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

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.


Introduction
In polarography and voltammetry of electrode reactions consisting of two-electron transfers, the responses depend on the stability of intermediates [1][2][3][4][5][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][8][9][10][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][19][20][21][22][23] and for impedance measurements [24][25][26], and in this paper the theory is extended to squarewave voltammetry [27][28][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][31][32][33].

The Model
The following reaction mechanism is analyzed: 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: International Journal of Electrochemistry (where Y = Ox (n+2)+ , Int (n+1)+ , and Red n+ ) is solved for the initial and boundary conditions: 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 is calculated as a function of electrode potential.The solutions are reported in the appendix.

Results and Discussion
Theoretical square-wave voltammogram of simple, reversible two-electron reaction depends on the difference in standard potentials of two charge transfers (E 0 2 − E 0 1 ), on square-wave amplitude (E SW ), and on potential increment (dE).If  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 E 0 1 .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 = 10 8 cm 3 /mol, as it was observed  in the anion-induced adsorption of some metal ions [35][36][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 −0.015V versus E 0 1 .The minimum in the forward component and the maximum in the backward component that correspond to the main peak appear at E 0 1 and −0.030 V versus E 0 1 , respectively.If βc * Ox = 150, the maximum of the main peak appears at E 0 1 , and the potentials of minimum and maximum of the forward and backward components are 0.005 V and −0.005 V versus E 0 1 , respectively.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 E 0 2 − E 0 1 = −0.050V.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 E 0 2 − E 0 1 = −0.100V, this difference increases to 250 mV.Oppositely, if E 0 2 − E 0 1 ≥ 0.050 V, the prepeak and postpeak disappear under the conditions of Figure 1(b), which means that they are less separated than for E 0 1 = E 0 2 .These results suggest that the difference E p,1 −E p,2 is an indicator of thermodynamic property of the reaction mechanism (1)- (3).
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.
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 × (E 0 2 − E 0 1 ) + 0.1255 V and intercept = −1.19× (E 0 2 − E 0 1 ) − 0.107 V. So, the relationships in Figure 4 can be described by the following formulae: However, as the difference E 0 2 − E 0 1 is generally unknown, the average values of the slope and the intercept can be used for the estimation of adsorption constant: 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 cal-culated using (17): log(βΓ max / √ D) = 1.5 ± 1.0.The precision of this estimation is obviously very modest.
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.

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,

Appendix
By the substitution ψ = c Ox + c Int + c Red , the system of differential equations ( 4) is reduced to the single differential equation with the following initial and boundary conditions: Using Laplace transforms, (A.1) is transformed into integral equation 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 = (50 f ) −1 .In this way the solution of (A.7) is obtained: ϕ l,m = F RT E m − E 0 l , (A.12) (where l = 1 and 2).The surface concentrations of electroactive species are defined by ( 11), ( 12), (15), and (A.The forward and backward components of square-wave voltammogram and their difference (−ΔΦ = −(Φ f − Φ b )) are calculated.

2 ,
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 = E 0 1 , Φ p,b = 0.6352, and E p,b = E 0 1 .Under the influence of adsorption of intermediate, the voltammogram depends

Figure 5 :
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.

Table 1 :
Meanings of symbols.
f , Φ b Dimensionless currents of the forward and backward components −ΔΦ p Dimensionless net peak currentΦ p, f , Φ p,bDimensionless peak currents of the forward and and βΓ max ( f /D) ≥ 100.If E 0 2 − E 0 1 = 0.1 V, the splitting occurs for βΓ max ( f /D) > 200.