и»;

(123Л7)
1/2

This condition gives diffusion-controlled behavior when 0.886/ATA ^ 0.05, or log
К = 1.25 - (1/2) log A; this represents the right boundary (line 2). The pure kinetic region is also defined by large A values, this time as К —> 0. One can set the boundary by
1/2
using A > 1.4 (line 3) and the condition that the second term on the right predomi172
nates in (12.3.17). Thus 0.886//ГА > 10 or log К = -1/2 log A - 1.05 (line 4). Note
that the exact locations of these boundaries depend on the levels of approximations
used. Moreover, in this pure kinetic region, (12.3.14) becomes
(12.3.18)
so that a plot of irm vs. i in this region is a straight line of slope -rr1/2/2K(kf + къ)т.
This behavior is evident in the plots shown in Figure 12.3.8.
For simple reversal chronopotentiometry, the ratio of reversal transition time TI to the
forward time T\ is 1/3, just as in the diffusion-controlled case, independent of the rate constants. However, for cyclic chronopotentiometry the transition times for the third (тз) and
subsequent reversals differ from those of the diffusion-controlled case (31).

12.3.2

Preceding Reaction—C r Ei
This scheme is the same as that in Section 12.3.1, except that the electron-transfer reaction
(12.3.2) is totally irreversible and is governed by the charge-transfer parameters a and k°.
The limiting current behavior in chronoamperometric and polarographic methods will not
be perturbed by irreversibility in the electron-transfer reaction; since the potential is stepped
to a value sufficiently beyond the equilibrium value that reaction (12.3.2) proceeds rapidly.
Thus the results will be the same as in Section 12.3.l(b). This case illustrates an important
advantage of chronoamperometric methods: that the potential can be chosen to eliminate
complexities in the analysis of the behavior caused by the heterogeneous electron-transfer

IT ""

mA-s1/2

Figure 12.3.8 Variation of irm
with /, for various values of
(kf + kb) (in s" 1 ). Calculated for

50
/ (mA/cm2)

£ = 0.1, C* = 0.11 mM, and
D = 10" 5 cm2/s. [Reprinted with
permission from P. Delahay and T.
Berzins, J. Am. Chem. Soc, 75,
2486 (1953). Copyright 1953,
American Chemical Society.]


12.3 Theory for Transient Voltammetry and Chronopotentiometry «< 495
step. On the other hand, once the rate constants of the homogeneous reactions have been deduced, potential steps to less extreme potentials can provide information about a and k°.
This requires solution of the more complex problem where the boundary condition for C o (0,
t) is governed by the heterogeneous reaction rate. This problem will not be examined here.
The CrEi case has been treated for linear sweep voltammetry (25). Because of the irreversibility of the electron transfer, no anodic current is observed on the reverse scan and
cyclic voltammetric behavior need not be considered. Typical i-E curves are shown in Figure
12.3.9. The limiting behavior again depends on the magnitude of the kinetic parameter KX\12,
where Aj is the A factor of Table 12.3.1, with n set to an = a for a one-electron "E" step:
(

(12.3.19)

\aF

When Ai is small, the behavior is the same as that of the unperturbed irreversible one-step,
one-electron reaction, as described in Section 6.3, except that the concentration of О is
given by C*[K/(l + K)]. This represents the limiting behavior at high scan rates. For large
Ai and large values of KX\12, the preceding reaction can be considered to be essentially at
equilibrium at all times, and again the i-E behavior becomes that of the unperturbed irreversible case in Section 6.3, with the wave shifted (from the position it would have had
without the preceding reaction) in a negative direction by an amount (RT/aF) ln[K/(l +
K)]. For small values of KX\12 (but with large Ai), the behavior depends on £°, as well as К
and Ai, and the i-E curve no longer shows a peak, but instead has an S-shape with a current
plateau. This is the pure kinetic region where the limiting current becomes independent of
v, as in the case of the C r E r scheme. Under these conditions the current is given by (25)
FAC*Dm K(k{

i =

1 +

\irDmafv(K + 1)1
L k°(kf + kh)m

къ)т

(12.3.20)

\

Nicholson and Shain (25) suggest that for all ranges of KX\12 the kinetic parameters can

be obtained by fitting the kinetic peak (or plateau) current for the CrEi case, i^ to that for
the diffusion-controlled peak current for an irreversible charge transfer, id (equation
6.3.12), by the empirical equation

V

1

1.02 + 0.531/KVxt

(12.3.21)

/ooV
0.4

/

1.0

\
\

0.3

//

0.2

ff/

0.1

nn

*f*^
120

I

3.0

10.0
I

60
0
Potential, mV

L

-60

Figure 12.3.9 Curves of current [plotted as 7rmx(bt),
where x(bt) is defined as in (6.3.6)] vs. potential at different values of {KX\l2yx (shown on curves). The potential
scale is a(E - E°) + (RT/F) \n[(7rDb)m/k°) ~ (RT/F)
ln[K/(l + K)].b = aFv/RT; X{ = (kf + kb)fh. [Reprinted
with permission from R. S. Nicholson and I. Shain, Anal.
Chem., 36, 706 (1964). Copyright 1964, American
Chemical Society.]