Theory of Square-Wave Voltammetry of Two-Electron Reduction with the Adsorption of Intermediate

Abstract

Thermodynamically unstable intermediate of fast and reversible two-electron electrode reaction can be stabilized by the adsorption to the electrode surface. In square-wave voltammetry of this reaction mechanism, the split response may appear if the electrode surface is not completely covered by the adsorbed intermediate. The dependence of the difference between the net peak potentials of the prepeak and postpeak on the square-wave frequency is analyzed theoretically. This relationship can be used for the estimation of adsorption constant.

1. Introduction

In polarography and voltammetry of electrode reactions consisting of two-electron transfers, the responses depend on the stability of intermediates [1–6]. In the case of fast and reversible electroreduction, two waves or peaks appear if the standard potential of the second electron transfer is two hundred millivolts or more lower than the standard potential of the first electron transfer [2, 3]. For the difference in standard potentials higher than −0.1 V, the response is a single wave or peak [6]. Thermodynamically unstable intermediate can be kinetically stabilized if the second charge transfer is slow [1, 3, 4]. Furthermore, the intermediate can be stabilized by the adsorption to the electrode surface [7–11]. This phenomenon was observed in electroreduction of hydrogen [8, 9], chloropyridines [12], and paraquet [13], in electrooxidation of bismuth [7], aluminum [14], and methanol [15], in the passivation of cobalt electrode [16], and in the anodic evolution of oxygen [17]. The theory of two-electron reaction complicated by the adsorption of intermediate is developed for pulse and differential pulse polarography [18–23] and for impedance measurements [24–26], and in this paper the theory is extended to square-wave voltammetry [27–29]. For this technique there are theories for two-step simple and catalytic surface reactions in which all electroactive species are strongly adsorbed to the electrode surface [30–33].

2. The Model

The following reaction mechanism is analyzed:OInxt^(𝑛+2)+_(𝑛+1)++𝑒⁻⇄Int^(𝑛+1)+,(1)Int_(𝑛+1)++𝑒⁻⇄Red^𝑛+,(2)⇄Int^(𝑛+1)+_ads,(3) where 𝑛=0, 1, 2, or more. It is assumed that only the reactant Ox⁽ⁿ⁺²⁾⁺ is initially present in the solution, that the mass transfer occurs through the stationary, planar, semi-infinite diffusion, that both electron transfers are fast and reversible, that all diffusion coefficients are equal, that the adsorption can be described by Langmuir isotherm, and that there are no interactions between molecules in the adsorbed monolayer. The system of differential equations for three electroactive species:𝜕𝑐_𝑌𝜕𝑡=𝐷𝜕²𝑐_𝑌𝜕𝑥²,(4) (where Y = Ox⁽ⁿ⁺²⁾⁺, Int⁽ⁿ⁺¹⁾⁺, and Redⁿ⁺) is solved for the initial and boundary conditions:𝑡𝑡>=0,0,𝑥𝑥⟶≥0∞∶𝑐∶_Ox𝑐=_Ox𝑐^∗_Ox,𝑐_Int=0,𝑐_Red=0,Γ_Int=0,(5)𝑥=0∶𝐷⟶𝜕𝑐𝑐^∗_OxOx,𝑐_Int⟶0,𝑐_Red⟶0,(6)𝜕𝑥_𝑥=0=−𝐼¹𝐷𝜕𝑐_Int𝐹𝑆,(7)𝜕𝑥_𝑥=0=𝑑Γ^Int𝑑𝑡+𝐼¹−𝐼₂𝐷𝜕𝑐_Red𝐹𝑆,(8)𝜕𝑥_𝑥=0=𝐼²𝐼=𝐼₁𝐹𝑆,(9)𝑐+𝐼₂,(10)_Ox𝑐_𝑥=0=𝑐_Int_𝑥=0exp𝜑₁,(11)_Int𝜑_𝑥=0=𝑐_Red_𝑥=0exp𝜑₂,(12)₁=𝐹𝜑𝑅𝑇𝐸−𝐸⁰¹,(13)₂=𝐹𝛽𝑐𝑅𝑇𝐸−𝐸⁰²,(14)_Int_𝑥=0=Γ_maxΓ−^IntΓ_Int.(15) The meanings of all symbols are given in Table 1. Equation (4) is transformed into the system of integral equations and solved by the numerical method of Olmstead and Nicholson [34]. A dimensionless current Φ=𝐼(𝐹𝑆𝑐^∗_Ox)⁻¹(𝐷𝑓)^−1/2 is calculated as a function of electrode potential. The solutions are reported in the appendix.

Table 1 

Meanings of symbols.

3. Results and Discussion

Theoretical square-wave voltammogram of simple, reversible two-electron reaction depends on the difference in standard potentials of two charge transfers (𝐸⁰²−𝐸⁰¹), on square-wave amplitude (𝐸_SW), and on potential increment (dE). If 𝐸⁰¹=𝐸⁰², 𝐸_SW=50 mV, and 𝑑𝐸=−5 mV, the voltammogram is a single peak with the maximum at the standard potential and the dimensionless net peak current −ΔΦ_𝑝=1.6835. The characteristics of the forward, reductive and the backward, oxidative components are the following: Φ_𝑝,𝑓=−1.0483, 𝐸_𝑝,𝑓=𝐸⁰¹, Φ_𝑝,𝑏=0.6352, and 𝐸_𝑝,𝑏=𝐸⁰¹. Under the influence of adsorption of intermediate, the voltammogram depends on two additional parameters: the dimensionless reactant concentration (𝛽𝑐^∗_Ox) and the dimensionless adsorption constant (𝛽Γ_max√𝑓/𝐷). Note that the adsorption constant of Henry isotherm is equal to the product 𝛽Γ_max.

Figure 1 shows the influence of the product 𝛽𝑐^∗_Ox. If this product is smaller than 25, the response is split into two peaks with the maxima at 0.075 V and −0.070 V versus 𝐸⁰¹. The net peak currents in Figure 1(a) are 0.88 and 1.36, respectively. These values change from 0.90 and 1.51, for 𝛽𝑐^∗_Ox=0.1, to 0.86 and 1.17 for 𝛽𝑐^∗_Ox=25. The splitting appears because the adsorption of intermediate facilitates the transfer of the first electron but requires an additional energy for the reduction of adsorbed intermediate [18, 20, 22]. So, the peaks 1 and 2 in Figure 1(a) are the prepeak and postpeak, respectively. The condition for the split response is that the electrode surface is not fully covered by the adsorbed monolayer. Assuming that 𝛽 = 10⁸ cm³/mol, as it was observed in the anion-induced adsorption of some metal ions [35–37], this condition is satisfied for 𝑐^∗_Ox<2.5×10⁻⁴ mol/L. At higher concentrations the main peak develops, as can be seen in Figure 1(b). Its net peak potential is −0.015 V versus 𝐸⁰¹. The minimum in the forward component and the maximum in the backward component that correspond to the main peak appear at 𝐸⁰¹ and −0.030 V versus 𝐸⁰¹, respectively. If 𝛽𝑐^∗_Ox=150, the maximum of the main peak appears at 𝐸⁰¹, and the potentials of minimum and maximum of the forward and backward components are 0.005 V and −0.005 V versus 𝐸⁰¹, respectively.

(a)

(b)

Figure 1 

Square-wave voltammetry (SWV) of electrode reactions (1)–(3). 𝐸02=𝐸01, 𝐸SW=50 mV, dE = −5 mV, 𝐸st=0.3 V versus 𝐸01, 𝛽Γmax√(𝑓/𝐷)=100, and 𝛽𝑐∗Ox=10 (a) and 40 (b). The net peak potentials are marked by the short vertical lines above the abscissa.

The separation between net peak potentials of the split response is independent of the reactant concentration but depends on the square-wave amplitude. If 𝐸_SW=20 mV, 𝐸_𝑝,1−𝐸_𝑝,2=175 mV, and if 𝐸_SW=70 mV, 𝐸_𝑝,1−𝐸_𝑝,2=110 mV. For this reason all analyses in this work are performed with the amplitude of 50 mV.

Figure 2 shows the voltammogram calculated for 𝐸⁰²−𝐸⁰¹=−0.050 V. All other conditions are as in Figure 1(b). It can be noted that the prepeak and the postpeak are more pronounced than in Figure 1(b). The difference in their net peak potentials is 195 mV, and if 𝐸⁰²−𝐸⁰¹=−0.100 V, this difference increases to 250 mV. Oppositely, if 𝐸⁰²−𝐸⁰¹≥0.050 V, the prepeak and postpeak disappear under the conditions of Figure 1(b), which means that they are less separated than for 𝐸⁰¹=𝐸⁰². These results suggest that the difference 𝐸_𝑝,1−𝐸_𝑝,2 is an indicator of thermodynamic property of the reaction mechanism (1)–(3).

Figure 2 

SWV of the reactions (1)–(3). 𝐸02−𝐸01=−0.050 V and 𝛽𝑐∗Ox=40. All other parameters are as in Figure 1.

The influence of adsorption constant is shown in Figure 3, which is to be compared with Figure 1(b). If neither 𝛽 nor the reactant concentration are changed, the parameter 𝛽Γ_max√𝑓/𝐷 can be augmented by increasing either Γ_max or frequency. In the first case the surface coverage is diminished, which explains why in Figure 3 the main peak is much smaller than in Figure 1(b). Also, the difference 𝐸_𝑝,1−𝐸_𝑝,2 increases to 165 mV, compared to 145 mV in Figure 1(a). The same can be achieved by the variation of frequency, which can be changed between 10 Hz and 2000 Hz.

Figure 3 

SWV of the reactions (1)–(3). 𝛽𝑐∗Ox=40 and 𝛽Γmax√(𝑓/𝐷)=150. All other parameters are as in Figure 1.

The relationships between the difference 𝐸_𝑝,1−𝐸_𝑝,2 and the logarithm of parameter 𝛽Γ_max√𝑓/𝐷 are shown in Figure 4 for various standard potentials and for the unsaturated electrode surface (𝛽𝑐^∗_Ox=10). All these functions are straight lines with the slopes and intercepts that depend on the difference in standard potentials. Figure 5 shows that these dependences are also linear: slope=0.075×(𝐸⁰²−𝐸⁰¹)+0.1255 V and intercept=−1.19×(𝐸⁰²−𝐸⁰¹)−0.107 V. So, the relationships in Figure 4 can be described by the following formulae:𝐸_𝑝,1−𝐸_𝑝,2=×0l.o0g75×𝛽Γ𝐸⁰²_max−𝐸⁰¹+0.1255−1.19×𝐸₀₂√𝑓/𝐷−𝐸⁰¹−0.107V.(16) However, as the difference 𝐸⁰²−𝐸⁰¹ is generally unknown, the average values of the slope and the intercept can be used for the estimation of adsorption constant:𝐸_𝑝,1−𝐸_𝑝,2=×(0.1lo2g55±𝛽Γ0.00_max75)−0.107±√0.1𝐷19V.+12log𝑓(17) For instance, if the difference 𝐸_𝑝,1−𝐸_𝑝,2=0.220 V is measured at 𝑓=144 Hz, the parameter 𝛽Γ_max/√𝐷 can be calculated using (17): log(𝛽Γ_max/√𝐷)=1.5±1.0. The precision of this estimation is obviously very modest.

Figure 4 

Dependence of the difference in net peak potentials of the prepeak and postpeak on the logarithm of dimensionless adsorption parameter (the symbols). 𝛽𝑐∗Ox=10 and (𝐸02−𝐸01)/V = 0.100 (1), 0.050 (2), 0.000 (3), −0.050 (4), and −0.100 (5). The lines are linear approximations. All other parameters are as in Figure 1.

(a)

(b)

Figure 5 

Dependence of the slopes (a) and the intercepts (b) of straight lines shown in Figure 4 on the difference in standard potentials of the second and the first electron transfers (the symbols). The lines are linear approximations.

Finally, our calculations show that the maximal currents of prepeak and postpeak depend linearly on the square-root of frequency and on 𝑓^0.583, respectively.

4. Conclusion

These results show that the adsorption of thermodynamically unstable intermediate can cause the splitting of square-wave voltammogram of two-electron electrode reaction. If 𝛽𝑐^∗_Ox<25, the difference between net peak potentials of the prepeak and postpeak is independent of reactant concentration and depends on the difference in standard potentials of two electron transfers and on the logarithm of the dimensionless adsorption parameter 𝛽Γ_max√𝑓/𝐷 as defined by (16). This equation applies for reversible charge transfers, equal diffusion coefficients, 𝐸_SW=50 mV, 𝑑𝐸=−5 mV, −0.1 V ≤𝐸⁰²−𝐸⁰¹<0.1 V, and 𝛽Γ_max√(𝑓/𝐷)≥100. If 𝐸⁰²−𝐸⁰¹=0.1 V, the splitting occurs for 𝛽Γ_max√(𝑓/𝐷)>200.

Appendix

By the substitution 𝜓=𝑐_Ox+𝑐_Int+𝑐_Red, the system of differential equations (4) is reduced to the single differential equation𝜕𝜓𝜕𝑡=𝐷𝜕²𝜓𝜕𝑥²(A.1) with the following initial and boundary conditions:𝑡=𝑡>0,0𝑥,≥𝑥0→∶∞𝜓=∶𝜓𝑐^∗_Ox→,Γ𝑐_Int∗Ox=0,(A.2)𝑥=0∶𝐷𝜕𝜓,(A.3)𝜕𝑥_𝑥=0=𝑑Γ^Int𝜓𝑑𝑡,(A.4)_𝑥=0=𝑐_Int𝛽𝜓_𝑥=0e_𝑥=0xp𝜑₁+1+exp−𝜑₂,(A.5)1+exp𝜑₁+exp−𝜑₂=Γ_maxΓ−^IntΓ_Int.(A.6) Using Laplace transforms, (A.1) is transformed into integral equation𝜓_𝑥=0=𝑐^∗_Ox−(𝐷𝜋)^−1/2𝜕𝜕𝑡^t₀Γ_Int𝑑𝜏√𝑡−𝜏.(A.7) The time 𝑡 is divided into 𝑚 increments (t = md), and it is assumed that within each increment the function Γ_Int can be replaced by the average value Γ_Int,𝑗 [34]. Furthermore, it is defined that each square-wave half-period is divided into 25 time increments: 𝑑=(50𝑓)⁻¹. In this way the solution of (A.7) is obtained:𝜓_𝑥=0,𝑚=𝑐^∗_Ox−10√2𝑓√𝐷𝜋Γ^Int,𝑚+^𝑚−1_𝑗=1Γ_Int,𝑗𝑆_{𝑚−𝑗+1}−𝑆_𝑚−𝑗,(A.8) where m = 1, 2, 3,… and 𝑆_𝑘=√𝑘−√𝑘−1. Equation (A.8) is introduced into (A.6) to obtain the solution for the surface coverage 𝜃_Int,𝑚=Γ_Int,𝑚/Γ_max:𝜃_Int,𝑚=−𝑧^1,𝑚2−𝑧^21,𝑚−4𝑧_2,𝑚𝑧2,(A.9)_1,𝑚=−1−𝛽𝑐^∗Ox+1+exp𝜑_1,𝑚+exp−𝜑_2,𝑚10√2/√𝜋⋅𝛽√𝑓/√𝐷Γ_max+^𝑚−1_𝑗=1𝑧𝜃_𝑗𝑆_{𝑚−𝑗+1}−𝑆_𝑚−𝑗,(A.10)_2,𝑚=𝛽𝑐^∗Ox10√2/√𝜋⋅𝛽√𝑓/√𝐷Γ_max−^𝑚−1_𝑗=1𝜑𝜃_𝑗𝑆_{𝑚−𝑗+1}−𝑆_𝑚−𝑗,(A.11)_𝑙,𝑚=𝐹𝑅𝑇𝐸^𝑚−𝐸^0𝑙,(A.12) (where l = 1 and 2). The surface concentrations of electroactive species are defined by (11), (12), (15), and (A.9):𝑐𝑐_Int_𝑥=0,𝑚=𝛽1⋅1−𝜃𝜃^Int,𝑚_Int,𝑚,(A.13)_Ox_𝑥=0,𝑚=exp𝜑^1,𝑚𝛽⋅1−𝜃𝜃^Int,𝑚_Int,𝑚𝑐,(A.14)_Red_𝑥=0,𝑚=exp−𝜑^2,𝑚𝛽⋅1−𝜃𝜃^Int,𝑚_Int,𝑚.(A.15) The currents 𝐼₁ and 𝐼₂ are determined by solving (4) for Y = Ox⁽ⁿ⁺²⁾⁺ and Y = Redⁿ⁺, respectively:𝐼₁𝐹𝑆√𝐷=−√𝑐𝜋^∗Ox𝑡+𝜕𝜕𝑡^𝑡0𝑐_Ox_𝑥=0𝐼√𝜋(𝑡−𝜏)𝑑𝜏,(A.16)₂𝐹𝑆√𝐷=−𝜕𝜕𝑡^𝑡0𝑐_Red_𝑥=0√𝜋(𝑡−𝜏)𝑑𝜏.(A.17) Using (A.14) and (A.15) and the numerical method described previously, the solution for the dimensionless current Φ_𝑚=(𝐼_1,𝑚+𝐼_2,𝑚)(𝐹𝑆𝑐^∗_Ox)⁻¹(𝐷𝑓)^−1/2 is obtained:Φ_1,𝑚=−5√2√𝑚𝜋+10√2√𝜋𝑐^Ox_𝑥=0,𝑚𝑐^∗_Ox+^𝑚−1_𝑗=1𝑐_Ox_𝑥=0,𝑗𝑐^∗_Ox𝑆_{𝑚−𝑗+1}Φ−𝑆_𝑚−𝑗,(A.18)_2,𝑚=−10√2√𝜋𝑐^Red_𝑥=0,𝑚𝑐^∗_Ox+^𝑚−1_𝑗=1𝑐_Red_𝑥=0,𝑗𝑐^∗_Ox𝑆_{𝑚−𝑗+1}−𝑆_𝑚−𝑗.(A.19) The forward and backward components of square-wave voltammogram and their difference (−ΔΦ=−(Φ_𝑓−Φ_𝑏)) are calculated.