**4.1.2 Experimental set-up**

A typical three electrode set-up is used for the LSV/RDE experiments (Bamford & Compton, 1986; Diard et al., 1996). The electrochemical cell contains a Ag/AgCl reference electrode (Schott-Geräte), a rotating disk working electrode and a platinum grid as counter electrode. A platinum RDE is used for the ferri/ferrocyanide system and a gold RDE for the hexaamineruthenium (III)/(II) system. The RDE electrodes are fabricated by embedding a 4 mm diameter polycrystalline platinum or gold rod in an insulating mantle of polyvinylidenefluoride.

The electrode is rotated by an RDE control system of Autolab. For the hexaammineruthenium (III)/(II) reaction, the rotation speed is set to 500 and 1000 rpm. In the experiments with the ferri/ferrocyanide system, the rotation speeds are 300, 1000, 1500 and 2000 rpm. The voltammograms are measured using a high resolution galvanostat/potentiostat PGSTAT100 (Autolab Instruments) of Ecochemie, controlled by the GPES 4.8 or the Nova 1.5 softwares. In the LSV experiments, the potential is swept from 0.3 to -0.4 V vs NHE for the Ru(NH3)+<sup>3</sup> <sup>6</sup> / Ru(NH3)+<sup>2</sup> <sup>6</sup> reaction and from 0.8 to 0.2 V vs NHE for the Fe(CN)−<sup>3</sup> <sup>6</sup> /Fe(CN)−<sup>4</sup> <sup>6</sup> reaction. The scan rate is taken constant at 1 mV/s. The step potential is set to 0.00015 V, in this way a maximum number of data points is measured.

All measurements are performed in a 200 ml glass electrolytic cell, thermostatted at 25 ± 0.5 ◦C using a water jacket connected to a thermostat bath (Lauda RE304). Prior to the measurements, the electrolyte is deoxygenated by bubbling with nitrogen gas (Air Liquide) for 10 min, while during the experiment a nitrogen blanket is maintained over the cell. This results in a substantial flattening of the reduction plateau of both the ferricyanide and the hexaamineruthenium (III).

#### **4.1.3 Electrode pretreatment**

The reproducibility of the measurements is strongly increased by means of applying the following pretreatment of the working electrode surface: 1) mechanical polishing of the electrode on a rotating disk (Struers DP10, on cloth), successively using diamond paste of 7 *μ*m and 1 *μ*m grain size (Struers) for the Pt electrode, and 9 *μ*m, 3 *μ*m diamond paste and 0.04 *μ*m Al2O3 paste (Struers) for the Au electrode; 2) ultrasonic rinsing with deionized water followed by degreasing with chloroform, also in an ultrasonic bath (Elma model T470/H); 3) four cyclic voltammograms are performed before each experiment in order to remove oxide and trace contaminants from the metallic surface (Robertson et al., 1988). During the cyclic voltammetry measurements, the potential is swept between -0.35 V to 1.45 V vs Ag/AgCl at a scan velocity of 50 mV/s on the Au electrode (hexaammineruthenium (III)/(II) system) and between +0.55 and -0.45 V vs Ag/AgCl at a scan velocity of 10 mV/s, on the Pt electrode (ferri/ferrocyanide system).

### **4.2 Modeling the ferri/ferrocyanide redox reaction**

12 Electrochemical Cells

The chemicals used for the study of the hexaammineruthenium (III)/(II) redox reaction are: [Ru(NH3)6]Cl3 (Sigma-Aldrich, 98%), Na2HPO4 · 2H2O and NaH2PO4 ·12H2O (both ProLabo A.R.). Solutions are made with once-distilled and deionized water. A 0.1 M phosphate buffer pH 7 solution is used as the supporting electrolyte and the concentration of the electroactive species hexaammineruthenium (III) is 0.001 M. The following chemicals are used for the ferri/ferrocyanide reaction: K4[Fe(CN)6] ·3H2O, K3[Fe(CN)6] and KCl (all Merck P.A.). The supporting electrolyte is a 1 M KCl solution and the concentrations of the electroactive components ferri/ferrocyanide are 0.005 M. In that way, a negligible migration flux, constant activity and diffusion coefficients of the electroactive species, a low electrolyte resistance and

A typical three electrode set-up is used for the LSV/RDE experiments (Bamford & Compton, 1986; Diard et al., 1996). The electrochemical cell contains a Ag/AgCl reference electrode (Schott-Geräte), a rotating disk working electrode and a platinum grid as counter electrode. A platinum RDE is used for the ferri/ferrocyanide system and a gold RDE for the hexaamineruthenium (III)/(II) system. The RDE electrodes are fabricated by embedding a 4 mm diameter polycrystalline platinum or gold rod in an insulating mantle

The electrode is rotated by an RDE control system of Autolab. For the hexaammineruthenium (III)/(II) reaction, the rotation speed is set to 500 and 1000 rpm. In the experiments with the ferri/ferrocyanide system, the rotation speeds are 300, 1000, 1500 and 2000 rpm. The voltammograms are measured using a high resolution galvanostat/potentiostat PGSTAT100 (Autolab Instruments) of Ecochemie, controlled by the GPES 4.8 or the Nova 1.5 softwares. In the LSV experiments, the potential is swept from 0.3 to -0.4 V vs NHE for the Ru(NH3)+<sup>3</sup>

scan rate is taken constant at 1 mV/s. The step potential is set to 0.00015 V, in this way a

All measurements are performed in a 200 ml glass electrolytic cell, thermostatted at 25 ± 0.5 ◦C using a water jacket connected to a thermostat bath (Lauda RE304). Prior to the measurements, the electrolyte is deoxygenated by bubbling with nitrogen gas (Air Liquide) for 10 min, while during the experiment a nitrogen blanket is maintained over the cell. This results in a substantial flattening of the reduction plateau of both the ferricyanide and the

The reproducibility of the measurements is strongly increased by means of applying the following pretreatment of the working electrode surface: 1) mechanical polishing of the electrode on a rotating disk (Struers DP10, on cloth), successively using diamond paste of 7 *μ*m and 1 *μ*m grain size (Struers) for the Pt electrode, and 9 *μ*m, 3 *μ*m diamond paste and

<sup>6</sup> /

<sup>6</sup> reaction. The

<sup>6</sup> /Fe(CN)−<sup>4</sup>

a uniform current distribution are assumed for both electrochemical systems.

<sup>6</sup> reaction and from 0.8 to 0.2 V vs NHE for the Fe(CN)−<sup>3</sup>

**4. Analytical modeling of LSV/RDE experiments**

**4.1 The experimental procedure**

**4.1.2 Experimental set-up**

of polyvinylidenefluoride.

hexaamineruthenium (III).

**4.1.3 Electrode pretreatment**

maximum number of data points is measured.

Ru(NH3)+<sup>2</sup>

**4.1.1 Composition of the electrolytes**

The simplest electrochemical reactions, which can be found among the different kinds of electrode processes, are those where electrons are exchanged across the interface by flipping oxidation states of transition metal ions in the electrolyte adjacent to the electrode surface (Bamford & Compton, 1986), i.e. an ET (electron transfer) mechanism. The electrode acts as the source or sink of electrons for the redox reaction and is supposed to be inert. The reduction of ferricyanide to ferrocyanide (Angell & Dickinson, 1972; Bamford & Compton, 1986; Bruce et al., 1994; Iwasita et al., 1983) is an example of such a mechanism:

$$\left(Fe(\text{CN})\_6^{-3} + e^- \right) \rightleftharpoons Fe(\text{CN})\_6^{-4} \tag{23}$$

#### **4.2.1 Results of the experimental study**

A set of equivalent polarization curves of the ferri/ferrocyanide reaction is used as the experimental data, which is the input to the model methodology. As advised in (Tourwé et al., 2006), 11 voltammograms are measured under identical conditions for every rotation speed. The resulting voltammograms for the Fe(CN)−<sup>3</sup> <sup>6</sup> /Fe(CN)−<sup>3</sup> <sup>6</sup> system are shown in Figure 3.

Fig. 3. Voltammograms of the reduction/oxidation of 0.005 M ferri/ferrocyanide in 1 M KCl, at 2000 rpm.

#### **4.2.2 The analytical expression for the current as a function of the potential**

The analytical expression is derived for the reduction/oxidation of ferri/ferrocyanide (reaction (23)). The basic equations that describe this mechanism, when studied under steady-state conditions on a rotating disk electrode, are:

$$w = K\_{\rm ox} \mathbb{C}\_{\rm red}(0) - K\_{\rm red} \mathbb{C}\_{\rm ox}(0) \tag{24}$$

$$I = iS = i\_fS = FSv \tag{25}$$

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5

(b)

5.00E−1 1.00E+001.50E+002.00E+002.50E+003.00E+003.50E+004.00E+00

(b) *kred* (m/s)

300 rpm 1000 rpm 1500 rpm 2000 rpm

(Iexp − Imodel) ±2σ

E (V vs NHE)

300 rpm 1000 rpm 1500 rpm 2000 rpm

−1 −0.5 0 0.5 1 1.5 x 10−5

I (A)

<sup>35</sup> Modeling and Quantification of Electrochemical Reactions in RDE (Rotating Disk Electrode) and IRDE (Inverted Rotating Disk Electrode) Based Reactors

Fig. 4. Modeling the reduction/oxidation of ferri/ferrocyanide: (a) Comparison of the modeled curve and the mean experimental curve, at 2000 rpm. (b) Difference between the

parameters are determined less accurately. This does not imply that the results at this lowest rotation speed have to be rejected. When calculating the best estimates of the parameters, the higher uncertainty is taken into account. The 95% confidence intervals for *kox*, *kred* and *αox* obtained for the intermediate rotation speeds (1000, 1500 and 2000 rpm) overlap, indicating

1.00E−08 1.50E−08 2.00E−08 2.50E−08 3.00E−08 3.50E−08

(c) *kox* (m/s)

Fig. 5. Comparison of the 95 % confidence intervals of the charge transfer parameters

Iexp Imodel Iexp ± 2σ

E (V vs NHE)

experiments and the model within the 95% confidence interval.

(a)

that the results are independent of the rotation speed.

300 rpm 1000 rpm 1500 rpm 2000 rpm

4.7E−01 4.8E−01 4.9E−01 5.0E−01 5.1E−01 5.2E−01 5.3E−01 5.4E−01

(a) *αox*

obtained at 300 rpm, 1000 rpm, 1500 rpm and 2000 rpm.

−4 −3 −2 −1 0 1 2 3 4 <sup>5</sup> x 10−4

I (A)

$$v = v\_{ox/red} = -m\_{ox}[\mathbb{C}\_{ox}^\* - \mathbb{C}\_{ox}(0)] = m\_{red}[\mathbb{C}\_{red}^\* - \mathbb{C}\_{red}(0)] \tag{26}$$

$$\mathbf{i}\_{\text{lim},\text{ox}/\text{red}} = \pm \mathbf{F} \mathbf{m}\_{\text{red}/\text{ox}} \mathbf{C}\_{\text{red}/\text{ox}}^{\*} = \pm 0.620 \mathbf{F} \mathbf{D}\_{\text{red}/\text{ox}}^{2/3} \boldsymbol{\nu}^{-1/6} \boldsymbol{\omega}^{1/2} \mathbf{C}\_{\text{red}/\text{ox}}^{\*} \tag{27}$$

$$K\_{\rm ox\,/red} = k\_{\rm ox\,/red} \exp \frac{\pm \alpha\_{\rm ox\,/red} nFE}{RT} \tag{28}$$

From this set of equations an expression for the current as a function of the potential, the experimental parameters and the model parameters can be derived. Considering *αox* + *αred* = 1, the model parameters are *Dred*, *Dox*, *kox*, *kred* and *αox*. The experimental parameters are *C*∗ *red*, *C*<sup>∗</sup> *ox*, *S*, *n* and *T*, which are all known. *ν* and *ω* are eliminated in the expression for the current, by using the limiting current density as a model parameter, rather than the diffusion coefficient. Solving this set of equations leads to the following current-potential relation (Tourwé et al., 2007):

$$I = \frac{nFS(k\_{ox} \exp\left(\frac{a\_{\text{eff}} nFE}{kT}\right) \mathbf{C}\_{red}^\* - k\_{red} \exp\left(\frac{-a\_{\text{ref}} nFE}{kT}\right) \mathbf{C}\_{ox}^\*)}{1 + nFS\left(\frac{k\_{ox} \exp\left(\frac{a\_{\text{eff}} nFE}{kT}\right) \mathbf{C}\_{red}^\*}{\mathbf{i}\_{lin,ox}} - \frac{k\_{red} \exp\left(\frac{-a\_{\text{ref}} nFE}{kT}\right) \mathbf{C}\_{ox}^\*}{\mathbf{i}\_{lin,red}}\right)}\tag{29}$$

#### **4.2.3 The fitting results and their statistical evaluation**

For the ferri/ferrocyanide system, the analytical model is based on 5 parameters: the rate constant of the reduction reaction, *kred*, the rate constant of the oxidation reaction, *kox*, the charge transfer coefficient, *αox*, the reduction limiting current density, *ilim*,*red* and the oxidation limiting current density, *ilim*,*ox*. The mathematical expression for the current as a function of the potential (Eq. (29)) is fitted to the mean of the equivalent experimental voltammograms, using the method described previously.

Four rotation speeds are considered for the study of the ferri/ferrocyanide redox reaction: 300, 1000, 1500 and 2000 rpm. Figure 4.a presents a comparison between the mean experimental voltammogram at 2000 rpm and the modeled voltammogram, calculated with the best-fit-parameters, and the 95% confidence interval ±2*σ*, with *σ* the standard deviation of the current in the set of experimental curves. The difference between the experimental and the modeled curve (Figure 4.b) lies in the 95% confidence band. This means that the model is able to describe the experiments appropiately.

In order to check the validity of the analytical modeling to estimate the model parameters, an evaluation of the parameter values determined at different rotation speeds is carried out. In Figure 5 the 95% confidence intervals for the best fit-parameters at four rotation speeds are compared. It can be seen that for all parameters the 95% confidence intervals obtained at the different rotation speeds overlap with the interval for the lowest rotation speed (300 rpm). The latter is, however, much larger than those obtained for the higher rotation speeds. This is due to the fact that at lower rotation speeds mass transfer rapidly becomes the rate determining step. The potential region where charge transfer influences the reaction rate is smaller than for the other rotation speeds and, as a consequence, the charge transfer 14 Electrochemical Cells

*ox* − *Cox*(0)] = *mred*[*C*<sup>∗</sup>

±*αox*/*rednFE*

*red*/*ox* <sup>=</sup> <sup>±</sup>0.620*FD*2/3

*ox*, *S*, *n* and *T*, which are all known. *ν* and *ω* are eliminated in the expression for the current, by using the limiting current density as a model parameter, rather than the diffusion coefficient. Solving this set of equations leads to the following current-potential relation

*red* − *kredexp*

From this set of equations an expression for the current as a function of the potential, the experimental parameters and the model parameters can be derived. Considering *αox* + *αred* = 1, the model parameters are *Dred*, *Dox*, *kox*, *kred* and *αox*. The experimental parameters are

> *RT* )*C*<sup>∗</sup> *red ilim*,*ox* <sup>−</sup> *kred exp*( <sup>−</sup>*αrednFE*

For the ferri/ferrocyanide system, the analytical model is based on 5 parameters: the rate constant of the reduction reaction, *kred*, the rate constant of the oxidation reaction, *kox*, the charge transfer coefficient, *αox*, the reduction limiting current density, *ilim*,*red* and the oxidation limiting current density, *ilim*,*ox*. The mathematical expression for the current as a function of the potential (Eq. (29)) is fitted to the mean of the equivalent experimental voltammograms,

Four rotation speeds are considered for the study of the ferri/ferrocyanide redox reaction: 300, 1000, 1500 and 2000 rpm. Figure 4.a presents a comparison between the mean experimental voltammogram at 2000 rpm and the modeled voltammogram, calculated with the best-fit-parameters, and the 95% confidence interval ±2*σ*, with *σ* the standard deviation of the current in the set of experimental curves. The difference between the experimental and the modeled curve (Figure 4.b) lies in the 95% confidence band. This means that the model is

In order to check the validity of the analytical modeling to estimate the model parameters, an evaluation of the parameter values determined at different rotation speeds is carried out. In Figure 5 the 95% confidence intervals for the best fit-parameters at four rotation speeds are compared. It can be seen that for all parameters the 95% confidence intervals obtained at the different rotation speeds overlap with the interval for the lowest rotation speed (300 rpm). The latter is, however, much larger than those obtained for the higher rotation speeds. This is due to the fact that at lower rotation speeds mass transfer rapidly becomes the rate determining step. The potential region where charge transfer influences the reaction rate is smaller than for the other rotation speeds and, as a consequence, the charge transfer

*Kox*/*red* = *kox*/*redexp*

 *<sup>α</sup>oxnFE RT C*∗

*kox exp*( *<sup>α</sup>oxnFE*

*v* = *KoxCred*(0) − *KredCox*(0) (24)

*I* = *iS* = *if S* = *FSv* (25)

*red*/*oxν*−1/6*ω*1/2*C*<sup>∗</sup>

 <sup>−</sup>*αrednFE RT*

*ilim*,*red*

*RT* )*C*<sup>∗</sup> *ox*

 *C*∗ *ox*)

*red* − *Cred*(0)] (26)

*RT* (28)

*red*/*ox* (27)

(29)

steady-state conditions on a rotating disk electrode, are:

*v* = *vox*/*red* = −*mox*[*C*<sup>∗</sup>

*ilim*,*ox*/*red* = ±*Fmred*/*oxC*<sup>∗</sup>

*<sup>I</sup>* <sup>=</sup> *nFS*(*koxexp*

**4.2.3 The fitting results and their statistical evaluation**

using the method described previously.

able to describe the experiments appropiately.

1 + *nFS*

*C*∗ *red*, *C*<sup>∗</sup>

(Tourwé et al., 2007):

Fig. 4. Modeling the reduction/oxidation of ferri/ferrocyanide: (a) Comparison of the modeled curve and the mean experimental curve, at 2000 rpm. (b) Difference between the experiments and the model within the 95% confidence interval.

parameters are determined less accurately. This does not imply that the results at this lowest rotation speed have to be rejected. When calculating the best estimates of the parameters, the higher uncertainty is taken into account. The 95% confidence intervals for *kox*, *kred* and *αox* obtained for the intermediate rotation speeds (1000, 1500 and 2000 rpm) overlap, indicating that the results are independent of the rotation speed.

Fig. 5. Comparison of the 95 % confidence intervals of the charge transfer parameters obtained at 300 rpm, 1000 rpm, 1500 rpm and 2000 rpm.

The best-fit estimates of the charge transfer parameters are calculated as the weighted mean of the values obtained at the four rotation speeds taken into consideration. Introducing weighting factors helps to adjust the impact of one result on the mean with regard to their level of uncertainty. The maximum likelihood estimate of normally distributed variables *xi* with different variances *σ*<sup>2</sup> *<sup>i</sup>* and the same mean value *μ* is given by (Sorenson, 1980):

$$\mu = \frac{\Sigma\_{\bar{l}} \frac{\chi\_{\bar{l}}}{\sigma\_{\bar{l}}^2}}{\Sigma\_{\bar{l}} \frac{1}{\sigma\_{\bar{l}}^2}} \tag{30}$$

−0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 −1.5

<sup>37</sup> Modeling and Quantification of Electrochemical Reactions in RDE (Rotating Disk Electrode) and IRDE (Inverted Rotating Disk Electrode) Based Reactors

of the model parameters must be done. The model parameters are fitted for the experimental data obtained at two different rotation speeds. The best-fit values and their standard deviation obtained for 500 rpm and 1000 rpm are shown in Table 1. The values of the charge transfer coefficient are slightly different. In the case of the rate constant, the best-fit values at different rotation speeds present significant differences. Thus, the estimated values are not consistent with the results expected for the proposed model: while the limiting current density is proportional to the square root of *ω*, the values of the charge transfer parameters must be

The different results of the estimated parameters at the two rotation speeds could be due to interfering reactions in the supporting electrolyte or, more likely, to the oxidation of hexaamineruthenium (II) to hexaamineruthenium (III), occurring at the initial potentials of the polarization curve. The oxidation half-reaction is not taken in consideration in the formulation of the analytical current-potential expression; this could have an influence on the parameter estimation. According to our modeling strategy, a better estimation of the model parameters needs further investigation on the treatment of the experimental data and/or the formulation

Model parameters 500 rpm 1000 rpm

Table 1. Best-fit parameters and their standard deviation for 500 rpm and 1000 rpm.

*αred* 7.67E-01 2.95E-03 7.49E-01 2.15E-03 *kred* (m/s) 6.17E-05 3.36E-07 7.14E-05 3.46E-07 *ilim*,*red* (A/m2) -3.29E+00 1.51E-03 -4.54E+00 2.20E-03

The combination of linear sweep voltammetry with an RDE is a powerful tool to study electrochemical reactions. One limitation to the technique is the presence of gas in some electrochemical systems. The inverted rotating disk electrode (IRDE) is designed to tackle

Estimate Std Estimate Std

of the reaction mechanism. Present research is devoted to this topic.

**5. The inverted rotating disk electrode (IRDE) reactor**

Fig. 6. Difference between the mean experimental and the modeled curves, at 500 rpm,

E (V vs NHE)

−1

−0.5

I (A)

within the 95% confidence band.

independent of the rotation speed.

0

0.5

1

1.5 x 10−6

Iexp − Imodel ±2σ

This results in the following values for the charge transfer coefficient and the rate constants: *αox* = 5.14E-01 ± 1.28E-02, *kox* = 1.41E-08 m/s ± 4.50E-09 m/s and *kred* = 1.61E+00 m/s ± 4.82E-01 m/s.

The diffusion coefficients can be calculated from the Levich equation (Eq. 8). Using the values of *ilim*,*red* and *ilim*,*ox*, determined for 300, 1000, 1500 and 2000 rpm, this results in *Dred* = 8.07E-10 m2/s <sup>±</sup> 2.23E-17 m2/s and *Dox* = 8.31E-10 m2/s <sup>±</sup> 2.75E-17 m2/s. The obtained parameter values compare well with the values presented in literature (Angell & Dickinson, 1972; Beriet & Pletcher, 1993; Bruce et al., 1994; Jahn & Vielstich, 1962).

#### **4.3 The analytical modeling of the hexaammineruthenium (III)/(II) redox reaction**

The reduction of hexaammineruthenium (III) to hexaammineruthenium (II) is extensively described in literature (Beriet & Pletcher, 1994; Deakin et al., 1985; Elson et al., 1975; Khoshtariya et al., 2003; Marken et al., 1995; Muzikar & Fawcett, 2006) as a one electron transfer reaction, i.e.,

$$\mathrm{Ru(NH\_3)\_6^{+3}} + e^- \rightleftharpoons \mathrm{Ru(NH\_3)\_6^{+2}} \tag{31}$$

Due to the fact that the hexaamineruthenium (II) complex is not stable in solution, the electrochemical reaction (31) is studied here only in the direction of the reduction, with the hexaammineruthenium (III) being the only electroactive species initially present in the electrolyte.

For the reduction of Ru(NH3)+<sup>3</sup> <sup>6</sup> to Ru(NH3)+<sup>2</sup> <sup>6</sup> , the mathematical expression describing the reaction mechanism is obtained, leaving out the oxidation parameters. The current-potential relation is formulated with the following expression:

$$I = \frac{-nFSk\_{red}\exp\left(\frac{-\alpha\_{red}nFE}{RT}\right)\mathbb{C}\_{ox}^\*}{i\_{lim, red} - nFSk\_{red}\exp\left(\frac{-\alpha\_{red}nFE}{RT}\right)\mathbb{C}\_{ox}^\*} \tag{32}$$

For the analytical modeling of the reaction, also 11 identical experimental curves are performed, at 500 and 1000 rpm. In the reduction of *Ru*(*NH*3)+<sup>3</sup> <sup>6</sup> , an unexpected variation of the current values in the region of the limiting current is observed. This behavior differs from the characteristic curve of an ET mechanism and might be due to additional reactions in the supporting electrolyte. This contribution of the supporting electrolyte is measured and subtracted from the experimental polarization curves.

In Figure 6 it can be seen that the difference between the experimental curve and the modeled curve, at 500 rpm, lies in the 95% confidence band. At this point we could accept that the model is appropiate to describe the experiments, but an evaluation of the estimated values 16 Electrochemical Cells

The best-fit estimates of the charge transfer parameters are calculated as the weighted mean of the values obtained at the four rotation speeds taken into consideration. Introducing weighting factors helps to adjust the impact of one result on the mean with regard to their level of uncertainty. The maximum likelihood estimate of normally distributed variables *xi*

> *μ* = Σ*i xi σ*2 *i* Σ*i* 1 *σ*2 *i*

This results in the following values for the charge transfer coefficient and the rate constants: *αox* = 5.14E-01 ± 1.28E-02, *kox* = 1.41E-08 m/s ± 4.50E-09 m/s and *kred* = 1.61E+00 m/s ±

The diffusion coefficients can be calculated from the Levich equation (Eq. 8). Using the values of *ilim*,*red* and *ilim*,*ox*, determined for 300, 1000, 1500 and 2000 rpm, this results in *Dred* = 8.07E-10 m2/s <sup>±</sup> 2.23E-17 m2/s and *Dox* = 8.31E-10 m2/s <sup>±</sup> 2.75E-17 m2/s. The obtained parameter values compare well with the values presented in literature (Angell & Dickinson,

The reduction of hexaammineruthenium (III) to hexaammineruthenium (II) is extensively described in literature (Beriet & Pletcher, 1994; Deakin et al., 1985; Elson et al., 1975; Khoshtariya et al., 2003; Marken et al., 1995; Muzikar & Fawcett, 2006) as a one electron

Due to the fact that the hexaamineruthenium (II) complex is not stable in solution, the electrochemical reaction (31) is studied here only in the direction of the reduction, with the hexaammineruthenium (III) being the only electroactive species initially present in the

reaction mechanism is obtained, leaving out the oxidation parameters. The current-potential

*ilim*,*red* <sup>−</sup> *nFSkredexp*( <sup>−</sup>*αrednFE*

For the analytical modeling of the reaction, also 11 identical experimental curves are

of the current values in the region of the limiting current is observed. This behavior differs from the characteristic curve of an ET mechanism and might be due to additional reactions in the supporting electrolyte. This contribution of the supporting electrolyte is measured and

In Figure 6 it can be seen that the difference between the experimental curve and the modeled curve, at 500 rpm, lies in the 95% confidence band. At this point we could accept that the model is appropiate to describe the experiments, but an evaluation of the estimated values

<sup>−</sup> *Ru*(*NH*3)+<sup>2</sup>

 <sup>−</sup>*αrednFE RT*

 *C*∗ *ox*

*RT* )*C*<sup>∗</sup> *ox*

<sup>6</sup> (31)

<sup>6</sup> , an unexpected variation

<sup>6</sup> , the mathematical expression describing the

1972; Beriet & Pletcher, 1993; Bruce et al., 1994; Jahn & Vielstich, 1962).

*Ru*(*NH*3)+<sup>3</sup>

<sup>6</sup> to Ru(NH3)+<sup>2</sup>

*<sup>I</sup>* <sup>=</sup> <sup>−</sup>*nFSkredexp*

performed, at 500 and 1000 rpm. In the reduction of *Ru*(*NH*3)+<sup>3</sup>

**4.3 The analytical modeling of the hexaammineruthenium (III)/(II) redox reaction**

<sup>6</sup> + *e*

*<sup>i</sup>* and the same mean value *μ* is given by (Sorenson, 1980):

(30)

(32)

with different variances *σ*<sup>2</sup>

4.82E-01 m/s.

transfer reaction, i.e.,

For the reduction of Ru(NH3)+<sup>3</sup>

relation is formulated with the following expression:

subtracted from the experimental polarization curves.

electrolyte.

Fig. 6. Difference between the mean experimental and the modeled curves, at 500 rpm, within the 95% confidence band.

of the model parameters must be done. The model parameters are fitted for the experimental data obtained at two different rotation speeds. The best-fit values and their standard deviation obtained for 500 rpm and 1000 rpm are shown in Table 1. The values of the charge transfer coefficient are slightly different. In the case of the rate constant, the best-fit values at different rotation speeds present significant differences. Thus, the estimated values are not consistent with the results expected for the proposed model: while the limiting current density is proportional to the square root of *ω*, the values of the charge transfer parameters must be independent of the rotation speed.

The different results of the estimated parameters at the two rotation speeds could be due to interfering reactions in the supporting electrolyte or, more likely, to the oxidation of hexaamineruthenium (II) to hexaamineruthenium (III), occurring at the initial potentials of the polarization curve. The oxidation half-reaction is not taken in consideration in the formulation of the analytical current-potential expression; this could have an influence on the parameter estimation. According to our modeling strategy, a better estimation of the model parameters needs further investigation on the treatment of the experimental data and/or the formulation of the reaction mechanism. Present research is devoted to this topic.


Table 1. Best-fit parameters and their standard deviation for 500 rpm and 1000 rpm.
