Definitions of original and derived parameters and the values for the simple reversible e− transfer reaction. Calculated for n = 1 and at 298 K.
New methods for analysis of current-potential curves in terms of their derivatives are presented for studying various types of electrode processes – such as simple electron transfer reactions (reversible, quasi-reversible, and irreversible electron transfer) as well as chemically coupled electron transfer reactions along with a diagnostic scheme for differentiating these various types of electrochemical reaction mechanisms. Expressions for first- and higher order derivatives are derived from theoretical analytical solutions for currents for the different types of electrode mechanisms. The derivative curves are analyzed in terms of various parameters which characterize peak shape or peak symmetry with an emphasis on the second derivatives with well-defined anodic and cathodic peaks. Second derivatives can yield, in a simpler manner, the symmetry ratios; i.e., a ratio of anodic to cathodic peak-currents (ipa/ipc), and a ratio of anodic to cathodic peak-widths (Wpa/wpc) and a ratio of anodic to cathodic peak potential differences (ΔEpa/ΔEpc) or a peak separation (Epa-Epc) are evaluated, and these ratio can be related to kinetic parameters associated with a particular types of electrode mechanisms. Peaks are found to be symmetrical for a simple reversible electron transfer process (Er). However, peaks become asymmetrical when the electron transfer become slower (namely, irreversible, Eirr) or e− transfer reaction is coupled with homogeneous chemical reactions such as a prior reaction (CEr) or a follower-up reaction (ECr). From measured values of such symmetry ratios above, one can gain insight to the nature of the electrochemical systems enabling us to determine various kinetic parameters associated with a system. A diagnostic criteria for assigning an electrode mechanism is devised based on the values of asymmetry parameters measured, which are unity for a simple reversible electron transfer process.
- Derivative Voltammetry
- Electrochemical Reaction Mechanism
- Electrode Kinetics
- and Irreversible Electron Transfer
- Chemically Couple Electron Transfer
- Diagnostic Criteria for Electrode Processes
The aim of this Chapter is several folds (1) to provide a brief background on the derivative approach in measurements and its various applications in science and engineering fields, (2) to review some of our research work on advances on derivatives voltammetry (DV) which have been scattered among dozens of different journls and conference presentations/proceedings in the past, together with additional new findings and development which have not been published yet, (3) to analyze and compare those characteristic results from each of different electrochemical reaction mechanisms, and to construct a master table of diagnostic criteria for differentiating various types of the electrode mechanisms. Namely, as the subtitle indicates, the primary goal of the project is, utilizing derivative voltammetry, to invent a general criteria for differentiating various types of electrode processes, in order to complement other methods such as cyclic voltammetry(CV), which is the most popular one, or Cyclic Square-Wave Voltammetry (CSWV).
Derivatives approaches in analyzing data signals, have attracted much interests producing numerous research articles in the past [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Although earlier pioneering reports on the derivative approach in electrochemistry are found in the reference sections in our previous reports in electronalyticalchemistry [11, 12, 13, 14, 15], the approach with derivatives is not only limited to electrochemical methods but also have extended to other various fields of science and engineering [23, 24, 25, 26, 27, 33]. These reviews and earlier articles introduces basics of the method and techniques of signal processing [2, 3, 4, 28, 34], as applied to spectroscopic and spectrophotometric data [2, 3, 4, 5, 6, 7, 8, 9, 34], electrochemical data [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], and other signals from seismic data  and data from biomechanics measurements [24, 25, 26, 27, 33]. Several group [3, 6] extended the derivatives to as high as fourth order for enhanced peak resolutions. For example, a single broad peak of UV absorption spectrum from a ternary mixture of amino acids (phenylalanine, tyrosine and tryptophan), were separated and resolved in its second and fourth derivatives ; the physiological states of certain algae have been assessed from an analysis of 4th derivative of absorption spectra of chlorophyll a and b ; a second order derivatives of an IR spectra of a DNA in malignant tissues exhibits some differences from those of a normal one . The ever increasing interests in the derivative approaches in analysis of signals stem from (a) the information hidden behind in the raw data can be easily exposed from analysis of finer features emerged in the shapes of derivatives, the higher order the more the revelation, and (b) the technical advances in digital signal enhancing methods [2, 14, 15, 23, 24, 25, 26, 27, 28, 29, 34] that improve the S/N ratio which was not possible from analog instruments. A review of signal processing in electrochemistry with an introduction of software packages and extensive references therein by Jakubowska  may provide further assistance for interested readers.
In the field of electrochemical analysis, the derivative approach has been applied to potentiometric [10, 16] as well as to voltammetric measurements [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] with the enhanced resolution and better signal-to-noise (S/N) ratios.
More recent applications for derivative techniques have also been reported for analysis of oxidation of methanol , for improved quantification of ill-defined peaks in determining Pt in water and sediments , for analysis of alloxan , and for quantitation of naringin, an antioxidant .
1.2 Application to study of electrochemical reaction mechanisms
Most reports of derivative approach on electrochemical system has been for its application in chemical analysis [10, 19, 20] while only few groups have attempted to apply the approach to the study of kinetics of electrode reactions [11, 12, 13, 14, 15, 16, 17, 18, 21, 22]. Elucidation of electrochemical reaction mechanisms on electrode surface is not only important in developing electroanalytical methods for determining an analyte in a system but also devising effective electrochemical energy storage cells and other applied fields of engineering.
1.2.1 Drawbacks of cyclic voltammetry
As far as method of studying electrode reaction mechanisms concerned [1, 38], the cyclic voltammetry (CV) has been dominant in the field [39, 40, 41]. However, we have developed this new derivative approach to investigate electrode kinetics [11, 12, 13, 14, 15] in order to help in elucidating electrochemical reaction mechanisms: this is because CV has drawbacks of complications arising from the interference of the capacitive current to charge the electrical double layer at the interface; as results those peaks from CV are not purely Faradaic, and giving problems associated with defining/evaluating peak currents (ip), and peak potentials (Ep) and peak widths (Wp). In present work, currents are taken in normal pulse voltammetric (NPP) mode by sampling the current near the end of a pulse in order to minimize the non-Faradaic currents: then, first-, second-, and third- derivatives are taken from the NPP data.
1.2.2 Several common types of electrochemical systems
The electrochemical systems under present investigation includes the simple electron transfer process; the reversible electron transfer reaction (
1.2.3 Experimental measurements of derivatives
The theoretical voltammogram (zeroth-order derivative) is basically to emulate a normal pulse voltammogram (NPV) obtained with a normal pulse mode from an electrochemical analyzer with a DME or HMDE or SMDE (static mercury drop electrode) ; typically, with a pulse period of 1.000 s, pulse width of 50 ms, current sampling time of 33 ms, and scan rate of 1 mV/sec or 2 mV/s with a commercial electrochemical analyzer such an instrument as BAS-100 and Tacussel PRG-5 Pulse Polarograph. In calculation of theoretical currents, the original currents are normalized by dividing them with the diffusion controlled current (id) to obtain a dimensionless currents.
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 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 .
|E1/4 (mV)||+28||Ep3a (mV)||59|
|E1/2 (mV)||0||Ep3m (mV)||0.00|
|E3/4 (mV)||-28||Ep3c (mV)||-59|
|ΔEqa (mV)||E1/4-E1/2||28||ΔE3p (mV)||Ep3a-Ep3c||118|
|ΔEqc (mV)||E1/2-E3/4||28||ΔE3a (mV)||Ep3a-Ep3m||59|
|ΔEh0 (mV)||E1/4-E3/4||56.4||ΔE3c (mV)||Ep3m-Ep3c||59|
|Ep2a (mV)||34||W½3c (mV)||54|
1.3.1 Symmetry parameters
A quantitative measure of symmetry
However, measurements of symmetry parameters from
As far as analysis of
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 : 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
1.3.3 Similar analysis from differential pulse polarography (DPP)
Our group has adopted a different approach focusing on the analysis of
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, 63, 64, 65], has been adopted by Bottomley group [66, 67, 68, 69, 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 pseudo-derivative 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 . Diagnostic criteria as a tool for studying electrode mechanisms based on CSWV scheme have been investigated by the group for reversible  and quasi-reversible  processes, EC type , ECE type , CE type  mechanisms, and other non-unity stoichiometric cases . 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, peak-potentials 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.
2.1 Advantages of derivative approach
In general, original signals (y) from a measurement can be given in a polynomials of a dependent (x) variable:
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 current-potential 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.
|E1/4 (mV)||+28||<28 (-205)|
|E1/2 (mV)||0||<0 (-253)||>0||<0|
|E3/4 (mV)||-28||<-28 (-295)|
|ΔEqa (mV)||E1/4-E1/2||28||>28 (48)||<28||>28|
|ΔEqc (mV)||E1/2-E3/4||28||>28 (42)||<28||>28|
|ΔEh0 (mV)||E1/4-E3/4||56.4||>56.4 (90)||<56.4||>51.5, <56.4|
|Ri0(normalized o Er)||1.0||<1.0 (0.64)||1.0||<1.0|
|ΔE1p (mV)||Ep-Eo||0.0||<0.0 (-265)||>0||<0.0|
|Ri1(normalized o Er)||1.00||<1.00 (0.67)||>1.00||<1.00|
|W½ (mV)||90.5||>90.5 (143)||<90.5||<90.5|
|Wqa (mV)||45.3||>45.3 (80)||<45.3||>42, <45.3|
|Wqc (mV)||45.3||>45.3 (63)||<45.3||>40, <45.3|
|Ep2a (mV)||34||<34 (-210)||< 34||> 34|
|Ep2c (mV)||-34||<-34 (-319)||<-34||>-34|
|ΔEp2 (mV)||Ep2a-Ep2c||68||>68 (108)||< 68||>59, <68|
|ip2a||14||>14 (51)||> 14||>10|
|W½2a (mV)||63||>63 (111)||>50, <63,||<63|
|W½2c (mV)||63||>63 (81 )||> 35, <63||<63|
|Ep3c (mV)||-59||<-59 (-353)||>-59||>-300|
|ΔE3p (mV)||Ep3a-Ep3c||118||>118 (188)||<118||<118|
|ΔE3a (mV)||Ep3a-Ep3m||59||>59 (110)||<59||<59|
|ΔE3c (mV)||Ep3m-Ep3c||59||>59 (78)||<59||<59|
|ip3c||2461||<2461 (1061)||< 8367||<2461|
|W½3a (mV)||54||>54 (102)||>50, <54||>42.5, <54|
|W½3m (mV)||41||>41 (64)||>32, <54||>35, <41|
|W½3c (mV)||54||>54 (63)||>32, <42||>50, <54|
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 . 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 . A value of ∆E = 1.0 mV was used for a finer resolution for all curves. The differentiation methods are summarized in Table 3.
|Types of Mechanisms||Current|
|CEr (Prior Chemical)||i||di’/dE||di”/dE||di”’/dE|
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 semi-infinite linear diffusion, is expressed by the following relationship for a simple reversible electron-transfer process [1, 38]
and E is an applied potential with respect to the reversible polarographic half-wave-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 .
Examination of Eq. 2 reveals that
This implies that
188.8.131.52 For the first derivatives
Eq. 5 shows
This means that the
184.108.40.206 For the second derivatives
One can examine the second derivative (i”), Eq. (6), to find
Thus, the second derivative (Figure 1C, solid line)
Following relationships are found for Er type.
1. Peak Potentials
2. Peak Currents
3. Peak Widths
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.
220.127.116.11 For the third derivatives
It’s symmetry is the same as that of the first derivative,
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 ko 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.
18.104.22.168 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 ; 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.
22.214.171.124 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 ko (or ks) 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 (ko < < (D/t)1/2) and quasi-reversible if those competing rates are comparable (ko
In terms of values of ko, by adopting typical values of D (=5x10−5 cm2/s) and t (=1.000 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, 55, 56, 72, 73], and has final forms of the following equations for a planar diffusion:
assumed DR = DO = D.
All symbols have their usual meanings and may be refer to the reference for details,
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 .
Successive differentiations of the current yield the following expressions for the first-, second- and third- derivatives respectively . Taking derivative of the current with respect to potential (E) yields the first derivative,
The analytical differentiations of i’ and expressions for the second- and third- derivatives are more involved and given elsewhere . Alternatively, it is also possible to obtain second and third derivative curves from successive
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 ko) (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.
126.96.36.199 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  of the author’s group: for the irreversible or quasi-reversible process, following relationships are found from plots
1. Peak Potentials
It is found that the peak potential for totally irreversible process (ko < 10−5 cm/s) is directly proportional to log (ko), shifting towards cathodic direction as ko decreases for a given αn, and is inversely proportional to αn at a constant ko .
<0 (because log(ko) 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 
The half-peak width (W½), the anodic half-peak width (Wqa), and the cathodic half-peak width (Wqc) at 298 K for a totally irreversible reaction are independent of ko, but inversely proportional to αn 
188.8.131.52 For the second derivative
1. Peak-Potentials and Peak Separation
Both peak potentials (anodic and cathodic) of the second derivatives for totally irreversible processes (ko < 10−5 cm/s) are directly proportional to log(ko) for a given αn, moving to cathodic direction as ko decreases, which is the same trend as for Ep in the first derivative.
and Ep2a-Ep2c = 2.3(RT/F)(1/αn) = 0.0591(1/αn) (mV)
Both peak potentials (Ep2a and Ep2a) are dependent on ko as well as αn values yielding −210 mV and -319 mV respectively, with ko = 10−5 cm/s and αn =0.5. However, the peak potential difference depends only on αn values, not on ko.
2. Peak-Heights and Their Ratios.
A plot of the two peak-heights (ipa for anodic and ipc cathodic side) of the second derivative depend heavily on αn, a magnitude of both heights increases with increasing αn. However, ipc is greater than ipa for most of the case unless αn = 0.3 which is strange.
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
Here again, it should be noted that although the two half-peak-widths depends on αn, their ratio is invariant.
184.108.40.206 For the third derivatives
Third derivative are obtained by differentiating the second derivative. The details of the derivation and results were given elsewhere . 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 half-peak ratios were analyzed from various plots. The results with possible ranges of the values are summarized in Table 2.
220.127.116.11 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 (W½a/W½c) 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. 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
18.104.22.168 Notes on computations for quasi-reversible and irreversible processes
All computations are carried out with ko = 1.0x101cm/s for a reversible case, ko = 1.0x10−3cm/s for a quasi-reversible case, ko = 1.0x10−5 cm/s for an irreversible case for simple e− transfer reactions. A value of 0.50 is used for all quasi-reversible and irreversible cases. For quasi-reversible processes, only the case with 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 ≠ 0.5 exhibits much complicated behaviors depending on values of αn . This is because depending on values of the kinetic parameters (ko and αn) the changes in some dependent variables become not monotonic,, often exhibiting discontinuity at around 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 quadruple-precision mode (32 significant digits) had to be employed for all calculation: namely, the function exp.(x2)erfc(x): namely F(Qst½) = exp.(Qs2t)erfc(Qst½)) (Eq. 28) had to be evaluated to the 32 digit-precision .
3.1.3 Reversible electron transfer coupled with a prior chemical equilibrium (CEr)
22.214.171.124 Theoretical expressions for and the derivatives
This mechanism that involves pre-chemical step can be given as follows.
Successive differentiations of the current yield the first-, second- and third- derivatives, respectively.
Second derivative becomes
Third derivative becomes
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.
|ΔEh0 (mV)||E1/4-E3/4||56.4||>56.4 (90)||<56.4||>51.5, <56.4|
|W½ (mV)||90.5||>90.5 (143)||<90.5||>90.5|
|ΔEp2 (mV)||Ep2a-Ep2c||68||>68 (108)||< 68||>59, <68|
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 , 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].
126.96.36.199 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 is more negative than Eo,
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
3. Peak Width
Half-peak-withs become less than 90.4/n (mV), approaching to 80/n (mV) as K (hence kf) decreases
The anodic part of a half-peak is always smaller than the cathodic part.
Wqa/Wqc < 1.00.
188.8.131.52 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.
Ep2a-Ep2c < 68.0/n (mV)
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.
3. Peak Widths and the Ratio
The anodic half-peak is always smaller than the cathodic one
Wqa/Wqc < 1.00
184.108.40.206 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.  for full details.
3.1.4 Reversible Electron transfer coupled with a follow-up chemical reaction (ErC)
220.127.116.11 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 .
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 . 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 ∆E being as small as 1 mV in order to increase a resolution of peaks . Such derivatives thus obtained are given in Figure 5. In general, all curves shift to the left (
18.104.22.168 For the first derivative
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,
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
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,
The anodic part of a half-peak is always larger than the cathodic part.
It should be noted again that these trends are the opposite of those from CEr type.
22.214.171.124 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.
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.
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.
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.
126.96.36.199 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 . 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, 39, 40, 41] and CSWV [66, 67, 68, 69, 70]. The present DV 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.
A quantitative measure of symmetry
From careful analysis of derivatives of voltammetric current-potential curves, it is possible to extract various parameter which are associated with asymmetry of the derivatives associated with various types of electrode mehanisms. Among those parameters, some of them are strongly influenced by electron transfer kinetics and electrode reaction mechanisms associated with a system, and these parameters are particularly useful in elucidating the reaction mechaisms. In particular, the ratios of the anodic to cathodic peak-currents (
Several parts of the work have been presented at various conferences including the 3rd Chemical Congress of North America (Toronto, Canada, 1988), the 57th Georgia Academy of Science Meeting (Lawrenceville, GA 1999), the 221st National Meeting of American Chemical Society (ACS, San Diego, 2001), and the 65th South East Regional Meeting (SERM of ACS, Atlanta, GA 2013). This work was partially supported by Academic Computing Services of Old Dominion University, and with Computer Services through Cornell National Super Computer Center (NSF supported), and by Faculty Fellowship Awards (2008-2009) from the Writers Institute of Georgia Perimeter College (GPC). The author acknowledges the work of all collaborators, Prof. Tae-Kee Hong (Haseo University, Korea) in particular. GPC has been consolidated to Georgia State University (GSU) as Perimeter College in 2016.