*3.1.1.4 Simple values of symmetry parameters of reversible electrochemical process (Er)*

The symmetric relationship among the symmetry parameters are found from the simple fast electron transfer process; namely, a reversible electron transfer (Erev, or Er) type in which the electrode process is basically Nernstian. What will happen to the shapes of voltammograms and its derivatives if the system involve non-Nernstian behavior with a slower irreversible electron transfer rate (Eirr type), or the electron transfer is coupled with a prior chemical equilibrium (CEr Type) or an Er coupled with a follow-up chemical reactions (ErC). These studies have been already done mostly and discussed in detail by our group [14]; to summarize, the symmetry exhibited in the derivatives in Er disappears for the non-Nernstian electron transfer (i.e., irreversible) system. Namely, the asymmetry strongly depends on those the electron transfer parameters (α and ko ) for Eirr type, and the kinetic parameters (i.e., k and Keq) for chemically coupled processes (CrEr and ErCirr Type). In this work, comparisons are made for the symmetry parameters for the different electrode mechanisms. Details are given in later sections for each of corresponding mechanisms.

#### *3.1.1.5 Reversibility in electrochemical reactions*

In general, reversibility in electrochemical reactions is divided into three kinds depending on the magnitude of the standard heterogeneous rate constant k<sup>o</sup> (or ks) *Advances in Derivative Voltammetry - A Search for Diagnostic Criteria of Several… DOI: http://dx.doi.org/10.5772/intechopen.96409*

with respect to the diffusion coefficient (D): an electron transfer reaction is considered reversible if e� transfer is much faster the diffusion (i.e., ko > > (D/t)1/2), and irreversible if e� transfer is much slower the diffusion (k<sup>o</sup> < < (D/t)1/2) and quasireversible if those competing rates are comparable (k<sup>o</sup> � (D/t)1/2).

In terms of values of k<sup>o</sup> , by adopting typical values of D (=5x10�<sup>5</sup> cm2 /s) and t (=1.000 s).

*reversible for ko > 0.020 cm/s, quasi-reversible for 0.020 cm/s > k<sup>o</sup> > 5x10*�*<sup>5</sup> cm/s, irreversible for k<sup>o</sup> < 5x10*�*<sup>5</sup> cm/s.*

In the following sections, other types of processes than the simple reversible process will be treated.

## *3.1.2 Theoretical expression for and the derivatives for Equasirev and Eirr types of electron transfer*

Currents (i) as a function of the applied potential (E) for non-Nernstian (i.e., irreversible and quasi-reversible) system with a slower electron transfer, have been previously derived and given in several references [44, 54–56, 72, 73], and has final forms of the following equations for a planar diffusion:

$$\mathbf{I} = \mathbf{n} \mathbf{F} \mathbf{A} \mathbf{C}\_{\mathbf{o}} \mathbf{k}\_{\mathbf{f}} \left[ \exp \left( \mathbf{Q}^{\mathbf{2}} \mathbf{t} \right) \mathbf{erfc} \left( \mathbf{Q} \mathbf{t}^{\mathbb{N}} \right) \right] \tag{22}$$

assumed DR = DO = D.

All symbols have their usual meanings and may be refer to the reference for details,

where,

�id nF ð Þ *<sup>=</sup>*RT <sup>3</sup> e e2 � 4e <sup>þ</sup> <sup>1</sup> � �*=*ð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup>

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derivative curves are symmetrical.

the same as in lower order derivatives.

equations (relationships) for those parameters.

on those the electron transfer parameters (α and ko

*3.1.1.5 Reversibility in electrochemical reactions*

corresponding mechanisms.

**12**

*3.1.1.3 For the third derivatives*

As given in the value of the unity (1.0) for those symmetry parameters (i.e. anodic to cathodic peak-current ratios, anodic to cathodic half-peak-width ratios, and anodic to cathodic peak potential ratios) for first or second derivatives, the

It's symmetry is the same as that of the first derivative, *being symmetrical with respect to E = 0*. One can find the ranges of parameter values in similar fashion for the third derivatives. Details of the work can be found elsewhere [15]: the key findings on the peak potentials, peak current and half peak widths and the various ratio for symmetry are given in **Tables 1** and **2**. It should be noted that all those values for the anodic to cathodic symmetry parameters (those ratios Rs) are unity,

As shown above, the values of parameters above for Er process are derived analytically from the equations which are much simpler than others. However, for an irreversible electron process (Eirr, or Equasirev) and chemically coupled electron transfer reactions such as ErC and CEr, the expressions for the current and their derivatives are so complicated that it is practically impossible to derive the values of those parameters analytically by solving algebraic equation (Refer to later section). Therefore, a numerical approach had to be adopted. Namely, (a) about 10 curves for derivatives vs. potentials at various values of independent variables (the heterogeneous rate constant k<sup>o</sup> and αn for simple electron transfer; the homogeneous constant Keq, kf, kb and k for chemically coupled electron transfer) had to be calculated, (b) then, all those curves are plotted out, and (c) those plots are analyzed graphically by examining the curves carefully in order to obtain values or

*3.1.1.4 Simple values of symmetry parameters of reversible electrochemical process (Er)*

parameters (i.e., k and Keq) for chemically coupled processes (CrEr and ErCirr Type). In this work, comparisons are made for the symmetry parameters for the different electrode mechanisms. Details are given in later sections for each of

In general, reversibility in electrochemical reactions is divided into three kinds depending on the magnitude of the standard heterogeneous rate constant k<sup>o</sup> (or ks)

The symmetric relationship among the symmetry parameters are found from the simple fast electron transfer process; namely, a reversible electron transfer (Erev, or Er) type in which the electrode process is basically Nernstian. What will happen to the shapes of voltammograms and its derivatives if the system involve non-Nernstian behavior with a slower irreversible electron transfer rate (Eirr type), or the electron transfer is coupled with a prior chemical equilibrium (CEr Type) or an Er coupled with a follow-up chemical reactions (ErC). These studies have been already done mostly and discussed in detail by our group [14]; to summarize, the symmetry exhibited in the derivatives in Er disappears for the non-Nernstian electron transfer (i.e., irreversible) system. Namely, the asymmetry strongly depends

<sup>4</sup> h i

¼ 0 (21)

) for Eirr type, and the kinetic

$$\mathbf{Q} = (\mathbf{k}\_{\mathbf{f}} + \mathbf{k}\_{\mathbf{b}}) / \mathbf{D}^{\mathsf{V}\_{\mathbf{f}}} \tag{23}$$

$$\mathbf{k}\_{\mathbf{f}} = \mathbf{k}^{\bullet} \exp\left\{-\mathbf{an} \mathbf{F} (\mathbf{E} - \mathbf{E}^{\bullet})/\mathbf{RT}\right\} \tag{24}$$

$$\mathbf{k}\_{\mathbf{b}} = \mathbf{k}^{\diamond} \exp\left\{ (\mathbf{1} - \mathbf{a}) \mathbf{n} \mathbf{F} (\mathbf{E} - \mathbf{E}^{\diamond}) / \mathbf{R} \mathbf{T} \right\} \tag{25}$$

$$\mathbf{E}\_{1/2} = \mathbf{E}^o + \left(\mathbf{RT} / \mathbf{nF}\right) \ln\left(\mathbf{D}\_{\mathbf{R}} / \mathbf{D}\_{\mathbf{O}}\right)^{\mathsf{V}\_{\mathbf{I}}} = \mathbf{E}^o \tag{26}$$

normalizing the current, Eq. (22), to the diffusion current, Eq. (3), yields

$$\mathbf{i\_n} = (\pi \mathbf{t} / \mathbf{D})^{1/2} \mathbf{k\_f} \left[ \exp \left( \mathbf{Q}^2 \mathbf{t} \right) \text{erfc} \left( \mathbf{Q} \mathbf{t}^{\mathbb{N}} \right) \right] \tag{27}$$

where,

$$\mathbf{F}\left(\mathbf{Qt}^{1/2}\right) = \exp\left(\mathbf{Q}^2 \mathbf{t}\right)\mathbf{erfc}\left(\mathbf{Qt}^{\vee}\right) \tag{28}$$

This normalized current, which is independent of the concentration and electrode surface area, mostly is dependent on the heterogeneous rate constants (ko ), the transfer coefficient (α) and n, and the curve found to exhibit an asymmetry [15].

Successive differentiations of the current yield the following expressions for the first-, second- and third- derivatives respectively [15]. Taking derivative of the current with respect to potential (E) yields the first derivative,

$$\mathbf{i}' = (\pi \mathbf{t}/\mathbf{D})^{1/2} \mathbf{k}\_{\mathbf{f}} \left[ (-\mathbf{n} \mathbf{F}/\mathbf{RT}) \mathbf{F} \left( \mathbf{Q} \mathbf{t}^{1/2} \right) + 2 \mathbf{Q}' \left[ \mathbf{Q} \mathbf{t} \mathbf{F} \left( \mathbf{Q} \mathbf{t}^{1/2} \right) - (\mathbf{t}/\pi \right)^{1/2} \right] \tag{29}$$

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where

$$\mathbf{Q'} = (-\alpha \mathbf{nF}/\mathbf{RT})\mathbf{k}\_{\mathbf{f}}/\mathbf{D}\_{\mathbf{o}}^{1/2} - ((\mathbf{1} - \alpha)\mathbf{nF}/\mathbf{RT})\mathbf{k}\_{\mathbf{b}}/\mathbf{D}\_{\mathbf{r}}^{1/2} \tag{30}$$

1. Peak Potentials

*DOI: http://dx.doi.org/10.5772/intechopen.96409*

constant ko [15].

<0 (because log(k<sup>o</sup>

9.6 (for Er) to 6.2 as ko decreases [15]

cathodic half-peak width (Wq

Wq

Wq

*3.1.2.2 For the second derivative*

relationships were found [15]:

(k<sup>o</sup>

and Ep

**15**

2a-Ep

1. Peak-Potentials and Peak Separation

Ep

Ep

same trend as for Ep in the first derivative.

Wq a *=*Wq

2. Peak Currents

3. Peak-Widths.

independent of ko

It is found that the peak potential for totally irreversible process

*Advances in Derivative Voltammetry - A Search for Diagnostic Criteria of Several…*

) is always negative for Eirr).

The half-peak width (W½), the anodic half-peak width (Wq

c

direction as k<sup>o</sup> decreases for a given αn, and is inversely proportional to αn at a

The normalized peak heights is directly proportional to αn, decreasing from

, but inversely proportional to αn [15]

<sup>a</sup> <sup>¼</sup> <sup>1</sup>*:*62 RT ð Þ *<sup>=</sup>*<sup>F</sup> ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ¼ <sup>V</sup> <sup>41</sup>*:*7 1ð Þ *<sup>=</sup>*α<sup>n</sup> ð Þ mV ¼ 83*:*4 ðmVÞ >45*:*3ðmVÞ for αn ¼ 0*:*5

<sup>c</sup> <sup>¼</sup> <sup>1</sup>*:*26 RT ð Þ *<sup>=</sup>*<sup>F</sup> ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ¼ <sup>V</sup> 32, 4 1ð Þ *<sup>=</sup>*α<sup>n</sup> ð Þ mV ¼ 64*:*8 ðmVÞ > 45*:*3ðmVÞ for αn ¼ 0*:*5

Second derivative was obtained by differentiating the first derivative, Eq. (29).

Both peak potentials (anodic and cathodic) of the second derivatives for totally irreversible processes (ko < 10�<sup>5</sup> cm/s) are directly proportional to log

) for a given αn, moving to cathodic direction as ko decreases, which is the

2a <sup>¼</sup> <sup>E</sup><sup>o</sup> <sup>þ</sup> ð Þ <sup>60</sup>*:*<sup>3</sup> log ko f g ð Þþ <sup>201</sup> ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ mV (37)

2c <sup>¼</sup> Eo <sup>þ</sup> ð Þ <sup>60</sup>*:*<sup>3</sup> log ko f g ð Þþ <sup>67</sup> ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ mV (38)

¼ 118 mV > 68mV for αn ¼ 0*:*5 (39)

The details of the derivation and the results are given elsewhere and following

2c = 2.3(RT/F)(1/αn) = 0.0591(1/αn) (mV)

W½ ¼ 2*:*80 RT ð Þ *=*F ð Þ 1*=*αn ð Þ¼ V 72 1ð Þ *=*αn ð Þ mV

Ep <sup>¼</sup> <sup>E</sup><sup>o</sup> <sup>þ</sup> ð Þ <sup>60</sup>*:*<sup>3</sup> log ko <sup>½</sup> ð Þþ <sup>165</sup>Þ�ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ mV (31)

ip*=*id ¼ 12*:*3ð Þ� αn 6*:*2<9*:*6 for αn ¼ 0*:*5 (32)

¼ 144 ðmVÞ >90*:*4 ðmVÞ for αn ¼ 0*:*5 (33)

), shifting towards cathodic

a

) at 298 K for a totally irreversible reaction are

<sup>c</sup> <sup>¼</sup> <sup>1</sup>*:*28<sup>&</sup>gt; <sup>1</sup>*:*00 for all <sup>α</sup><sup>n</sup> (36)

), and the

(34)

(35)

(k<sup>o</sup> < 10�<sup>5</sup> cm/s) is directly proportional to log (k<sup>o</sup>

The analytical differentiations of i' and expressions for the second- and thirdderivatives are more involved and given elsewhere [15]. Alternatively, it is also possible to obtain second and third derivative curves from successive *numerical* differentiations starting from the values of first derivative.

As shown above, the current expression for or an irreversible e�transfer and quasi-reversible much complicated than that of reversible case, because the current depends not only on the applied potential but also ko and αn which introduces much differences in the peak shapes and peak symmetry. Comparisons of the cases are made in **Figure 2** for a contrast for Er and Eirr processes.

It can be readily seen from the curves that as the reduction becomes more difficult (i.e., with smaller k<sup>o</sup> ) (a) all curves shifts to negative (cathodic) direction for the quasi-reversible and irreversible process, (b) the curves and peaks become widened, and (c) the curve becomes asymmetric which is much pronounced in the second and third derivatives although it is not readily seen from the current and its first derivative. By analyzing the formula and the curves graphically, one can find possible values and their ranges for the following variables or asymmetric parameters listed on **Table 1**: such values of the irreversible process are summarized in **Table 2** along with those of the reversible process for comparisons.

#### *3.1.2.1 For the first derivative*

First derivative is obtained by differentiating Eq. (27) and (28), the details of differentiation and the results were given elsewhere in original work [15] of the author's group: for the irreversible or quasi-reversible process, following relationships are found from plots

#### **Figure 2.**

*Comparison of currents (normalized at 100, black), and their first (blue), second (green), and third derivatives (red). Those for a simple reversible electron transfer are given in solid lines on the left, and those for an irreversible electron transfer reaction (with k<sup>o</sup> = 10*�*<sup>5</sup> cm/s, n = 1, and α = 0.5 at T = 298 K) are in dotted lines on the right. The derivatives were scaled appropriately in order to bring all of them in the same plotting area: Scaling factors are given in the parentheses.*

*Advances in Derivative Voltammetry - A Search for Diagnostic Criteria of Several… DOI: http://dx.doi.org/10.5772/intechopen.96409*

1. Peak Potentials

where

Q' ¼ �ð Þ αnF*=*RT kf*=*Do

*Analytical Chemistry - Advancement, Perspectives and Applications*

differentiations starting from the values of first derivative.

made in **Figure 2** for a contrast for Er and Eirr processes.

**Table 2** along with those of the reversible process for comparisons.

difficult (i.e., with smaller k<sup>o</sup>

*3.1.2.1 For the first derivative*

ships are found from plots

**Figure 2.**

**14**

*area: Scaling factors are given in the parentheses.*

1*=*2

The analytical differentiations of i' and expressions for the second- and thirdderivatives are more involved and given elsewhere [15]. Alternatively, it is also possible to obtain second and third derivative curves from successive *numerical*

As shown above, the current expression for or an irreversible e�transfer and quasi-reversible much complicated than that of reversible case, because the current depends not only on the applied potential but also ko and αn which introduces much differences in the peak shapes and peak symmetry. Comparisons of the cases are

It can be readily seen from the curves that as the reduction becomes more

for the quasi-reversible and irreversible process, (b) the curves and peaks become widened, and (c) the curve becomes asymmetric which is much pronounced in the second and third derivatives although it is not readily seen from the current and its first derivative. By analyzing the formula and the curves graphically, one can find possible values and their ranges for the following variables or asymmetric parameters listed on **Table 1**: such values of the irreversible process are summarized in

First derivative is obtained by differentiating Eq. (27) and (28), the details of differentiation and the results were given elsewhere in original work [15] of the author's group: for the irreversible or quasi-reversible process, following relation-

*Comparison of currents (normalized at 100, black), and their first (blue), second (green), and third derivatives (red). Those for a simple reversible electron transfer are given in solid lines on the left, and those for an irreversible electron transfer reaction (with k<sup>o</sup> = 10*�*<sup>5</sup> cm/s, n = 1, and α = 0.5 at T = 298 K) are in dotted lines on the right. The derivatives were scaled appropriately in order to bring all of them in the same plotting*

–ð Þ ð Þ 1 � α nF*=*RT kb*=*Dr

) (a) all curves shifts to negative (cathodic) direction

<sup>1</sup>*=*<sup>2</sup> (30)

It is found that the peak potential for totally irreversible process (k<sup>o</sup> < 10�<sup>5</sup> cm/s) is directly proportional to log (k<sup>o</sup> ), shifting towards cathodic direction as k<sup>o</sup> decreases for a given αn, and is inversely proportional to αn at a constant ko [15].

$$\mathbf{E\_{p}} = \mathbf{E^{o}} + [(60.3)\log(\mathbf{k^{o}}) + 165)](\mathbf{1/am})\,\mathrm{(mV)}\tag{31}$$

<0 (because log(k<sup>o</sup> ) is always negative for Eirr).

2. Peak Currents

The normalized peak heights is directly proportional to αn, decreasing from 9.6 (for Er) to 6.2 as ko decreases [15]

$$\mathbf{i}\_{\rm p}/\mathbf{i}\_{\rm d} = \mathbf{12.3}(\boldsymbol{\alpha}\mathbf{n}) \sim \mathbf{6.2} < \mathbf{9.6} \text{ for } \boldsymbol{\alpha}\mathbf{n} = \mathbf{0.5} \tag{32}$$

3. Peak-Widths.

The half-peak width (W½), the anodic half-peak width (Wq a ), and the cathodic half-peak width (Wq c ) at 298 K for a totally irreversible reaction are independent of ko , but inversely proportional to αn [15]

$$\begin{array}{l} \text{(W}\_{\text{/2}} = 2.80 \text{(RT/F)} (1/\text{cm}) (\text{V}) = 72 (1/\text{cm}) (\text{mV}) \\ = 144 \ (\text{mV}) > 90.4 \ (\text{mV}) \qquad \text{for } \text{on} \ = \text{ 0.5} \end{array} \tag{33}$$

$$\begin{aligned} \text{(\textbf{W}\_q\textsuperscript{a} = 1.62(\text{RT/F})(1/\text{m})(\text{V}) = 41.7(1/\text{m})(\text{mV})} \\ = \\$3.4 \ (\text{mV}) > 45.3(\text{mV}) \text{ for } \text{on} = \text{0.5} \end{aligned} \tag{34}$$

$$\begin{aligned} \mathbf{W\_{q}}^{c} &= \mathbf{1.26}(\mathbf{RT/F}) (\mathbf{1/am}) (\mathbf{V}) = \mathbf{32}, \mathbf{4(1/am)(mV)} \\ &= \mathbf{64.8 } (\mathbf{mV}) > \mathbf{45.3} (\mathbf{mV}) \text{ for } \mathbf{on} \end{aligned} \tag{35}$$

$$\text{W}\_{\text{q}}\,^{\text{a}}\,/\text{W}\_{\text{q}}\,^{\text{c}} = \textbf{1.28} > \textbf{1.00} \text{ for all } \text{on} \tag{36}$$

#### *3.1.2.2 For the second derivative*

Second derivative was obtained by differentiating the first derivative, Eq. (29). The details of the derivation and the results are given elsewhere and following relationships were found [15]:

1. Peak-Potentials and Peak Separation

Both peak potentials (anodic and cathodic) of the second derivatives for totally irreversible processes (ko < 10�<sup>5</sup> cm/s) are directly proportional to log (k<sup>o</sup> ) for a given αn, moving to cathodic direction as ko decreases, which is the same trend as for Ep in the first derivative.

$$\mathbf{E\_p}^{\ 2\mathbf{a}} = \mathbf{E^o} + \{ (60.3) \log \left( \mathbf{k^o} \right) + 20 \mathbf{1} \} (\mathbf{1/au}) \left( \mathbf{mV} \right) \tag{37}$$

$$\mathbf{E\_{p}}^{\ 2\mathbf{c}} = \mathbf{E^{o}} + \{ (60.3) \log \left( \mathbf{k^{o}} \right) + 67 \} \left( \mathbf{1/on} \right) \left( \mathbf{mV} \right) \tag{38}$$

$$\text{and } \mathbf{E\_p}^{2a} \text{-} \mathbf{E\_p}^{2c} = 2.3 \text{(RT/F)} \,\text{(1/cm)} = 0.0591 \,\text{(1/cm) (mV)}$$

$$= 118 \text{ mV} > 68 \text{mV for am} = 0.5 \tag{39}$$

Both peak potentials (Ep 2a and Ep 2a) are dependent on ko as well as αn values yielding �210 mV and -319 mV respectively, with ko = 10�<sup>5</sup> cm/s and <sup>α</sup>n =0.5. However, the peak potential difference depends only on αn values, not on ko .

$$
\Delta \mathbf{E\_p}^{\text{2a}} / \Delta \mathbf{E\_p}^{\text{2c}} = 210 \text{mV} / 319 \text{mV} = 0.66 < 1.00 \tag{40}
$$

These changes (or sensitivity) of symmetry parameter values with increasing order of

cm/s for a quasi-reversible case, ko = 1.0x10�<sup>5</sup> cm/s for an irreversible

cm/s for a reversible case,

½) = exp.(Qs

O Keq ¼ kf*=*kb ð Þ in solution (45)

� �<sup>1</sup>*=*<sup>2</sup>

ð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup> *<sup>=</sup>*Keq � �<sup>t</sup>

O þ ne� ! R Nernstian at the electrode ð Þ (46)

t

<sup>i</sup>' <sup>¼</sup> idð Þ nF*=*RT 2kf ð Þ<sup>t</sup> e 1 <sup>þ</sup> <sup>e</sup>*=*Keq � �<sup>i</sup> � <sup>1</sup> � � (48)

ð Þ 2kf <sup>t</sup> e 1 <sup>þ</sup> <sup>e</sup>*=*Keq � �<sup>i</sup> <sup>þ</sup> ð Þ RT*=*nF <sup>1</sup> <sup>þ</sup> <sup>e</sup>*=*Keq � �i' � <sup>1</sup> � � (49)

2

t)erfc(Qst

½))

½Þ� (47)

a derivative are also observed in CEr as well as ErC. In short, the values of the symmetry parameters with a ratio of anodic values to cathodic values (all **R's**) are 1.00 for reversible electron transfer reactions; however it will deviates from the unity

(<1.00 or > 1.00) for other types of processes as shown in later sections.

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*3.1.2.5 Notes on computations for quasi-reversible and irreversible processes*

case for simple e� transfer reactions. A value of *αnof* 0.50 is used for all quasireversible and irreversible cases. For quasi-reversible processes, only the case with *αn* ¼ 0.5 is reported. The values for all parameters for quasi-reversible process (with αn = 0.5) lies in between those for reversible and those for irreversible case. Those values for quasi-reversible case with αn 6¼ 0.5 exhibits much complicated behaviors depending on values of αn [15]. This is because depending on values of the kinetic parameters (k<sup>o</sup> and αn) the changes in some dependent variables become not

Propagations of round-off errors, associated with differentiations (or subtractions) which result in a loss of significant figures, encountered in evaluating the derivatives (Eq. 27–30): the error were so severe that an extended quadrupleprecision mode (32 significant digits) had to be employed for all calculation:

)erfc(x): namely F(Qst

*3.1.3 Reversible electron transfer coupled with a prior chemical equilibrium (CEr)*

[1, 42]: the basic rection scheme and final forms of formula are presented here. This mechanism that involves pre-chemical step can be given as follows.

At a more interesting case of Keq< < 1, the currents [1, 13] is given by

Successive differentiations of the current yield the first-, second- and third-

exp Keqkfð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup> *<sup>=</sup>*Keq � �<sup>2</sup>

h �erfc Keqkf

Full derivations of the derivatives from a current expression are given elsewhere

All computations are carried out with k<sup>o</sup> = 1.0x10<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.96409*

monotonic,, often exhibiting discontinuity at around *αn* ¼ 0.3.

(Eq. 28) had to be evaluated to the 32 digit-precision [13].

*3.1.3.1 Theoretical expressions for and the derivatives*

Y ⇌ kf kb

ko = 1.0x10�<sup>3</sup>

namely, the function exp.(x<sup>2</sup>

<sup>i</sup> <sup>¼</sup> id ð Þ <sup>π</sup><sup>t</sup> <sup>1</sup>*=*<sup>2</sup> Keqkf

derivatives, respectively.

<sup>i</sup>" <sup>¼</sup> idð Þ nF*=*RT <sup>2</sup>

**17**

Second derivative becomes

Third derivative becomes

� �<sup>1</sup>*=*<sup>2</sup>

2. Peak-Heights and Their Ratios.

A plot of the two peak-heights (ip <sup>a</sup> for anodic and ip <sup>c</sup> cathodic side) of the second derivative depend heavily on αn, a magnitude of both heights increases with increasing αn. However, ip <sup>c</sup> is greater than ip <sup>a</sup> for most of the case unless αn = 0.3 which is strange.

$$|\mathbf{i\_{p}}^{\ a}/\mathbf{i\_{p}}^{c}| = 0.78 < \mathbf{1.00} \text{ for } \mathfrak{an} = \mathbf{0.5} \tag{41}$$

3. Peak-Widths and Their Ratio.

Two half-peak-width (anodic and cathodic) of the second derivatives, and the ratio for Eirr processes are given below

$$\begin{aligned} \text{W}\_{\text{V}}{}^{\text{2a}} &= 2.10 \text{(RT/F)} (1/\text{cm}) \text{ (V)} = 54.0 (1/\text{cm}) \text{ (mV)} \\ &= 108 \text{ mV} > 64 \text{ mV for } \text{cm} = \text{0.5} \end{aligned} \tag{42}$$

$$\begin{aligned} \text{W}\_{\text{V}}{}^{\text{2c}} &= 1.53 (\text{RT/F}) (1/\text{cm}) \text{ (V)} = 39.4 (1/\text{cm}) \text{ (mV)} \\ &= 78.8 \text{ mV} > 64 \text{ mV for } \text{cm} = \text{0.5} \end{aligned} \tag{43}$$

$$\begin{aligned} \text{W}\_{\text{V}}{}^{\text{2a}} / \text{W}\_{\text{V}}{}^{\text{2c}} &= 1.37 > 1.00 \text{ for all } \text{cm} \end{aligned} \tag{44}$$

Here again, it should be noted that although the two half-peak-widths depends on αn, their ratio is invariant.

## *3.1.2.3 For the third derivatives*

Third derivative are obtained by differentiating the second derivative. The details of the derivation and results were given elsewhere [15]. The number of parameters increases, and results on the parameters increase as three peaks are available, and those parameters (peak separations, peak-current ratios, and halfpeak ratios were analyzed from various plots. The results with possible ranges of the values are summarized in **Table 2**.

#### *3.1.2.4 Sensiivity increases with increasing order of derivatives*

In **Table 2**, the key results from irreversible process (as well as two chemically coupled processes) are juxtaposed here for a ready comparison of the differences observed among the types of mechanisms. Basically, the symmetry observed in the curves for the reversible process, disappears as the electron transfer rates become slower (i.e., irreversible type): the unity (1.00) values of the ratio parameters becomes no longer 1.00 for irreversible (Eirr) process and for other types of electrode mechanisms. Namely, the ratio of anodic to cathodic peak-widths (W½ a /W½ c ) increases to 1.27 (an increase of 27%) for the first derivative, and it increases to 1.37 (an increase of 37%) for the second derivative, while it increases to1.62 (an increase of 62%) for the third derivative. In general these ratios increases systematically, as an order of a derivative increases; the higher the order is, the larger the changes are.

*Advances in Derivative Voltammetry - A Search for Diagnostic Criteria of Several… DOI: http://dx.doi.org/10.5772/intechopen.96409*

These changes (or sensitivity) of symmetry parameter values with increasing order of a derivative are also observed in CEr as well as ErC. In short, the values of the symmetry parameters with a ratio of anodic values to cathodic values (all **R's**) are 1.00 for reversible electron transfer reactions; however it will deviates from the unity (<1.00 or > 1.00) for other types of processes as shown in later sections.

#### *3.1.2.5 Notes on computations for quasi-reversible and irreversible processes*

All computations are carried out with k<sup>o</sup> = 1.0x10<sup>1</sup> cm/s for a reversible case, ko = 1.0x10�<sup>3</sup> cm/s for a quasi-reversible case, ko = 1.0x10�<sup>5</sup> cm/s for an irreversible case for simple e� transfer reactions. A value of *αnof* 0.50 is used for all quasireversible and irreversible cases. For quasi-reversible processes, only the case with *αn* ¼ 0.5 is reported. The values for all parameters for quasi-reversible process (with αn = 0.5) lies in between those for reversible and those for irreversible case. Those values for quasi-reversible case with αn 6¼ 0.5 exhibits much complicated behaviors depending on values of αn [15]. This is because depending on values of the kinetic parameters (k<sup>o</sup> and αn) the changes in some dependent variables become not monotonic,, often exhibiting discontinuity at around *αn* ¼ 0.3.

Propagations of round-off errors, associated with differentiations (or subtractions) which result in a loss of significant figures, encountered in evaluating the derivatives (Eq. 27–30): the error were so severe that an extended quadrupleprecision mode (32 significant digits) had to be employed for all calculation: namely, the function exp.(x<sup>2</sup> )erfc(x): namely F(Qst ½) = exp.(Qs 2 t)erfc(Qst ½)) (Eq. 28) had to be evaluated to the 32 digit-precision [13].
