*3.1.3.3 For the second derivative*

Following relationships can be obtained from analysis of the graphs for the second derivative.

1. Peak-Potentials and Peak Separation and the Ratio The separation of the two peaks decreases as K (hence kf) from 68.0/n (mV), the value for Er.

Ep 2a-Ep 2c < 68.0/n (mV)

$$
\Delta \mathbf{E\_p}^{\;2\mathbf{a}} / \Delta \mathbf{E\_p}^{\;2\mathbf{c}} > \mathbf{1} \tag{55}
$$

*3.1.3.4 For the third derivative*

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

Ref. [42] for full details.

the rate constant (k) increases.

the analysis for the first derivative.

increases to a value of 12.3, thus

ip (normalized) > 1.00. which is opposite of CEr.

*3.1.4.2 For the first derivative*

1. Peak Potentials

2. Peak Currents

3. Peak Width

**21**

than Eo ,

follows [11].

Parameters associated with the third derivatives can be graphically analyzed in a similar fashion, and the results are given in the **Table 2** for the various values for three peaks, peak separations, peak-current ratios, and half-peak ratios. Refer to

This e� transfer reaction with a post kinetic process mechanism can be given as

The closed form of the analytical solution of the concentration gradient and the current at the specific boundary conditions for this ErC mechanism can be found elsewhere, but the solutions for the current include Dawson Integrals are too lengthy and complicated be reproduced here and can be found elsewhere [11]. Nonetheless, we managed to obtain the first derivatives by analytically differentiating the current equation. However, further analytical differentiation of the first derivative in order to obtain second and third derivatives were nearly impossible; thus, a numerical approach had to be adopted; namely, the two higher order derivatives i.e., (second and third derivatives) were obtained with ΔE being as small as 1 mV in order to increase a resolution of peaks [14]. Such derivatives thus obtained are given in **Figure 5**. In general, all curves shift to the left (*i.e*., anodic direction) as

For an ErC type mechanism, following relationships have been observed from

Peak potentials always shift to anodic direction as the rate constant (k) for the follow up chemical reaction increases. Namely, Ep is more positive

Peak currents increase from 9.6 (the value for Er) as the rate constant (k)

Half peak-with decreases from 90.5/n (mV) as the rate constant (k) increases,

O þ ne� ! R Nernstian at the electrode ð Þ (58)

<sup>k</sup> P in solution ð Þ (59)

Eo � Ep <0*:*0 (60)

ip >9*:*6 (61)

*3.1.4 Reversible Electron transfer coupled with a follow-up chemical reaction (ErC)*

*3.1.4.1 The mechanism and derivation of the current and it's derivatives*

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R !

2. Peak Currents and the Ratio

It is observed that the anodic peak currents are always larger than the cathodic ones, but smaller than 1.25.

$$\mathbf{1.00} < |\mathbf{i\_p}^{\mathbf{a}} / \mathbf{i\_p}^{\mathbf{a}}| < \mathbf{1.25} \tag{56}$$

3. Peak Widths and the Ratio

The anodic half-peak is always smaller than the cathodic one

$$\text{°W}\_{1/2}\text{°} < \text{W}\_{1/2}\text{°} < \text{W}\_{1/2}\text{ °} \left(\text{rev}\right) < \text{W}\_{1/2}\text{°} \left(\text{rev}\right) = \text{64/n}\left(\text{mV}\right) \tag{57}$$

Wq a /Wq <sup>c</sup> < 1.00 *Advances in Derivative Voltammetry - A Search for Diagnostic Criteria of Several… DOI: http://dx.doi.org/10.5772/intechopen.96409*

#### *3.1.3.4 For the third derivative*

Parameters associated with the third derivatives can be graphically analyzed in a similar fashion, and the results are given in the **Table 2** for the various values for three peaks, peak separations, peak-current ratios, and half-peak ratios. Refer to Ref. [42] for full details.

*3.1.4 Reversible Electron transfer coupled with a follow-up chemical reaction (ErC)*

#### *3.1.4.1 The mechanism and derivation of the current and it's derivatives*

This e� transfer reaction with a post kinetic process mechanism can be given as follows [11].

$$\text{O} + \text{ne}^- \rightarrow \text{R} \quad \text{Nernstein} \quad (\text{at the electrode}) \tag{58}$$

$$\mathbf{R} \xrightarrow{\mathbf{k}} \mathbf{P} \quad \text{(in solution)}\tag{59}$$

The closed form of the analytical solution of the concentration gradient and the current at the specific boundary conditions for this ErC mechanism can be found elsewhere, but the solutions for the current include Dawson Integrals are too lengthy and complicated be reproduced here and can be found elsewhere [11]. Nonetheless, we managed to obtain the first derivatives by analytically differentiating the current equation. However, further analytical differentiation of the first derivative in order to obtain second and third derivatives were nearly impossible; thus, a numerical approach had to be adopted; namely, the two higher order derivatives i.e., (second and third derivatives) were obtained with ΔE being as small as 1 mV in order to increase a resolution of peaks [14]. Such derivatives thus obtained are given in **Figure 5**. In general, all curves shift to the left (*i.e*., anodic direction) as the rate constant (k) increases.

#### *3.1.4.2 For the first derivative*

*3.1.3.3 For the second derivative*

the value for Er.

2c < 68.0/n (mV)

2. Peak Currents and the Ratio

ones, but smaller than 1.25.

3. Peak Widths and the Ratio

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

<sup>c</sup> < 1.00

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

second derivative.

**Figure 4.**

*with Keq = 10*�*<sup>4</sup>*

Ep 2a-Ep

Wq a /Wq

**20**

Following relationships can be obtained from analysis of the graphs for the

*Three-dimensional plots of normalized current-potential (i-E) curves at various forward rate constants (kf)*

*(C) and third derivatives (D) for a better view of valleys present in second and third derivatives.*

*. The perspectives of the current (A) and first derivative(B) is different from those of the second*

The separation of the two peaks decreases as K (hence kf) from 68.0/n (mV),

It is observed that the anodic peak currents are always larger than the cathodic

2c > 1 (55)

2cj<1*:*<sup>25</sup> (56)

<sup>c</sup> ð Þ¼ rev <sup>64</sup>*=*n mV ð Þ (57)

2a*=*ΔEp

2a*=*ip

<sup>a</sup> ð Þ rev <sup>&</sup>lt;W1*<sup>=</sup>*<sup>2</sup>

1. Peak-Potentials and Peak Separation and the Ratio

*Analytical Chemistry - Advancement, Perspectives and Applications*

ΔEp

1*:*00<jip

The anodic half-peak is always smaller than the cathodic one

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

For an ErC type mechanism, following relationships have been observed from the analysis for the first derivative.

1. Peak Potentials

Peak potentials always shift to anodic direction as the rate constant (k) for the follow up chemical reaction increases. Namely, Ep is more positive than Eo ,

$$\mathbf{E\_o - E\_p < 0.0} \tag{60}$$

2. Peak Currents

Peak currents increase from 9.6 (the value for Er) as the rate constant (k) increases to a value of 12.3, thus

$$\mathbf{i}\_{\mathbf{p}} > \mathbf{9.6} \tag{61}$$

ip (normalized) > 1.00. which is opposite of CEr.

3. Peak Width

Half peak-with decreases from 90.5/n (mV) as the rate constant (k) increases,

#### **Figure 5.**

*Theoretical normalized current (A), and its first (B), second (C), and third derivatives (D) at various values of the homogeneous rate constant(k) for the ErC type of electrode reaction. Calculated for k values of (a) 0 (b) 0.3, (c) 0.562, (d) 1.0, (e) 1.78, (f) 3.0, (g) 5.62, (h) 10.0 (i) 30.0 s*�*<sup>1</sup> , and T = 298 K and t = 0.952 s. the normalized currents are dimensionless; thus, the first derivatives are in a unit of V*�*<sup>1</sup> , second derivatives in V*�*<sup>2</sup> , third derivatives in V*�*<sup>3</sup> .*

$$\mathbf{W}\_{1/2} < \mathbf{90.5}\,(\text{mV})\tag{62}$$

The anodic part of a half-peak is always larger than the cathodic part.

$$\text{W}\_{\text{q}}\,^{\text{c}} < \text{W}\_{\text{q}}\,^{\text{a}} < \text{W}\_{\text{q}}\,^{\text{a}}\,(\text{rev}) = \text{W}\_{\text{q}}\,^{\text{c}}\,(\text{rev}) = \text{45.3/n}\,(\text{mV})\tag{63}$$

$$\mathbf{W\_q } ^\text{a}\prime\prime\prime\_q > \mathbf{1.00} \tag{64}$$

0*:*71<∣ip

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3. Peak-Widths and their Ratio

*2a-Ep 2c).*

**Figure 6.**

**23**

*peak separation (Ep*

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

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

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

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

2a*=*ip

*Effect of k (A) on the (a) the anodic, (b) cathodic peak potentials of the second derivative, (B) effect of k on the*

This implies that the rate constant, k. can be directly determined from the

As shown in **Figure 8**, as a follow-up chemical reaction occurs faster, both anodic and cathodic half-peak-width decreases (i. e., become sharper), but cathodic one become sharper than the anodic one at a higher rate, resulting in an increase in the half-peak ratio of larger than 1 as illustrated in the **Figure 8**.

<sup>a</sup> ð Þ¼ rev W1*<sup>=</sup>*<sup>2</sup>

<sup>c</sup> > 1*:*00

measurements of the peak currents ratio of the second derivatives.

2c∣< 1*:*00 (66)

<sup>c</sup> ð Þ¼ rev <sup>63</sup>*=*n mV ð Þ (67)

It should be noted again that these trends are the opposite of those from CEr type.

#### *3.1.4.3 For the second derivative*

Following relationships have been observed from the analysis of the second derivative.

1. Peak-Potentials and Peak Separation

As shown in **Figure 6**, both peak potentials shift anodically, and the separations of the two peaks decrease from 68.0n (mV) and approach to 54 mV as the rate constant (k) increases.

$$\text{54/n (mV)} < \text{E}\_{\text{p}}{}^{2\text{a}} - \text{E}\_{\text{p}}{}^{2\text{c}} < \text{68.0/n (mV)} \tag{65}$$

This suggests that the homogeneous rate constant, k. can be directly determined from the measurements of the peak separation of the second derivative.

2. Peak-Currents and their Ratio

As shown in **Figure 7**, normalized values of both peak currents increase with inceasing k. However, the peak current ratios always degreases from 1.0 to about 0.71 as k increases.

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#### **Figure 6.**

W1*<sup>=</sup>*<sup>2</sup> <90*:*5 mV ð Þ (62)

<sup>c</sup> ð Þ¼ rev <sup>45</sup>*:*3*=*n mV ð Þ (63)

*, and T = 298 K and t = 0.952 s. the*

*, second derivatives in V*�*<sup>2</sup>*

*,*

<sup>c</sup> >1*:*00 (64)

2c <sup>&</sup>lt; <sup>68</sup>*:*0*=*n mV ð Þ (65)

The anodic part of a half-peak is always larger than the cathodic part.

*Theoretical normalized current (A), and its first (B), second (C), and third derivatives (D) at various values of the homogeneous rate constant(k) for the ErC type of electrode reaction. Calculated for k values of (a) 0 (b)*

It should be noted again that these trends are the opposite of those from CEr type.

Following relationships have been observed from the analysis of the second

2a � Ep

As shown in **Figure 6**, both peak potentials shift anodically, and the separations of the two peaks decrease from 68.0n (mV) and approach to 54 mV as

This suggests that the homogeneous rate constant, k. can be directly determined from the measurements of the peak separation of the second derivative.

As shown in **Figure 7**, normalized values of both peak currents increase with inceasing k. However, the peak current ratios always degreases from 1.0 to

<sup>a</sup> ð Þ¼ rev Wq

Wq a *=*Wq

Wq

*.*

*3.1.4.3 For the second derivative*

derivative.

**22**

**Figure 5.**

*third derivatives in V*�*<sup>3</sup>*

<sup>c</sup> <Wq

1. Peak-Potentials and Peak Separation

the rate constant (k) increases.

2. Peak-Currents and their Ratio

about 0.71 as k increases.

<sup>a</sup> <Wq

*0.3, (c) 0.562, (d) 1.0, (e) 1.78, (f) 3.0, (g) 5.62, (h) 10.0 (i) 30.0 s*�*<sup>1</sup>*

*Analytical Chemistry - Advancement, Perspectives and Applications*

*normalized currents are dimensionless; thus, the first derivatives are in a unit of V*�*<sup>1</sup>*

54*=*n mV ð Þ< Ep

*Effect of k (A) on the (a) the anodic, (b) cathodic peak potentials of the second derivative, (B) effect of k on the peak separation (Ep 2a-Ep 2c).*

$$0.71 < |\mathbf{i}\_{\mathbf{p}}|^{2\mathbf{a}} / |\mathbf{i}\_{\mathbf{p}}| < 1.00\tag{66}$$

This implies that the rate constant, k. can be directly determined from the measurements of the peak currents ratio of the second derivatives.

3. Peak-Widths and their Ratio

As shown in **Figure 8**, as a follow-up chemical reaction occurs faster, both anodic and cathodic half-peak-width decreases (i. e., become sharper), but cathodic one become sharper than the anodic one at a higher rate, resulting in an increase in the half-peak ratio of larger than 1 as illustrated in the **Figure 8**.

$$\mathbf{W\_{1/2}}^{\ c} < \mathbf{W\_{1/2}}^{\ a} < \mathbf{W\_{1/2}}^{\ a} \left(\text{rev}\right) = \mathbf{W\_{1/2}}^{\ c} \left(\text{rev}\right) = \mathbf{63}/\text{n} \left(\text{mV}\right) \tag{67}$$

$$\mathbf{W\_{1/2}}^{\ a}/\mathbf{W\_{1/2}}^{\ c} > \mathbf{1.00}$$

#### **Figure 7.**

*Effect of k (A) on the (a) the anodic, (b) cathodic peak currents of the second derivative, which are normalized with respect to the values of reversible process. (B) Effect of k on the ratio of the anodic to cathodic peak current ratio, ip 2a/ip 2c.*

only requires, in general, a single scan for a system under investigation: few experimental variables need to be changed although some experimental variables (such as duration of the pulse) can be optimized at an initial stage of measurement. Mostly, a single scan at a optimized condition will suffice while other methods such as CV and CSWV requires multiple scans (mostly six or more). In present DV method, analysis of the second derivative can be enough for most cases: however, for a system requires a better resolution and a higher sensitivity, one may resort to the third

*Effect of k (A) on the (a) the anodic, (b) cathodic half peak-widths of the second derivative, an those values normalized to the values for the reversible case whose scales are given at the right side, (B) on the anodic to*

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A quantitative measure of symmetry *in the original current are not readily available*, can be found indirectly from a ratio of difference in several potentials defined from a voltammogram: namely, a quarter-wave potentials (E1/4), half-wave potentials (E1/2) and a three-quarter-wave potentials); then a ratio of an anodic to

is very limited. Nevertheless, the ratio is unity for a symmetrical curve (for a reversible case), but it deviates from one if the curve becomes asymmetrical, yielding 1.14 for Eirr (**Table 2**). On the other hand, asymmetry *in first derivatives is more readily*

*found* in terms of a ratio of anodic-to-cathodic quarter-peak ratio (**Wq**

**a /ΔEq**

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

**a /Wq c** ): this

derivatives for more diagnostic parameters.

*2a/W<sup>½</sup> 2c.*

cathodic quarter-wave potential differences (i.e., **ΔEq**

**4. Conclusions**

**25**

**Figure 8.**

*cathodic half-peak ratios, W<sup>½</sup>*

This suggets that the homogeneous rate constant, k. can be directly determined from the measurements of a ratio of the two half-peak-width of the second derivatives.

#### *3.1.4.4 For the third derivative*

Third derivatives can be also analyzed graphically in the same fashion, and the details of the graphic analysis and descriptions can be found elsewhere [14]. Only key results are summarized in **Table 2** for the various symmetry parameters, and **Table 4** for key parameters of ratios (several peak-current ratios, half-peak ratios, peak separations or their ratios).

#### **3.2 Advantages of derivative voltammetric method**

The scheme of derivative voltammetric (DV) approach is simpler, more straight-forward and faster compared with other methods of studying electrode reaction mechanisms, namely, CV [1, 38–41] and CSWV [66–70]. The present DV *Advances in Derivative Voltammetry - A Search for Diagnostic Criteria of Several… DOI: http://dx.doi.org/10.5772/intechopen.96409*

#### **Figure 8.**

This suggets that the homogeneous rate constant, k. can be directly determined from the measurements of a ratio of the two half-peak-width of the second derivatives.

*Effect of k (A) on the (a) the anodic, (b) cathodic peak currents of the second derivative, which are normalized with respect to the values of reversible process. (B) Effect of k on the ratio of the anodic to cathodic peak current*

Third derivatives can be also analyzed graphically in the same fashion, and the details of the graphic analysis and descriptions can be found elsewhere [14]. Only key results are summarized in **Table 2** for the various symmetry parameters, and **Table 4** for key parameters of ratios (several peak-current ratios, half-peak ratios,

The scheme of derivative voltammetric (DV) approach is simpler, more straight-forward and faster compared with other methods of studying electrode reaction mechanisms, namely, CV [1, 38–41] and CSWV [66–70]. The present DV

*3.1.4.4 For the third derivative*

**Figure 7.**

*ratio, ip*

**24**

*2a/ip 2c.*

peak separations or their ratios).

**3.2 Advantages of derivative voltammetric method**

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*Effect of k (A) on the (a) the anodic, (b) cathodic half peak-widths of the second derivative, an those values normalized to the values for the reversible case whose scales are given at the right side, (B) on the anodic to cathodic half-peak ratios, W<sup>½</sup> 2a/W<sup>½</sup> 2c.*

only requires, in general, a single scan for a system under investigation: few experimental variables need to be changed although some experimental variables (such as duration of the pulse) can be optimized at an initial stage of measurement. Mostly, a single scan at a optimized condition will suffice while other methods such as CV and CSWV requires multiple scans (mostly six or more). In present DV method, analysis of the second derivative can be enough for most cases: however, for a system requires a better resolution and a higher sensitivity, one may resort to the third derivatives for more diagnostic parameters.
