A new approach to a first order irreversible homogeneous chemical reaction followed by an electrochemical process (C_iE mechanism): theory, simulation and application

Abstract

Chronoamperometric profiles were derived for systems in which an irreversible, homogeneous chemical reaction generates an electroactive species that readily reacts at the surface of a planar electrode. From the resulting current (i) equations, a semi-operation (convolution voltammetry and semi-differentiation) approach was proposed to extract the rate constant of the chemical reaction through non-linear fitting. These equations were validated using digital simulations, and the semi-derivative approach successfully recovered the simulated parameters. The proposed kinetic method was then experimentally applied to a well described chemical reaction, the oxidation of iodide by hydrogen peroxide in acidic medium, followed by the electrochemical reduction of triiodide. The rate constants obtained from the method were in agreement with existing literature across various temperatures, and an Arrhenius plot led to a reasonable value for the activation energy of the process.

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Introduction

Electrochemical processes are essential for sustaining the global economy while aligning with the principles of a circular economy to minimize environmental harm.^1,2 As a result, traditional processes are increasingly being adapted to use electricity from clean energy sources.³ Understanding the interplay between chemical reactions and electrochemical processes is pivotal for advancing technologies in sensing,^4,5 environmental electrochemistry^6,7 and industrial catalysis.^8,9 Methods capable of accurately analysing coupled chemical-electrochemical systems are essential for decoding complex reaction mechanisms, optimizing industrial processes, and developing advanced diagnostic and monitoring tools.

The coupling between chemical and electrochemical reactions leads to characteristic voltammetric or chronoamperometric current profiles, which are entangled with the reaction mechanism.^10–13 A well-known example is the “ECE” (electrochemical–chemical–electrochemical) mechanism, where a chemical reaction involving the product of an electrochemical reaction generates a new electroactive species, and therefore is followed by another electrochemical oxidation or reduction.^14,15 The electrochemistry of many organic molecules has been elucidated and characterized using adequate models, simulations, and mathematical treatment of voltammetric or chronoamperometric data according to this class of mechanisms.^16–19

On the other hand, a “CE” (chemical–electrochemical) mechanism occurs when an electroactive species is generated by a homogeneous chemical reaction. Traditionally, studies of CE mechanisms assume that the chemical reaction has reached equilibrium before any electrochemical measurement begins, with voltammetric profiles dependent on the equilibrium constant of the reaction.^12,20–22 However, in certain situations—such as in situ monitoring of biologically generated electroactive species or kinetic characterization of homogeneous chemical reactions—the assumption of chemical equilibrium does not hold. Despite the wide range of relevant CE mechanisms, the current profiles of chronoamperometric experiments following irreversible chemical reactions are still lacking. This mechanism is essential, for instance, to pharmaceutical industries with a class of remedies called pro-drugs.²³ In another example, this system is commonly found in interactions between neurotransmitters and drugs.¹³ Studies involving these molecules may present a particularly untrivial variation of CE mechanism which is the irreversible homogeneous chemical reaction followed by an electrochemical process (C_iE mechanism). The description of this cases is also not found in classical electrochemical literature, such as Bard's,²⁴ Compton's,¹⁰ Brett's,²⁵ Pletcher's,²⁶ and Saveant's¹² textbooks.

The irreversibility of the chemical step is of the utmost importance since it significantly improves the systems in (a) in sensing, as it increases the electrochemical response, hence the sensibility by providing a greater formation of the electroactive species, (b) in catalysis a bidirectional catalyst is usually prone to a lack of efficiency, as the generated product can also be converted into the reagents, (c) in (electro)synthetic routes the irreversibility diminishes the generation of by-products.^27,28 Moreover, the practical and theoretical comprehension of the C_iE case enables the use of versatile electrochemical techniques like chronoamperometry in a wider range of kinetic studies, potentially benefiting pharmaceutical, biotechnological, industrial and academic research. Henceforth, the theoretical and practical description to a case which not only is lacking in textbooks due to its untriviality but also presents several applications, as aforementioned, is relevant.

From a technical perspective, when a chronoamperometric experiment is conducted under conditions that lead to a steady-state current (such as with microelectrodes²⁹ or under laminar flow³⁰), the current is predictably proportional to the concentration of the electroactive species generated by the chemical reaction, enabling straightforward kinetic studies. However, in scenarios where linear diffusion predominates—such as with macroelectrodes or microelectrodes encountering hasty concentration changes—current profiles become more complex. Thus, these cases require a more detailed theoretical framework to accurately extract kinetic information. In an analogous way to how convolution techniques and semi-differentiation^31–33 simplify asymmetric voltammetric profiles, allowing the use of non-linear fits^34,35 for the full characterization of the voltammetric peaks, these techniques can be used to simplify chronoamperometric profiles.

In our research, we address this critical gap by deriving novel equations that describe chronoamperometric current profiles for irreversible chemical reactions within CE mechanisms, particularly under linear diffusion conditions. Our theoretical framework not only provides insights on CE mechanisms but also undergoes validation through digital simulations. The kinetic method was applied in a well-described reaction,^36,37 the iodide oxidation by hydrogen peroxide in acidic medium, to validate the model in comparison to the literature. The chemical and electrochemical steps for such a system can be represented as follows:

3I⁻ + H₂O₂ + 2H⁺ → I₃⁻ + 2H₂O (Chemical)

I₃⁻ + 2e⁻ → 3I⁻ (Electrochemical)

Our study has two purposes: firstly, to present a comprehensive theoretical derivation of current profiles in such systems, and second, to validate these predictions. By doing so, we establish a robust framework applicable to diverse electrochemical systems where traditional equilibrium assumptions fall short. Secondly, our findings advance the understanding of chronoamperometric profiles in systems with irreversible chemical reactions preceding electrochemical processes, providing a solid foundation for research and practical applications in electrochemistry, which can be further explored in academic and industrial applications.

Experimental

Digital simulations

All the digital simulations were performed with DigiElch 7 software from Gamry Instruments (Warminster, PA, USA). The number of electrons involved in the electrochemical reaction was set as 1, the formal reduction potential of the electroactive species was set as 0 V, and its standard electrochemical rate constant as 10⁴ cm s⁻¹ (very fast electrode reaction). The electrode potential was set as 0.4 V, so that the concentration of A (generated from R through a chemical reaction) at the electrode surface is virtually null throughout the simulation. The diffusion coefficient of A was set as 1 × 10⁻⁵ cm² s⁻¹.

For 1D, semi-infinite diffusion simulations, the electrode area was set as 1 cm². The number of simulated points was not the same for all simulations, and the time step is described in the text. In the 2D simulations, a disk with a 3 mm diameter was considered, and 4 × 10⁴ points were simulated.

Reagents and solutions

Analytical grade hydrochloric acid (HCl) was acquired from Synth (Brazil). Potassium iodide (KI) and 30% hydrogen peroxide (H₂O₂) were purchased from Merck (Germany). All the solutions were prepared with deionized water (Gehara Co., model OS10LX ultra-pure system, water resistivity ≥ 18 MΩ cm). Hydrogen peroxide stock solution was titrated with potassium permanganate from Merck (Germany), and its concentration was determined to be 5.27 mol L⁻¹. This stock solution was diluted before being used in the kinetic experiments. HCl 2.0 mol L⁻¹ and KI 1.0 mol L⁻¹ stock solutions were prepared.

Apparatus

All the electrochemical measurements were performed with potentiostat/galvanostat PGSTAT 101 (Metrohm Autolab), controlled with NOVA 1.11 (Metrohm Autolab). A platinum disk (3 mm in diameter), platinum wire and Ag/AgCl, KCl_(sat) electrode were used, respectively, as working, auxiliary and reference electrodes. Chronoamperometric measurements were performed with an applied potential of 0.2 V vs. Ag/AgCl, KCl_(sat.), and a time step of 0.05 s. For temperature control (±0.1 °C), a Polyscience circulating water bath model 911 was used.

Preparation of the electrochemical cells

The kinetics experiments required the fast mixing of three reactants: HCl, KI, and H₂O₂. To achieve this, a 10 mL plastic syringe was adapted to become a reactor or an electrochemical cell, as represented in Fig. 1. Five holes were made in the syringe to accomodate three electrodes while the two remaining holes served as injection points for the solutions. The cylindrical syringe was 1.6 cm in internal diameter and the injection holes were carved 2.0 cm below the surface where the working electrode was inserted (Fig. 1). In all experiments the piston was positioned so that the bottom of the cell was 2.5 cm below the injection holes. The surface of the working electrode was parallel to the bottom of the cell and approximately 1 cm below the injection holes. The cell was immersed in a thermostatic bath so that, after the injection, the solution level was below the bath level. To avoid loss of material by splashing and to promote the circular motion of the solutions, the injections were made close to the cell wall and not perpendicularly to the bottom. Because the experiments were conducted manually, the rate and angle of injection were not thoroughly controlled in our experiments A volume of 0.6 mL of a mixture of HCl (0.5 or 1.0 mol L⁻¹) and H₂O₂ (26.4 mmol L⁻¹) was hastily injected, as 2.4 mL of KI (1.0 mol L⁻¹) solution also injected. Both solutions were thermostatically controlled at 15, 25 and 40.3 °C before the injections. After the solutions were mixed, the concentrations of iodide and H₂O₂ were 0.8 mol L⁻¹ and 5.27 mmol L⁻¹, respectively, whilst the concentration of H⁺ was either 0.1 or 0.2 mol L⁻¹. HCl and KI served both as reactants and as supporting electrolytes.

Fig. 1 Schematic representation of the cell built with a syringe. 1 – working electrode; 2 – auxiliary electrode; 3 – reference electrode; 4 – injection point 1; 5 – injection point 2; 6 – piston.

The syringe's piston was used to regulate the volume of the cell so that all the electrodes were adequately immersed in the solution resulting from the simultaneous injection of reactants. The working electrode was placed so that its surface was parallel to the bottom of the cell. The electric potential of 0.2 V vs. Ag/AgCl, KCl_(sat) was applied to the working electrode before the reactant injections were made.

Numerical calculations and data analysis

Numerical calculations involving the Dawson function (or Dawson integral), discussed in the Results and discussion section, were made with Wolfram Mathematica 12 using the function DawsonF. Semi-derivatives and semi-integrals of the simulated chronoamperograms were obtained with the software eL-Chem-Viewer.³⁸ Semi-derivatives and semi-integrals of experimental data were performed using the Riemann-Liouville algorithm available in NOVA 1.11 software.

All the non-linear fits were performed in OriginLab 2022b with the Levenberg–Marquardt method.

Results and discussion

Theoretical approach

For a first-order, irreversible, homogeneous chemical reaction preceding an electrochemical reduction reaction (CE) involving n electrons, it is possible to define the derivative of the concentrations of the reactants R and A with respect to time as:where t is the time, D_R and D_A are the diffusion coefficients of, respectively, R and A, k_f is the rate constant of the first order homogeneous chemical reaction and A and R (in italics) are, respectively, the concentrations of species A and R.

Eqn (1) and (2) consider the contributions of diffusion as stated by Fick's 2nd law and the chemical reaction, with the inclusion of a term that corresponds to the kinetic law for a first order reaction. For a one-dimensional diffusion regime, eqn (1) and (2) are simplified to:

The boundary conditions considered are:

A(x,0) = A_i, ∀x ≥ 0 (5)

A(0,t) = 0, ∀t > 0 (6)

R(x,0) = R_i (7)

A(x,t) = A_i + R_i(1 − exp(−k_ft)), for x → + ∞ (8)

where A_i and R_i are the initial concentrations of reactants A and R, respectively, and j is the flux of A at the electrode surface. If the chemical reaction is homogeneous and irreversible, the concentration gradient of R equals zero for any x. Therefore, the concentration of R can be written as a function of time t and does not depend on the position x:

R(t) = R_i exp(−k_ft) (10)

Substituting in eqn (4), we have:

To solve this partial differential equation presented, the Laplace Transform method is applied.³⁹ The detailed solution to the equation is found in Appendix. From the Laplace transform method, eqn (12), which relates the concentration gradient of A with the time and other physical and chemical constants of the system, is obtained:

The integral in the equation above relates directly to the semi-integral⁴⁰ of the exponential function:

The semi-integral of the function exp(−x) is given by:⁴¹

where daw(z) is the Dawson function of the variable z, which is defined as:⁴²

Substituting in eqn (12):

To convert this equation in terms of electrical current, the following equation must be considered:

where i is the current, n is the number of electrons involved in the electrochemical reaction, F is the Faraday constant and S is the area of the electrode.

Therefore, we have:

The expected current profiles when the ratio between the coefficients that multiply the terms t^−1/2 (Cottrell component) and (Dawson component) of eqn (18) is varied are shown in Fig. 2.

Fig. 2 (a) Numerical plot of the Dawson function of the square root of the independent variable x, with the division of its behavior in areas I, II and III meaning different stages of the modeled physicochemical processes. (b) Numerical plots of the Dawson function with the addition of crescent amounts of “pure Cottrelian” behavior, representing electrochemical measures starting at different reaction stages.

In Fig. 2a, a “pure Dawson” profile is observed, where the coefficient that multiplies the Cottrell component is zero. In the chemical system modelled, this case corresponds to a chronoamperometric measure that starts at the same time as the homogeneous chemical reaction. Under these conditions, for t = 0, the current is also 0, since daw(0) = 0. Right after the beginning of the chemical reaction (interval I in Fig. 2a), the current increases as a consequence of the formation of electroactive species A. The continuous generation of this species throughout the system increases the concentration gradient (and the current, according to eqn (9) and (17)).

Simultaneously, if the concentration of A is 0 at the electrode surface (stated as a boundary condition in eqn (6)), a diffusion layer takes place. The effect of the growth of this diffusion layer is the decrease of the concentration gradient of A. However, the composition of the diffusion layer is also affected by the chemical generation of A, which tends to increase the concentration gradient, as aforementioned. The competition between these two contradictory effects is fully captured by the integral on the right-hand side of eqn (12), which resulted in the Dawson function in Fig. 2a.

As the chemical reaction proceeds, it becomes slower, and therefore the effect of increasing the concentration gradient becomes progressively less intense with time. Meanwhile, the diffusion layer continues to grow until, at some point (at the end of interval I or the beginning of interval II), the effect of its expansion matches the effect of A generation. At this point, the concentration gradient, which had been increasing, reaches its maximum. Subsequently (interval II), the concentration gradient decreases, and after the homogeneous chemical reaction is virtually completed, the current profile becomes equivalent to the one predicted by the Cottrell equation (interval III).

If the initial concentration of the electroactive species A (A_i) is 0, eqn (18) becomes:

Evaluating the numerical values of the Dawson function of the square root of a real number x (Fig. 1a), one finds that the maximum value of the function is approximately 0.541, reached when x = 0.854. Then the following equations for the peak current, i_p, and peak time, t_p, are obtained:

With these equations, it would be possible to estimate the rate constant of the chemical reaction that converts the reactant R to the electroactive species A, as the peak time depends on no variables besides k_f.

In Fig. 2b, the plots show the expected current profiles as the coefficient that multiplies the Cottrell term of eqn (18) increases from 0 to 0.25, while the Dawson term coefficient is maintained as 1. As the Cottrell's coefficient increases, the peak loses definition, and the profile tends, to a cottrelian behavior. These are the current profiles that would be observed if the chronoamperometric measurements are started after the beginning of the homogeneous chemical reaction, when species A is already present in solution. In this situation, eqn (20) and (21) are no longer valid.

Initiating chronoamperometric measurements precisely at the moment the chemical reaction begins can be challenging. To address this issue and enhance the accuracy of kinetic parameter extraction by leveraging multiple data points from the measurement curves, the following sections discuss transforming complex predicted chronoamperogram profiles into simple exponential equations through the application of semi-operations, such as semi-integration and semi-differentiation.

Semi-operations

By taking the semi-integral of both sides of eqn (18), the following expression is obtained:³²

Eqn (22) is valid for chronoamperometric measurements starting at any extension of the coupled chemical reaction, and implies that a measured current, which is originally of a complicated form, involving the Dawson function, can be transformed into simple exponential functions. This transformation enables three-parameter (a, b, c) non-linear fits (eqn (23)) for the extraction of k_f from the chronoamperograms using the c parameter. Moreover, the extension of reaction at the beginning of the chronoamperometric measurement can also be inferred from the parameters a and b in the equation.

y = a + b exp(−cx) (23)

where a, b and c are the fit parameters that represent , and k_f, respectively, if the semi-integral current is being fitted.

The m(t) profiles (not necessarily the numerical values) can be said to reflect the total concentration of species A in the system. This statement can be easily verified by comparing eqn (8) and (22).

By differentiating m(t) with respect to t, the semi-derivative of the current, e(t), is obtained:

The expression above allows non-linear fits with only two parameters (a and b), according to eqn (24). However, when the semi-derivative of the current is taken, the information about the extension of the chemical reaction at t = 0 is lost.

y = a exp(−bx) (25)

where a and b are the fit parameters that represent and k_f, respectively, if the semi-derivative current is being fitted. Although the value of k_f is a factor in parameter a, the value of the latter depends on other variables, so that a and b are independent parameters.

To verify the validity of all the mathematical predictions made for the chronoamperometric profiles and their semi-derivatives and semi-integrals, digital simulations were performed.

Simulations

A system including the boundary conditions defined in eqn (1) to (9) was simulated with the commercial software DigiElch 7. An appropriate interpretation of the simulated systems is that if the chemical reaction starts at the same time as the electroanalytical measurement, R_i = 1 mmol L⁻¹ and A_i = 0 mmol L⁻¹. If the measurement starts when 20% of R has already been consumed, then R_i = 0.8 mmol L⁻¹ and A_i = 0.2 mmol L⁻¹, and so on. Fig. 3a shows the chronoamperograms simulated considering that the measurement is started (t = 0) at different reaction extents, keeping the analytical concentration of the reactant R, given by R_i + A_i, constant (1 mmol L⁻¹ or 1 μmol cm⁻³). All the chronoamperograms were simulated considering k_f = 1 s⁻¹.

Fig. 3 Simulated chronoamperograms starting when 0 (black), 20% (red), 40% (blue), 60% (green), 80% (lilac) and 100% (yellow) of the reactant R has been consumed (a) and their semi-integrals (b) and semi-derivatives (c) with respect to time. Simulation parameters: k_f = 1 s⁻¹; n = 1; D_A = 1 × 10⁻⁵ cm² s⁻¹; S = 1 cm²; A_i + R_i = 1 mmol L⁻¹; time step: 0.001 s. Other parameters in Experimental section.

Fig. 3a demonstrates that the mathematically predicted profiles in Fig. 2 are in agreement with the simulated chronoamperograms. Additionally, the black solid line of Fig. 3a reaches a peak at i_p = 1.863 × 10⁻⁴ A and t_p = 0.854 s, which are almost the same values that are calculated using eqn (20) and (21), which are 1.861 × 10⁻⁴ A and 0.854 s, respectively, considering D_A = 1 × 10⁻⁵, k_f = 1 s⁻¹, R_i = 1 × 10⁻⁶ mol cm⁻³ and S = 1 cm². This demonstrates strong agreement between both mathematical results and the digital simulations, implying that eqn (18), (20) and (21) properly describe the system proposed.

Interestingly, in Fig. 3a (inset) an “isosbestic point” (referred here as an isoamperic point) can be observed. This means that, with a defined value of k_f and for any values of A_i and R_i, keeping the sum A_i + R_i constant, at a certain time, the measured current will always be the same. Visual inspection indicated that this point that seems to be the maximum point of the “pure Dawson” graph (black solid line). Indeed, by postulating the existence of such an isoamperic point and manipulating eqn (18), one can find that this point occurs exactly at t_p defined in eqn (21). These mathematical manipulations are detailed in the Appendix section.

The occurrence of the isoamperic point can be explained by considering the interplay of multiple phenomena affecting the concentration gradient of the electroactive species A. The growth of the diffusion layer tends to decrease the concentration gradient. The gradient would decay with t^−0.5, as stated by eqn (18) and implicitly contemplated in eqn (12). However, there is also an opposite effect caused by the chemical reaction, which changes the composition of the diffusion layer by generating A throughout the solution, countering the effect of the growth of the diffusion layer on the concentration gradient.

A variation in the value of A_i affects the gradient with the same intensity, but with an opposite effect, that a variation in the value of R_i affects it. Because A_i + R_i is constant, then at the point that these two tendencies balance, i.e., at the peak of the pure Dawson curve, the variation of A_i is balanced by a variation of R_i and vice-versa. Therefore, these variations will have no net effect on the concentration gradient, originating the isoamperic point.

Fig. 3b and c show the time semi-integral and semi-derivative plots of the chronoamperograms in Fig. 3a, highlighting the transformation of the current profiles. These curves appear to be consistent with the exponential forms predicted in eqn (22) and (24). This consistency is further validated by the non-linear fits performed with eqn (23) and (25). These fits successfully recover the rate constant of the chemical reaction, the initial A/R ratio, and the pre-exponential factor of each case, shown in Table 1.

Table 1 Results of the non-linear fits of the graphs in Fig. 3b and c. Shaded columns are referent to the fits of the semi-integrals, and the non-shaded region is referent to the fits of the semi-derivatives

Simulated Ai/Ri^a	Semi-integral				Semi-derivative
	Recovered Ai/Ri^b	Fitted kf (s^−1)	Error in kf (10^−3 s^−1)	R ^2	Fitted kf (s^−1)	Error in kf (10^−3 s^−1)	Calculated (10^−4 A s^−1/2)^c	Fitted (10^−4 A s^−1/2)	Error in (10^−7 A s^−1/2)	R ^2
a The ratios, from top to bottom, are equivalent to proportions Ai : Ri of 0 : 1; 1 : 9; 2 : 8; 3 : 7; 4 : 6; 5 : 5; 6 : 4; 7 : 3; 8 : 2 and 9 : 1. b The ratios were recovered by calculating , where a and b are the fit parameters in eqn (23). c Calculated considering n = 1, F = 96485 C mol^−1, S = 1 cm^2, Ri from 0.1 to 1 μmol cm^−3, DA = 1 × 10^−5 cm^2 s^−1 and kf = 1 s^−1.
0	<0.001	0.9999	0.026	>0.99999	0.99995	0.0058	3.051	3.052	0.031	>0.99999
0.111	0.1015	0.9965	0.036	>0.99999	0.99713	0.035	2.746	2.752	0.17	>0.99999
0.250	0.2281	0.9921	0.078	0.99999	0.99366	0.074	2.441	2.452	0.32	0.99998
0.429	0.3920	0.9872	0.13	0.99996	0.98926	0.12	2.136	2.153	0.48	0.99996
0.667	0.5942	0.9804	0.19	0.99992	0.98352	0.19	1.831	1.853	0.62	0.9999
1.00	0.8978	0.9625	0.28	0.99982	0.97571	0.27	1.526	1.554	0.75	0.99979
1.50	1.3255	0.9452	0.40	0.99962	0.96448	0.39	1.220	1.255	0.87	0.99955
2.33	2.0526	0.9091	0.57	0.99917	0.94692	0.58	0.915	0.956	0.97	0.99901
4.00	4.3433	0.7521	0.95	0.99726	0.91566	0.8	0.610	0.658	1.0	0.99757
9.00	9.0441	0.5988	1.1	0.99538	0.84459	1.6	0.305	0.362	0.91	0.99229

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The fitted values in Table 1 show a strong agreement with the quantities defined in the simulations for low values of A_i/R_i, and tend to deviate as this ratio increases. This deviation is also reflected in the R² values, that become progressively lower. These results must be read under the consideration that the ratio A_i/R_i represents the extent of the homogeneous chemical reaction at the moment that the electrochemical reaction starts. In practice, this moment corresponds to the instant when the potential is applied to the working electrode or when an already polarized electrode is immersed in the solution in which the reactions are taking place. In this case, it becomes clear that starting the electrochemical measurements too late will cause deviations in the results of the kinetic analysis, since the information regarding the chemical reaction is lost, and the current measured will be mostly due to the electrochemical reaction of the electroactive species formed before the measurements starts. This increasing inaccuracy is reflected in the statistical error in k_f and the pre-exponential factor, which grows nearly 3 orders of magnitude.

Another reason for the accuracy loss is related to the numerical methods used to calculate the semi-derivatives and semi-integrals of discrete data. Theoretically, the time semi-derivative of a Cottrell-like curve should result in 0, and its semi-integral must be a constant. However, when these operations are conducted numerically, these analytical values are not obtained. Instead, the curves shown in Fig. S1, in ESI,† result from the numerical operations. Moreover, a more detailed discussion regarding the sources of this errors and the algorithms for semi-operations are presented in the ESI.† Overall deviations from the analytical results can be minimized by using shorter time steps, which increase the number of calculation points and consequently, improves the accuracy.

The issues related to the numerical methods for fractional operations apply to the Cottrell component of eqn (18). Thus, as the Cottrell component becomes increasingly relevant with the increase of the ratio A_i/R_i, the fitted and recovered values reported in Table 1 tend to deviate from the values defined in the simulations. Moreover, a “trade-off” between the accuracies of the fitted k_f and of the fitted pre-exponential factor is observed when eqn (25) is used for the fits instead of eqn (23). These results are compared in Table S1 (ESI†).

By varying k_f and keeping A_i = 0, the simulated chronoamperograms in Fig. 4a are achieved. Once again, one may notice that the “pure Cottrell” curve intersects all the “pure Dawson” curves at their peaks, as priorly discussed. The peak currents obtained in the simulations are proportional to , as predicted by eqn (20), and the slope of the linear fit in Fig. 4b is 0.1863 mA s^−1/2, in agreement with this equation. The simulated peak times match the predictions of eqn (21).

Fig. 4 Simulated chronoamperograms with k_f set as 0.05 (black), 0.1 (red), 0.2 (blue), 0.5 (green), 1 s⁻¹ (violet) and without any chemical reaction taking place, with all of R_i = 0 A_i = 1 mmol L⁻¹ (yellow) (a), linear regression of i_pvs. (b) and the semi-integrals (c) and semi-derivatives (d) of the chronoamperograms with respect to time. Simulation parameters: n = 1; D_A = 1 × 10⁻⁵ cm² s⁻¹; S = 1 cm²; R_i = 1 mmol L⁻¹; A_i = 0; time step: 0.01 s. Other parameters are presented in Experimental section.

The non-linear fits of the semi-derivatives and semi-integrals of the chronoamperograms were performed with eqn (23) and (25), respectively. Table 2 presents the k_f values extracted from the non-linear fits, further validating the method's utility for deriving kinetic information from the homogeneous reaction.

Table 2 Results of the non-linear fits of the graphs in Fig. 4b and c. All the fits considered presented R² > 0.99999

Simulated kf (s^−1)	k f from SD^a (s^−1)	Error (10^−3 s^−1)	k f from SI^b (s^−1)	Error (10^−3 s^−1)	calculated^c (10^−4 A s^−1/2)	from SD^a (10^−4 A s^−1/2)
a Fitted from semi-derivative with eqn (23). b Fitted from semi-integral with eqn (25). c Calculated considering n = 1, F = 96 485 C mol^−1, S = 1 cm^2, Ri = 1 μmol cm^−3, DA = 1 × 10^−5 cm^2 s^−1.
0.05	0.05003	0.00080	0.05003	0.00050	0.1526	0.1526
0.1	0.1001	0.0032	0.1001	0.0012	0.3051	0.3056
0.2	0.2002	0.0069	0.2002	0.0042	0.6102	0.6119
0.5	0.5005	0.018	0.4991	0.038	1.5256	1.5332
1	0.9987	0.10	1.0039	0.29	3.0511	3.0743

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All the fitted parameters from Table 2 match the values defined in the simulations with good accuracy. Unlike the fits presented in Table 1, there are no major deviations due to the issues related to the numerical semi-operations applied to the Cottrell components of the current, since A_i = 0, even though the number of points used for the simulations was 5 times smaller than the number of points of the simulations in Fig. 3. The fitted curves and parameters are presented in Fig. S4 (ESI†).

By setting A_i = 0.5 mmol L⁻¹ and R_i = 0.5 mmol L⁻¹, the chronoamperograms in Fig. 5a are obtained. Most chronoamperograms in Fig. 5a are barely distinguishable from pure Cottrell profiles, since the peak of the Dawson component is not evident in these conditions. On the other hand, the time semi-derivatives and semi-integrals (Fig. 5b and c) can still provide information of the chemical kinetics of the homogeneous reaction.

Fig. 5 Simulated chronoamperograms with k_f set as 0.05 (black), 0.1 (red), 0.2 (blue), 0.5 (green), 1 s⁻¹ (violet) and without any chemical reaction taking place, with R_i = 0 A_i = 1 mmol L⁻¹ (yellow) (a), and their semi-integrals (b) and semi-derivatives (c) with respect to time. In figure (d), the semi-integrals of the chronoamperograms simulated with k_f set as 0.05 s⁻¹ (red) and 0.5 s⁻¹ (black) using 0.025 s (dashed) and 0.005 s (solid) time steps are presented. Simulation parameters: n = 1; D_A = 1 × 10⁻⁵ cm² s⁻¹; S = 1 cm²; R_i = 0.5 mmol L⁻¹; A_i = 0.5 mmol L⁻¹. Other parameters in Experimental.

For the simulations shown in Fig. 5, a time step of 0.005 s was adopted, which is 5 times shorter than that used in the simulations presented in Fig. 4. Because of the imprecision in the semi-differentiation and semi-integration of the cottrellian component of the current, in this case there is a significant difference between the results of the non-linear fits performed with time-steps of 0.025 and 0.005 s. These differences are detailed in Table 3, reflected in the values of the error in k_f.

Table 3 Results of the non-linear fits of the graphs in Fig. 5 with different time steps. More fit information and parameters in Fig. S3 (ESI)

Simulated kf (s ^−1)	Results from semi-derivative, time step 0.025 s		Results from semi-integral, time step 0.025 s
	Fitted kf (s^−1)	Error (10^−3 s^−1)	Fitted kf (s^−1)	Error (10^−3 s^−1)
0.05	0.0505	0.014	0.0502	0.0083
0.1	0.0999	0.047	0.0988	0.031
0.2	0.2021	0.18	0.1898	0.18
0.5	0.5064	0.67	0.4237	1.4
1	1.0125	1.8	0.7722	4.7

---

Simulated kf (in s ^−1)	Results from semi-derivative, time step 0.005 s		Results from semi-integral, time step 0.005 s
	Fitted kf (s^−1)	Error (10^−3 s^−1)	Fitted kf (s^−1)	Error (10^−3 s^−1)
0.05	0.0502	0.0022	0.0501	0.0010
0.1	0.1000	0.010	0.0994	0.0065
0.2	0.2009	0.037	0.1951	0.039
0.5	0.5003	0.21	0.4630	0.36
1	1.0132	0.67	0.8795	1.3

---

The discrepancy between the simulated k_f and the ones recovered from the semi-integrals increases with the value of k_f. This occurs because the numerical issues of the semi-operations applied to the Cottrell curve are more pronounced at short times (Fig. 5d). Large values of k_f imply that the kinetic information is extracted from these same short times which are impaired by numerical imprecisions. For small k_f values, the semi-integrated and semi-differentiated current values are less affected by numerical issues and can still be effectively used.

When k_f is recovered from semi-derivatives, their values are more consistent with the simulated values if the time interval for the non-linear fits is carefully chosen. Kinetic information for faster chemical processes is found at the beginning of the simulation, where numerical instability is more relevant. However, the Dawson component of the chronoamperograms is “amplified” as k_f increases (eqn (24)), while the numerical instability caused by the Cottrelian component remains unaffected. Therefore, performing the exponential fits of semi-derivative graphs in adequate intervals enables accurate measurements of k_f, as shown in Table 3.

As extensively discussed, the simulations showed to be in accordance with the equations derived analytically with the Laplace Transform method. In addition, it is clear that the semi-derivative and semi-integral methods allow to perform the extraction of crucial kinetic information of the kinetics of the homogeneous chemical reaction. Nevertheless, careful adjustment of the length of the time-steps according to the initial concentration of the electroactive species is required. Following, the practical application of the theory proposed in this work is discussed, as well as the experimental limitations and eventual deviations from the theory and the simulations.

Proof of concept

The reaction selected to proof the concept for the developed kinetic model was the chemical oxidation of iodide by hydrogen peroxide in an acidic solution, generating iodine (or, in excess iodide, triiodide):

H₂O₂ + 3I⁻ + 2H⁺ → I₃⁻ + 2H₂O

The reaction can undergo two pathways: the catalyzed (with rate constant k₁) and the uncatalyzed (with rate constant k₂) pathways. In acidic media, a pre-equilibrium is established between the protonated and unprotonated forms of H₂O₂ before the rate-determining step of the reaction, which is the nucleophilic attack of I⁻ on the oxygen in hydrogen peroxide.^36,37,43 The rate constant for the nucleophilic attack on the protonated species is greater than that for the attack on the non-protonated H₂O₂. The rate law for the reaction in acidic media is given by eqn (26):

If the reaction proceeds in a medium containing large excess of I⁻ and H⁺ ([I⁻] ≫ [H₂O₂] and [H⁺] ≫ [ H₂O₂]), it can be considered to follow a typical pseudo-first order rate law, with (k₁ + k₂[H⁺])[I⁻] serves as an apparent rate constant k_ap. This constant may be modulated by varying the concentration of H⁺ or I⁻ in the system.

Placing a polarized electrode in the solution where the reaction occurs is expected to yield a current profile similar to those predicted in the theoretical and simulation sections. The electrode potential provides a driving force for the reduction of triiodide that is enough to make the concentration of I₃⁻ at the surface of the electrode virtually null. The electrochemical reduction (the E step from CE) of triiodide may be written as:

I₃⁻ + 2e⁻ → 3I⁻

In this scenario, I₃⁻ and H₂O₂ correspond, respectively, to the species A and R of eqn (1), with k_ap corresponding to k_f. Even though the product of the electrochemical reaction also serves as the reactant in the homogeneous chemical reaction, the concentration gradient of iodide can be assumed to be negligible throughout the solution. This assumption is valid since the amount of electrolytically generated iodide is insignificant compared to the excess of iodide initially present in the electrochemical cell.

A chronoamperometric experiment was performed in an electrochemical cell in which the chemical oxidation of iodide by hydrogen peroxide was taking place. In our experimental arrangement, iodide and H⁺ are present in large excess (0.8 and 0.1 mol L⁻¹, respectively) with respect to H₂O₂ (5.27 mmol L⁻¹), leading the reaction to proceed under a pseudo-first order regime. Moreover, the excess of iodide ensures that all the iodine formed in the reaction with hydrogen peroxide is readily converted to triiodide, which is soluble in water. The potential applied to the working electrode was 0.2 V, enough for the concentration of triiodide at the surface of the electrode to be virtually null throughout the experiment. The voltammogram used to observe the redox process of the pair I₃⁻/I⁻ is presented in Fig. S6 (ESI†).

Efficient reactant homogenization is essential for achieving accurate, reliable, and meaningful results in all chemical kinetics experiments. In the experimental approach proposed in the present work, the heterogeneity of the system would lead to the concentration gradients along the solution, which should affect directly the concentration gradient at the electrode surface, leading to unexpected deviations from the current profiles proposed in the theoretical section. We attempted to mitigate these effects by preparing the electrochemical cell in the way described in the Experimental section.

In Fig. 6a, a chronoamperogram measured at 25 °C under the conditions described above and its semi-derivative with respect to time are presented. The current and semi-derivative current profiles obtained experimentally are very similar to the calculated and simulated profiles (Fig. 3 and 4, for example). However, notable differences are the fact that semi-derivative current does not decay to zero as would be expected for the pure exponential decay predicted in eqn (24) and the current drop after the peak looks less accentuated than in the simulations. Both these effects are explained by the contribution of radial diffusion at the scale of time in which the experiment was performed, and will be discussed in greater detail later in this manuscript.

Fig. 6 (a) Chronoamperogram obtained with Pt disk electrode at 25 °C in a mixture of 0.8 mol L⁻¹ of KI, 0.1 mol L⁻¹ of HCl and 5.27 mmol L⁻¹ of H₂O₂, generating triiodide (black solid line) and its semi-derivative with respect to time (red dots). The applied potential was 0.2 V and the time step was 0.05 s. (b) Semi-derivative shown in figure (a) (blue dots), non-linear fit according to y = A exp(−k_fx) + y₀ (red solid line) and fit residual (red dots and inset). Fitted parameters: A = − 2.77 μA s^−1/2; k = 0.0219 s⁻¹; y₀ = −0.240 μA s^−1/2; R² = 0.9999.

Since the semi-derivative current in Fig. 6a does not decay to zero, eqn (23) was used for the exponential non-linear fit. For the same reason, the semi-integral approach could not be used for our experiments. The semi-integral chronoamperograms are found in Fig. S7 (ESI†), and the reason for the deviations are also discussed in ESI.† The non-linear fit lead to k_ap or k_f of (2.19 ± 0.02) × 10⁻² s⁻¹, which is in great agreement with the value of (2.30 ± 0.07) × 10⁻² s⁻¹, reported previously by other researchers.^36,37

Furthermore, Fig. 6b shows the non-linear fit of the semi-derivative current presented in Fig. 6a and the residual analysis of the fit (inset). The residuals are in the order of 0.01 μA s^−1/2, which corresponds to the noise of the semi-derivative current. This is also visually evident from the thickness of the red and black dispersions in Fig. 6b. Also, the residual is almost uniformly distributed around the y = 0 curve, demonstrating the convergence of the theory proposed in this work with the experimental results.

It was aforementioned that the experimental results in Fig. 6 deviate from the theoretical predictions. These apparent inconsistencies are related to the contribution of radial diffusion of the electroactive species (triiodide) to the measured current in the time scale of the experiment. Eqn (3) and (4) apply only to semi-infinite linear diffusion, hereafter referred to as linear diffusion. This diffusion regime is predominant in disk macroelectrodes, like the one used in our experiments. However, for a disk electrode, the contributions of radial diffusion become more significant as the diffusion layer becomes thicker and its geometry tends to assume spherical form. Therefore, if the time scale of the experiment is long, even for macroelectrodes there may be significant contributions of radial diffusion to the current.

If radial diffusion is the predominant regime, the stationary state is reached. In this condition, the concentration gradient does not change due to the growth of the diffusion layer, but due to the variation of the concentration of electroactive species in the bulk solution or at the surface of the electrode. The profiles observed in Fig. 6 reflect a mixed regime: for large values of time, both linear and radial diffusion are taking place.

Therefore, as the experiment time increases, the current tends to the stationary state, and the current decay after the peak is less accentuated than it was predicted by eqn (18). This equation predicts that the current should decay with the square root of time after the chemical reaction is over, and its semi-derivative should be virtually null. However, the effect of radial diffusion causes the semi-derivative current to diverge from the expected value for long times.

In Fig. 7, a 2D simulation performed with parameters that correspond to the experimental conditions is compared to the experimental results and to a 1D simulation. In contrast with the 1D simulation, which considers only linear diffusion, in the 2D simulation considers the effects of radial diffusion. These effects resulted in a greater resemblance between the experimental and simulated currents than the previous 1D simulations.

Fig. 7 Chronoamperogram obtained with Pt disk electrode at 25 °C in a mixture of 0.8 mol L⁻¹ of KI, 0.1 mol L⁻¹ of HCl and 5.27 mmol L⁻¹ of H₂O₂, generating triiodide (blue solid line) with an applied potential of 0.2 V and a time step of 0.05 s, 1D simulation of semi-infinite linear diffusion (red solid line) and 2D simulation of a disk with 0.15 cm in radius. The relevant simulation parameters are: D_A = 1 × 10⁻⁵ cm² s⁻¹ and k_f = 0.0219 s⁻¹.

Both time and peak current are normalized in Fig. 7 for the sake of comparison of the current profiles. The diffusion coefficient was set according to previous reports by Darral and Oldham.⁴⁴ The experimental and 2D simulation curves are nearly identical until k_ft ∼ 2.5. Considering that the value of k_f defined in the 2D simulation is the one obtained from the semi-derivative of the experimental current, we here reinforce that the semi-derivative approach proposed in this work estimates correctly the rate constant of the reaction.

Finally, the effect of the variation of k_f (or k_ap) was evaluated experimentally. For this purpose, the concentration of hydrochloric acid was varied at 25 °C, and for a fixed concentration of the reactants, the effect of temperature variation was also examined. In Fig. 8a, the peaks become narrower and that t_p decreases with the increase of temperature, reflecting the increase of k_ap. Moreover, when the concentration of H⁺ is halved at 25 °C, the measured peak is broader and t_p is greater (compare red and green solid lines in Fig. 8a).

Fig. 8 (a) Chronoamperogram obtained with Pt disk electrode in a mixture of 0.8 mol L⁻¹ of KI, 0.2 mol L⁻¹ of HCl (0.1 mol L⁻¹ for green line) and 5.27 mmol L⁻¹ of H₂O₂, generating triiodide at 15 (black), 25 (red and green) and 40.3 °C (blue). (b) Semi-derivative of the chronoamperogram obtained at 40.3 °C (blue solid line) and non-linear fit of y = A exp(−k_fx) + y₀ (red solid line). (c) Linearized Arrhenius plot with the rate constants found in the three experimental temperatures (R² = 0.9931).

By calculating the semi-derivative of these chronoamperograms and fitting the resulting curves the k_ap values for [H⁺] = 0.2 mol L⁻¹ at 15, 25 and 40.3 °C are, respectively, (2.09 ± 0.04) × 10⁻², (3.60 ± 0.02) × 10⁻² and (9.34 ± 0.03) × 10⁻² s⁻¹. These values align closely with those reported by Liebhafsky and Mohammad:³⁶ (1.93 ± 0.06) × 10⁻², (3.69 ± 0.11) × 10⁻² and (9.0 ± 0.3) × 10⁻² s⁻¹ (this last value is reported at 40.0 °C). The chronoamperogram measured in [H⁺] = 0.1 mol L⁻¹ showed in Fig. 8 is the same one of Fig. 6 and 7, and its accordance to literature has already been discussed.

Additionally, an Arrhenius plot is included in Fig. 8c, according to the linearized Arrhenius equation:

In eqn (27), k is the rate constant, P is a pre-exponential factor with the dimensions of k, E_a is the activation energy in J mol⁻¹, R is the universal gas constant (8.3145 J K⁻¹ mol⁻¹) and T is the temperature in K.

When applying the Arrhenius equation, one must note that it is an empiric law and is useful for limited ranges of temperature. Furthermore, for the reaction studied in this work, there would be formally two activation concomitant energies, the one related to k₁ and the other one related to k₂ in eqn (26), and a more elaborated approach would be required for the calculation of both these energies. However, for simplicity, we consider an “overall” activation energy, which was found to be (52.4 ± 3.1 kJ mol⁻¹), being compatible with the values reported by Liebhafsky and Mohammad.²⁰

The peak currents in Fig. 8 were not mathematically analyzed because the temperature of the system not only affects k_ap, but also the diffusion coefficient of triiodide (or iodine). In this case, a comprehensive study of the diffusion kinetics of these species in the studied media would be necessary for the adequate corrections to be performed. In addition, the homogenization methods employed occasionally caused air bubbles on the platinum disk electrode, altering its electroactive area and directly impacting the measured peak current.

An experimental artifact is evident in the blue solid line of Fig. 8a. This was likely due to a brief instability of the potential supply, which caused a variation in the concentration of triiodide at the electrode surface. As a consequence, a sudden increase of flux is observed right before the current starts to decay, eventually aligning with the expected chronoamperometric profile prior to the mass transport disruption.

Although the semi-derivative current diverts from the exponential decay predicted by eqn (24) when the potential supply is affected (Fig. 8b), it unquestionably returns to the exponential profile right after it. This return is confirmed when the non-linear fit of eqn (23) is applied, discarding the points that clearly deviate too much of the exponential profile. This non-linear fit is also shown in Fig. 8b.

The possibility of the performance of such a fit even after an experimental problem is a demonstration of the versatility of the semi-derivative approach for the kinetic analysis. It also demonstrates an analysis of a reaction which is that has already started before the current measurement begins, like in the case of the simulations of Fig. 5.

From the experimental results presented in this work, potential limitations for the kinetic method proposed in this study can be inferred: convection and heterogeneity effects at short time scale and radial diffusion at long time scale. The convection and heterogeneity effects can be minimized with the use of efficient methods of homogenization of the reactants at the beginning of the reaction. On the other hand, if radial diffusion effects are too intense, it may be overcome with the use of electrodes with wider diameters to avoid major contributions of radial diffusion or narrower diameters that provide pure stationary state currents.

Practical guide

The main steps that must be followed for the application of the proposed method to chemical kinetics studies are summarized in Fig. 9. Step 1 consists of starting a chronoamperometric measurement at the desired potential, in which the electrochemical oxidation or reduction of the product of the homogeneous chemical reaction of interest occurs. If possible, the potential should set the concentration of this species as virtually 0 at the electrode surface. This condition is usually valid if the potential is set 150 – 200 mV below (for reduction) or above (for oxidation) the peak potential observed for the species in cyclic voltammetry.

Fig. 9 Practical workflow of the kinetic approach proposed in this work.

During step 1, the electrical circuit is not closed, because there is no electrical contact between the electrodes, and the measured current is close to 0. The circuit is closed in step 2, in which the solutions containing all the reactants of the homogeneous reaction and the supporting electrolyte are injected simultaneously in the electrochemical cell. At this point, the chemical reaction begins to generate the electroactive species that is consumed at the electrode surface. Steps 1 and 2 could be switched if, for some reason, the operator chooses to start the measure after the reaction has already started. Step 3 consists basically on monitoring the current profiles generated. The operator can stop the chronoamperometric measure when the time is close to 10 times the peak time, when the chemical reaction is most likely over.

Once the chronoamperogram is measured, step 4 is carried out: the semi-derivative of the current with respect to time is calculated in order to transform the complicated current profile (described by eqn (18)) into an exponential function. This operation can be found in some software used for control of potentiostats, but can also be made with eL-Chem Viewer²⁹ and some scripts available for Matlab and Python. In step 5, the exponential profile obtained is fitted with eqn (23), and the fit parameters provide kinetic information of the chemical reaction (step 6).

Conclusions

The fundamental equations for the chronoamperometry of irreversible CE mechanisms were derived—eqn (18), (20), and (21)—and a semi-operation (semi-derivative and semi-integral) approach for extracting kinetic information was proposed. Using digital simulations, these equations were validated, and both semi-derivative and semi-integral operations, combined with exponential non-linear fits, successfully recovered the physicochemical parameters set in the simulations.

The validity of the proposed theoretical framework was further confirmed through its application to a real CE reaction: the chemical oxidation of iodide coupled with the electrochemical reduction of iodine (or triiodide). The semi-derivative approach was particularly effective for measuring the rate constant of the homogeneous reaction, though deviations from the pure linear diffusion regime were identified. Another potential limitation of the method is related to the homogenization of the reactants once they are mixed, which tends to be more critical to fast reactions. Despite these deviations, the calculated rate constants across different temperatures and H⁺ concentrations showed strong agreement with previously reported values in the literature, reinforcing the reliability of the derived equations.

This study not only advances the theoretical understanding of chronoamperometric profiles for irreversible CE mechanisms but also demonstrates the practical applicability of the derived equations in real electrochemical systems and opens the perspective for the studies addressing different reaction orders and mechanisms, such as second order reactions and kinetic analysis of reaction intermediate formation. Also, in future works, further improvement of the experimental setup in order to optimize the homogenization of the reactants could minimize deviations from the proposed equations and make this method suitable for the study of reactions of sub-second time scale.

Finally, this work provides a robust foundation for the chronoamperometric analysis of CE mechanisms, with the potential to enhance our understanding of complex chemical systems, serving as a useful tool for kinetic analysis with potential applications in biosensing, in situ monitoring and optimization of chemical reactions. Our results also reinforce the power and versatility of electroanalytical techniques and the usefulness of fractional calculus methods in electrochemistry.

Appendix

Deduction of current as a function of time

The Laplace transform is applied to both sides of eqn (11):

This way, the problem transforms into and ordinary differential equation (ODE) in the variable x. Solving the ODE, one obtains the following equation:

Now it is necessary to bring the functions back to the t domain by taking the inverse Laplace transform of both sides, leading to:

The second term on the right-hand side of the equation above is a multiplication of two functions, and the inverse Laplace transform of the multiplication of two functions in the s domain is the convolution these functions back in the t domain. Therefore, eqn (12) is obtained:

Finding the isoamperic point

An “ad hoc” demonstration that the isoamperic point is exactly at the peak of the “pure Dawson” chronoamperogram is discussed in this section. It is so said ad hoc because it is necessary to assume the existence of an isoamperic point for the deduction provided, and the occurrence of such a point is not necessarily proved mathematically.

In Section 3.2, the analytical concentration of the reactant R is said to be constant in all the simulated chronoamperograms of Fig. 3. This means that the sum of the initial concentration of R and the initial bulk concentration of A is α:

R_i + A_i = α

where α is a constant.

If the isoamperic point is at a time t_iso, eqn (18) reduces to:

where β is also a constant.

Algebraic manipulation leads to:

The isoamperic point is also a point of the “pure Cottrell” curve when A_i = α and R₀ = 0. Therefore, we have:

The equation above may be solved with Wolfram Mathematica, resulting in:

Author contributions

Rafael A. L. Chioquetti: conceptualization, methodology, validation, investigation, data curation, writing – original draft, writing – review & editing, visualization. Raphael P. Bacil: conceptualization, methodology, writing – review & editing, funding acquisition. Silvia H. P. Serrano: conceptualization, methodology, writing – review & editing, supervision, project administration, funding acquisition.

Data availability

The data supporting the findings of this study are available in the ESI† of this article.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors would like to acknowledge Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for funding, Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for scholarship no. 88887.965396/2024-00, Financiadora de Estudos e Projetos (FINEP) for project no. 01.23.0237.00, and scholarship with Fundação de Apoio à Universidade de São Paulo (FUSP) grant no. 4052. We also thank Prof. Dr Mauro Bertotti for the DigiElch 7 Software license.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp04092a

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