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

#### *3.1.3.1 Theoretical expressions for and the derivatives*

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

$$\mathbf{Y} \underset{\mathbf{k\_b}}{\overset{\text{k\_f}}{\rightleftharpoons}} \mathbf{O} \quad \mathbf{K\_{eq}} = \mathbf{k\_f}/\mathbf{k\_b} \quad (\text{in solution}) \tag{45}$$

$$\text{O} + \text{ne}^- \rightarrow \text{R} \quad \text{Nernstein} \quad (\text{at the electrode}) \tag{46}$$

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

$$\mathbf{i} = \mathbf{i}\_d \left[ (\pi \mathbf{t})^{1/2} \left( \mathbf{K\_{eq}} \mathbf{k\_f} \right)^{1/2} \exp \left( \mathbf{K\_{eq}} \mathbf{k\_f} (\mathbf{1} + \mathbf{e}) / \mathbf{K\_{eq}} \right)^2 \mathbf{t} \right] \text{erfc} \left( \left( \mathbf{K\_{eq}} \mathbf{k\_f} \right)^{1/2} (\mathbf{1} + \mathbf{e}) / \mathbf{K\_{eq}} \right) \mathbf{t}^{\natural} \text{i} \tag{47}$$

Successive differentiations of the current yield the first-, second- and thirdderivatives, respectively.

$$\mathbf{i}' = \mathbf{i}\_d (\mathbf{n} \mathbf{F} / \mathbf{R} \mathbf{T}) (2 \mathbf{k}\_\mathbf{f} \ \mathbf{t}) \mathbf{e} \left[ (\mathbf{1} + \mathbf{e} / \mathbf{K}\_\mathbf{eq}) \mathbf{i} - \mathbf{1} \right] \tag{48}$$

Second derivative becomes

$$\mathbf{i}"\rightarrow\mathbf{i}\_{\mathrm{d}}(\mathbf{nF}/\mathbf{RT})^{2}(2\mathbf{k}\_{\mathrm{f}}\,\mathrm{t})\mathbf{e}\Big[(\mathbf{1}+\mathbf{e}/\mathbf{K}\_{\mathrm{eq}})\mathbf{i}+(\mathbf{RT}/\mathbf{nF})\Big(\mathbf{1}+\mathbf{e}/\mathbf{K}\_{\mathrm{eq}})\mathbf{i}'-\mathbf{1}\Big]\tag{49}$$

Third derivative becomes

Both peak potentials (Ep

ΔEp

2. Peak-Heights and Their Ratios. A plot of the two peak-heights (ip

with increasing αn. However, ip

αn = 0.3 which is strange.

3. Peak-Widths and Their Ratio.

W½

W½

*3.1.2.3 For the third derivatives*

values are summarized in **Table 2**.

**16**

2a*=*ΔEp

*Analytical Chemistry - Advancement, Perspectives and Applications*

∣ip a *=*ip c

ratio for Eirr processes are given below

W½

depends on αn, their ratio is invariant.

2a*=*W½

*3.1.2.4 Sensiivity increases with increasing order of derivatives*

2a and Ep

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

second derivative depend heavily on αn, a magnitude of both heights increases

Two half-peak-width (anodic and cathodic) of the second derivatives, and the

2a <sup>¼</sup> <sup>2</sup>*:*10 RT ð Þ *<sup>=</sup>*<sup>F</sup> ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ¼ <sup>V</sup> <sup>54</sup>*:*0 1ð Þ *<sup>=</sup>*α<sup>n</sup> ð Þ mV

2c <sup>¼</sup> <sup>1</sup>*:*53 RT ð Þ *<sup>=</sup>*<sup>F</sup> ð Þ <sup>1</sup>*=*α<sup>n</sup> ð Þ¼ <sup>V</sup> <sup>39</sup>*:*4 1ð Þ *<sup>=</sup>*α<sup>n</sup> ð Þ mV

¼ 108 mV >64 mV for αn ¼ 0*:*5

¼ 78*:*8 mV >64 mV for αn ¼ 0*:*5

Here again, it should be noted that although the two half-peak-widths

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

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

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.

mechanisms. Namely, the ratio of anodic to cathodic peak-widths (W½

<sup>c</sup> is greater than ip

<sup>a</sup> for anodic and ip

2a) are dependent on ko as well as αn values

<sup>c</sup> cathodic side) of the

<sup>a</sup> for most of the case unless

2c <sup>¼</sup> 210mV*=*319mV <sup>¼</sup> <sup>0</sup>*:*66<1*:*<sup>00</sup> (40)

∣ ¼ 0*:*78< 1*:*00 for αn ¼ 0*:*5 (41)

2c <sup>¼</sup> <sup>1</sup>*:*<sup>37</sup> <sup>&</sup>gt;1*:*00 for all <sup>α</sup><sup>n</sup> (44)

.

(42)

(43)

a /W½ c )

$$\mathbf{i}\prime\prime = \mathbf{i}\_{\mathrm{d}}(\mathbf{nF}/\mathbf{RT})^{3}(2\mathbf{k}\_{\mathrm{f}}\,\mathrm{t})\mathrm{e}\left[\left(\mathbf{1}+\mathbf{4e}/\mathbf{K\_{\mathrm{q}}}\right)\mathbf{i}+\left(2\mathbf{RT}/\mathbf{nF}\right)\left(\mathbf{1}+2\mathbf{e}/\mathbf{K\_{\mathrm{q}}}\right)\mathrm{i}\prime+\left(\mathbf{RT}/\mathbf{nF}\right)^{2}\left(\mathbf{1}+\mathbf{e}/\mathbf{K\_{\mathrm{q}}}\right)\mathrm{i}\prime\prime-\mathbf{1}\right].\tag{50}$$

Such curves according to the derivative formula are given in **Figure 3**. Full details of derivations of the three derivatives above and the analysis of the all parameters from those graphs were presented elsewhere [12, 15, 42], and only important key results are presented here, and summarizedin the **Tables 2** and **4**.

Typical i-E curves at various kf value for CE process are given **Figure 3A** in two dimensions, and **Figure 4A** in three dimensions with an additional axis for log(kf). The plots of the original currents and three derivatives were calculated according to the Eq. (47), (48), (49), and (50). As it can be expected the entire i-E curve moves upward with the diffusion-controlled current as kf increases. Asymmetry is not readily noticeable in the current and first derivatives, but it becomes more visible in the second and third derivatives: namely, anodic side of the peak height is larger than the cathodic side in both second and the third derivatives, This is more evident for larger kf where currents are larger in all potentials. All the relationships are obtained graphically [42], because finding the analytical solution of the dependent variables and parameters from the theoretical equations are too involved to be calculated. Graphic analysis of the curves yields the following results: the full account for the analysis of the derivative curves can be found in Ref. [12, 42].

#### *3.1.3.2 For the first derivative*

For a CEr process, following relationships can be obtained from analysis of the graphs for the first derivative.

> Peak potentials always shifts to more cathodic direction as the equilibrium constant for the prior chemical step, Keq (or kf), becomes smaller. Namely, Ep

**Parameters Definitions Er Eirr ErC CEr**

W½ (mV) 90.5 >90.5 (143) <90.5 >90.5

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

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

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

**<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**

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

**3a W½3a/W½3m 1.31 >1.31 (1.60) >1.31, <1.6 >1.19, <1.31**

**3c W½3c/W½3m 1.31 <1.31 (0.98) >1.0, <1.31 >1.31,<1.43**

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

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

**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

As expected, ip deceases as K (hence kf) decreases from the reversible value of 9.6

Half-peak-withs become less than 90.4/n (mV), approaching to 80/n (mV) as

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

<sup>a</sup> ð Þ rev <sup>&</sup>lt;Wq

Eo � Ep >0*:*0 (51)

ip <9*:*6 (52)

W1*<sup>=</sup>*<sup>2</sup> <90*:*5*=*n mV ð Þ (53)

<sup>c</sup> ð Þ¼ rev <sup>90</sup>*:*5*=*n mV ð Þ (54)

is more negative than Eo

K (hence kf) decreases

Wq

<sup>c</sup> < 1.00.

<sup>a</sup> <Wq

<sup>c</sup> <Wq

2. Peak Current

*Current*

*1st Der.*

*2nd Der.*

**3***rd Der.*

**Table 4.**

ΔEh

**Rq**

**Rw**

**R**Δ**<sup>E</sup>**

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

**Rw**

**Ri**

**Ri**

**Ri**

**Rw**

**Rw**

**Rw**

ΔEp

**<sup>0</sup> ΔEq**

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

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

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

**<sup>2</sup>** Δ**Ep**

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

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

**3a |ip**

**3c |ip**

**a /ΔEq**

**a /Wq**

2a-Ep

**2a/**Δ**Ep**

**2a/ip**

**3a/ip**

**3a/ip**

**3c/ip**

*Notes: All values calculated with the same conditions as in Table 2.*

3. Peak Width

Wq a /Wq

**19**

,

*Summary of asymmetry parameters which are more sensitive to the kinetics.*

Normalized peak currents found to be smaller than 9.6.

1. Peak-Potentials

#### **Figure 3.**

*Normalized currents (a), their first (B), second (C) and third (D) derivatives at various values of forward rate constant at Keq =10*�*<sup>2</sup> for CEr type mechanism: (a) kf = 10 s*�*<sup>1</sup> (for the lowest current and derivatives), (b) 5.62, (c) 3.16, (d) 1.78, (e) 1.00 (for the highest current and derivatives), calculated for T = 298, t = 1,0, and n = 1.*


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

#### **Table 4.**

<sup>i</sup>"' <sup>¼</sup> idð Þ nF*=*RT <sup>3</sup> 2kf ð Þ<sup>t</sup> e 1 <sup>þ</sup> 4e*=*Keq

*3.1.3.2 For the first derivative*

graphs for the first derivative.

1. Peak-Potentials

**Figure 3.**

*n = 1.*

**18**

� �<sup>i</sup> <sup>þ</sup> ð Þ 2RT*=*nF <sup>1</sup> <sup>þ</sup> 2e*=*Keq

*Analytical Chemistry - Advancement, Perspectives and Applications*

Such curves according to the derivative formula are given in **Figure 3**. Full details of derivations of the three derivatives above and the analysis of the all parameters from those graphs were presented elsewhere [12, 15, 42], and only important key results are presented here, and summarizedin the **Tables 2** and **4**. Typical i-E curves at various kf value for CE process are given **Figure 3A** in two dimensions, and **Figure 4A** in three dimensions with an additional axis for log(kf). The plots of the original currents and three derivatives were calculated according to the Eq. (47), (48), (49), and (50). As it can be expected the entire i-E curve moves upward with the diffusion-controlled current as kf increases. Asymmetry is not readily noticeable in the current and first derivatives, but it becomes more visible in the second and third derivatives: namely, anodic side of the peak height is larger than the cathodic side in both second and the third derivatives, This is more evident for larger kf where currents are larger in all potentials. All the relationships are obtained graphically [42], because finding the analytical solution of the dependent variables and parameters from the theoretical equations are too involved to be calculated. Graphic analysis of the curves yields the following results: the full account for the analysis of the derivative curves can be found in Ref. [12, 42].

For a CEr process, following relationships can be obtained from analysis of the

*Normalized currents (a), their first (B), second (C) and third (D) derivatives at various values of forward rate constant at Keq =10*�*<sup>2</sup> for CEr type mechanism: (a) kf = 10 s*�*<sup>1</sup> (for the lowest current and derivatives), (b) 5.62, (c) 3.16, (d) 1.78, (e) 1.00 (for the highest current and derivatives), calculated for T = 298, t = 1,0, and*

� �i' <sup>þ</sup> ð Þ RT*=*nF <sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup>*=*Keq

h i

� �i" � <sup>1</sup>

(50)

*Summary of asymmetry parameters which are more sensitive to the kinetics.*

Peak potentials always shifts to more cathodic direction as the equilibrium constant for the prior chemical step, Keq (or kf), becomes smaller. Namely, Ep is more negative than Eo ,

$$\rm E\_o - E\_p > 0.0 \tag{51}$$

2. Peak Current

Normalized peak currents found to be smaller than 9.6. As expected, ip deceases as K (hence kf) decreases from the reversible value of 9.6

$$\mathbf{i}\_{\mathfrak{p}} < \mathfrak{P}. \mathbf{6} \tag{52}$$

3. Peak Width

Half-peak-withs become less than 90.4/n (mV), approaching to 80/n (mV) as K (hence kf) decreases

$$\mathbf{W}\_{1/2} < \mathbf{90.5/n} \text{ (mV)}\tag{53}$$

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

$$\text{W}\_{\text{q}}\text{ }^{\text{a}} < \text{W}\_{\text{q}}\text{ }^{\text{c}} < \text{W}\_{\text{q}}\text{ }^{\text{a}}\text{ }(\text{rev}) < \text{W}\_{\text{q}}\text{ }^{\text{c}}(\text{rev}) = \Re \text{0.5}/\text{n}\text{ }(\text{mV})\tag{54}$$

Wq a /Wq <sup>c</sup> < 1.00.

**Figure 4.**

*Three-dimensional plots of normalized current-potential (i-E) curves at various forward rate constants (kf) with Keq = 10*�*<sup>4</sup> . The perspectives of the current (A) and first derivative(B) is different from those of the second (C) and third derivatives (D) for a better view of valleys present in second and third derivatives.*
