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

Authors: Milivoj Lovrić; Šebojka Komorsky-Lovrić
Journal: International Journal of Electrochemistry
 (2012)

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## 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.


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## Body

## 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:(1)Ox(n+2)++e-⇄Int(n+1)+,(2)Int(n+1)++e-⇄Redn+,(3)Int(n+1)+⇄(Int(n+1)+)ads,
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:(4)∂cY∂t=D∂2cY∂x2,
(where Y = Ox(n+2)+, Int(n+1)+, and Redn+) is solved for the initial and boundary conditions:(5)t=0,x≥0:cOx=cOx*,cInt=0,cRed=0,ΓInt=0,(6)t>0,x⟶∞:cOx⟶cOx*,cInt⟶0,cRed⟶0,(7)x=0:D(∂cOx∂x)x=0=-I1FS,(8)D(∂cInt∂x)x=0=dΓIntdt+I1-I2FS,(9)D(∂cRed∂x)x=0=I2FS,(10)I=I1+I2,(11)(cOx)x=0=(cInt)x=0exp⁡(φ1),(12)(cInt)x=0=(cRed)x=0exp⁡(φ2),(13)φ1=FRT(E-E10),(14)φ2=FRT(E-E20),(15)β(cInt)x=0=ΓIntΓmax⁡-ΓInt.
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(FScOx*)-1(Df)-1/2 is calculated as a function of electrode potential. The solutions are reported in the appendix.Table 1
Meanings of symbols.
βAdsorption constantcOx, cInt, cRedConcentrations of the reactant, intermediate and the productcOx*Concentrations of the reactant in the bulk of solutionDCommon diffusion coefficientdESquare-wave potential incrementEElectrode potentialE10, E20Standard potentials of the first and the second electron transfersEswSquare-wave amplitudeEpPeak potentialEp,1,Ep,2Peak potentials of the prepeak and postpeakEp,f,Ep,bPeak potentials of the forward and backward componentsEstStarting potentialFFaraday constantfSquare-wave frequencyφDimensionless potential differenceΦDimensionless currentΦf,ΦbDimensionless currents of the forward and backward components-ΔΦpDimensionless net peak currentΦp,f,Φp,bDimensionless peak currents of the forward and back wardcomponentsΓIntSurface concentration of adsorbed intermediateΓmaxMaximum surface concentration of adsorbed intermediateRGas constantSElectrode surface areatTimeTTemperatureθSurface coveragexDistance perpendicular to the electrode surface


## 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 (E20-E10), on square-wave amplitude (ESW), and on potential increment (dE). If E10=E20, ESW=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 -ΔΦp=1.6835. The characteristics of the forward, reductive and the backward, oxidative components are the following: Φp,f=-1.0483, Ep,f=E10, Φp,b=0.6352, and Ep,b=E10. Under the influence of adsorption of intermediate, the voltammogram depends on two additional parameters: the dimensionless reactant concentration (βcOx*) and the dimensionless adsorption constant (βΓmaxf/D). Note that the adsorption constant of Henry isotherm is equal to the product βΓmax.Figure1 shows the influence of the product βcOx*. 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 E10. 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 βcOx*=0.1, to 0.86 and 1.17 for βcOx*=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 cOx*<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 E10. The minimum in the forward component and the maximum in the backward component that correspond to the main peak appear at E10 and −0.030 V versus E10, respectively. If βcOx*=150, the maximum of the main peak appears at E10, and the potentials of minimum and maximum of the forward and backward components are 0.005 V and −0.005 V versus E10, respectively.Square-wave voltammetry (SWV) of electrode reactions (1)–(3). E20=E10, ESW=50 mV, dE = −5 mV, Est=0.3 V versus E10, βΓmax(f/D)=100, and βcOx*=10 (a) and 40 (b). The net peak potentials are marked by the short vertical lines above the abscissa.
(a)(b)The separation between net peak potentials of the split response is independent of the reactant concentration but depends on the square-wave amplitude. IfESW=20 mV, Ep,1-Ep,2=175 mV, and if ESW=70 mV, Ep,1-Ep,2=110 mV. For this reason all analyses in this work are performed with the amplitude of 50 mV.Figure2 shows the voltammogram calculated for E20-E10=-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 E20-E10=-0.100 V, this difference increases to 250 mV. Oppositely, if E20-E10≥0.050 V, the prepeak and postpeak disappear under the conditions of Figure 1(b), which means that they are less separated than for E10=E20. These results suggest that the difference Ep,1-Ep,2 is an indicator of thermodynamic property of the reaction mechanism (1)–(3).Figure 2
SWV of the reactions (1)–(3). E20-E10=-0.050 V and βcOx*=40. All other parameters are as in Figure 1.The influence of adsorption constant is shown in Figure3, which is to be compared with Figure 1(b). If neither β nor the reactant concentration are changed, the parameter βΓmaxf/D can be augmented by increasing either Γmax 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 Ep,1-Ep,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). βcOx*=40 and βΓmax(f/D)=150. All other parameters are as in Figure 1.The relationships between the differenceEp,1-Ep,2 and the logarithm of parameter βΓmaxf/D are shown in Figure 4 for various standard potentials and for the unsaturated electrode surface (βcOx*=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×(E20-E10)+0.1255 V and intercept=-1.19×(E20-E10)-0.107 V. So, the relationships in Figure 4 can be described by the following formulae:(16)Ep,1-Ep,2=[0.075×(E20-E10)+0.1255]×log⁡(βΓmax⁡f/D)-1.19×(E20-E10)-0.107V.
However, as the difference E20-E10 is generally unknown, the average values of the slope and the intercept can be used for the estimation of adsorption constant:(17)Ep,1-Ep,2=(0.1255±0.0075)×[log⁡(βΓmax⁡D)+12log⁡f]-0.107±0.119V.
For instance, if the difference Ep,1-Ep,2=0.220 V is measured at f=144 Hz, the parameter βΓmax/D can be calculated using (17): log(βΓmax/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).βcOx*=10 and (E20-E10)/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.Dependence of the slopes (a) and the intercepts (b) of straight lines shown in Figure4 on the difference in standard potentials of the second and the first electron transfers (the symbols). The lines are linear approximations.
(a)(b)Finally, our calculations show that the maximal currents of prepeak and postpeak depend linearly on the square-root of frequency and onf0.583, 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. IfβcOx*<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 βΓmaxf/D as defined by (16). This equation applies for reversible charge transfers, equal diffusion coefficients, ESW=50 mV, dE=-5 mV, −0.1 V ≤E20-E10<0.1 V, and βΓmax(f/D)≥100. If E20-E10=0.1 V, the splitting occurs for βΓmax(f/D)>200.



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*Source: 596268-2012-03-22.xml*