**1.3 Basic scheme with symmetry in the peaks associated with derivative voltammetry and definitions of various parameters from the plots of currents and it's derivatives**

Please refer to the **Figure 1** for the definitions of all the dependent variables associated with the current-potential curve (zeroth order derivative) and it's first, second and third derivatives. All measured variable and derived variables associated

#### **Figure 1.**

*Normal pulse voltammograms (A) and their higher order derivatives: 1st order (B), 2nd order (C), and 3rd derivatives (D). Several important experimental variables (parameters) such as peak-potentials, peak-currents and peak-widths are defined on the graph,, for two different electrode mechanisms; (1) for a simple Erev type mechanism (solid lines), and (2) for a reversible electron transfer coupled with a follow-up chemical reaction ErC type mechanism (dotted line). Potentials (x-axis) are with respect to a formal potential, E<sup>o</sup> , or E1/2 for the Erev reaction. Calculated for T = 298 K and t = 0.952 s. the currents are normalized with respect to the diffusion currents and dimensionless; thus, the unit of first derivatives is V<sup>1</sup> , second derivatives is V<sup>2</sup> , third derivatives is V<sup>3</sup> .Note: For the original currents (A), only the half-wave potential (E1/2) is graphically defined. Graphic definitions of the quarter-wave potential (E1/4) and three-quarter-wave potential (E3/4), are omitted in order to make the figure simple.*

with current and the derivatives are listed in **Table 1** along with those values for a reversible e- transfer process, which is the most simple type [15].

#### *1.3.1 Symmetry parameters*

A quantitative measure of symmetry *in the original current are not readily available*, but can be found indirectly (and inconveniently) from a ratio of difference in several potentials defined from an original voltammogram (zeroth order deivatives): namely, a quarter-wave potentials (E1/4, at which a current reaches a quarter of the diffusion current, id), half-wave potentials (E1/2 or Eh) and a threequarter-wave potentials, E3/4, at which a current becomes (3/4)id); then a ratio of an anodic to cathodic quarter-wave potential differences from a half-wave potentials (i.e., **ΔEq a /ΔEq <sup>c</sup>** = (E1/2-E1/4)/(E1/2-E3/4) = **Rq o** ) is calculated. It is very limited.

three peak currents an their ratios, and three half-peaks widths and their ratios. Among the ratios from third derivative, the ratio of anodic to cathodic peak poten-

*Definitions of original and derived parameters and the values for the simple reversible e transfer reaction.*

as symmetry parameters: and *all of these values of symmetry parameters are unity (1.00) for an Er type*. Unlike Er that exhibits a well-defined perfect symmetry in the derivative curves, these parameter values become no longer unity for other type of slower electron transfer mechanisms (Eirr, and Equasirev) or a fast e transfer reaction (Erev) is chemically coupled homogeneously (namely, CEr and ErC type),

**3c**), and a ratio of anodic to cathodic half-peak widths (**Rw**

**3c**) are most notable. All of these ratios are particularly noticeable being utilized

As far as analysis of *the derivatives* concerned, most of the work have dealt with only the linear diffusion (on a planar electrode) or semi-infinite liner diffusion (on DME, or SMDE), not with spherical diffusion (on a spherical electrode), even

**<sup>3</sup> = ΔE3a/ΔE3c**), the anodic to cathodic peak currents ratio

**Parameters Definitions Er Parameters Definitions Er**

**<sup>c</sup> 1.0 RΔ<sup>E</sup>**

**<sup>c</sup> 1.00** W½

2c 68 **Rw**

**2c 1.00 Rw**

3a (mV) 59

3m (mV) 0.00

3c (mV) -59

3a-Ep

3a-Ep

3m-Ep

**<sup>3</sup> ΔE3a/ ΔE3c 1.00**

3a 2461

3m -7382

3c 2461

**3a/ip**

**3a/ip**

**3c/ip**

3a (mV) 54

3m (mV) 41

3c (mV) 54

**<sup>3</sup> W½3a/W½3c 1.00**

**3a W½3a/W½3m 1.32**

**3c W½3c/W½3m 1.32**

3c 118

3m 59

3c 59

**3c 1.00**

**3m| 0.32**

**3m| 0.32**

<sup>p</sup> (mV) Ep

<sup>a</sup> (mV) Ep

<sup>c</sup> (mV) Ep

**<sup>3</sup> ip**

**3a |ip**

**3c |ip**

*Current* **3***rd Der.*

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

ΔEq

ΔEq

ΔEh

ΔE1

Wq

Wq

Ep

Ep

Ep

**R**Δ**<sup>E</sup>**

ip

ip

**R <sup>i</sup>**

**Rw**

W½

W½

*Calculated for n = 1 and at 298 K.*

**Rq**

Ri

**Rw**

**<sup>0</sup> ΔEq**

**<sup>1</sup> Wq**

<sup>2</sup> (mV) Ep

**<sup>2</sup>** Δ**Ep**

**<sup>2</sup> |ip**

**a /ΔEq**

**a /Wq**

*2nd Der.* W½

2a-Ep

**2a/**Δ**Ep**

2a 14

2c -14

2a (mV) 64

2c (mV) 64

**<sup>2</sup> W½2a/W½2c 1.00**

**2a/ip**

*1st Der.* ip

E1/4 (mV) +28 Ep

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

E1/2 (mV) 0 Ep

E3/4 (mV) -28 Ep

<sup>a</sup> (mV) E1/4-E1/2 28 ΔE3

<sup>c</sup> (mV) E1/2-E3/4 28 ΔE<sup>3</sup>

<sup>0</sup> (mV) E1/4-E3/4 56.4 ΔE3

<sup>p</sup> (mV) Ep-Eo 0.0 ip

W½ (mV) 90.5 **Ri**

<sup>a</sup> (mV) 45.3 **Ri**

<sup>c</sup> (mV) 45.3 **Ri**

2a (mV) 34 W½

2c (mV) -34 **Rw**

**2c| 1.00**

<sup>1</sup> ip/id 9,7 ip

**<sup>3</sup> = W½**

**3a/**

tial difference (**RΔ<sup>E</sup>**

**3a/ip**

introducing *asymmetry* in those curves.

(**Ri <sup>3</sup> = ip**

**Table 1.**

**W½**

**5**

However, measurements of symmetry parameters from *the first derivatives* are much easier than those from the oignal current, yielding a variety of (half a dozen) of variables. Namely, in addition to that based on peak potential differences (with respect to Eo ), a peak current, half-peak width and anodic and cathodic and quarter-peak widths, and a ratio of anodic to cathodic quarter-peak width (**Rw <sup>1</sup> = Wq a /Wq c** ) are also available. *The second derivative* can be characterized by more (about ten) variables, such as peak potentials and their ratio (**RΔ<sup>E</sup> <sup>2</sup> = ΔEp 2a/ ΔEp 2c**), peak currents and their ratio (**Ri <sup>2</sup> = |ip 2a/ip 2c|**), and half-peak widths and a ratio of anodic to cathodic half-peak widths (**Rw <sup>2</sup> = W½ 2a/W½ 2c**). For *the third derivative*, with appearance of an additional third peak, the number of variables doubles to about twenty: three peak potentials, the differences among them, and


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

#### **Table 1.**

with current and the derivatives are listed in **Table 1** along with those values for a

*Normal pulse voltammograms (A) and their higher order derivatives: 1st order (B), 2nd order (C), and 3rd derivatives (D). Several important experimental variables (parameters) such as peak-potentials, peak-currents and peak-widths are defined on the graph,, for two different electrode mechanisms; (1) for a simple Erev type mechanism (solid lines), and (2) for a reversible electron transfer coupled with a follow-up chemical reaction*

*Erev reaction. Calculated for T = 298 K and t = 0.952 s. the currents are normalized with respect to the diffusion*

*.Note: For the original currents (A), only the half-wave potential (E1/2) is graphically defined. Graphic definitions of the quarter-wave potential (E1/4) and three-quarter-wave potential (E3/4), are omitted in order*

*ErC type mechanism (dotted line). Potentials (x-axis) are with respect to a formal potential, E<sup>o</sup>*

A quantitative measure of symmetry *in the original current are not readily available*, but can be found indirectly (and inconveniently) from a ratio of difference in

**o**

) are also available. *The second derivative* can be characterized by

**2a/ip**

**<sup>2</sup> = W½**

**2a/W½**

*, second derivatives is V<sup>2</sup>*

) is calculated. It is very limited.

**2c|**), and half-peak widths and a

**2c**). For *the third*

**<sup>2</sup> = ΔEp**

*, or E1/2 for the*

*, third derivatives*

**2a/**

deivatives): namely, a quarter-wave potentials (E1/4, at which a current reaches a quarter of the diffusion current, id), half-wave potentials (E1/2 or Eh) and a threequarter-wave potentials, E3/4, at which a current becomes (3/4)id); then a ratio of an anodic to cathodic quarter-wave potential differences from a half-wave poten-

However, measurements of symmetry parameters from *the first derivatives* are much easier than those from the oignal current, yielding a variety of (half a dozen) of variables. Namely, in addition to that based on peak potential differences (with

), a peak current, half-peak width and anodic and cathodic and

**<sup>2</sup> = |ip**

*derivative*, with appearance of an additional third peak, the number of variables doubles to about twenty: three peak potentials, the differences among them, and

reversible e- transfer process, which is the most simple type [15].

*currents and dimensionless; thus, the unit of first derivatives is V<sup>1</sup>*

*Analytical Chemistry - Advancement, Perspectives and Applications*

several potentials defined from an original voltammogram (zeroth order

**<sup>c</sup>** = (E1/2-E1/4)/(E1/2-E3/4) = **Rq**

quarter-peak widths, and a ratio of anodic to cathodic quarter-peak width

more (about ten) variables, such as peak potentials and their ratio (**RΔ<sup>E</sup>**

*1.3.1 Symmetry parameters*

*to make the figure simple.*

**a /ΔEq**

**2c**), peak currents and their ratio (**Ri**

ratio of anodic to cathodic half-peak widths (**Rw**

tials (i.e., **ΔEq**

**Figure 1.**

*is V<sup>3</sup>*

respect to Eo

**<sup>1</sup> = Wq a /Wq c**

(**Rw**

**ΔEp**

**4**

*Definitions of original and derived parameters and the values for the simple reversible e transfer reaction. Calculated for n = 1 and at 298 K.*

three peak currents an their ratios, and three half-peaks widths and their ratios. Among the ratios from third derivative, the ratio of anodic to cathodic peak potential difference (**RΔ<sup>E</sup> <sup>3</sup> = ΔE3a/ΔE3c**), the anodic to cathodic peak currents ratio (**Ri <sup>3</sup> = ip 3a/ip 3c**), and a ratio of anodic to cathodic half-peak widths (**Rw <sup>3</sup> = W½ 3a/ W½ 3c**) are most notable. All of these ratios are particularly noticeable being utilized as symmetry parameters: and *all of these values of symmetry parameters are unity (1.00) for an Er type*. Unlike Er that exhibits a well-defined perfect symmetry in the derivative curves, these parameter values become no longer unity for other type of slower electron transfer mechanisms (Eirr, and Equasirev) or a fast e transfer reaction (Erev) is chemically coupled homogeneously (namely, CEr and ErC type), introducing *asymmetry* in those curves.

As far as analysis of *the derivatives* concerned, most of the work have dealt with only the linear diffusion (on a planar electrode) or semi-infinite liner diffusion (on DME, or SMDE), not with spherical diffusion (on a spherical electrode), even

though there are many studies on the original currents (not derivatives) with spherical electrodes from earlier days of polarographic/voltammetric works as given in general monographs [30, 38, 43, 44] as well as in articles on particular systems and methods [45–55]. These include studies on EC mechanisms [45, 46], CE mechanisms [46, 47, 54], on various pulse polarographic methods [48, 55], on DC polarography [49], on AC polarography [72], on cyclic voltammetry [51], on catalytic EC mechanism with the square-wave voltammetry [52], on a double potential method [53] and on DPV at spherical and microelectrodes [56].

convention [12–15, 42]. In addition to the analysis of peak in term of peak-currents and peak-potentials [57–60], an analysis in terms of half-peak-widths was introduced later [61]. In present study a focus is given more to the *second derivative* because (a) it generates two well-defined peaks of anodic and cathodic that parallel to two peaks of DPP, (b) it is more sensitive and practical than first derivative which has only one peak and fewer visible parameters, and (c) it is less complicated and more practical than third derivative even though it is less sensitive than the

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

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

*1.3.4 Other approaches for diagnostic criteria with cyclic square-wave voltammetry*

Cyclic square-wave voltammetry (CSWV), which is an extension of SWV [62–65], has been adopted by Bottomley group [66–70]. Like pulse voltammetric techniques, SWV is effective in removing the interference from the capacitive current that is inherent in cyclic voltammetry(CV). SWV, like DPV, is a pseudoderivative technique in which a derivative-shaped peak is obtained from the current with electronic control of input of square-wave potentials and current sampling times [62]. Diagnostic criteria as a tool for studying electrode mechanisms based on CSWV scheme have been investigated by the group for reversible [66] and quasireversible [67] processes, EC type [68], ECE type [69], CE type [70] mechanisms, and other non-unity stoichiometric cases [71]. Basically, the impact of experimental SW parameters - such as pulse heights (or amplitude), step heights (or potential increments), switching potentials, and period – on the peak-currents, peakpotentials and separations, peak-widths were examined. It was found that those observed peak-related quantities changes characteristically according to each particular types of the electrochemical mechanisms, which make it feasible to be

In general, original signals (y) from a measurement can be given in a polyno-

*<sup>y</sup>* <sup>¼</sup> <sup>a</sup>*x*<sup>n</sup> <sup>þ</sup> <sup>b</sup>*x*ð Þ <sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>c</sup>*x*ð Þ <sup>n</sup>�<sup>2</sup> þ���þ <sup>f</sup>*x*<sup>2</sup> <sup>þ</sup> <sup>g</sup>*<sup>x</sup>* <sup>þ</sup> <sup>h</sup>

Taking its firstst derivative of a signal above removes a constant component (h) which represent a background process yielding the signals, and taking 2nd derivative removes a linear component (gx term) which represent a secondary process for a signal component, thus revealing the hidden higher order processes. In other words, from these operations, (a) undesired components in the signal (a constant term, a linier component, and quadratic component) can be removed successively upon repeated differentiations) leaving higher order terms only. Thus, upon differentiations, changes (in- or de-creases) in the signal can be developed to yields peaks which were hidden behind the original signal; hence the better-defined peaks, makes the analysis of curve much easier than the original curves; thus revealing concealed features in the original function. Basically, higher order non-linear components in the function (original signals) transforms into peaks in the process of

third derivative.

*(CSWV)*

utilized as diagnostic parameters.

mials of a dependent (x) variable:

**2.1 Advantages of derivative approach**

**2. Theory**

differentiations.

**7**

#### *1.3.2 Effects of electrode sphericity*

Current-potentials curves depends not only on types of the electrode mechanisms but also shape (geometry) of electrodes. However, present study focuses on a planar electrode with a linear diffusion. The effect of electrode sphericity can be minimized on planar electrode in pulse or ultrafast voltammetry because the diffusion layer thickness near the electrode is also depend on the duration of the applied potentials [43]: with application of short pulses, diffusion onto a spherical electrode (for cases with DME, HMDE, and SMDE) is approximated to a linear diffusion by minimizing the diffusion layer thickness, Nevertheless, effects of electrode sphericity *on the derivatives* of currents have been addressed by other groups [18, 54, 56] and author's group [31, 32]. Recently, Molina and coworkers [18] took a different approach that examine the dependence of peak-potentials (Ep) and peak-currents (ip) in first derivatives on the dimensionless rate constant (χ = kτ) and on the dimensionless electrode sphericity (ξ = (Dt)1/2/r); and found that, as the sphericity increases, the peak-potentials (Ep) moves towards more negative potentials for EC and moves oppositely towards more positive potentials for CE mechanism, while peak currents (ip) for both types of mechanisms decreases. As far as DV concerned, our recent work [31, 32] show that even slower electron transfer processes exhibits symmetry when electrode sphericity increases: this suggess that planar electrode is much more effective than the spherical electrode with the present DP method.

#### *1.3.3 Similar analysis from differential pulse polarography (DPP)*

Our group has adopted a different approach focusing on the analysis of *peak shapes in terms of symmetry parameters* as reported in our earlier studies on the firstand higher- order derivatives [12–15, 42], instead of examining shifts in peak potentials and changes in the magnitude of peak currents which is adopted by the other group [18, 54, 56]. Namely, in our previous studies [57–61], we have introduced the approach of analyzing peak asymmetry *found in the differential pulse polarography/voltammetry (DPP/DPV)*: namely, the differential pulse voltammograms can be viewed as a *pseudo-derivative of the i-E curve* which is emulated mechanically and/or electronically with the voltammetric/polarographic analyzer. In the earlier works, two peaks are generated from DPP; namely, a cathodic peak generated from a reduction process with a pulse going with the same direction as the scan direction (with a pulse amplitude of ΔE = -50 mV), and an anodic peak generated with a pulse going against the scan direction (with ΔE = +50 mV) to drive oxidation of reduced species back to the oxidized form, then the two peaks are compared and analyzed. In the original normal pulse program, the currents are assigned as positive for the reduction process. In the derivative voltammetry, however, if values of the derivative are positive or the peak appears in more positive potentials, it is labeled as "anodic", if the values are negative, or the peak appears in more negative potentials, it is labeled as "cathodic" regardless of actual redox reaction occurring on the electrode surface, following the previous

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

convention [12–15, 42]. In addition to the analysis of peak in term of peak-currents and peak-potentials [57–60], an analysis in terms of half-peak-widths was introduced later [61]. In present study a focus is given more to the *second derivative* because (a) it generates two well-defined peaks of anodic and cathodic that parallel to two peaks of DPP, (b) it is more sensitive and practical than first derivative which has only one peak and fewer visible parameters, and (c) it is less complicated and more practical than third derivative even though it is less sensitive than the third derivative.

## *1.3.4 Other approaches for diagnostic criteria with cyclic square-wave voltammetry (CSWV)*

Cyclic square-wave voltammetry (CSWV), which is an extension of SWV [62–65], has been adopted by Bottomley group [66–70]. Like pulse voltammetric techniques, SWV is effective in removing the interference from the capacitive current that is inherent in cyclic voltammetry(CV). SWV, like DPV, is a pseudoderivative technique in which a derivative-shaped peak is obtained from the current with electronic control of input of square-wave potentials and current sampling times [62]. Diagnostic criteria as a tool for studying electrode mechanisms based on CSWV scheme have been investigated by the group for reversible [66] and quasireversible [67] processes, EC type [68], ECE type [69], CE type [70] mechanisms, and other non-unity stoichiometric cases [71]. Basically, the impact of experimental SW parameters - such as pulse heights (or amplitude), step heights (or potential increments), switching potentials, and period – on the peak-currents, peakpotentials and separations, peak-widths were examined. It was found that those observed peak-related quantities changes characteristically according to each particular types of the electrochemical mechanisms, which make it feasible to be utilized as diagnostic parameters.

#### **2. Theory**

though there are many studies on the original currents (not derivatives) with spherical electrodes from earlier days of polarographic/voltammetric works as given in general monographs [30, 38, 43, 44] as well as in articles on particular systems and methods [45–55]. These include studies on EC mechanisms [45, 46], CE mechanisms [46, 47, 54], on various pulse polarographic methods [48, 55], on DC polarography [49], on AC polarography [72], on cyclic voltammetry [51], on catalytic EC mechanism with the square-wave voltammetry [52], on a double potential

Current-potentials curves depends not only on types of the electrode mechanisms but also shape (geometry) of electrodes. However, present study focuses on a planar electrode with a linear diffusion. The effect of electrode sphericity can be minimized on planar electrode in pulse or ultrafast voltammetry because the diffusion layer thickness near the electrode is also depend on the duration of the applied potentials [43]: with application of short pulses, diffusion onto a spherical electrode (for cases with DME, HMDE, and SMDE) is approximated to a linear diffusion by minimizing the diffusion layer thickness, Nevertheless, effects of electrode sphericity *on the derivatives* of currents have been addressed by other groups [18, 54, 56] and author's group [31, 32]. Recently, Molina and coworkers [18] took a different approach that examine the dependence of peak-potentials (Ep) and peak-currents (ip) in first derivatives on the dimensionless rate constant (χ = kτ) and on the dimensionless electrode sphericity (ξ = (Dt)1/2/r); and found that, as the sphericity increases, the peak-potentials (Ep) moves towards more negative potentials for EC and moves oppositely towards more positive potentials for CE mechanism, while peak currents (ip) for both types of mechanisms decreases. As far as DV concerned, our recent work [31, 32] show that even slower electron transfer processes exhibits symmetry when electrode sphericity increases: this suggess that planar electrode is much more effective than the spherical electrode with the present DP method.

method [53] and on DPV at spherical and microelectrodes [56].

*Analytical Chemistry - Advancement, Perspectives and Applications*

*1.3.3 Similar analysis from differential pulse polarography (DPP)*

*polarography/voltammetry (DPP/DPV)*: namely, the differential pulse

voltammograms can be viewed as a *pseudo-derivative of the i-E curve* which is emulated mechanically and/or electronically with the voltammetric/polarographic analyzer. In the earlier works, two peaks are generated from DPP; namely, a cathodic peak generated from a reduction process with a pulse going with the same direction as the scan direction (with a pulse amplitude of ΔE = -50 mV), and an anodic peak generated with a pulse going against the scan direction (with

currents are assigned as positive for the reduction process. In the derivative voltammetry, however, if values of the derivative are positive or the peak appears in more positive potentials, it is labeled as "anodic", if the values are negative, or the peak appears in more negative potentials, it is labeled as "cathodic" regardless of actual redox reaction occurring on the electrode surface, following the previous

ΔE = +50 mV) to drive oxidation of reduced species back to the oxidized form, then the two peaks are compared and analyzed. In the original normal pulse program, the

Our group has adopted a different approach focusing on the analysis of *peak shapes in terms of symmetry parameters* as reported in our earlier studies on the firstand higher- order derivatives [12–15, 42], instead of examining shifts in peak potentials and changes in the magnitude of peak currents which is adopted by the other group [18, 54, 56]. Namely, in our previous studies [57–61], we have introduced the approach of analyzing peak asymmetry *found in the differential pulse*

*1.3.2 Effects of electrode sphericity*

**6**

#### **2.1 Advantages of derivative approach**

In general, original signals (y) from a measurement can be given in a polynomials of a dependent (x) variable:

$$\mathbf{y} = \mathbf{a}\mathbf{x}^{\mathbf{n}} + \mathbf{b}\mathbf{x}^{(\mathbf{n}-1)} + \mathbf{c}\mathbf{x}^{(\mathbf{n}-2)} + \dots - - + \mathbf{f}\mathbf{x}^2 + \mathbf{g}\mathbf{x} + \mathbf{h}$$

Taking its firstst derivative of a signal above removes a constant component (h) which represent a background process yielding the signals, and taking 2nd derivative removes a linear component (gx term) which represent a secondary process for a signal component, thus revealing the hidden higher order processes. In other words, from these operations, (a) undesired components in the signal (a constant term, a linier component, and quadratic component) can be removed successively upon repeated differentiations) leaving higher order terms only. Thus, upon differentiations, changes (in- or de-creases) in the signal can be developed to yields peaks which were hidden behind the original signal; hence the better-defined peaks, makes the analysis of curve much easier than the original curves; thus revealing concealed features in the original function. Basically, higher order non-linear components in the function (original signals) transforms into peaks in the process of differentiations.

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

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 and a ruler. Those values of parameters, found analytically or numerically/ graphically are summarized (refer to the results sections for details) in **Table 2** for four types of the mechanisms for comparisons.

**Parameters Definitions Er Eirr ErC CEr**

E1/2 (mV) 0 <0 (-253) >0 <0

<sup>a</sup> (mV) E1/4-E1/2 28 >28 (48) <28 >28

<sup>c</sup> (mV) E1/2-E3/4 28 >28 (42) <28 >28

(normalized o Er) 1.0 <1.0 (0.64) 1.0 <1.0

(normalized o Er) 1.00 <1.00 (0.67) >1.00 <1.00 W½ (mV) 90.5 >90.5 (143) <90.5 <90.5

<sup>a</sup> (mV) 45.3 >45.3 (80) <45.3 >42, <45.3

<sup>c</sup> (mV) 45.3 >45.3 (63) <45.3 >40, <45.3

2a (mV) 34 <34 (-210) < 34 > 34

2c (mV) -34 <-34 (-319) <-34 >-34

2a 14 >14 (51) > 14 >10

2c -14 <14 (-72) >-14 >10

2a (mV) 63 >63 (111) >50, <63, <63

2c (mV) 63 >63 (81 ) > 35, <63 <63

**3a (mV) 59 <59 (-165) >59 >-180**

**3m (mV) 0.00 <0.00 (-275) >0.00 >-230**

3c (mV) -59 <-59 (-353) >-59 >-300

**<sup>R</sup>Δ<sup>E</sup> <sup>3</sup> <sup>Δ</sup>E3a/ <sup>Δ</sup>E3c 1.00 <sup>&</sup>gt;1.00 (1.41) <sup>&</sup>gt;1.0, <sup>&</sup>lt;1.6 <sup>&</sup>lt;1.00**

3a 2461 <2641 (449) <3692 <2461

3m -7382 >-7382 (-1961) >-15500 >-7382

3c 2461 <2461 (1061) < 8367 <2461

3a (mV) 54 >54 (102) >50, <54 >42.5, <54

3m (mV) 41 >41 (64) >32, <54 >35, <41

**<sup>2</sup> W½2a/W½2c 1.00 >1.00 (1.37) >1.00, <1.45 <1.0**

<sup>p</sup> (mV) Ep-Eo 0.0 <0.0 (-265) >0 <0.0 ip 9.6 <9.6 (6.4) >9.6 <9.6

<sup>0</sup> (mV) E1/4-E3/4 56.4 >56.4 (90) <56.4 >51.5, <56.4

**<sup>c</sup> 1.0 >1.00 (1.14) >1.00 <1.00**

**<sup>c</sup> 1.00 >1.00 (1.27) > 1.00 <1.00**

2c 68 >68 (108) < 68 >59, <68

**2c 1.00 <1.0 (0.66) >1.0 >1.0**

**2c| 1.00 <1.00 (0.71) >0.71, <1.00 >1.0, <1.25**

3c 118 >118 (188) <118 <118

3m 59 >59 (110) <59 <59

3c 59 >59 (78) <59 <59

**3c 1.00 <1.00 (0.42) >0.4, <1.0** >1.0, <1.27

**3m| 0.32 <0.32 (0.22) >0.2, <0.32** >0.32, <0.36

**3m| 0.32 >0.52 (0.52) >0.32,<0.60** >0.28,<0.32

*Current* E1/4 (mV) +28 <28 (-205)

ΔEq

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

ΔEq

ΔEh

Ri 0

Ri 1

Wq

Wq

Ep

ΔEp

**Rw**

**RE**

ip

ip

**R <sup>i</sup>**

**Rw**

W½

W½

**Ep**

Ep

ΔE<sup>3</sup>

ΔE<sup>3</sup>

ΔE<sup>3</sup>

ip

ip

ip

**Ri**

**Ri**

**Ri**

W½

W½

**9**

*1st Der.* ΔE1

*2nd Der.* Ep

**3***rd Der.* **Ep**

**R0**

**qw ΔEq**

**<sup>1</sup> Wq**

<sup>2</sup> (mV) Ep

**<sup>2</sup>** Δ**Ep**

**<sup>2</sup> |ip**

<sup>p</sup> (mV) Ep

<sup>a</sup> (mV) Ep

<sup>c</sup> (mV) Ep

**<sup>3</sup> ip**

**3a |ip**

**3c |ip**

E3/4 (mV) -28 <-28 (-295)

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

**a /ΔEq**

**a /Wq**

2a-Ep

**2a/**Δ**Ep**

**2a/ip**

3a-Ep

3a-Ep

3m-Ep

**3a/ip**

**3a/ip**

**3c/ip**

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

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**.
