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International Journal of Electrochemistry
Volume 2012 (2012), Article ID 596268, 7 pages
Research Article

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

Department of Marine and Environmental Research, Ruđer Bošković Institute, 10000 Zagreb, Croatia

Received 9 December 2011; Revised 13 January 2012; Accepted 13 January 2012

Academic Editor: Sheng S. Zhang

Copyright © 2012 Milivoj Lovrić and Šebojka Komorsky-Lovrić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 [16]. 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 [711]. 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 [1823] and for impedance measurements [2426], and in this paper the theory is extended to square-wave voltammetry [2729]. 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 [3033].

2. The Model

The following reaction mechanism is analyzed:Ox(𝑛+2)++𝑒Int(𝑛+1)+,(1)Int(𝑛+1)++𝑒Red𝑛+,(2)Int(𝑛+1)+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:𝜕𝑐𝑌𝜕𝜕𝑡=𝐷2𝑐𝑌𝜕𝑥2,(4) (where Y = Ox(n+2)+, Int(n+1)+, and Redn+) is solved for the initial and boundary conditions:𝑡=0,𝑥0𝑐Ox=𝑐Ox,𝑐Int=0,𝑐Red=0,ΓInt=0,(5)𝑡>0,𝑥𝑐Ox𝑐Ox,𝑐Int0,𝑐Red0,(6)𝑥=0𝐷𝜕𝑐Ox𝜕𝑥𝑥=0𝐼=1𝐷𝐹𝑆,(7)𝜕𝑐Int𝜕𝑥𝑥=0=𝑑ΓInt+𝐼𝑑𝑡1𝐼2𝐷𝐹𝑆,(8)𝜕𝑐Red𝜕𝑥𝑥=0=𝐼2𝐹𝑆,(9)𝐼=𝐼1+𝐼2𝑐,(10)Ox𝑥=0=𝑐Int𝑥=0𝜑exp1𝑐,(11)Int𝑥=0=𝑐Red𝑥=0𝜑exp2𝜑,(12)1=𝐹𝑅𝑇𝐸𝐸01𝜑,(13)2=𝐹𝑅𝑇𝐸𝐸02𝛽𝑐,(14)Int𝑥=0=ΓIntΓmaxΓInt.(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(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.

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 (𝐸02𝐸01), on square-wave amplitude (𝐸SW), and on potential increment (dE). If 𝐸01=𝐸02, 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=𝐸01, Φ𝑝,𝑏=0.6352, and 𝐸𝑝,𝑏=𝐸01. 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.

Figure 1 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 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 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 [3537], this condition is satisfied for 𝑐Ox<2.5×104 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 𝛽𝑐Ox=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.

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 𝛽𝑐Ox=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 𝐸SW=20 mV, 𝐸𝑝,1𝐸𝑝,2=175 mV, and if 𝐸SW=70 mV, 𝐸𝑝,1𝐸𝑝,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 𝐸02𝐸01=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 𝐸02𝐸01=0.100 V, this difference increases to 250 mV. Oppositely, if 𝐸02𝐸010.050 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 𝐸𝑝,1𝐸𝑝,2 is an indicator of thermodynamic property of the reaction mechanism (1)–(3).

Figure 2: SWV of the reactions (1)–(3). 𝐸02𝐸01=0.050 V and 𝛽𝑐Ox=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 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,1𝐸𝑝,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 difference 𝐸𝑝,1𝐸𝑝,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×(𝐸02𝐸01)+0.1255 V and intercept=1.19×(𝐸02𝐸01)0.107 V. So, the relationships in Figure 4 can be described by the following formulae:𝐸𝑝,1𝐸𝑝,2=𝐸0.075×02𝐸01+0.1255×log𝛽Γmax𝐸𝑓/𝐷1.19×02𝐸010.107V.(16) However, as the difference 𝐸02𝐸01 is generally unknown, the average values of the slope and the intercept can be used for the estimation of adsorption constant:𝐸𝑝,1𝐸𝑝,2×=(0.1255±0.0075)log𝛽Γmax𝐷+12log𝑓0.107±0.119V.(17) For instance, if the difference 𝐸𝑝,1𝐸𝑝,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 (𝐸02𝐸01)/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.
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 𝑓0.583, 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 𝛽𝑐Ox<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 𝐸02𝐸01<0.1 V, and 𝛽Γmax(𝑓/𝐷)100. If 𝐸02𝐸01=0.1 V, the splitting occurs for 𝛽Γmax(𝑓/𝐷)>200.


By the substitution 𝜓=𝑐Ox+𝑐Int+𝑐Red, the system of differential equations (4) is reduced to the single differential equation𝜕𝜓𝜕𝜕𝑡=𝐷2𝜓𝜕𝑥2(A.1) with the following initial and boundary conditions:𝑡=0,𝑥0𝜓=𝑐Ox,ΓInt=0,(A.2)𝑡>0,𝑥𝜓𝑐Ox,(A.3)𝑥=0𝐷𝜕𝜓𝜕𝑥𝑥=0=𝑑ΓInt𝜓𝑑𝑡,(A.4)𝑥=0=𝑐Int𝑥=0𝜑exp1+1+exp𝜑2,(A.5)𝛽𝜓𝑥=0𝜑1+exp1+exp𝜑2=ΓIntΓmaxΓInt.(A.6) Using Laplace transforms, (A.1) is transformed into integral equation𝜓𝑥=0=𝑐Ox(𝐷𝜋)1/2𝜕𝜕𝑡t0ΓInt𝑑𝜏𝑡𝜏.(A.7) The time 𝑡 is divided into 𝑚 increments (t = md), and it is assumed that within each increment the function ΓInt can be replaced by the average value ΓInt,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:𝜓𝑥=0,𝑚=𝑐Ox102𝑓Γ𝐷𝜋Int,𝑚+𝑚1𝑗=1ΓInt,𝑗𝑆𝑚𝑗+1𝑆𝑚𝑗,(A.8) where m = 1, 2, 3,… and 𝑆𝑘=𝑘𝑘1. Equation (A.8) is introduced into (A.6) to obtain the solution for the surface coverage 𝜃Int,𝑚=ΓInt,𝑚/Γmax:𝜃Int,𝑚𝑧=1,𝑚2𝑧21,𝑚4𝑧2,𝑚2,𝑧(A.9)1,𝑚=1𝛽𝑐Ox+𝜑1+exp1,𝑚+exp𝜑2,𝑚102/𝛽𝜋𝑓/𝐷Γmax+𝑚1𝑗=1𝜃𝑗𝑆𝑚𝑗+1𝑆𝑚𝑗,𝑧(A.10)2,𝑚=𝛽𝑐Ox102/𝛽𝜋𝑓/𝐷Γmax𝑚1𝑗=1𝜃𝑗𝑆𝑚𝑗+1𝑆𝑚𝑗,𝜑(A.11)𝑙,𝑚=𝐹𝐸𝑅𝑇𝑚𝐸0𝑙,(A.12) (where l = 1 and 2). The surface concentrations of electroactive species are defined by (11), (12), (15), and (A.9):𝑐Int𝑥=0,𝑚=1𝛽𝜃Int,𝑚1𝜃Int,𝑚,𝑐(A.13)Ox𝑥=0,𝑚=𝜑exp1,𝑚𝛽𝜃Int,𝑚1𝜃Int,𝑚𝑐,(A.14)Red𝑥=0,𝑚=exp𝜑2,𝑚𝛽𝜃Int,𝑚1𝜃Int,𝑚.(A.15) The currents 𝐼1 and 𝐼2 are determined by solving (4) for Y = Ox(n+2)+ and Y = Redn+, respectively:𝐼1𝐹𝑆𝐷𝑐=Ox+𝜕𝜋𝑡𝜕𝑡𝑡0𝑐Ox𝑥=0𝐼𝜋(𝑡𝜏)𝑑𝜏,(A.16)2𝐹𝑆𝐷𝜕=𝜕𝑡𝑡0𝑐Red𝑥=0𝜋(𝑡𝜏)𝑑𝜏.(A.17) Using (A.14) and (A.15) and the numerical method described previously, the solution for the dimensionless current Φ𝑚=(𝐼1,𝑚+𝐼2,𝑚)(𝐹𝑆𝑐Ox)1(𝐷𝑓)1/2 is obtained:Φ1,𝑚5=2+𝑚𝜋102𝜋𝑐Ox𝑥=0,𝑚𝑐Ox+𝑚1𝑗=1𝑐Ox𝑥=0,𝑗𝑐Ox𝑆𝑚𝑗+1𝑆𝑚𝑗,Φ(A.18)2,𝑚=102𝜋𝑐Red𝑥=0,𝑚𝑐Ox+𝑚1𝑗=1𝑐Red𝑥=0,𝑗𝑐Ox𝑆𝑚𝑗+1𝑆𝑚𝑗.(A.19) The forward and backward components of square-wave voltammogram and their difference (ΔΦ=(Φ𝑓Φ𝑏)) are calculated.


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