**3. Results and discussions**

#### **3.1 Derivation of the original current expression for various types of electrode mechanisms**

#### *3.1.1 Simple reversible electron transfer reaction (Erev or Er)*

A theoretical current (i) as a function of the applied potential (E), under semiinfinite linear diffusion, is expressed by the following relationship for a simple reversible electron-transfer process [1, 38]

$$\text{O} + \text{ne}^- \rightarrow \text{R (Nernstein)}\tag{1}$$

$$\text{id}(\mathbf{E}) = \text{id}\left[\mathbf{1}/(\mathbf{1} + \mathbf{e})\right] \tag{2}$$

where


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

**2.2 Peak currents, peak potentials, and peak widths and the ratios**

*Analytical Chemistry - Advancement, Perspectives and Applications*

Theoretical expressions for the first- second- and third- derivatives of currentpotential curves were analytically derived from successive differentiations of the original current expression (i.e., zeroth order derivative) obtained; then the derivatives as function of potential were computed and plotted. The raw and derived parameters (**Figure 1**) for the simple reversible process (Erev), calculated from the theoretical equations (Eq. 1–21; refer to Section 3.1.1) for lower order derivatives and those values given in the last column of **Table 1**; this analytical approach requires only a pencil and papers. However, for third derivatives for Erev and all derivatives for other types (Eiir, CEr, and ErC), it is difficult to derive analytical solutions (equations) to calculate values of the parameter. Thus, numerical

approaches had to be employed; namely, the values of a parameter as a function of an independent variable (such as a rate constant, k) had to be plotted on a graphs first, then a possible value or ranges of values were read from the graphs, and an equation for the relationships to calculate the parameters had to be found from such graphs, and this numerical approach requires much of computations, graph papers

graphically are summarized (refer to the results sections for details) in **Table 2** for

Theoretical equations for first, second and third derivatives for irreversible (and quasi reversible) processes are obtained with successive differentiations of analytical expressions of the current [15]. The derivatives of the CE type of mechanism are obtained with successive differentiations in a similar fashion [12, 13, 42]. Whenever possible, analytical solutions are sought after first because it is computationally less expensive. Numeral differentiations had to be adopted when analytical solution cannot be found. Thus, the analytical solutions of current expression and it's first derivative for the EC type of mechanisms are so complicated that analytical differentiations for the second and third derivatives were practically impossible [11, 14], numerical differentiations had to be adopted [14]. A value of ΔE = 1.0 mV was used for a finer resolution for all curves. The differentiation methods are summarized in **Table 3**.

**3.1 Derivation of the original current expression for various types of electrode**

A theoretical current (i) as a function of the applied potential (E), under semiinfinite linear diffusion, is expressed by the following relationship for a simple

> O þ ne� ! R Nernstian ð Þ (1) i Eð Þ¼ id 1½ � *=*ð Þ 1 þ e (2)

*3.1.1 Simple reversible electron transfer reaction (Erev or Er)*

reversible electron-transfer process [1, 38]

and a ruler. Those values of parameters, found analytically or numerically/

**2.3 Obtaining theoretical derivatives for various types of electrode mechanisms - analytical method vs. numerical (digital) method**

four types of the mechanisms for comparisons.

**3. Results and discussions**

**mechanisms**

where

**8**


*Notes: (a) Calculated for n=1and T=298<sup>o</sup> K for the reversible e*� *transfer rection (Third Column).*

*(b) All values for the irreversible case were calculated for ko =1.0x10*�*<sup>5</sup> cm/s and αn=0.50;*

*For the case with αn* 6¼ *0.50, these values are not restricted to the ranges given above. those values the parentheses are calculated with ks=1.0x10*�*<sup>5</sup> cm/s*

*(c) Values for quasi-reversible processes, calculated for αn=0.5, lie in between those for reversible processes and those for irreversible processes, hence the column for quasi-reversible cases is omitted.*

#### **Table 2.**

*Comparisons of values independent variable and derived parameters for various mechanisms.*

$$\mathbf{i}\_d = \mathbf{n} \mathbf{F} \mathbf{A} \mathbf{D}\_o^{-1/2} \mathbf{C}\_o / (\pi \mathbf{t})^{1/2} \text{ (the diffusion-controlled current)} \tag{3}$$

$$\mathbf{e} = \exp\left(\mathbf{nFE/RT}\right) \tag{4}$$

*3.1.1.1 For the first derivatives*

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

Wq

*3.1.1.2 For the second derivatives*

1. Peak Potentials

2. Peak Currents

3. Peak Widths

**11**

W1/2

<sup>a</sup> <sup>¼</sup> Wq

Wq a *=*Wq i 0

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

**Figure 1B**) and their ratios. For a reversible process it was found [14].

One can examine the second derivative (i"), Eq. (6), to find

ated with the peaks. The two peak heights for the anodic, ip

2a � Ep

∣Ep 2a*=*Ep

ip

∣ip 2a*=*ip

2a ¼ �ip

Following relationships are found for Er type.

Ep

peaks are defined at two peak potentials, Ep

W1*<sup>=</sup>*<sup>2</sup>

2a <sup>¼</sup> W1*<sup>=</sup>*<sup>2</sup>

W1*<sup>=</sup>*<sup>2</sup>

2a*=*W1*<sup>=</sup>*<sup>2</sup>

The peak-potentials in Eq. 15 can be found by solving i'"(E) = 0 (Eq. 7) for E.

Thus, the second derivative (**Figure 1C**, solid line) *has the same symmetry as the*

2c which are the peak-widths for the anodic (positive) and cathodic (negative) peaks in the second derivative. It should be pointed out that the adjectives "anodic" and "cathodic" in this context are *not* related to the actual redox processes associ-

2a and Ep

*current with respect to its inflection point*. One can define shape parameters W1/2

negative (cathodic) parts to define two semi-peak widths (Wq

ð Þ¼ �E i 0

This means that the *first derivative is symmetrical with respect to an axis E = 0* (**Figure 1B**, solid line). The value of the peak-width at a half-height for the reversible process proved to be 90.5/n mV at 298 K [18]. A shape parameter from the first derivative can be defined; the peak widths are divided into positive (anodic) and

ð Þ E (10)

W1*<sup>=</sup>*<sup>2</sup> ¼ 90*:*5mV (11)

<sup>c</sup> <sup>¼</sup> 1 at any temperature (13)

2c <sup>¼</sup> <sup>68</sup>*=*n mV at 298 K ð Þ (15)

2c<sup>∣</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>00</sup> (16)

2c<sup>∣</sup> <sup>¼</sup> <sup>l</sup>*:*<sup>00</sup> (18)

2c <sup>¼</sup> <sup>1</sup>*:*<sup>00</sup> (20)

2c <sup>¼</sup> <sup>62</sup>*:*5*=*n mV at 298 K ð Þ or (19)

2c (17)

<sup>c</sup> <sup>¼</sup> ð Þ <sup>1</sup>*=*<sup>2</sup> W1*<sup>=</sup>*<sup>2</sup> <sup>¼</sup> <sup>45</sup>*:*3*=*n mV at 298 K (12)

i}ð Þ¼� �E i}ð Þ E and i}ð Þ¼ 0 0 (14)

<sup>a</sup> and Wq

c ,

2a and

2c,

2a, and the cathodic, ip

2c, respectively (**Figure 1C**).

Eq. 5 shows

and E is an applied potential with respect to the reversible polarographic halfwave-potential, E1/2 (rev) or Eh. Other parameters and variables have their usual meanings. Eq. 2 is derived assuming DO = DR. Successive differentiations of Eq. 2 with respect to E yields [15].

$$\mathbf{i}'(\mathbf{E}) = -\mathbf{i}\_d(\mathbf{nF}/\mathbf{RT}) \left[ \mathbf{e}/(\mathbf{1} + \mathbf{e})^2 \right] \tag{5}$$

$$\mathbf{i}''(\mathbf{E}) = -\mathrm{id}(\mathbf{nF}/\mathbf{RT})^2 \left[ \mathbf{e} (\mathbf{e} - \mathbf{1})/(\mathbf{1} + \mathbf{e})^3 \right] \tag{6}$$

$$\mathbf{i}\text{ "}\text{ (E) = }-\text{id}(\text{nF/RT})^3 \left[ \mathbf{e} \left( \mathbf{e}^2 - 4\mathbf{e} + \mathbf{1} \right) / \left( \mathbf{1} + \mathbf{e} \right)^4 \right] \tag{7}$$

Examination of Eq. 2 reveals that

$$\dot{\mathbf{i}}(-\mathbf{E}) = \dot{\mathbf{i}}\_d - \dot{\mathbf{i}}(\mathbf{E}) \tag{8}$$

$$\mathbf{i}(\mathbf{0}) = \mathbf{i}\_{\mathbf{d}}/2 \tag{9}$$

This implies that *the i-E curve (solid line, Figure 1A) is symmetrical with respect to its inflection point (0, id/2)*.


**Table 3.**

*Methods of differentiation for various electrode mechanisms. Derivatives are obtained from the analytical solutions of currents for each cases, except for i", i"' with the post kinetics system (ErC).*

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

#### *3.1.1.1 For the first derivatives*

Eq. 5 shows

$$\dot{\mathbf{i}}'(-\mathbf{E}) = \dot{\mathbf{i}}'(\mathbf{E}) \tag{10}$$

This means that the *first derivative is symmetrical with respect to an axis E = 0* (**Figure 1B**, solid line). The value of the peak-width at a half-height for the reversible process proved to be 90.5/n mV at 298 K [18]. A shape parameter from the first derivative can be defined; the peak widths are divided into positive (anodic) and negative (cathodic) parts to define two semi-peak widths (Wq <sup>a</sup> and Wq c , **Figure 1B**) and their ratios. For a reversible process it was found [14].

$$\mathbf{W}\_{1/2} = \mathbf{90.5mV} \tag{11}$$

$$\mathbf{W\_q^a} = \mathbf{W\_q^c} = (\mathbf{1}/2)\mathbf{W\_{1/2}} = \mathbf{45.3/n}\text{ mV at 298 K} \tag{12}$$

$$\mathbf{W\_q \triangleq / W\_q}^\* = \mathbf{1} \text{ at any temperature} \tag{13}$$

#### *3.1.1.2 For the second derivatives*

id ¼ nFADo

W½

*Notes: (a) Calculated for n=1and T=298<sup>o</sup>*

*calculated with ks=1.0x10*�*<sup>5</sup>*

**Table 2.**

**Rw**

**Rw**

**Rw**

*(b) All values for the irreversible case were calculated for ko*

*cm/s*

i}0

Examination of Eq. 2 reveals that

**Types of Mechanisms Current**

with respect to E yields [15].

*its inflection point (0, id/2)*.

**Table 3.**

**10**

1*=*2

*for irreversible processes, hence the column for quasi-reversible cases is omitted.*

*Analytical Chemistry - Advancement, Perspectives and Applications*

i 0

<sup>i</sup>}ð Þ¼� <sup>E</sup> id nF ð Þ *<sup>=</sup>*RT <sup>2</sup>

**i**

ErC(Post Chemical) i di'/dE *Δi"/ΔE*

*solutions of currents for each cases, except for i", i"' with the post kinetics system (ErC).*

Co*=*ð Þ <sup>π</sup><sup>t</sup> <sup>1</sup>*=*<sup>2</sup> ð Þ the diffusion‐controlled current (3) e ¼ exp nFE ð Þ *=*RT (4)

<sup>2</sup> h i

e eð Þ � <sup>1</sup> *<sup>=</sup>*ð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup> <sup>3</sup> h i

<sup>4</sup> h i

ið Þ¼ �E id � i Eð Þ (8) i 0ð Þ¼ id*=*2 (9)

> **2nd Der. i"**

di'/dE di"/dE di"'/dE

*(numerical)*

(5)

(6)

(7)

**3rd Der. i"'**

*Δi"'/ΔE (numerical)*

and E is an applied potential with respect to the reversible polarographic halfwave-potential, E1/2 (rev) or Eh. Other parameters and variables have their usual meanings. Eq. 2 is derived assuming DO = DR. Successive differentiations of Eq. 2

**Parameters Definitions Er Eirr ErC CEr**

3c (mV) 54 >54 (63) >32, <42 >50, <54

**<sup>3</sup> W½3a/W½3c 1.00 >1.00 (1.62) >1.0, <1.7 >1.00,<1.19**

**3a W½3a/W½3m 1.32 >1.32 (1.60) >1.3, <1.6 >1.19,<1.32**

**3c W½3c/W½3m 1.32 <1.32 (0.98) >1.0, <1.3 >1.32, <1.43**

*K for the reversible e*� *transfer rection (Third Column).*

*cm/s and αn=0.50;*

*=1.0x10*�*<sup>5</sup>*

*For the case with αn* 6¼ *0.50, these values are not restricted to the ranges given above. those values the parentheses are*

*(c) Values for quasi-reversible processes, calculated for αn=0.5, lie in between those for reversible processes and those*

*Comparisons of values independent variable and derived parameters for various mechanisms.*

ð Þ¼� E idð Þ nF*=*RT e*=*ð Þ 1 þ e

ð Þ¼� <sup>E</sup> id nF ð Þ *<sup>=</sup>*RT <sup>3</sup> e e<sup>2</sup> � 4e <sup>þ</sup> <sup>1</sup> � �*=*ð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup>

This implies that *the i-E curve (solid line, Figure 1A) is symmetrical with respect to*

**1sit Der. i'**

Er(rev) i di'/dE di"/dE di"'/dE Eirr(irreversible) i di'/dE di"/dE di"'/dE CEr (Prior Chemical) i di'/dE di"/dE di"'/dE

*Methods of differentiation for various electrode mechanisms. Derivatives are obtained from the analytical*

One can examine the second derivative (i"), Eq. (6), to find

$$\mathbf{i}\,\mathbf{i}''(-\mathbf{E}) = -\mathbf{i}\,\mathbf{i}''(\mathbf{E}) \text{ and } \mathbf{i}\,\mathbf{i}''(\mathbf{0}) = \mathbf{0} \tag{14}$$

Thus, the second derivative (**Figure 1C**, solid line) *has the same symmetry as the current with respect to its inflection point*. One can define shape parameters W1/2 2a and W1/2 2c which are the peak-widths for the anodic (positive) and cathodic (negative) peaks in the second derivative. It should be pointed out that the adjectives "anodic" and "cathodic" in this context are *not* related to the actual redox processes associated with the peaks. The two peak heights for the anodic, ip 2a, and the cathodic, ip 2c, peaks are defined at two peak potentials, Ep 2a and Ep 2c, respectively (**Figure 1C**).

Following relationships are found for Er type.

1. Peak Potentials

$$\mathbf{E\_p}^{\text{2a}} - \mathbf{E\_p}^{\text{2c}} = \mathbf{68} / \mathbf{n} \text{ mV (at 298 K)}\tag{15}$$

$$|\mathbf{E\_p}^{\text{2a}}/\mathbf{E\_p}^{\text{2c}}| = \mathbf{1.00} \tag{16}$$

2. Peak Currents

$$\mathbf{i\_p}^{\text{2a}} = -\mathbf{i\_p}^{\text{2c}} \tag{17}$$

$$|\mathbf{i\_p}^{\text{2a}}/\mathbf{i\_p}^{\text{2c}}| = \text{l.00} \tag{18}$$

3. Peak Widths

$$\left| \mathbf{W\_{1/2}}^{\text{2a}} = \mathbf{W\_{1/2}}^{\text{2c}} = \mathbf{62.5/n} \text{ mV (at 298 K) or} \right. \tag{19}$$

$$\mathbf{W\_{1/2}}^{\text{Ta}} / \mathbf{W\_{1/2}}^{\text{2c}} = \mathbf{1.00} \tag{20}$$

The peak-potentials in Eq. 15 can be found by solving i'"(E) = 0 (Eq. 7) for E.

$$-\text{id}(\mathbf{nF}/\mathbf{RT})^3 \left[ \mathbf{e} \left( \mathbf{e}^2 - 4\mathbf{e} + \mathbf{1} \right) / \left( \mathbf{1} + \mathbf{e} \right)^4 \right] = \mathbf{0} \tag{21}$$

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 quasi-

In the following sections, other types of processes than the simple reversible

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

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

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

E1*<sup>=</sup>*<sup>2</sup> <sup>¼</sup> Eo <sup>þ</sup> ð Þ RT*=*nF ln Dð Þ <sup>R</sup>*=*DO

F Qt<sup>1</sup>*=*<sup>2</sup> � � <sup>¼</sup> exp Q<sup>2</sup>

surface area, mostly is dependent on the heterogeneous rate constants (ko

current with respect to potential (E) yields the first derivative,

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

in <sup>¼</sup> ð Þ <sup>π</sup>t*=*<sup>D</sup> <sup>1</sup>*=*<sup>2</sup>

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

kf exp Q<sup>2</sup>

This normalized current, which is independent of the concentration and electrode

kf ð Þ �nF*=*RT F Qt1*<sup>=</sup>*<sup>2</sup> � � <sup>þ</sup> 2Q' QtF Qt1*<sup>=</sup>*<sup>2</sup> � � � ð Þ <sup>t</sup>*=*<sup>π</sup>

<sup>h</sup> h i (29)

, by adopting typical values of D (=5x10�<sup>5</sup> cm2

*cm/s,*

t � �erfc Qt½ � � � � (22)

<sup>Q</sup> <sup>¼</sup> ð Þ kf <sup>þ</sup> kb *<sup>=</sup>*D<sup>½</sup> (23)

t � �erfc Qt½ � � � � (27)

t � �erfc Qt½ � � (28)

<sup>½</sup> <sup>¼</sup> Eo (26)

), the

1*=*2 Þ

kf <sup>¼</sup> ko exp �αnF E � Eo f g ð Þ*=*RT (24)

kb <sup>¼</sup> <sup>k</sup><sup>o</sup> exp 1ð Þ � <sup>α</sup> nF E � Eo f g ð Þ*=*RT (25)

/s) and t

reversible if those competing rates are comparable (k<sup>o</sup> � (D/t)1/2).

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

*cm/s.*

<sup>I</sup> <sup>¼</sup> nFACokf exp Q<sup>2</sup>

In terms of values of k<sup>o</sup>

process will be treated.

assumed DR = DO = D.

*transfer*

details, where,

where,

<sup>i</sup>' <sup>¼</sup> ð Þ <sup>π</sup>t*=*<sup>D</sup> <sup>1</sup>*=*<sup>2</sup>

**13**

*reversible for ko > 0.020 cm/s,*

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

*irreversible for k<sup>o</sup> < 5x10*�*<sup>5</sup>*

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

forms of the following equations for a planar diffusion:

(=1.000 s).

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 derivative curves are symmetrical.

#### *3.1.1.3 For the third derivatives*

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, the same as in lower order derivatives.

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 equations (relationships) for those parameters.
