Theory of Square-Wave Voltammetry of Two-Step Electrode Reaction Using an Inverse Scan Direction

Abstract

A theory of square-wave voltammetry of two-step electrode reaction with kinetically controlled electron transfers is developed, and a special case of thermodynamically unstable intermediate is analyzed. If the first reaction step is reversible and the second one is quasireversible, the response splits into two peaks if the scan direction is inverted. The separation of these peaks increases with frequency.

1. Introduction

In square-wave voltammetry (SWV), it is usual that only the reactant is initially present in the solution and that at the starting potential, no electrode reaction occurs [1–3]. However, there is a variation of SWV in which the measurement starts at the potential at which the electrode reaction is controlled by the diffusion of reactant and the scan direction is reversed comparing to the classical SWV [4–6]. This inverse mode was used for the determination of kinetic parameters of electrode reactions [7–10]. In the present paper, it is demonstrated that the inverse SWV can be used for the identification of quasireversible two-step electrode reaction with thermodynamically unstable intermediate. It is well known that the response of this type of mechanism depends on standard rate constants and transfer coefficients of both electron transfer steps [11–16]. If the second charge transfer is slower than the first one, the response may consist of a single two-electron voltammetric peak at the lowest square-wave frequency and of two peaks at the highest frequency. Nevertheless, there are only a limited number of reactions that can undergo such transformation within the available range of frequencies. It is shown here that in many of other cases, the separation of the response into two peaks can be obtained by inverting the scan direction in SWV.

2. The Model

An electrode reaction that occurs through two consecutive steps is analyzedOx^(𝑚+2)++e⁻⇄Int^(𝑚+1)+𝐸⁰₁,𝛼₁,𝑘_𝑠1,Int^(𝑚+1)++e⁻⇄Red^𝑚+𝐸⁰₂,𝛼₂,𝑘_𝑠2. (1)(2) The symbols 𝐸⁰_𝑖, 𝛼_𝑖, and 𝑘_𝑠𝑖 stand for the standard potential, the transfer coefficient, and the standard rate constant of the first (𝑖=1) and the second (𝑖=2) electron transfer step and 𝑚≥0. It is supposed that all three redox species may be initially present in the solution. For the mass transfer realized by the planar, semi-infinite diffusion, the system of differential equations and its solution are reported in the appendix. The meanings of symbols are explained after the appendix.

3. Results and Discussion

Square-wave voltammogram of fast and reversible two-step electrode reaction depends on standard potentials of individual electron transfers. If 𝐸⁰₂−𝐸⁰₁≥0.3 V, 𝐸_st−𝐸⁰₁=0.4 V, 𝐸_sw=50 mV, and d𝐸=−5 mV, the response is a single peak appearing at the potential (𝐸⁰₁+𝐸⁰₂)/2. Its dimensionless net peak current ΔΦ_p=Δ𝐼_p(𝐹𝑆𝑐^∗_tot)⁻¹(𝐷𝑓)^−1/2 is equal to 1.9115. The voltammograms start to split into two peaks if 𝐸⁰₂−𝐸⁰₁<−0.1 V. For 𝐸⁰₂−𝐸⁰₁<−0.3 V and the SWV parameters as above, the response consists of two peaks characterized by ΔΦ_p,1=ΔΦ_p,2=0.752,𝐸_p,1=𝐸⁰₁, and 𝐸_p,2=𝐸⁰₂. The voltammograms of kinetically controlled electrode reactions are generally smaller and depend on the dimensionless kinetic parameter 𝜆=𝑘_𝑠(𝐷𝑓)^−1/2. The reaction appears quasireversible if 1>𝜆>0.03 [1]. These boundaries correspond to 3 × 10⁻² cm/s > 𝑘_𝑠 > 9 × 10⁻⁴ cm/s if 𝐷=9×10⁻⁶ cm²/s and 𝑓=100 Hz. Under these conditions, the electrode reaction appears fast and reversible if 𝜆>1 and 𝑘_𝑠>3×10⁻² cm/s.

Figure 1 shows SWV response of two-step electrode reaction in which the first charge transfer is reversible (𝜆₁=100) and the second one is quasireversible (𝜆₂=0.1). The difference between standard potentials of the second and the first electron transfer steps is 0.1 V, and the electrode potential is expressed in relation to 𝐸⁰₁. If the starting potential is 0.4 V versus 𝐸⁰₁ and the scan is directed towards lower potentials, a negative value of dimensionless net current is in maximum at 0.030 V versus 𝐸⁰₁, which is close to the median potential (𝐸⁰₁+𝐸⁰₂)/2=0.050 V. The potential of minimum of the forward, reductive component and the potential of maximum of the backward, oxidative component are 0.025 V and 0.040 V, respectively. If, however, the starting potential is −0.4 V and the scan direction is positive, the net current exhibits a shoulder at about 0.15 V, and the potential of maximum of the forward, oxidative component is 0.140 V. The peak potentials of the net response and its backward component are 0.055 V and 0.015 V, respectively. The difference in voltammograms shown in Figures 1(a) and 1(b) is the consequence of the kinetics of electrode reaction. If both electron transfers are reversible (𝜆₁=1000 and 𝜆₂=1), the influence of the starting potential is negligible, and the voltammograms are similar to the one in Figure 1(a): 𝐸_p=0.055 V, 𝐸_p,f=0.060 V, and 𝐸_p,b=0.045 V for 𝐸_st=0.4 V and 𝐸_p=0.065 V, 𝐸_p,f=0.065 V, and 𝐸_p,b=0.060 V for 𝐸_st=−0.4 V. Furthermore, the response of two-step reaction consisting of reversible charge transfer (𝜆₁=10) and irreversible one (𝜆₂=0.01) is split into two peaks if 𝐸_st=−0.4 V but not if 𝐸_st=0.4 V. This is shown in Figure 2. In the case of negative scan direction, a single peak appears with the following peak potentials: 𝐸_p=0.000 V, 𝐸_p,f=−0.010 V, and 𝐸_p,b=0.015 V. The potentials of maxima of the split response are 0.015 V and 0.265 V. The forward component consists of two peaks, at 0.040 V and 0.275 V, but the backward component exhibits only one minimum at −0.015 V. For 𝜆₂=10⁻², there is a linear relationship between the second peak potential and the reciprocal of the transfer coefficient of the product oxidation: 𝐸_p,2−𝐸⁰₁=0.085+0.090/(1−𝛼₂) (V). Also, if 𝜆₂≤10⁻², the second peak potential depends linearly on the logarithm of the second dimensionless kinetic parameter: 𝐸_p,2−𝐸⁰₁=0.029−2.3×𝑅𝑇×log(𝜆₂)/(1−𝛼₂)𝐹 (V). These indicate that the second peak in Figure 2(b) originates from totally irreversible oxidation of the product of the second step: Red^𝑚+→Int^(𝑚+1)++e⁻. This product is created at potentials lower than −0.2 V by irreversible reduction of intermediate. The separation between reduction and oxidation processes in the second electrode reaction can be estimated from the difference between the second peak potential and the second standard potential: 𝐸_p,2−𝐸⁰₂=0.165 V. So, the reduction peak can be expected at −0.065 V versus 𝐸⁰₁, where it is covered by the response of the first electrode reaction. This explanation was confirmed by the simulation for 𝜆₁=1 and 𝜆₂=0.001. If 𝐸_st=−0.4 V, the second peak potential was 0.385 V, which meant that 𝐸_p,2−𝐸⁰₂=0.285 V and that the reduction peak had to be expected at −0.185 V. Indeed, if 𝐸_st=0.4 V, the response was split in two peaks with maxima at −0.005 V and −0.185 V. The second peak, was ascribed to irreversible reduction of the intermediate. Additional support was obtained by the variation of the second standard potential. If the second reaction step was quasireversible (𝜆₂=0.1 and 𝜆₁=100), and 𝐸⁰₂−𝐸⁰₁≥0.15 V, the shoulder shown in Figure 1(b) was developed as a separate peak, and its peak potential depended linearly on the difference between standard potentials: 𝐸_p,2=𝐸⁰₂+0.040 V versus 𝐸⁰₁. Also, if 𝐸⁰₂−𝐸⁰₁≤0.05 V, the shoulder disappeared. In the case of irreversible second reaction (𝜆₂=0.01 and 𝜆₁=10), the separation between cathodic and anodic peaks of the second electrode reaction was 0.230 V, and they both appeared in voltammograms if 𝐸⁰₂=𝐸⁰₁. This is shown in Figure 3. The peak potentials in this figure are the following: 𝐸_p,1=0.000 V and 𝐸_p,2^∗=−0.165 V for 𝐸_st=0.4 V and 𝐸_p,1=0.015 V and 𝐸_p,2=0.165 V for 𝐸_st=−0.4 V. If 𝐸⁰₂−𝐸⁰₁=0.05 V, the cathodic peak 2* merges with the peak 1 and appears as a shoulder at about −0.1 V. This is a transition to the response shown in Figure 2. The change of responses shown in Figures 1(b) and 2(b) can be achieved by the variation of square-wave frequency. The parameter 𝜆 is diminished ten times if the frequency is increased hundred times.

(a)

(b)

Figure 1 

Theoretical square-wave voltammograms of two-step electrode reaction (1) and (2). 𝐸02−𝐸01=0.1 V, 𝑐∗Ox/𝑐∗tot=1, 𝑐∗Int=0, 𝑐∗Red=0, 𝜆1=100, 𝜆2=0.1, 𝛼1=0.5, 𝛼2=0.5, 𝐸sw=50 mV, ∣d𝐸∣=5 mV, and 𝐸st−𝐸01=0.4 V (a) and −0.4 V (b). A dimensionless net current (ΔΦ) and its forward (Φf) and backward (Φb) components are shown.

(a)

(b)

Figure 2 

SWV of electrode reactions (1) and (2); 𝜆1=10, 𝜆2=0.01, and 𝐸st−𝐸01=0.4 V (a) and −0.4 V (b). All other data are as in Figure 1.

(a)

(b)

Figure 3 

SWV of reactions (1) and (2). 𝐸02=𝐸01, 𝜆1=10, 𝜆2=0.01, and 𝐸st−𝐸01=0.4 V (a) and −0.4 V (b). All other data are as in Figure 1. Irreversible reduction of the intermediate is marked by 2*.

The influence of the kinetics of the first electrode transfer is shown in Figure 4. If both reaction steps are equally fast and quasireversible (𝜆₁=𝜆₂=0.1) and 𝐸_st=0.4 V, the SWV response is a single peak with the maximum at −0.055 V. The resolution of the reaction steps can be achieved by inverting the scan direction, as can be seen in Figure 4(b). Under this condition, the voltammogram consists of two peaks with maxima at −0.093 V and 0.140 V, respectively. The second peak potential is equal to the potential of maximum of the forward component of the response shown in Figure 1(b). So, the second peak, originating from the second reaction step, is independent of the kinetics of the first reaction step, but the separation between two peaks is better in Figure 4(b) than in Figure 1(b) because of the quasireversible nature of the first reaction step in Figure 4(b).

(a)

(b)

Figure 4 

SWV of reactions (1) and (2). 𝜆1=0.1, 𝜆2=0.1, and 𝐸st−𝐸01=0.4 V (a) and −0.4 V (b). All other data are as in Figure 1.

4. Conclusions

These results show that individual electron transfers of two-step electrode reaction with thermodynamically unstable intermediate can be resolved by square-wave voltammetry with inverse scan direction. The method applies to the combination of reversible and quasireversible or two quasireversible steps. If the second charge transfer is irreversible, the second peak potential is a linear function of the logarithm of square-wave frequency with the slope Δ𝐸_p,2/Δlog𝑓=2.3×𝑅𝑇/2(1−𝛼₂)𝐹.

Appendix

For the electrode reactions (1) and (2), under the conditions of the planar, semi-infinite diffusion, the following system of differential equations has to be solved:𝜕𝑐_Ox𝜕𝜕𝑡=𝐷²𝑐_Ox𝜕𝑥²,𝜕𝑐_Int𝜕𝜕𝑡=𝐷²𝑐_Int𝜕𝑥²,𝜕𝑐_Red𝜕𝜕𝑡=𝐷²𝑐_Red𝜕𝑥²,𝑡=0,𝑥≥0:𝑐_Ox=𝑐^∗_Ox,𝑐_Int=𝑐^∗_Int,𝑐_Red=𝑐^∗_Red,𝑐^∗_Ox+𝑐^∗_Int+𝑐^∗_Red=𝑐^∗_tot,𝑡>0,𝑥⟶∞:𝑐_Ox⟶𝑐^∗_Ox,𝑐_Int⟶𝑐^∗_Int,𝑐_Red⟶𝑐^∗_Red,𝑥=0:𝐷𝜕𝑐_Ox𝜕𝑥_𝑥=0𝐼=−¹,𝐷𝐹𝑆𝜕𝑐_Int𝜕𝑥_𝑥=0=𝐼₁−𝐼₂,𝐷𝐹𝑆𝜕𝑐_Red𝜕𝑥_𝑥=0=𝐼₂,𝐼𝐹𝑆₁𝐹𝑆=−𝑘_𝑠1exp−𝛼₁𝜑₁×𝑐_Ox𝑥=0−𝑐_Int𝑥=0exp𝜑₁,𝐼₂𝐹𝑆=−𝑘_𝑠2exp−𝛼₂𝜑₂×𝑐_Int𝑥=0−𝑐_Red𝑥=0exp𝜑₂,𝜑₁=𝐹𝑅𝑇𝐸−𝐸⁰₁,𝜑₂=𝐹𝑅𝑇𝐸−𝐸⁰₂. (A.1)(A.2) The meanings of symbols are as follows: 𝑐_Ox,  𝑐_Int, and 𝑐_Red are concentrations of species Ox^(𝑚+2)+, Int^(𝑚+1)+, and Red^𝑚+, respectively, 𝑐^∗_Ox, 𝑐^∗_Int, and 𝑐^∗_Red are concentrations of these species in the bulk of solution, 𝐷 is a common diffusion coefficient, 𝐼₁ and 𝐼₂ are currents, 𝑆 is electrode surface area, and 𝑡, 𝑥, 𝑇, 𝑅, and 𝐹 have their usual meanings. Differential equations (A.1) are solved by Laplace transformation and by the method of numerical integration proposed by Olmstead and Nicholson [17]. The solution is the system of recursive formulae for the dimensionless current Φ_𝑖=𝐼_𝑖(𝐹𝑆𝑐^∗_tot)⁻¹(𝐷𝑓)^−1/2, where 𝑖=1 or 2 and 𝑓 is square-wave frequency. Note that Φ=Φ₁+Φ₂. Each square-wave half-period is divided into 25 time increments. It is assumed that within each time increment, the function 𝐼_𝑖 can be replaced by the average value 𝐼_𝑖,𝑗Φ_1,𝑚=𝑍−5𝑐^∗_Ox√𝜋exp𝜑_1,𝑚𝑐^∗_tot√2+𝑐^∗_Int𝑃𝑐^∗_tot+𝑐^∗_Red𝑄𝑐^∗_tot+𝑉^𝑚−1_𝑗=1Φ_1,𝑗𝑠_{𝑚−𝑗+1}+𝑊^𝑚−1_𝑗=1Φ_2,𝑗𝑠_{𝑚−𝑗+1},Φ_2,𝑚=Φ_1,𝑚×5√𝜋exp−1−𝛼₁𝜑_1,𝑚𝜆₁√2+1+exp−𝜑_1,𝑚+5𝑐^∗_Ox√𝜋exp𝜑_1,𝑚𝑐^∗_tot√2−5𝑐^∗_Int√𝜋𝑐^∗_tot√2+1+exp𝜑_1,𝑚×exp−𝜑_1,𝑚^𝑚−1_𝑗=1Φ_1,𝑗𝑠_{𝑚−𝑗+1}−^𝑚−1_𝑗=1Φ_2,𝑗𝑠_{𝑚−𝑗+1},𝑍=𝑁⁻¹𝜆₁exp1−𝛼₁𝜑_1,𝑚√2×5√𝜋+𝜆₂exp−𝛼₂𝜑_2,𝑚1+exp𝜑_2,𝑚√2,𝑁=5√𝜋+𝜆₁exp−𝛼₁𝜑_1,𝑚1+exp𝜑_1,𝑚√2×5√𝜋+𝜆₂exp−𝛼₂𝜑_2,𝑚1+exp𝜑_2,𝑚√2−2𝜆₁𝜆₂exp1−𝛼₁𝜑_1,𝑚exp−𝛼₂𝜑_2,𝑚,𝑃=25𝜋+5𝜆₂√2𝜋exp1−𝛼₂𝜑_2,𝑚5√2𝜋+2𝜆₂exp−𝛼₂𝜑_2,𝑚1+exp𝜑_2,𝑚,5√𝑄=𝜋𝜆₂exp1−𝛼₂𝜑_2,𝑚5√𝜋+𝜆₂√2exp−𝛼₂𝜑_2,𝑚1+exp𝜑_2,𝑚,𝑉=−1+exp𝜑_1,𝑚5√𝜋−𝜆₂√2exp−𝛼₂𝜑_2,𝑚−𝜆₂√21+exp𝜑_1,𝑚exp1−𝛼₂𝜑_2,𝑚×5√𝜋+𝜆₂√2exp−𝛼₂𝜑_2,𝑚1+exp𝜑_2,𝑚⁻¹×exp−𝜑_1,𝑚,5√𝑊=𝜋5√𝜋+𝜆₂√2exp−𝛼₂𝜑_2,𝑚1+exp𝜑_2,𝑚,𝜆_𝑖=𝑘_𝑠𝑖𝐷𝑓^−1/2(𝑖=1,2). (A.3) Here, 𝑠_𝑘=√𝑘−√𝑘−1, 1≤𝑚≤𝑀, 𝑀=50 (𝐸_start−𝐸_final)/d𝐸, and d𝐸=5 mV is the step increment of square-wave signal. In all calculations, the square-wave amplitude 𝐸_sw=50 mV was used.

Meanings of Symbols

𝛼1, 𝛼2:	The transfer coefficients of the first and the second electron transfer steps
𝑐Ox, 𝑐Int, 𝑐Red:	The concentrations of the reactant, intermediate and the product
𝑐∗Ox, 𝑐∗Int, 𝑐∗Red:	The concentrations of the reactant, intermediate and the product in the bulk of solution
𝑐∗tot:	The sum of bulk concentrations of the reactant, intermediate and the product
(𝑐Ox)𝑥=0, (𝑐Int)𝑥=0, (𝑐Red)𝑥=0:	The concentrations of the reactant, intermediate and the product near the electrode surface
𝐷:	The common diffusion coefficient
d𝐸:	The square-wave potential increment
𝐸:	The electrode potential
𝐸01, 𝐸02:	The standard potentials of the first and the second electron transfer steps
𝐸sw:	The square-wave amplitude
𝐸st:	The square-wave starting potential
𝐸p:	The peak potential
𝐸p,f, 𝐸p,b:	The peak potentials of the forward and backward components
𝐹:	The Faraday constant
𝑓:	The square-wave frequency
Φ:	The dimensionless current
ΔΦp:	The dimensionless net peak current
𝐼:	The current
Δ𝐼:	The net square-wave voltammetric response
𝐼f, 𝐼b:	The forward and backward currents
Δ𝐼p:	The net peak current
𝑘𝑠1, 𝑘𝑠2:	The standard rate constants of the first and the second electron transfer steps
𝜆1, 𝜆2:	The dimensionless kinetic parameters of the first and the second electrons
𝑛:	The number of electrons
𝑅:	The gas constant
𝑆:	The electrode surface area
𝑡:	The time
𝑥:	The distance perpendicular to the electrode surface.

Acknowledgment

This work was supported by the Croatian Ministry of Science, Education, and Sport.