Bent linear dependence ln*k* vs. *E* from Fig. 4B is not surprising. It is well-known that such a processed electrochemical responses in form of bent straight lines reveal the complexity of the original response of consecutive process (Ružić, 1970, 1974). It means that system requires another kinetic model with elementary kinetic parameters. Similar situation can be found for some normal pulse polarography data (Sanecki & Lechowicz, 1997), some overlapped voltammetric data (Rusling, 1983) and in our simulation results (Fig. 4, 5).

The Fig. 5 data indicate, that linear, non-linear and various complex app plots of 2D type are only a special cases of 3D dependences. The dependences app on *k*f as well as app on *k*2/*k*1 presented by us in (Sanecki & Skitał, 2002a) can be considered as the members of the same category.

In the light of presented facts, it is clear that variation of app, visualized by 2D and 3D plots is only an intriguing picture with no consequence for electrode kinetics since no change of mechanism was proven. As discussed earlier, the only cause of such plots origin is the fact, that paradigm of eqs. (5)-(6) as well as (7)-(8) (Bard & Faulkner, 2001; Greef et al., 1985) does not contain any chemical step.

In the process of modeling applied here (CV curves in Figs. 2, 4), no linear variability of ETC was either assumed or introduced into the model (i=0.5=const) i.e. normal Butler–Volmer's kinetics was applied. Therefore, the output app dependences (Figs. 3, 4 and 5) cannot contain *el* variability (Sanecki & Skitał, 2002a).

## **Conclusions**

62 Recent Trend in Electrochemical Science and Technology

boundary cases of ECE mechanisms as E or EE type with *k*f from 109 value to *k*f=1098 as

Fig. 4. (A) The influence of *k*f on the shape of CV responses and ln*k*app=*f*(*E*) dependences for simulated ECE mechanism. The interdependence between input CV curves and convolution results is shown. The length of non-linear curve fragment increases when *k*f increases (from the bottom to the top). The rate constants *k*app were calculated by convolution method (eq. (7) and (8)). (B) The influence of *k*2/*k*1 ratio on the shape of ln*k*app=*f*(*E*) dependence obtained from convoluted CV responses for simulated ECE mechanism. Parameters of CV curves are as

0

*E* (V)

20

40

60

80

*i* (A) A

100

Note the increase of complexity of structure from the left to the right. For values of parameters and details see original paper (Skitał & Sanecki, 2009). [Reprinted from *Polish J. Chem.*, Vol. 83, Skitał P. & Sanecki P., The ECE Process in Cyclic Voltammetry. ..., 1127–1138, Copyright

Fig. 5. The result of modeling of ECE process in cyclic voltammetry. An example of 3D plot:

0.46

0.48

0.50

0.52

0.54

app

0.4550 0.4725 0.4900 0.5075 0.5250 0.5425 0.5600 app

3D

on 3D and 2D plots. Scan rate 10 V/s, *k*1=0.03 cm s-1; the values of kinetic parameters are in original paper (Skitał & Sanecki, 2009). For *k*f value less than 0.1 s–1 (not included), where process can be simplified to E one, the respective 3D plot is flat. The 2D plot can be obtained by cutting the 3D one with a respective plane. [Reprinted from *Polish J. Chem.*, Vol. 83, Skitał P. & Sanecki P., The ECE Process in Cyclic Voltammetry. ..., 1127–1138, Copyright (2009),

app, determined by convolution method, as a function of log(*k*2/*k*1) and *k*f

2=0.5, *v*= 1 Vs-1. The *k*2/*k*1 and *k*f values are presented on the plot.

*v* = 10 V s-1

5 4 3 2 1 0 -1 -2

*k*f = 0.1 *k*f = 10 *k*f = 50 *k*f = 200 *k*f = 500 *k*f = 5000 *k*f = 30 000

2D

log(*k*2 /*k*1 )

*k*2 /*k*1 =1000 *k*f =5000

*E* (V)

B

*k*2 /*k*1 =10 *k*f =5000

*k*2 /*k*1 =0.333 *k*f =5000

ln*k*app

infinity, similarly as in Fig. 3 data.

CV response

*k*2 /*k*1 = 10

plots determined from convoluted CV responses

ln*k*app

follows: *k*1=0.03 cm s-1,

0

5

10

general parameter

0.1 10 50<PHONE_NUMBER> 30000

0

*v* = 10 V s-1

0

1

2

426

log(*k*2

4

3.

426

4.

.426

3

/*k*1

)

.426

2

.426

1

with permission from Polish Chemical Society.]

1=

(2009), with permission from Polish Chemical Society.]

*k*f (*s*

)

*k*f = 5 *k*f = 50

*k*f = 180000 *k*f = 1098

*k*f = 5000

## **6. The application of EC, ECE and ECE-ECE models with potential dependent transfer coefficient to selected electrode processes**

The classical Butler-Volmer electrochemical kinetics with a priori assumed invariability of (/*E*=0) provides approach which adequately describes the majority of observed

Mathematical Modeling of Electrode Processes

2-nitro-2-methylpropane Eq. (9):

presented in Table 2 and in Fig. 6 (

by statistical data (Sanecki & Skitał, 2007a).

permission of The Electrochemical Society.]

Kaczmarski, 1999) (Δ

difference between the

that a model without Δ

2 and

**(reduction of 1,3-benzenedisulfonyl difluoride)**

BDF), the ECE kinetic model without

**variability case**

of

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 65

Corrigan and Evans (Corrigan & Evans, 1980) showed that inclusion of variability in form

We obtained the result similar to that but for more extended scan rate range on GC and Hg electrodes in DMF solution, where the effect turned out to be more distinct. Our results are

are in full agreement with the results obtained by Corrigan and Evans. In the light of presented facts the need of introducing variability into kinetic model of the considered

/*E*, V–1 0.37± 0.03 0.40 0.42±0.02 0.42±0.03

results with analogical data published by Corrigan and Evans. [The Table reproduced by

A comparison of two identical reduction processes located at different potentials may be considered as a classical model for determination of variability. The approach was first

determined by the comparison of two –SO2F group reduction on Hg electrode (Sanecki &

method – the two different values of

The results (Fig. 7) indicate that for electroreduction of 1,3-benzenedisulfonyl difluoride (1,3-

4 (note also the difference in a shape of the peaks). Fig. 7, the left column plots indicate

's for both reduction stages i.e. between

estimation). At present, the same system was investigated using the GCE electrode.

**6.2 Comparison of two ECE-ECE reduction steps with large potential difference**

/*E* =0 and

/*E* = 0.42 V–1 was obtained as kinetic parameter by estimation for both electrodes.

/*E* parameter into kinetic model of one-electron quasireversible reduction process of

2 2 2

(Corrigan & Evans, 1980) (Sanecki & Skitał, 2007a) Hg Pt GCE Hg

e (9)

/*E* 0 for comparison) where the value

/*E* included is confirmed

variability is not possible. There is a significant

1 and

variability provides crossing experimental and theoretical plots. Both

variability

variability was

were optimized by

3 as well as between

/*E* values,

**6.1 Repetition of the important literature example of the observed elementary**

1 1

*k k t tt*

,

scan rate range 10-100 Vs–1 0.1-100 Vs–1 medium AN, TBAP AN, TEAP DMF, 0.3 M TBAP

Table 2. CV reduction of 2-methyl-2-nitropropane. The comparison of our

presented by Pearce and Geiger (Pierce & Geiger,1992). In turn, the

kinetic model E EC

The data shown in Table 2 indicate that our results, including the obtained

leads to better fit between experimental and theoretical responses.

system is clear (Fig. 6). The better fit obtained for model with

electrode processes. The problem of variability appears in some special cases of kinetic analysis e.g. during investigation of multi-stage electron transfer separated by a large potential interval, for series of substituted compounds with the same reactive group as well as for experimental long distance current-potential dependences of quasi-reversible processes (Corrigan & Evans, 1980; Matsuda & Tamamushi, 1979; Sanecki, 2001; Sanecki & Kaczmarski, 1999; Sanecki & Skitał, 2007b). The first case of the type was described by Pierce and Geiger as *alpha kinetic discrimination* (Pierce & Geiger, 1992). Currently, extended and still growing application of variety of solid electrodes (McDermott et al., 1992), including chemically modified and semiconductor electrodes, makes the problem of diversification and variability of transfer coefficient much more important than ever before.

The significance of the discussed problem is confirmed by simple calculations, based on relevant literature data (Angell & Dickinson, 1972; Garreau et al., 1979; Savéant & Tessier, 1982; Savéant & Tessier et al., 1977), which leads to the conclusion that ETC variability effect is usually 5-10 times stronger than the double layer correction effect (see also Sanecki & Skitał, 2007a).

Transfer coefficient is one of the fundamental concepts in electrode kinetics. The extensive theoretical support, including all modern theories of charge transfer, predicts the potential variability of ETC (Dogonadze, 1971; Hush, 1958; Levich, 1966; Marcus, 1956; Marcus, 1960; Marcus, 1977). The variation of ETC for reduction and oxidation processes of individual compounds and ions has been systematically investigated by Savéant's group, Corrigan and Evans and others (Angell & Dickinson, 1972; Bindra et al., 1975; Corrigan & Evans, 1980; Dogonadze, 1971; Frumkin, 1932; Garreau et al., 1979; Hush, 1958; Levich, 1966; Marcus, 1956; Marcus, 1960; Marcus, 1977; Matsuda & Tamamushi, 1979; McDermott et al., 1992; Nagy et al., 1988; Parsons & Passeron, 1966; Rifi & Covitz, 1974; Samec & Weber, 1973; Sanecki, 1986; Savéant & Tessier, 1975; Savéant & Tessier, 1982; Tyma & Weaver, 1980). The studies have been focused on the experimental detection of the ETC variability. The research goal has been achieved with minimal further continuation. In refs. (Angell & Dickinson, 1972; Bindra et al., 1975; Corrigan & Evans, 1980; Garreau et al., 1979; Matsuda & Tamamushi, 1979; McDermott et al., 1992; Nagy et al., 1988; Parsons & Passeron, 1966; Samec & Weber, 1973; Savéant & Tessier, 1975; Savéant & Tessier, 1982; Tyma & Weaver, 1980) one can easily find or calculate the /*E* 0 values for the one electron processes and for resolved two electron reduction (Sanecki, 2001; Sanecki & Kaczmarski, 1999). The investigation of a problem of the potential dependent has been continued by several authors (Chidsey, 1991; Finklea, 2001a, 2001b; Finklea & Haddox, 2001; Finklea et al., 2001; Haddox & Finklea, 2003; Miller, 1995; Smalley et al., 1995). What is more, well known analogy of and to *B* and *B* of the BrØnsted parameters (Albery, 1975; Frumkin, 1932; Rifi & Covitz, 1974), the ETC has also been interpreted as a Hammett-type reaction constant (Sanecki, 1986)*.* This interpretation, due to similarity of the homogenous and heterogeneous kinetics, allows the more flexible approach to the ETC value and is easy to understand by non electrochemists (Rifi & Covitz, 1974).

At present, the majority of electrochemists do not apply the ETC variability or consider it meaningless. Therefore, there is a need to indicate the special cases in which the effect may be of importance (Sanecki & Skitał, 2007a, 2007b). Our choice of respective experimental cases was focused on the following four examples (points 6.1-6.4). In turn, point 6.5 is devoted to comparison of IRu drop and ∂/∂*E* parameter influence on CV responses.

analysis e.g. during investigation of multi-stage electron transfer separated by a large potential interval, for series of substituted compounds with the same reactive group as well as for experimental long distance current-potential dependences of quasi-reversible processes (Corrigan & Evans, 1980; Matsuda & Tamamushi, 1979; Sanecki, 2001; Sanecki & Kaczmarski, 1999; Sanecki & Skitał, 2007b). The first case of the type was described by Pierce and Geiger as *alpha kinetic discrimination* (Pierce & Geiger, 1992). Currently, extended and still growing application of variety of solid electrodes (McDermott et al., 1992), including chemically modified and semiconductor electrodes, makes the problem of diversification

The significance of the discussed problem is confirmed by simple calculations, based on relevant literature data (Angell & Dickinson, 1972; Garreau et al., 1979; Savéant & Tessier, 1982; Savéant & Tessier et al., 1977), which leads to the conclusion that ETC variability effect is usually 5-10 times stronger than the double layer correction effect (see also Sanecki &

Transfer coefficient is one of the fundamental concepts in electrode kinetics. The extensive theoretical support, including all modern theories of charge transfer, predicts the potential variability of ETC (Dogonadze, 1971; Hush, 1958; Levich, 1966; Marcus, 1956; Marcus, 1960; Marcus, 1977). The variation of ETC for reduction and oxidation processes of individual compounds and ions has been systematically investigated by Savéant's group, Corrigan and Evans and others (Angell & Dickinson, 1972; Bindra et al., 1975; Corrigan & Evans, 1980; Dogonadze, 1971; Frumkin, 1932; Garreau et al., 1979; Hush, 1958; Levich, 1966; Marcus, 1956; Marcus, 1960; Marcus, 1977; Matsuda & Tamamushi, 1979; McDermott et al., 1992; Nagy et al., 1988; Parsons & Passeron, 1966; Rifi & Covitz, 1974; Samec & Weber, 1973; Sanecki, 1986; Savéant & Tessier, 1975; Savéant & Tessier, 1982; Tyma & Weaver, 1980). The studies have been focused on the experimental detection of the ETC variability. The research goal has been achieved with minimal further continuation. In refs. (Angell & Dickinson, 1972; Bindra et al., 1975; Corrigan & Evans, 1980; Garreau et al., 1979; Matsuda & Tamamushi, 1979; McDermott et al., 1992; Nagy et al., 1988; Parsons & Passeron, 1966; Samec & Weber, 1973; Savéant & Tessier, 1975; Savéant & Tessier, 1982; Tyma & Weaver,

for resolved two electron reduction (Sanecki, 2001; Sanecki & Kaczmarski, 1999). The

authors (Chidsey, 1991; Finklea, 2001a, 2001b; Finklea & Haddox, 2001; Finklea et al., 2001; Haddox & Finklea, 2003; Miller, 1995; Smalley et al., 1995). What is more, well known

Rifi & Covitz, 1974), the ETC has also been interpreted as a Hammett-type reaction constant

 (Sanecki, 1986)*.* This interpretation, due to similarity of the homogenous and heterogeneous kinetics, allows the more flexible approach to the ETC value and is easy to

At present, the majority of electrochemists do not apply the ETC variability or consider it meaningless. Therefore, there is a need to indicate the special cases in which the effect may be of importance (Sanecki & Skitał, 2007a, 2007b). Our choice of respective experimental cases was focused on the following four examples (points 6.1-6.4). In turn, point 6.5 is

variability appears in some special cases of kinetic

/*E* 0 values for the one electron processes and

has been continued by several

*B* of the BrØnsted parameters (Albery, 1975; Frumkin, 1932;

/∂*E* parameter influence on CV responses.

and variability of transfer coefficient much more important than ever before.

electrode processes. The problem of

1980) one can easily find or calculate the

devoted to comparison of IRu drop and ∂

investigation of a problem of the potential dependent

understand by non electrochemists (Rifi & Covitz, 1974).

Skitał, 2007a).

analogy of

 and to *B* and

#### **6.1 Repetition of the important literature example of the observed elementary variability case**

Corrigan and Evans (Corrigan & Evans, 1980) showed that inclusion of variability in form of /*E* parameter into kinetic model of one-electron quasireversible reduction process of 2-nitro-2-methylpropane Eq. (9):

$$\text{t-BuNO}\_2 + \text{e}^- \xrightarrow{k\_{\text{s}\_1} \text{at}\_{\text{s}\_1}} \text{t-BuNO}\_2 \text{•} \xrightarrow{k\_{\text{d}\_1}} \text{t-Bu}^\bullet + \text{NO}\_2^- \tag{9}$$

leads to better fit between experimental and theoretical responses.

We obtained the result similar to that but for more extended scan rate range on GC and Hg electrodes in DMF solution, where the effect turned out to be more distinct. Our results are presented in Table 2 and in Fig. 6 (/*E* =0 and /*E* 0 for comparison) where the value /*E* = 0.42 V–1 was obtained as kinetic parameter by estimation for both electrodes.

The data shown in Table 2 indicate that our results, including the obtained /*E* values, are in full agreement with the results obtained by Corrigan and Evans. In the light of presented facts the need of introducing variability into kinetic model of the considered system is clear (Fig. 6). The better fit obtained for model with /*E* included is confirmed by statistical data (Sanecki & Skitał, 2007a).

Table 2. CV reduction of 2-methyl-2-nitropropane. The comparison of our variability results with analogical data published by Corrigan and Evans. [The Table reproduced by permission of The Electrochemical Society.]

## **6.2 Comparison of two ECE-ECE reduction steps with large potential difference (reduction of 1,3-benzenedisulfonyl difluoride)**

A comparison of two identical reduction processes located at different potentials may be considered as a classical model for determination of variability. The approach was first presented by Pearce and Geiger (Pierce & Geiger,1992). In turn, the variability was determined by the comparison of two –SO2F group reduction on Hg electrode (Sanecki & Kaczmarski, 1999) (Δ method – the two different values of were optimized by estimation). At present, the same system was investigated using the GCE electrode.

The results (Fig. 7) indicate that for electroreduction of 1,3-benzenedisulfonyl difluoride (1,3- BDF), the ECE kinetic model without variability is not possible. There is a significant difference between the 's for both reduction stages i.e. between 1 and 3 as well as between 2 and 4 (note also the difference in a shape of the peaks). Fig. 7, the left column plots indicate that a model without Δvariability provides crossing experimental and theoretical plots. Both

Mathematical Modeling of Electrode Processes

equivalent (Sanecki & Skitał, 2007a).

**reductions. The substituent effect**

*I* 2 *I* 1

> *I* 2 *I* 1

0.5 Vs-1

electroreduction (Fig. 7).

*I* / A

*I* / A

0

0 -100 -200 -300

1=2=3=

*k*4=0.0065±0.0005 cms–1,

(D)– with included

(F)– with included

1=2=3=

*E* / V (vs. SCE)

*I*

*E* / V (vs. SCE)

*I* 4 *I* 3

(B) 100 Vs-1

*I* 4 *I* 3

(A)

*I*

variability (

variability (

identical reducible group) compounds are compared (Fig. 8).

difference in peak potential is under 0.1 V, the difference in

1=2, 3=4; ∆

0

0 -100 -200 -300

*I* / A

*I* / A

*=f(E)* variability along the CV response is shown on the plot. The scale for

vertical axis. [Reproduced by permission of The Electrochemical Society.]

kinetic parameters are quite reasonable but due to the small difference in

The effect of substituent on elementary

Δ*E*p≈0.1V to show the limit of possibility of

The two substituted compounds values of ∂

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 67

introduction lead to almost the same dependences which suggests that the models are

When the two consecutive steps of two-electron process are not naturally split and *k*2<*k*1 or *k*2≈*k*1, it is possible to determine the elementary transfer coefficients with satisfactory accuracy. For such cases the difference of peak potentials is seen on calculated elementary steps parameters (Sanecki & Skitał, 2007a). When *k*2>*k*1, the respective confidence intervals for second electrochemical step are wider than those for the first one in case of 1,3-BDF

Fig. 7. Comparison of experimental (red points) and theoretically obtained curves (black lines) of the CV electroreduction of 1,3-BDF in 0.3 M TBAP/DMF on GCE. Estimated

*I* 2 *I* 1

parameters for ECE-ECE model are: (A) & (B)– with variability not included (/E=0 and

*I I* 3 2 *I* 1

(C) 0.5 Vs-1

*I*

*E* / V (vs. SCE)

*E* / V (vs. SCE)

*I* 4 *I* 3

*I*

(D) 100 Vs-1

*I* 4

2001; Sanecki & Kaczmarski, 1999). Now we compare the reduction of two compounds with

charged with significant error. It suggests that for two substituted compounds, when

4=0.51±0.01, *k*f1=1100±200 s–1, *k*f2=8000±900 s–1; (C) &

0 -100 -200 -300

0

*I* / A *I*

Estart

*I* / A

Estart

/E≠0) in (Sanecki & Skitał, 2007a). The estimated linear

/∆*E*≠0) in (Sanecki & Skitał, 2007a); (E) &

was considered in our previous papers (Sanecki,

variability detection when two similar (with

/∂*E* obtained by comparison of respective

 = *f* (*E*)

 = *f* (*E*)

*I*

*I* 2 *I* 1

values is unrecognizable and

is the right

value must be

4): *k*1=0.017±0.001 cms–1, *k*2=0.064±0.01 cms–1, *k*3=0.041±0.003 cms–1,

(E) 0.5 Vs-1

*E* / V (vs. SCE)

*E* / V (vs. SCE)

*I* 4 *I* 3

(F) 100 Vs-1

*I* 4 *I* 3 *I* 2 *I* 1

variability

0.0

0.0

0.3

0.6

0.3

0.6

included and not included is insignificant here. Moreover, the two ways of

**6.4 Comparison of the** *p***-toluenesulfonyl fluoride and benzenesulfonyl fluoride**

Fig. 6. CV reduction of 2-methyl-2-nitropropane. The comparison of two EC models: without (top row) and with (bottom row) variability included. For parameters and details see original paper (Sanecki & Skitał, 2007a). [Reproduced by permission of The Electrochemical Society.]

of the approaches, i.e. introducing of different values (Δ variability) and continuous variability (/*E*), gave very similar results. The small potential difference and respective confidence intervals do not allow determination of the difference between *'*s within each of the two considered reduction stages (two-electron reduction of SO2F). Moreover, the comparison of various electron transfers steps: anion radical creating (1) and radical reducing (2) is not appropriate for variability determination.

The results and comparison of the two applied methods of variability determination are presented in Table 3.

Table 3. The comparison of variability values for the electroreduction of 1,3-BDF. [The Table reproduced by permission of The Electrochemical Society.]

## **6.3 Comparison of two reduction steps within ECE process with a small potential difference. Reduction of** *p***-toluenesulfonyl fluoride**

The comparison of experimental CV responses for ECE reduction of *p*-toluenesulfonyl fluoride (TSF) (dissociative electron transfer process) with respective Δ variability model and variable with ∂/∂*E* parameter included model is presented in paper (Sanecki & Skitał, 2007a). Due to small potential difference between the steps (56 mV at *v*=1 Vs–1), where the steps are not naturally resolved, the difference between models with variability

Fig. 6. CV reduction of 2-methyl-2-nitropropane. The comparison of two EC models:

 = *f* (*E*)

10 Vs-1

*E*start

variability determination.

the two considered reduction stages (two-electron reduction of SO2F). Moreover, the comparison of various electron transfers steps: anion radical creating (1) and radical

values (Δ

/*E*), gave very similar results. The small potential difference and respective

*E* / V (vs. SCE)

200 0 -200 -400

200 0 -200 -400

0.3

0.6

2 1

*E* / V (vs. SCE)

0.9

**Hg electrode** (Sanecki & Skitał,

Δ*E*p=0.749 V

/∂*E* = 0.51

/∂*E* = 0.41

variability values for the electroreduction of 1,3-BDF. [The

/∂*E* parameter included model is presented in paper (Sanecki &

2007a)

± 0.09

± 0.07

∂

∂

see original paper (Sanecki & Skitał, 2007a). [Reproduced by permission of The

10 Vs-1

*I* / A

*I* / A

50 0 -50 -100 -150

50 0 -50 -100 -150

0.3

0.6

0.9

 1 Vs-1 *E* / V (vs. SCE)

confidence intervals do not allow determination of the difference between

**GC electrode** (Sanecki & Skitał,

Δ*E*p=0.705 V

/∂*E* = 0.20

/∂E = 0.20 ± 0.03 V-1

**6.3 Comparison of two reduction steps within ECE process with a small potential**

The comparison of experimental CV responses for ECE reduction of *p*-toluenesulfonyl

Skitał, 2007a). Due to small potential difference between the steps (56 mV at *v*=1 Vs–1),

2007a)

± 0.07

The results and comparison of the two applied methods of

**difference. Reduction of** *p***-toluenesulfonyl fluoride**

∂

∂

Table reproduced by permission of The Electrochemical Society.]

fluoride (TSF) (dissociative electron transfer process) with respective Δ

where the steps are not naturally resolved, the difference between models with

variability included. For parameters and details

100 Vs-1

 = *f* (*E*)

*I* / A

*I* / A

variability) and continuous

100 Vs-1

*E*start

variability determination are

**Hg electrode**  (Sanecki & Kaczmarski, 1999)

/∂*E* = 0.44

∂

± 0.04

variability model

variability

0.3

*E* / V (vs. SCE)

2 1

*E* / V (vs. SCE)

0.6

0.9

*'*s within each of

without (top row) and with (bottom row)

*E*start

of the approaches, i.e. introducing of different

*E* / V (vs. SCE)

2 1

 **variability**

Electrochemical Society.]

1 Vs-1

 = *f* (*E*)

*I* / A

*I* / A

presented in Table 3.

**The method of**

**determination**

Estimation of the ∂

values

parameter

and variable

reducing (2) is not appropriate for

/∂E

Estimation of the different

Table 3. The comparison of

with ∂

variability (

20 0

20 0

## **6.4 Comparison of the** *p***-toluenesulfonyl fluoride and benzenesulfonyl fluoride reductions. The substituent effect**

When the two consecutive steps of two-electron process are not naturally split and *k*2<*k*1 or *k*2≈*k*1, it is possible to determine the elementary transfer coefficients with satisfactory accuracy. For such cases the difference of peak potentials is seen on calculated elementary steps parameters (Sanecki & Skitał, 2007a). When *k*2>*k*1, the respective confidence intervals for second electrochemical step are wider than those for the first one in case of 1,3-BDF electroreduction (Fig. 7).

Fig. 7. Comparison of experimental (red points) and theoretically obtained curves (black lines) of the CV electroreduction of 1,3-BDF in 0.3 M TBAP/DMF on GCE. Estimated parameters for ECE-ECE model are: (A) & (B)– with variability not included (/E=0 and 1=2=3=4): *k*1=0.017±0.001 cms–1, *k*2=0.064±0.01 cms–1, *k*3=0.041±0.003 cms–1, *k*4=0.0065±0.0005 cms–1, 1=2=3=4=0.51±0.01, *k*f1=1100±200 s–1, *k*f2=8000±900 s–1; (C) & (D)– with included variability (1=2, 3=4; ∆/∆*E*≠0) in (Sanecki & Skitał, 2007a); (E) & (F)– with included variability (/E≠0) in (Sanecki & Skitał, 2007a). The estimated linear *=f(E)* variability along the CV response is shown on the plot. The scale for is the right vertical axis. [Reproduced by permission of The Electrochemical Society.]

The effect of substituent on elementary was considered in our previous papers (Sanecki, 2001; Sanecki & Kaczmarski, 1999). Now we compare the reduction of two compounds with Δ*E*p≈0.1V to show the limit of possibility of variability detection when two similar (with identical reducible group) compounds are compared (Fig. 8).

The two substituted compounds values of ∂/∂*E* obtained by comparison of respective kinetic parameters are quite reasonable but due to the small difference in value must be charged with significant error. It suggests that for two substituted compounds, when difference in peak potential is under 0.1 V, the difference in values is unrecognizable and

Mathematical Modeling of Electrode Processes

determination of exact values not only

Hence, the low scan rate data are safe for

and the peak height with small influence on *E*p.

case of undercompensation), the obtained ∂

**7. Alkyl iodides electroreduction and**

are constant with *v*.

distinct for potential dependent

1996, 1997; Jakobsen et al., 1999)).

(the case of overcompensation), the obtained ∂

height and passes *E*p towards negative potentials. The

compensation (not recommended). The

compounds, were compared.

and start.

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 69

The IRu compensation problems have been resolved by the method of estimation with the use of theoretical peaks (Sanecki et al., 2003). The series of data in 0.1–100 V/s scan rate range for EC, ECE, and ECE–ECE mechanisms, which correspond to the investigated

At first, series of theoretical EC responses were generated, for input set of kinetic parameters. The obtained responses were treated as pseudo-experimental ones and the output kinetic parameters were evaluated from them by a standard estimation procedure. The obtained results indicate that generally, for all kinetic models, IRu attenuates the

/*E,* but all input values of kinetic parameters. For

/*E* effect decreases both the slope

) increases with *v*. For IRu<0

) decreases with v and its

/*E* determination, even in case of lack of IRu

/*E*≠0 effect, however, is generally observed as a

/*E*

/∂*E* = 0.01–0.04 V–1.

effect,

/∂*E* effect.

/∂*E*

/∂*E*

instance in the range 5–100 Vs–1 Ru=100 Ω produces non existing ∂

EC model, the influence of IRu effect is expressed on *k*1 to a higher degree, and less on

The IRu effect depends on a scan rate value. It is practically invisible at low *v* up to 1–2 Vs-1 (Fig. 9). The effect increases gradually with *v* and at 25, 50 and 100 V/s is clearly visible, for

visible gap in fits for all scan rate peaks (examples: Fig. 9). The influence on the peak shape is different for both effects. The IRu effect decreases the slope without change of the peak

/∂*E* (and itself

 **variability**

/∂*E* (and itself

parameter due to small sensitivity of their kinetic systems.

An overcorrection done on IRu affected responses is not able to provide a variable

uncompensated IRu, which can be erroneously considered as comprising an ∂

since after IRu correction CV peaks are steeper which is exactly opposite to observed ∂

effect on CV response (Fig. 9). On the contrary, the real risk is the CV response charged with

Other important results obtained from extensive simulations are as follows. For IRu>0 (the

The recognized rules can be treated as diagnostic criteria. For the other ECE and ECE–ECE models under consideration, the above rules and conclusions hold, although they are less

The process of the electroreduction of organic iodides is an example of two-electron reductive cleavage reaction and a subject of extensive investigation in the organic electrochemistry (Andrieux et al., 1979; Caldwell & Hacobian, 1968a, 1968b; Colichman & Kung Liu, 1954; Hush, 1957; Hussey & Diefenderfer, 1967; Jaworski et al., 1992; Mairanovskii, 1969; Mairanovskii & Rubinskaya, 1972; Mairanovskii et al., 1975; Pause et al., 2000; Peters, 1991; Sanecki, 2001; Sawyer et al., 1995; Sease et al., 1968; Stackelberg, 1949;). The established mechanism is relevant for the cleavage of carbon–halogen bond and has been successfully applied also for the other categories of compounds (e.g. (Andrieux et al.,

negative values can be obtained. For IRu=0 (the proper compensation), the obtained ∂

variability is not a problem to consider. The similar picture as presented in Fig. 8 and the respective calculation results for two substituted compounds as well as for the series of iodobenzenes examples of variability can be found in paper (Sanecki, 2001).

Fig. 8. The comparison of CV reduction of benzenesulfonyl fluoride (BSF) (red line) and TSF (blue line) in 0.3M TBAP/DMF on: (A) GC and (B) Hg electrodes. The curves on plots (A) and (B) were shifted to compare the slopes: plot (A) on GCE by Δ*E*= –0.122 V, plot (B) on Hg by Δ*E*= –0.098 V. Note the small difference of slopes which corresponds to a difference of . The same effect is seen on estimated values: 1=0.67±0.02 for BSF and 1=0.62±0.02 for TSF both on GCE. On Hg electrode for BSF 1=0.67±0.02, for TSF 1=0.62±0.02. It results in ∂1/∂E = 0.41±V–1 for GCE and ∂1/∂*E*=0.61 V-1 for Hg electrode. For other parameters see original paper (Sanecki & Skitał, 2007a). [Reproduced by permission of The Electrochemical Society.]

#### **6.5 Comparison of IRu drop and ∂/∂***E* **parameter influence on CV responses**

The influence of IRu effect and non-constant transfer coefficient effect on current response shape can be similar and there is a real risk of making a mistake of detecting a non existing ∂/∂*E* effect. Therefore, the comparison of both effects by simulations is presented in Fig. 9.

Fig. 9. Theoretical CV current responses for EC kinetic model. The influence of uncompensated resistance Ru and included variability for 1Vs-1 and 100Vs-1. The scan rate value is the one difference between left and right plot. Curves (1),(2),(3),(4): variable model with /*E=*0 and Ru=0Ω, 100Ω, -100Ω, 200Ω, respectively; curve (5): variable model with /*E=*0.4 and Ru=0Ω. For other parameters see the paper (Sanecki & Skitał, 2007a). [Reproduced by permission of The Electrochemical Society.]

variability is not a problem to consider. The similar picture as presented in Fig. 8 and the respective calculation results for two substituted compounds as well as for the series of

A

Fig. 8. The comparison of CV reduction of benzenesulfonyl fluoride (BSF) (red line) and TSF (blue line) in 0.3M TBAP/DMF on: (A) GC and (B) Hg electrodes. The curves on plots (A) and (B) were shifted to compare the slopes: plot (A) on GCE by Δ*E*= –0.122 V, plot (B) on Hg by Δ*E*= –0.098 V. Note the small difference of slopes which corresponds to a difference of

original paper (Sanecki & Skitał, 2007a). [Reproduced by permission of The Electrochemical

The influence of IRu effect and non-constant transfer coefficient effect on current response shape can be similar and there is a real risk of making a mistake of detecting a non existing

/∂*E* effect. Therefore, the comparison of both effects by simulations is presented in Fig. 9.

1=0.67±0.02, for TSF

0

1=0.67±0.02 for BSF and

**/∂***E* **parameter influence on CV responses**

*I* / A

0.5 Vs-1

1/∂*E*=0.61 V-1 for Hg electrode. For other parameters see

3

variability for 1Vs-1 and 100Vs-1. The scan rate

100 Vs-1

5

1=0.62±0.02. It results in

TSF

BSF

values:

Fig. 9. Theoretical CV current responses for EC kinetic model. The influence of

*E* / V (vs. SCE)

200

0

value is the one difference between left and right plot. Curves (1),(2),(3),(4): variable

/*E=*0 and Ru=0Ω, 100Ω, -100Ω, 200Ω, respectively; curve (5): variable

/*E=*0.4 and Ru=0Ω. For other parameters see the paper (Sanecki & Skitał, 2007a).

TSF

BSF

*E* / V (vs. SCE)

.

1=0.62±0.02 for TSF

*E* / V (vs. SCE)

B

model

model with

*E* / V (vs. SCE)

5

2 1 4

variability can be found in paper (Sanecki, 2001).

*I* / A

iodobenzenes examples of

*I* / A

0

The same effect is seen on estimated

1/∂E = 0.41±V–1 for GCE and ∂

**6.5 Comparison of IRu drop and ∂**

uncompensated resistance Ru and included

1 Vs-1

5

*I* / A 1

3

5

2

4

[Reproduced by permission of The Electrochemical Society.]

∂

∂

with  20

0

Society.]

both on GCE. On Hg electrode for BSF

0.5 Vs-1

The IRu compensation problems have been resolved by the method of estimation with the use of theoretical peaks (Sanecki et al., 2003). The series of data in 0.1–100 V/s scan rate range for EC, ECE, and ECE–ECE mechanisms, which correspond to the investigated compounds, were compared.

At first, series of theoretical EC responses were generated, for input set of kinetic parameters. The obtained responses were treated as pseudo-experimental ones and the output kinetic parameters were evaluated from them by a standard estimation procedure. The obtained results indicate that generally, for all kinetic models, IRu attenuates the determination of exact values not only /*E,* but all input values of kinetic parameters. For EC model, the influence of IRu effect is expressed on *k*1 to a higher degree, and less on /*E* and start.

The IRu effect depends on a scan rate value. It is practically invisible at low *v* up to 1–2 Vs-1 (Fig. 9). The effect increases gradually with *v* and at 25, 50 and 100 V/s is clearly visible, for instance in the range 5–100 Vs–1 Ru=100 Ω produces non existing ∂/∂*E* = 0.01–0.04 V–1. Hence, the low scan rate data are safe for /*E* determination, even in case of lack of IRu compensation (not recommended). The /*E*≠0 effect, however, is generally observed as a visible gap in fits for all scan rate peaks (examples: Fig. 9). The influence on the peak shape is different for both effects. The IRu effect decreases the slope without change of the peak height and passes *E*p towards negative potentials. The /*E* effect decreases both the slope and the peak height with small influence on *E*p.

An overcorrection done on IRu affected responses is not able to provide a variable effect, since after IRu correction CV peaks are steeper which is exactly opposite to observed ∂/∂*E* effect on CV response (Fig. 9). On the contrary, the real risk is the CV response charged with uncompensated IRu, which can be erroneously considered as comprising an ∂/∂*E* effect.

Other important results obtained from extensive simulations are as follows. For IRu>0 (the case of undercompensation), the obtained ∂/∂*E* (and itself ) increases with *v*. For IRu<0 (the case of overcompensation), the obtained ∂/∂*E* (and itself ) decreases with v and its negative values can be obtained. For IRu=0 (the proper compensation), the obtained ∂/∂*E* are constant with *v*.

The recognized rules can be treated as diagnostic criteria. For the other ECE and ECE–ECE models under consideration, the above rules and conclusions hold, although they are less distinct for potential dependent parameter due to small sensitivity of their kinetic systems.

#### **7. Alkyl iodides electroreduction and variability**

The process of the electroreduction of organic iodides is an example of two-electron reductive cleavage reaction and a subject of extensive investigation in the organic electrochemistry (Andrieux et al., 1979; Caldwell & Hacobian, 1968a, 1968b; Colichman & Kung Liu, 1954; Hush, 1957; Hussey & Diefenderfer, 1967; Jaworski et al., 1992; Mairanovskii, 1969; Mairanovskii & Rubinskaya, 1972; Mairanovskii et al., 1975; Pause et al., 2000; Peters, 1991; Sanecki, 2001; Sawyer et al., 1995; Sease et al., 1968; Stackelberg, 1949;). The established mechanism is relevant for the cleavage of carbon–halogen bond and has been successfully applied also for the other categories of compounds (e.g. (Andrieux et al., 1996, 1997; Jakobsen et al., 1999)).

Mathematical Modeling of Electrode Processes

**8. Theoretical discovery of** *isoalpha points*

**/***E* **or non-continuous change of**

treated as electrochemical analog of *isosbestic* point (Sanecki & Skitał, 2008).

Fig. 11. The discovery of *isoalpha points* as a result of including of

0.8; 1.0; 1.2; 1.4 increasing from the top to the bottom curves; (B):

current response. The one-electron CV responses for Eirr process (A):

(*E*iap, *I* iap)

*E*iap *E* / V *E*0

of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

*E* / V

**particular points on CV curve**

groups. 3. The determined ∂

The

different

0

5

10

convoluted current

*I* / A

15

20

**8.1 Parameter**

inclusion of the

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 71

by a small difference of reduction potentials caused by weak substituent effect of alkyl

times lower than that obtained in literature for aromatic compound electroreduction on Hg (Sanecki & Kaczmarski, 1999). For the other elementary processes the values in the range 0.174-0.4 V–1 were obtained (Savéant & Tessier, 1982). On the other hand, the respective values obtained for monoalkyl iodides are higher (Sanecki & Skitał, 2007b). 4. The respective comparison of elementary kinetic parameters for two-stage diiodomethane electroreduction on GCE indicates the presence of alpha kinetic discrimination with

/*E*=0.15 V-1 according to Pearce and Geiger nomenclature (Pierce & Geiger, 1992).

 variability was firstly used as a tool to reveal the particular properties of the investigated system in (Sanecki & Skitał, 2008). It was shown that respective simulations lead to discovery of the new meaningful phenomenon called *isoalpha* effect (Fig. 11,12). The

model of E, EC, ECE, ECE-ECE and of ErCErC–ErCErC mechanisms results in the change of the single CV curve into the set of CV curves comprising specific intersection point(s), called

 independent current-potential point or *isoalpha point (iap).* The place of *iap* appearance depends on *v* and/or *k* value (Fig. 12). The *isoalpha point* (*E*iap, *I*iap) can be, to some extend,

A

values (B) into kinetic model of one electron reduction step on the theoretical

convoluted current

decreasing from the top to the bottom curves. The insets show the convoluted form of main plot with revealed *isoalpha point*. Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models

 **(∆**

/*E*≠0 parameter or non-continuous variability of (Δ) into kinetic

*E* / V

**) as a source of revealing**

(*E*iap, *I* iap)

/*E≠*0 parameter (A) or

= 1; 0.9; 0.8; 0.7; 0.6; 0.5

/*E* = 0; 0.2; 0.4; 0.6;

*E* / V

B

/∂*E* value for diiodomethane is equal to 0.20 V–1 and is about two

The influence of carbon chain length and the number of iodine atoms in the molecule was examined. The two-electron reductive cleavage with iodine elimination was numerically resolved into one-electron consecutive steps. The mechanism of the process was discussed in the frames of the two mathematical models: EC(C)E (with radical dimerisation) for one stage electroreduction of RI as well as ECE-EC(C)E kinetic model for two stage electroreduction of RI2. In both models transfer coefficient variability was included. For electroreduction of diiodomethane the discrimination of elementary between two reduction stages was determined (Sanecki & Skitał, 2007b). The variability was determined using estimation procedure in which ∂/∂E was treated as other kinetic parameters and included into kinetic model (Corrigan & Evans,1980; McDermott et al.,1992).

The experimental facts and generally accepted relevant literature data suggested EC(E)E model with *k*f of very high value (concerted process). Additionally, the estimation results denoted inclusion of ETC variability into the model. The obtained values of ∂/∂E parameter for monoalkyl iodides were in range between 0.33–0.36 V–1 with maximum confidence interval equal to ±0.09. The respective value for diodomethane was 0.20±0.04 V–1 (Fig. 10).

Fig. 10. (A) The CV electroreduction of five alkyl iodides in 0.3 M TBAP in DMF on GCE. The concentration of substrate was 4 mM. Scan rate *v* = 5 Vs–1. Other details are specified in the original paper (Sanecki & Skitał, 2007b). (B) The CV electroreduction of CH2I2 in 0.3 M TBAP/DMF solution on GCE. The normalized CV current responses *Iv*–0.5. The *I*p*v*–0.5 values for both reduction stages are shown as respective points on maximum current or, for clarity, separately in the sub-window. Note the different slope of kinetic *I*p*v*–0.5 vs *v* dependence in subwindow. [Reprinted from *Electrochim. Acta*, Vol. 52, Sanecki, P. & Skitał P., The electroreduction of alkyl iodides ..., 4675-4684, Copyright (2007), with permission from Elsevier.]

### **Conclusions**

The influence of carbon chain length and the number of iodine atoms in the molecule was examined. The two-electron reductive cleavage with iodine elimination was numerically resolved into one-electron consecutive steps. The mechanism of the process was discussed in the frames of the two mathematical models: EC(C)E (with radical dimerisation) for one stage electroreduction of RI as well as ECE-EC(C)E kinetic model for two stage electroreduction of RI2. In both models transfer coefficient variability was included. For

parameters and included into kinetic model (Corrigan & Evans,1980; McDermott et al.,1992). The experimental facts and generally accepted relevant literature data suggested EC(E)E model with *k*f of very high value (concerted process). Additionally, the estimation results denoted inclusion of ETC variability into the model. The obtained values of ∂

parameter for monoalkyl iodides were in range between 0.33–0.36 V–1 with maximum confidence interval equal to ±0.09. The respective value for diodomethane was 0.20±0.04 V–1

Fig. 10. (A) The CV electroreduction of five alkyl iodides in 0.3 M TBAP in DMF on GCE. The concentration of substrate was 4 mM. Scan rate *v* = 5 Vs–1. Other details are specified in the original paper (Sanecki & Skitał, 2007b). (B) The CV electroreduction of CH2I2 in 0.3 M TBAP/DMF solution on GCE. The normalized CV current responses *Iv*–0.5. The *I*p*v*–0.5 values for both reduction stages are shown as respective points on maximum current or, for clarity, separately in the sub-window. Note the different slope of kinetic *I*p*v*–0.5 vs *v* dependence in

0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30

*E* / V (vs. SCE)

A

subwindow. [Reprinted from *Electrochim. Acta*, Vol. 52, Sanecki, P. & Skitał P., The electroreduction of alkyl iodides ..., 4675-4684, Copyright (2007), with permission from

1. The electroreduction of mono alkyl iodides and diiodomethane on glassy carbon electrode can be described by EC(C)E and ECE-EC(C)E numerical model, respectively. The second

electrochemical step of the EC(C)E process is generally slower then the first one. 2. The comparison of CV responses and the values of determined kinetic parameters for a sub-series of five mono iodides indicates that their slope and elementary transfer coefficient do not vary with a change of the alkyl substituent. The fact can be explained

/∂E was treated as other kinetic

*I*

v

p

 CH -0.15 2 I 2 - 0.1 V/s CH2 I 2 - 0.5 V/s CH2 I 2 - 1 V/s CH2 I 2 - 5 V/s CH2 I 2 - 25 V/s CH2 I 2 - 50 V/s

*I* v -0.5

0.1 1 10

*v* / Vs-1

1st peak 2nd peak

between two

variability was

*E* / V (vs. SCE)

/∂E

B

electroreduction of diiodomethane the discrimination of elementary

reduction stages was determined (Sanecki & Skitał, 2007b). The

determined using estimation procedure in which ∂

CH3 I C2 H5 I C3 H7 I C4 H9 I C5 H11I

(Fig. 10).

Elsevier.]

**Conclusions**

0

*I* / A

by a small difference of reduction potentials caused by weak substituent effect of alkyl groups.

## **8. Theoretical discovery of** *isoalpha points*

#### **8.1 Parameter /***E* **or non-continuous change of (∆) as a source of revealing particular points on CV curve**

The variability was firstly used as a tool to reveal the particular properties of the investigated system in (Sanecki & Skitał, 2008). It was shown that respective simulations lead to discovery of the new meaningful phenomenon called *isoalpha* effect (Fig. 11,12). The inclusion of the /*E*≠0 parameter or non-continuous variability of (Δ) into kinetic model of E, EC, ECE, ECE-ECE and of ErCErC–ErCErC mechanisms results in the change of the single CV curve into the set of CV curves comprising specific intersection point(s), called independent current-potential point or *isoalpha point (iap).* The place of *iap* appearance depends on *v* and/or *k* value (Fig. 12). The *isoalpha point* (*E*iap, *I*iap) can be, to some extend, treated as electrochemical analog of *isosbestic* point (Sanecki & Skitał, 2008).

Fig. 11. The discovery of *isoalpha points* as a result of including of /*E≠*0 parameter (A) or different values (B) into kinetic model of one electron reduction step on the theoretical current response. The one-electron CV responses for Eirr process (A): /*E* = 0; 0.2; 0.4; 0.6; 0.8; 1.0; 1.2; 1.4 increasing from the top to the bottom curves; (B): = 1; 0.9; 0.8; 0.7; 0.6; 0.5 decreasing from the top to the bottom curves. The insets show the convoluted form of main plot with revealed *isoalpha point*. Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

Mathematical Modeling of Electrode Processes

values of

of

value.

13A,B.

As a result of including of the ETC variation either through

point in which cathodic current is independent of

distinguishable electroactive species interconvert.

high *k*f value e.g. *k*f = 1×1010 s–1 (Fig. 12 D,E,F).

maintains all parameters constant except for

towards more negative potentials becoming flatter as

well established theory and provides its completion.

**8.2 ECE and ECEC mechanisms with included**

/*E*≠0 parameter or different

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 73

result is similar to isosbestic point (Berlett et al., 2000; IUAPAC, 1997; Nakajima et al., 2004). The system resembling *iap* and isosbestic point, called *isopotential point* has been described in electrochemistry (Edens et al., 1991; Eichhorn & Speiser, 1994; Fitch & Edens, 1989). Its appearance is associated with the linked multi-sweep experiments when two

The model for totally irreversible process applied with the use of *ESTYM\_PDE* was described in paper by Nicholson and Shain (Nicholson & Shain, 1964) and is given in Bard and Faulkner's monograph (Bard & Faulkner, 2001). It is not included in *DigiSim®* software. In *DigiSim®,* however, the Eirr mechanism response was easily generated as ErC with very

A question arises why other authors dealing with the effect of potential-dependent

voltammetry (Arun & Sangaranarayanan, 2004; Bieniasz & Speiser, 1998a; Bond & Mahon, 1997; Delahay, 1953; Nahir et al., 1994; Tender et al., 1994; Weber & Creager, 1994) did not observe so far such a phenomenon as *iap*, even if Marcus' λ parameter was discussed instead

. Most probably there are three reasons. Firstly, the range of the phenomenon is not very wide e.g. it appears for 0.05 > *k* > 1×10–4, at *v* =1Vs–1 (for *k*=0.05, Fig. 12 A,D the *iap* is diffused). At lower *k* values *iap* occurs but is invisible unless a magnification method is applied (Fig. 12 C,F). Secondly, is the phenomenon to be observed requires at least three (or more) superimposed curves and that was not fulfilled by the other authors. Even if one

The third reason in form of examples was shown in Fig. 12 C,F. The data indicate that it is not possible to find *iap* without magnification of the plot and therefore some authors could not notice such a hidden effect. The observed *iap* phenomenon is not contradictory to the

The similar, although less distinct *isoalpha points* are obtained for ErC process when

influence of scan rate and *E*0 value on position *iap* is discussed in (Sanecki & Skitał, 2008). It is seen that *iap* moves down when scan rate increases and its value becomes closer to *E*0

The kinetics should concern the steps and elementary parameters, which have clear physical meaning as it was performed earlier for Eirr process (Fig. 11,12). On the other hand, data applied for ECE process, allowed to generate dependences corresponding to those from Fig. 2 and may be important for ECE process kinetics. Examples of *isoalpha points* for both general (two-electron) peaks and elementary (one-electron) sub-peaks are presented in Fig.

 **variability**

or not visible *iap* can be found in paper by Delahay (Fig. 2 in (Delahay, 1953)).

, a single curve becomes the set of curves which intersect in one characteristic

/*E* parameter or by different

in

(Fig. 11 A and B, respectively). The

*,* no *iap* is observed and CV peak moves

values mode is introduced into the model (Fig. 12).The

decreases. Examples of the lacking

Parameter /*E* into respective kinetic equation was previously applied by Corrigan and Evans (Corrigan & Evans, 1980) as well as McDermott and coauthors (McDermott et al., 1992). The application of the non-continuous ETC variability parameter (Δ), in which each curve is generated for different value was applied (Sanecki & Kaczmarski, 1999; Sanecki & Skitał, 2007a, 2007b; Sanecki et al., 2006b).

Theoretical curves visualized in Fig. 11A were obtained by the algorithm which includes /*E* in the range 0 - 1.4 V-1. Despite the fact, that experimental values of /*E* do not exceed 0.5 V-1, the application of wider range of /E results in more extended set of curves. For the same reason =1 as a starting value was chosen (Fig. 11B). When the variability starts at =0.5, the intersection point (*iap*) remains the same but the plot is less convincing.

Fig. 12. The effect of including of /*E≠*0 parameter (plots (A), (B), (C)) or different values (plots (D), (E), (F)) into theoretical one-electron kinetic model Eirr as well as the influence of *k* value on isoalpha point current response. Plots (A), (B), (C) with /*E* = 0; 0.2; 0.4; 0.6; 0.8; 1.0; 1.2; 1.4. (A) *k* =0.05, (B) *k* =0.005, (C) *k* =0.00001. Plots (D), (E), (F) with = 1; 0.9; 0.8; 0.7; 0.6; 0.5; /*E* = 0, (D) *k* =0.05, (E) *k* =0.005, (F) *k* =0.00001. The calculations were done with the use of both *ESTYM\_PDE* (black lines, Eirr) and *DigiSim®* (red lines, ErC with *k*f=1×1010 s–1). Notice, that the both programs provide identical current responses (plots (D), (E), (F)). The insets ((C) and (F)) show the enlarged form of background fragment. Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

Parameter /*E* into respective kinetic equation was previously applied by Corrigan and Evans (Corrigan & Evans, 1980) as well as McDermott and coauthors (McDermott et al.,

Theoretical curves visualized in Fig. 11A were obtained by the algorithm which includes

=0.5, the intersection point (*iap*) remains the same but the plot is less

/*E≠*0 parameter (plots (A), (B), (C)) or different

/*E* = 0, (D) *k* =0.05, (E) *k* =0.005, (F) *k* =0.00001. The calculations

/*E* in the range 0 - 1.4 V-1. Despite the fact, that experimental values of

/E results in more extended set of

value was applied (Sanecki & Kaczmarski, 1999; Sanecki &

=1 as a starting value was chosen (Fig. 11B). When the

), in which each

/*E* do not

/*E* = 0;

=

1992). The application of the non-continuous ETC variability parameter (Δ

curve is generated for different

curves. For the same reason

Fig. 12. The effect of including of

(2008), with permission from Elsevier.]

1; 0.9; 0.8; 0.7; 0.6; 0.5;

values (plots (D), (E), (F)) into theoretical one-electron kinetic model Eirr as well as the influence of *k* value on isoalpha point current response. Plots (A), (B), (C) with

0.2; 0.4; 0.6; 0.8; 1.0; 1.2; 1.4. (A) *k* =0.05, (B) *k* =0.005, (C) *k* =0.00001. Plots (D), (E), (F) with

were done with the use of both *ESTYM\_PDE* (black lines, Eirr) and *DigiSim®* (red lines, ErC with *k*f=1×1010 s–1). Notice, that the both programs provide identical current responses (plots (D), (E), (F)). The insets ((C) and (F)) show the enlarged form of background fragment. Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright

variability starts at

convincing.

Skitał, 2007a, 2007b; Sanecki et al., 2006b).

exceed 0.5 V-1, the application of wider range of

As a result of including of the ETC variation either through /*E* parameter or by different values of , a single curve becomes the set of curves which intersect in one characteristic point in which cathodic current is independent of (Fig. 11 A and B, respectively). The result is similar to isosbestic point (Berlett et al., 2000; IUAPAC, 1997; Nakajima et al., 2004). The system resembling *iap* and isosbestic point, called *isopotential point* has been described in electrochemistry (Edens et al., 1991; Eichhorn & Speiser, 1994; Fitch & Edens, 1989). Its appearance is associated with the linked multi-sweep experiments when two distinguishable electroactive species interconvert.

The model for totally irreversible process applied with the use of *ESTYM\_PDE* was described in paper by Nicholson and Shain (Nicholson & Shain, 1964) and is given in Bard and Faulkner's monograph (Bard & Faulkner, 2001). It is not included in *DigiSim®* software. In *DigiSim®,* however, the Eirr mechanism response was easily generated as ErC with very high *k*f value e.g. *k*f = 1×1010 s–1 (Fig. 12 D,E,F).

A question arises why other authors dealing with the effect of potential-dependent in voltammetry (Arun & Sangaranarayanan, 2004; Bieniasz & Speiser, 1998a; Bond & Mahon, 1997; Delahay, 1953; Nahir et al., 1994; Tender et al., 1994; Weber & Creager, 1994) did not observe so far such a phenomenon as *iap*, even if Marcus' λ parameter was discussed instead of . Most probably there are three reasons. Firstly, the range of the phenomenon is not very wide e.g. it appears for 0.05 > *k* > 1×10–4, at *v* =1Vs–1 (for *k*=0.05, Fig. 12 A,D the *iap* is diffused). At lower *k* values *iap* occurs but is invisible unless a magnification method is applied (Fig. 12 C,F). Secondly, is the phenomenon to be observed requires at least three (or more) superimposed curves and that was not fulfilled by the other authors. Even if one maintains all parameters constant except for *,* no *iap* is observed and CV peak moves towards more negative potentials becoming flatter as decreases. Examples of the lacking or not visible *iap* can be found in paper by Delahay (Fig. 2 in (Delahay, 1953)).

The third reason in form of examples was shown in Fig. 12 C,F. The data indicate that it is not possible to find *iap* without magnification of the plot and therefore some authors could not notice such a hidden effect. The observed *iap* phenomenon is not contradictory to the well established theory and provides its completion.

The similar, although less distinct *isoalpha points* are obtained for ErC process when /*E*≠0 parameter or different values mode is introduced into the model (Fig. 12).The influence of scan rate and *E*0 value on position *iap* is discussed in (Sanecki & Skitał, 2008). It is seen that *iap* moves down when scan rate increases and its value becomes closer to *E*0 value.

#### **8.2 ECE and ECEC mechanisms with included variability**

The kinetics should concern the steps and elementary parameters, which have clear physical meaning as it was performed earlier for Eirr process (Fig. 11,12). On the other hand, data applied for ECE process, allowed to generate dependences corresponding to those from Fig. 2 and may be important for ECE process kinetics. Examples of *isoalpha points* for both general (two-electron) peaks and elementary (one-electron) sub-peaks are presented in Fig. 13A,B.

Mathematical Modeling of Electrode Processes

slightly different electronic properties.

continuous

0

10

20

30

40 *I* / A

comparison of

remains

system the effect of

/*E* or Δ

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 75

and *isoalpha* point appearance may be disfigured. Despite of the above, experimental confirmation of existence of *isoalpha point* is possible by the use of the set of electrodes with

A

Fig. 14. (A) The theoretical CV current responses of ErCErC–ErCErC process with non-

from the top curve to the bottom one. The calculations were done with the use of both *ESTYM\_PDE* (black lines) and *DigiSim®* (red lines). Notice that both programs provide

exactly the same current responses. Analogical picture is observed when

of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

1=2=3=

*E* / V

0

150

300

*I* / A

of *isoalpha point.* (B) The appearance of another iso-current point for pure diffusion kinetic case. The one-electron CV responses for Er process were generated. The *v* Vs-1 decreases from the top to the bottom. Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models

The Hg/Zn system, widely investigated in both aqueous and nonaqueous media, was chosen as a test ((Manzini & Lasia, 1994; Sanecki et al., 2006a) and literature therein). It turned out that Hg drop, after one or more CV cycles, behaves somehow differently even in

Fig. 15 presents the effect of Hg drop electrode pretreatment on CV response; for details reader is referred to the caption and original paper. The experimental results indicate (Fig. 15D) that an attempt for finding *iap* was successful. The experimental data, selected from Fig. 15 A,B,C, namely curves scanned for 0 and 1 minute intervals (Fig. 15A), provide the possibility of fitting experimental and simulated CV data which exploit *iap* property on

In our case each scan starts with the same concentration of both redox forms. Such conditions are not sufficient for the isopotential point formation of the type described in the literature (Edens et al., 1991; Eichhorn & Speiser, 1994; Fitch & Edens, 1989). In absence of changes of supporting electrolyte concentration, the only source of the *iap* appearance

to the change of electrode properties and not to solution parameters in Marcus' sense.

. Therefore, the occurrence of *iap* indicates that in the investigated

variability takes place. Another experimental example of isoalpha

4=1, 0.9, 0.8, 0.7, 0.6, 0.5 decreases

100 V s-1 B

0.05 V s-1

variability parameter obtained by a

/∂*E*=0.13. The observed variability of

/*E* is the source

*E* / V

is due

variability included. Parameter

the situation when metallic Zn was removed from it.

rising part of the curve (Fig. 15D plot). The

for both curves equals to ∂

Fig. 13. (A) The *isoalpha* plots for EirrCEirr mechanism. The /*E* value is 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6 from the top to the bottom. (B) The *isoalpha* plots for ErCErC mechanism. The value is 0.1, 0.3, 0.5, 0.7, 0.9 from the bottom to the top. The *isoalpha point* appears on both two-electron and one-electron CV responses. The calculations were done with the use of both *ESTYM\_PDE* (black lines) and *DigiSim®* (red lines). Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

Inclusion of /*E* into ECE kinetic model, with other parameters being constant, results in three sets of curves: one for two-electron and two for one-electron stages (Fig. 13A, B). Each set has characteristic intersect point *E*iap as it was observed earlier for one-electron process (Fig. 11,12).

#### **8.3 ErCErC–ErCErC mechanisms with included variability**

The data presented in Fig. 13A for ECE and ECEC mechanism indicate one *E*iap point on cathodic part of CV curve. For ECE-ECE and ECEC-ECEC sequences at least two *E*iap points appear for both continuous (not presented) and non-continuous (Fig. 14A) variability of , respectively. For the both mechanisms the position of *isoalpha points* is close to *E*0,i values.

It is worth to notice that apart from potential dependent kinetic cases considered here, the, to some extent similar, intersection point appears for pure diffusion kinetics, non sensitive to the value at all (Fig. 14B). The comparable diffusion kinetic case is presented in literature (Greef et al., 1985), Figure 6.5, p. 184) but without discussion in the context presented here.

## **9. Experimental confirmation of** *isoalpha* **points reality**

This sub-chapter deals with variability resulting from the change of electrode material properties caused by its modification. Under experimental conditions the change of electrode properties results in both: (1) a change of i.e. the slope of log *k* vs. *E* dependence (electrochemical reaction constant = –*ρ*); (2) a change of electrochemical rate constant *k*.

Therefore, the experimental confirmation of predicted by theory *isoalpha point* requires a system with change separately. For the most of experimental systems, the changes of and *k* are coupled together and therefore a shift of CV curves along potential axis is possible

A

1.0, 1.2, 1.4, 1.6 from the top to the bottom. (B) The *isoalpha* plots for ErCErC mechanism. The

0

10

20

30

*I* / A

*E* / V

 value is 0.1, 0.3, 0.5, 0.7, 0.9 from the bottom to the top. The *isoalpha point* appears on both two-electron and one-electron CV responses. The calculations were done with the use of both *ESTYM\_PDE* (black lines) and *DigiSim®* (red lines). Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright (2008), with permission from

three sets of curves: one for two-electron and two for one-electron stages (Fig. 13A, B). Each set has characteristic intersect point *E*iap as it was observed earlier for one-electron process

The data presented in Fig. 13A for ECE and ECEC mechanism indicate one *E*iap point on cathodic part of CV curve. For ECE-ECE and ECEC-ECEC sequences at least two *E*iap points appear for both continuous (not presented) and non-continuous (Fig. 14A) variability of

to some extent similar, intersection point appears for pure diffusion kinetics, non sensitive to

properties caused by its modification. Under experimental conditions the change of

Therefore, the experimental confirmation of predicted by theory *isoalpha point* requires a

and *k* are coupled together and therefore a shift of CV curves along potential axis is possible

change separately. For the most of experimental systems, the changes of

 value at all (Fig. 14B). The comparable diffusion kinetic case is presented in literature (Greef et al., 1985), Figure 6.5, p. 184) but without discussion in the context presented here.

respectively. For the both mechanisms the position of *isoalpha points* is close to *E*0,i values.

**9. Experimental confirmation of** *isoalpha* **points reality**

electrode properties results in both: (1) a change of

*E*iap

= 0.9

/*E* into ECE kinetic model, with other parameters being constant, results in

 **variability**

potential dependent kinetic cases considered here, the,

variability resulting from the change of electrode material

= –*ρ*); (2) a change of electrochemical rate constant *k*.

i.e. the slope of log *k* vs. *E* dependence

/*E* value is 0, 0.2, 0.4, 0.6, 0.8,

= 0.1

*E* / V

B

,

Fig. 13. (A) The *isoalpha* plots for EirrCEirr mechanism. The

/d*E* = 0 (*E*p

, *I* p )

**8.3 ErCErC–ErCErC mechanisms with included**

0

10

20

30 *I* / A

Elsevier.]

Inclusion of

(Fig. 11,12).

the

system with

d/d*E* = 1.6

*E*iap

d

It is worth to notice that apart from

This sub-chapter deals with

(electrochemical reaction constant

and *isoalpha* point appearance may be disfigured. Despite of the above, experimental confirmation of existence of *isoalpha point* is possible by the use of the set of electrodes with slightly different electronic properties.

Fig. 14. (A) The theoretical CV current responses of ErCErC–ErCErC process with noncontinuous variability included. Parameter 1=2=3=4=1, 0.9, 0.8, 0.7, 0.6, 0.5 decreases from the top curve to the bottom one. The calculations were done with the use of both *ESTYM\_PDE* (black lines) and *DigiSim®* (red lines). Notice that both programs provide exactly the same current responses. Analogical picture is observed when /*E* is the source of *isoalpha point.* (B) The appearance of another iso-current point for pure diffusion kinetic case. The one-electron CV responses for Er process were generated. The *v* Vs-1 decreases from the top to the bottom. Other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

The Hg/Zn system, widely investigated in both aqueous and nonaqueous media, was chosen as a test ((Manzini & Lasia, 1994; Sanecki et al., 2006a) and literature therein). It turned out that Hg drop, after one or more CV cycles, behaves somehow differently even in the situation when metallic Zn was removed from it.

Fig. 15 presents the effect of Hg drop electrode pretreatment on CV response; for details reader is referred to the caption and original paper. The experimental results indicate (Fig. 15D) that an attempt for finding *iap* was successful. The experimental data, selected from Fig. 15 A,B,C, namely curves scanned for 0 and 1 minute intervals (Fig. 15A), provide the possibility of fitting experimental and simulated CV data which exploit *iap* property on rising part of the curve (Fig. 15D plot). The variability parameter obtained by a comparison of for both curves equals to ∂/∂*E*=0.13. The observed variability of is due to the change of electrode properties and not to solution parameters in Marcus' sense.

In our case each scan starts with the same concentration of both redox forms. Such conditions are not sufficient for the isopotential point formation of the type described in the literature (Edens et al., 1991; Eichhorn & Speiser, 1994; Fitch & Edens, 1989). In absence of changes of supporting electrolyte concentration, the only source of the *iap* appearance remains /*E* or Δ. Therefore, the occurrence of *iap* indicates that in the investigated system the effect of variability takes place. Another experimental example of isoalpha

Mathematical Modeling of Electrode Processes

increasing of

found by other investigators.

any discussion of quantitative type.

space obtained from experimental responses.

**10. Conclusions**

(Greef et al., 1985; Sanecki et al., 2006a; Sawyer et al., 1995).

*I*/v0.5

2

0

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 77

anymicroscopic inhomogeneity may significantly affect the Fermi level of electrons and, therefore, the kinetics of electrode process. The similar effect of the change of electrode properties caused by the electrodeposition process is well recognized for solid electrodes

> 0.002 V/s 0.005 V/s 0.01 V/s 0.02 V/s 0.05 V/s 0.1 V/s 0.2 V/s 0.5 V/s

Fig. 16. The normalized complete CV responses *Iv*–0.5 vs. potential *E* for hydrogen ions – reduction in 0.1M HCl solution. Electrode system: GCE/100 monomolecular Pd layers. The isopotential crossing point is indicated with the arrow (experimental data from ref. (Skitał et al., 2010)). Note the almost vertical position of rising part of CV curves which suggests high

0.00 -0.25 -0.50

increasing concentration of atomic hydrogen with scan rate decrease. [Reprinted from *Electrochim. Acta*, Vol. 55, Skitał P. & Sanecki P., The mathematical model of the stripping

thin films dominating in both theoretical and applied electrochemistry (Swain, 2004).

Extended and still growing theory and application of solid electrodes (Horrocks et al., 1994; Kuznetsov & Ulstrup, 1999; Levich, 1966; Wantz et al., 2005) prove that presented results can be useful as an element of theory and practice concerning advanced solid electrodes and

We expect that another experimental *isoalpha points*, analogous to described above, will be

1. Acquisition of rigorous mathematical model of considered mechanism comprising elementary kinetic parameters of electrochemical and chemical steps is fundamental for

MPE method with a proper mathematical model of the considered mechanism, applied to a series of electrochemical responses, is the simple, clear and rigorous way to determine the values of kinetic parameters with their confidence intervals. Additional error space for all estimated kinetic parameters, characteristic for considered mechanism like a fingerprint, can be easily determined basing on obtained theoretical responses. Every mechanism together with its mathematical model can generate electrochemical theoretical responses and its own error space, independently of error

voltammetry ..., 5604–5609, Copyright (2010), with permission from Elsevier.]

values. Modification of active electrode surface can be explained by an

*E* / V (vs. SCE)

Fig. 15. The effect of non-linked multi scans CV experiment on the same mercury drop. The experimental system: 1mM Zn2+ in 1M NaClO4. Plots (A) and (B) the scans were repeated in 1 minute intervals, at open circuit without polarization; *E*0,exp= -0.970 V for Zn/Zn2+ system, *E*iap= -1.000V. Plot (C) – the nine scans were repeated in 1 minute intervals, at starting potential -0.700 V. At the interval metallic Zn was removed from the drop by electrolysis. Plot (D) the comparison of Zn2+ electroreduction on pure Hg drop (curve 1, scan 0 min., red points) with the same process on the drop with minor trace of metallic Zn (curve 2, scan 1 min., red points) as well as the respective theoretical curves (black lines) according to extended EE║Hg(Zn) kinetic model (Sanecki et al., 2006a). The curves 1 and 2 were selected from plot (A). The estimated alpha values: (scan 0) 1=2=0.66±0.02, (scan 1) 1=2=0.785±0.015. The observed *isopotential points* and scan rates are indicated on the plots. The other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719, Copyright (2008), with permission from Elsevier.]

point occurrence is provided by hydrogen ions reduction (Fig. 16). Increasing concentration of atomic hydrogen with scan rate decrease is expected to be a source of modification of active electrode surface. In consequence, the *isoalpha point* appeares.

The presented experimental results (Fig. 15, 16) together with theoretical ones (Fig. 11-14) suggest that the change of electrode properties should be considered as an origin of occurrence of the *iap*. It can be supported by a quotation from Corrigan and Evans (Corrigan & Evans, 1980): "*For all of these cases, eq. d/dE = F/2 λ does not adequately describe the potential dependence of because λ is a solution parameter which should not depend on surface conditions."*). The results by Mattheiss and Warren (Matthesiss & Warren, 1977) showed that anymicroscopic inhomogeneity may significantly affect the Fermi level of electrons and, therefore, the kinetics of electrode process. The similar effect of the change of electrode properties caused by the electrodeposition process is well recognized for solid electrodes (Greef et al., 1985; Sanecki et al., 2006a; Sawyer et al., 1995).

Fig. 16. The normalized complete CV responses *Iv*–0.5 vs. potential *E* for hydrogen ions – reduction in 0.1M HCl solution. Electrode system: GCE/100 monomolecular Pd layers. The isopotential crossing point is indicated with the arrow (experimental data from ref. (Skitał et al., 2010)). Note the almost vertical position of rising part of CV curves which suggests high increasing of values. Modification of active electrode surface can be explained by an increasing concentration of atomic hydrogen with scan rate decrease. [Reprinted from *Electrochim. Acta*, Vol. 55, Skitał P. & Sanecki P., The mathematical model of the stripping voltammetry ..., 5604–5609, Copyright (2010), with permission from Elsevier.]

Extended and still growing theory and application of solid electrodes (Horrocks et al., 1994; Kuznetsov & Ulstrup, 1999; Levich, 1966; Wantz et al., 2005) prove that presented results can be useful as an element of theory and practice concerning advanced solid electrodes and thin films dominating in both theoretical and applied electrochemistry (Swain, 2004).

We expect that another experimental *isoalpha points*, analogous to described above, will be found by other investigators.

## **10. Conclusions**

76 Recent Trend in Electrochemical Science and Technology

0

100

*I* / A

50 Vs-1

 0 min. 1 min. 2 min. 3 min. 4 min. 5 min. 6 min. 7 min. 8 min. 9 min. 10 min. *E* / V (vs. SCE)

A

Fig. 15. The effect of non-linked multi scans CV experiment on the same mercury drop. The experimental system: 1mM Zn2+ in 1M NaClO4. Plots (A) and (B) the scans were repeated in 1 minute intervals, at open circuit without polarization; *E*0,exp= -0.970 V for Zn/Zn2+ system, *E*iap= -1.000V. Plot (C) – the nine scans were repeated in 1 minute intervals, at starting potential -0.700 V. At the interval metallic Zn was removed from the drop by electrolysis. Plot (D) the comparison of Zn2+ electroreduction on pure Hg drop (curve 1, scan 0 min., red points) with the same process on the drop with minor trace of metallic Zn (curve 2, scan 1 min., red points) as well as the respective theoretical curves (black lines) according to extended EE║Hg(Zn) kinetic model (Sanecki et al., 2006a). The curves 1 and 2 were selected

0

10

20

*I* / A

cathodic

anodic

 0 min. 1 min. 2 min. 3 min. 4 min. 5 min. 6 min. 7 min. 8 min. 9 min. *E* / V (vs. SCE)

C

1=

 *because λ is a solution parameter which should not depend on surface conditions."*).

2=0.785±0.015. The observed *isopotential points* and scan rates are indicated on the plots. The other details are in original paper (Sanecki & Skitał, 2008). [Reprinted from *Electrochim. Acta*, Vol. 53, Sanecki, P. & Skitał P., The mathematical models of kinetics ..., 7711-7719,

point occurrence is provided by hydrogen ions reduction (Fig. 16). Increasing concentration of atomic hydrogen with scan rate decrease is expected to be a source of modification of

The presented experimental results (Fig. 15, 16) together with theoretical ones (Fig. 11-14) suggest that the change of electrode properties should be considered as an origin of occurrence of the *iap*. It can be supported by a quotation from Corrigan and Evans (Corrigan

The results by Mattheiss and Warren (Matthesiss & Warren, 1977) showed that

2=0.66±0.02, (scan 1)

1 2

1 Vs-1

 0 min. 1 min. 2 min. 3 min. 4 min. 5 min. 6 min. 7 min. 8 min. 9 min. *E* / V (vs. SCE)

D

 0 min. 1 min.

*E* / V (vs. SCE)

B

*/dE = F/2 λ does not adequately describe the potential*

from plot (A). The estimated alpha values: (scan 0)

0.05 Vs-1

1 Vs-1

Copyright (2008), with permission from Elsevier.]

& Evans, 1980): "*For all of these cases, eq. d*

active electrode surface. In consequence, the *isoalpha point* appeares.

1=

0

5

0

10

20

*I* / A

cathodic

anodic

*I* / A

*dependence of*

1. Acquisition of rigorous mathematical model of considered mechanism comprising elementary kinetic parameters of electrochemical and chemical steps is fundamental for any discussion of quantitative type.

MPE method with a proper mathematical model of the considered mechanism, applied to a series of electrochemical responses, is the simple, clear and rigorous way to determine the values of kinetic parameters with their confidence intervals. Additional error space for all estimated kinetic parameters, characteristic for considered mechanism like a fingerprint, can be easily determined basing on obtained theoretical responses. Every mechanism together with its mathematical model can generate electrochemical theoretical responses and its own error space, independently of error space obtained from experimental responses.

Mathematical Modeling of Electrode Processes

*E*iap – potential in the isoalpha point (V)

*k*f – chemical first order rate constant (s–1)

r; irr (in subscript) – reversible; irreversible

TBAP – tetrabutylammonium perchlorate TEAP – tetraethylammonium perchlorate

 *–* transfer coefficient of anodic process

to start with

*B –* Brnsted acid-base catalysis coefficients *λ –* solvent reorganisation factor in Marcus' theory

85, pp. 27-46, ISSN: 0022-0728.

101, pp. 3431-3441, ISSN: 0002-7863.

114, pp. 6892-6904, ISSN: 0002-7863.

MPE – multi-parameter estimation PDE – partial differential equation(s)

parameters

TSF – *p*-toluenesulfonyl fluoride

v – scan rate

*k*2, *,* 1,

start – the value of

**13. References**

101.

*B ,*  ) – elementary transfer coefficient

*E*start – the value of potential where

GCE – glassy carbon electrode *I*iap – current in the isoalpha point (A)

*E*p – peak potential (V)

*iap –* isoalpha point

ETC (or

– Potential Dependent Transfer Coefficient in Electrochemical Kinetics 79

variability starts

*k*i (*i*=1,2,3,...) – electrochemical rate constant for a step of multi electron process (cm s–1)

ESTYM\_PDE - program for solving the models expressed by PDE and estimating their

 *–* transfer coefficient of cathodic process (in electrode kinetics we use elementary kinetic parameters relating to, or describing the single step of kinetic sequence (*k*1,

Albery, W. J. (1975). *Electrode Kinetics*, Clarendon Press, ISBN 0-19-855433-8, Oxford, pp. 100-

Amatore, C. & Saveant, J. M. (1977). ECE and disproportionation: Part V. Stationary state

Andrieux, C. P., Blocman, C., Dumas-Bouchiat, J.-M. & Savéant J. M. (1979). Heterogeneous

Andrieux, C. P., Le Gorande, A. & Savéant, J. M., (1992). Electron transfer and bond

general solution application to linear sweep voltammetry. *J. Electroanal. Chem.,* Vol.

and homogeneous electron transfers to aromatic halides. An electrochemical redox catalysis study in the halobenzene and halopyridine series. *J. Am. Chem. Soc.*, Vol.

breaking. Examples of passage from a sequential to a concerted mechanism in the electrochemical reductive cleavage of arylmethyl halides. *J. Am. Chem. Soc.,* Vol.

2) as well as apparent (general) kinetic parameters i.e. relating to or

app))

*k* – electrochemical rate constant for one electron process (cm s–1)

*k*f1, *k*f2 – chemical first order rate constant of multi stage process (s–1)

Ru – uncompensated resistance between the working and reference electrode

describing the complex kinetic course or mechanism as a whole (e.g. *k*app,

variability

## **11. Acknowledgment**

The authors are grateful to professor Christian Amatore for valuable explanations, discussions and cooperation.

## **12. Definitions, abbreviations and symbols**

A mechanism can be defined as an system of chemical and electrochemical elementary steps with a set of respective elementary kinetic parameters.

A model is a mathematical description of the considered mechanism or system. Both the mechanism (system) and the model provide an electrochemical response, experimental and theoretical, respectively.

The mathematical model as well as respective electrochemical current or concentration response is a function of multiple variables. Among them there are system variables as electrochemical and chemical rate constants, equilibrium constants, transfer coefficients, standard potentials, etc. as well as variables of experimental type as scan rate, starting potential, concentration, temperature.

AN – acetonitryle 1,3-BDF – 1,3-benzenedisulfonyl difluoride BSF – benzenesulfonyl fluoride CV – cyclic voltammetry DMF – dimethylformamide *E*0,i –standard potential of step *i* (V)

*E*iap – potential in the isoalpha point (V)

*E*p – peak potential (V)

78 Recent Trend in Electrochemical Science and Technology

2. The common problems with extracting of kinetic parameters basing on selected characteristic points of complex electrochemical responses like half peak method or processed responses (e.g. convolution method) can be avoided by estimation of kinetic parameters on the whole original responses. The actual numerical possibilities and availability of respective software makes it possible to solve practically each of met in practice mechanisms by every researcher. The presented and quoted successful analysis of complex, multi-electron electrode processes with chemical step(s) provides the

3. MPE method makes it possible to judge controversial or confused cases of mechanisms. A good example is here a stepwise/concerted mechanism. A uniform approach to understanding of the stepwise/concerted systems was confirmed by respective modeling for ECE mechanism where fluent passage from stepwise to concerted mechanism has been obtained. The only difference between them is the value of *k*f. 4. The mathematical modeling of electrode processes linked with experimental results may become an inspiration for new results and deeper understanding of the nature of electrochemical processes and its description. A good example is here an including of

5. The occurrence of *isoalpha point* indicates that, under constant composition of the solution, in investigated system *α*/d*E* or Δ*α* type variability takes place. The origin of experimentally found *isoalpha point* can be explained by the change of electrode surface

The authors are grateful to professor Christian Amatore for valuable explanations,

A mechanism can be defined as an system of chemical and electrochemical elementary steps

A model is a mathematical description of the considered mechanism or system. Both the mechanism (system) and the model provide an electrochemical response, experimental and

The mathematical model as well as respective electrochemical current or concentration response is a function of multiple variables. Among them there are system variables as electrochemical and chemical rate constants, equilibrium constants, transfer coefficients, standard potentials, etc. as well as variables of experimental type as scan rate, starting

alpha variability into kinetic models and discovery of *isoalpha points*.

respective examples.

conditions.

**11. Acknowledgment**

theoretical, respectively.

AN – acetonitryle

potential, concentration, temperature.

BSF – benzenesulfonyl fluoride CV – cyclic voltammetry DMF – dimethylformamide *E*0,i –standard potential of step *i* (V)

1,3-BDF – 1,3-benzenedisulfonyl difluoride

discussions and cooperation.

**12. Definitions, abbreviations and symbols**

with a set of respective elementary kinetic parameters.

*E*start – the value of potential where variability starts

ETC (or ) – elementary transfer coefficient

GCE – glassy carbon electrode

*I*iap – current in the isoalpha point (A)

*iap –* isoalpha point

*k* – electrochemical rate constant for one electron process (cm s–1)

*k*i (*i*=1,2,3,...) – electrochemical rate constant for a step of multi electron process (cm s–1)

*k*f – chemical first order rate constant (s–1)

*k*f1, *k*f2 – chemical first order rate constant of multi stage process (s–1)

MPE – multi-parameter estimation

PDE – partial differential equation(s)

ESTYM\_PDE - program for solving the models expressed by PDE and estimating their parameters

r; irr (in subscript) – reversible; irreversible

Ru – uncompensated resistance between the working and reference electrode

TBAP – tetrabutylammonium perchlorate

TEAP – tetraethylammonium perchlorate

TSF – *p*-toluenesulfonyl fluoride

v – scan rate

 *–* transfer coefficient of cathodic process (in electrode kinetics we use elementary kinetic parameters relating to, or describing the single step of kinetic sequence (*k*1, *k*2, *,* 1, 2) as well as apparent (general) kinetic parameters i.e. relating to or describing the complex kinetic course or mechanism as a whole (e.g. *k*app, app))

 *–* transfer coefficient of anodic process

start – the value of to start with variability

*B , B –* Brnsted acid-base catalysis coefficients

*λ –* solvent reorganisation factor in Marcus' theory

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**4**

*Japan*

**Electron-Transfer-Induced**

*Tokyo University of Agriculture and Technology*

Kazuhiro Chiba\* and Yohei Okada

**Intermolecular [2 + 2] Cycloaddition**

**Reactions Assisted by Aromatic "Redox Tag"**

Electron transfer is a fundamental and ubiquitous process in chemical and biological reactions and thus has been extensively studied. A wide variety of unique molecular systems have been developed that connect electron donor and electron acceptor using different types of bridges to investigate electron-transfer reactions. Typically, laser flash photolysis techniques are used in combination with photo-excitable electron donors to provide insights into the chemical and physical properties of electron-transfer reactions. Electron donors are photochemically activated to induce electron transfer toward the electron acceptor, which is then detected spectroscopically. One recent achievement in this field is the discovery of long-range photo-induced electron transfer through a DNA helix,

In organic synthesis, electron-transfer-induced reactions have been used extensively to achieve various chemical transformations and construct a wide variety of organic compounds, including natural products, pharmaceutical products, and functional materials. To trigger these electron-transfer-induced reactions, photochemical processes are widely employed. Oneelectron oxidants and reductants also can be used to initiate electron-transfer-induced reactions. In this context, electrochemical approaches have been utilized to trigger either oneor two-electron transfers through electrode processes that afford electron-transfer-induced reactions. Based on electrochemical processes, various functional group transformations and a wide variety of carbon-carbon bond formation reactions can be accomplished in a controlled manner. For example, Kolbe electrolysis is a well-established process for forming carboncarbon bonds (Fig. 1). In this reaction, decarboxylation is anodically induced to generate

carbon free radicals, which are then homocoupled to make a new carbon-carbon bond.

In addition to free radicals, electrochemical processes also are efficient for the generation of several reactive organic species, including ions and radical ions that can be introduced into organic syntheses as intermediates. In this chapter, we describe electron-transfer-induced carbon-carbon bond formation reactions based on the generation of carbon radical cations as reactive intermediates through electrode processes. Electrochemical studies of the reaction mechanisms have led to the development of new intermolecular [2 + 2] cycloaddition

which offers interesting applications as novel conductive materials.

**1. Introduction**

reactions.

Yu, Q. & Wang, N. H. L. (1989). Computer simulations of the dynamics of multicomponent ion exchange and adsorption in fixed beds—gradient-directed moving finite element method. *Comput. Chem. Eng.*, Vol. 13, pp. 915-926, ISSN: 0098-1354.

## **Electron-Transfer-Induced Intermolecular [2 + 2] Cycloaddition Reactions Assisted by Aromatic "Redox Tag"**

Kazuhiro Chiba\* and Yohei Okada *Tokyo University of Agriculture and Technology Japan*

## **1. Introduction**

90 Recent Trend in Electrochemical Science and Technology

Yu, Q. & Wang, N. H. L. (1989). Computer simulations of the dynamics of multicomponent

element method. *Comput. Chem. Eng.*, Vol. 13, pp. 915-926, ISSN: 0098-1354.

ion exchange and adsorption in fixed beds—gradient-directed moving finite

Electron transfer is a fundamental and ubiquitous process in chemical and biological reactions and thus has been extensively studied. A wide variety of unique molecular systems have been developed that connect electron donor and electron acceptor using different types of bridges to investigate electron-transfer reactions. Typically, laser flash photolysis techniques are used in combination with photo-excitable electron donors to provide insights into the chemical and physical properties of electron-transfer reactions. Electron donors are photochemically activated to induce electron transfer toward the electron acceptor, which is then detected spectroscopically. One recent achievement in this field is the discovery of long-range photo-induced electron transfer through a DNA helix, which offers interesting applications as novel conductive materials.

In organic synthesis, electron-transfer-induced reactions have been used extensively to achieve various chemical transformations and construct a wide variety of organic compounds, including natural products, pharmaceutical products, and functional materials. To trigger these electron-transfer-induced reactions, photochemical processes are widely employed. Oneelectron oxidants and reductants also can be used to initiate electron-transfer-induced reactions. In this context, electrochemical approaches have been utilized to trigger either oneor two-electron transfers through electrode processes that afford electron-transfer-induced reactions. Based on electrochemical processes, various functional group transformations and a wide variety of carbon-carbon bond formation reactions can be accomplished in a controlled manner. For example, Kolbe electrolysis is a well-established process for forming carboncarbon bonds (Fig. 1). In this reaction, decarboxylation is anodically induced to generate carbon free radicals, which are then homocoupled to make a new carbon-carbon bond.

In addition to free radicals, electrochemical processes also are efficient for the generation of several reactive organic species, including ions and radical ions that can be introduced into organic syntheses as intermediates. In this chapter, we describe electron-transfer-induced carbon-carbon bond formation reactions based on the generation of carbon radical cations as reactive intermediates through electrode processes. Electrochemical studies of the reaction mechanisms have led to the development of new intermolecular [2 + 2] cycloaddition reactions.

Electron-Transfer-Induced Intermolecular [2 + 2]

R1

*-2e- , -2H+*

OH

OH

Hydroquinones

OH

OMe

*-2e- , -H+*

Phenols

membered ring have been reported.

Cycloaddition Reactions Assisted by Aromatic "Redox Tag" 93

+

Chiba, K.; Jinno, M.; Kuramoto, R.; Tada, M. *Tetrahedron Lett.* **1998**, *39*, 5527-5530.

Olefin nucleophiles

R1

O

O

Quinones

O

OMe

Phenoxonium cations

immediate trapping by olefin nucleophiles was required for the reactions.

Fig. 3. Electron-transfer-induced intermolecular [2 + 2] cycloaddition reactions

+

Chiba, K.; Fukuda, M.; Kim, S.; Kitano, Y.; Tada, M. *J. Org. Chem.* **1999**, *64*, 7654-7656.

Electron-transfer-induced cycloreversion reactions of four-membered rings also have been represented through radical ion intermediates. Many mechanistic studies on these electrontransfer-induced [2 + 2] cycloaddition reactions and cycloreversion reactions of four-