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copper cathode interface process
Chemical Engineering Science 56 (2001) 2695}2702
E!ectiveness factors in an electrochemical reactor with rotating cylinder electrode for the acid-cupric/copper cathode interface process JoseH Luis Nava-M. de Oca, E. Sosa, Carlos Ponce de LeoH n, M. T. Oropeza* Departament of Chemistry, Universidad Auto& noma Metropolitana-Iztapalapa Av. Michoaca& n y la Purn& sima, A.P. 55-534, C.P. 09340, Me& xico D.F., Me& xico, USA Received 2 March 2000; received in revised form 9 October 2000; accepted 17 October 2000
Abstract An expression for the global electrochemical deposition rate of copper in a rotating cylinder electrode (RCE) was developed through the e!ectiveness factor. The e!ectiveness factor of an electrochemical interface process, calculated using the DamkoK hler number, leads to identify the competition between intrinsic reaction (charge transfer) and bulk mass transport. The charge transfer constant was evaluated by electrochemical impedance spectroscopy and the speci"c mass transfer coe$cients were determined by electrolysis at controlled potential. Catalytic e!ectiveness factors were found between 0.08 and 0.22 establishing that the mass transport of Cu (II) ion controls the global electrochemical process at an imposed electrolysis potential of !0.9 V vs. SSE in turbulent #ow. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Rotating cylinder electrode; Electrolysis; Mass transfer; E!ectiveness factor
1. Introduction It is well known that in the chemical reaction engineering, the di!usion and intrinsic reaction are the main phenomena that determine the global reaction rate within a heterogeneous catalytic reactor. For this reason, it is interesting to study an electrochemical transformation as a heterogeneous catalytic process since the bulk mass transport and charge transfer (intrinsic reaction) participate in a global electrochemical transformation. There are di!erent factors in the electrochemical reactor operation that determine its behavior, such as polarization of the working electrode necessary to assure quantitative transformation and the relationship between mass transfer phenomena and the rotation speed of the polarized electrode (Gabe & Walsh, 1984a,b, 1985; Gabe, Wilcox, Gonzalez-Garcia, & Walsh, 1998). In particular, the reactor with rotating cylinder electrode (RCE) is one of the most studied models regarding the mass transfer phenomena and hydrodynamics. In the literature, besides dimensionless correlation concern* Corresponding author. Tel.: #52-5-724-4670; fax: #52-5-7244666. E-mail address: [email protected] (M. T. Oropeza).
ing mass transfer coe$cient variation and hydrodynamic regimen at a RCE (Gabe, 1974; Gabe & Walsh, 1984, 1985; Gabe & Weimberg, 1998), some authors have used an equation that predicts the concentration of a reactant species versus time within a simple batch type reactor (Gabe & Walsh, 1984; Genders & Weimberg, 1992; Walsh, 1993). In all the above papers, it was assumed that the mass transfer, determined by the limiting current, governs the global electrochemical process. The foregoing approach suggests a detailed analysis of the global electrochemical process kinetics from the heterogeneous catalysis viewpoint, since the operating principles of catalytic reactors can be helpful in the understanding of the operation of an RCE. The purpose of this paper is to show the competition between the charge transfer and bulk mass transport at an imposed electrolysis potential using the e!ectiveness factors that group together the contributions of each phenomenon involved. It is important to mention that in order to obtain a better estimate of each transfer coe$cient, this study analyzes the charge transfer (intrinsic reaction within double layer) by electrochemical impedance spectroscopy and the bulk mass transport by copper ion depletion in solution during electrolysis.
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 1 4 - 5
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Fig. 1. Concentration pro"le as a function of the distance under steady-state conditions for an electrochemical process.
2. Interface model of copper deposition process Since the Cu(II)/Cu(0) system is widely utilized as a reaction model for the characterization of RCE processes (Gabe & Walsh, 1984a,b, 1985; Pletcher & Walsh, 1993) it was selected to be used in an acid aqueous environment. The concentration pro"le, as a distance function (Fig. 1), helps the identi"cation of bulk mass transfer phenomena. In the Helmholtz plane (HP), the reaction can take place in two consecutive steps (Reid & Allan, 1987): Cu(II)#1e\Cu(I), Cu(I)#1e\Cu(0). However, if we impose a potential corresponding to the limiting current plateau (electrolysis potential), the Cu(II) ion transport from the bulk to the Nernst di!usion layer, can limit the global electrochemical process. The aforementioned implies that in order to reach HP, the Cu(II) species early needs to cross a -thick "lm (Nernst di!usion layer).
3. E4ectiveness factor in rotating cylinder reactor Two central parts in heterogeneous transformation process engineering are the catalytic reaction and the di!usion of species involved in the chemical system. In the electrodeposition of a metallic species both of them are present. The "rst one as a need for enough surface energy to perform the desired transformation (electronic transfer reaction) and the second, as a resistance that an electroactive species has to overcome in order to reach the electrode surface. One of these phenomena may control the global rate, depending on the potential imposed in the electrode processes (Gabe, 1974; Gabe & Walsh, 1984a,b, 1985).
In this work, we "rst identify the catalytic surface where an electroactive species is transformed and which is going to be called the electrocatalyst. Then the interface di!usion and reaction at isothermal conditions will be discussed as contributions to the global electrochemical transformation rate within the RCE. The heat transport phenomena are not mentioned, however, it does not mean they are undervalued. Finally, an expression for global cathodic process rate is determined using general e!ectiveness factor assumptions. E!ectiveness factor, , of an interfacial chemical process is de"ned by the relationship between the rate of an interface process R (real rate), when the concentration of an electroactive species at the surface is C and the rate of an ideal process, R , when the surface concentration is equal to that of the species in `bulka C (Cassiere & Carberry, 1973): R " (1) R considering "rst a order reaction, the equation is transformed to
k C C " " , k C C where k is the intrinsic reaction rate constant. In steady-state conditions, the electroactive species arriving by a mass transport mechanism disappears through the intrinsic reaction (Cassiere & Carberry, 1973): k a(C !C)"k C, (2) K where the surface concentration (the unobservable, C) in terms of bulk concentration (the observable, C ) is (Car berry, 1976): 1 C" C . (2) 1#k /k a K In this equation, the DamkoK hler dimensionless number appears in an explicit form, Da , which is the ratio of the intrinsic reaction rate constant with regard to the speci"c mass transfer constant: k (3) Da " . k a K It is important to mention that in steady-state conditions, mass transport and intrinsic-reaction rate are equal, but it does not mean that mass and charge transfer coe$cients have the same value. Furthermore, the global reaction rate is equal to the rate of the lowest step in the mechanism (Carberry, 1976; Fogler, 1992). The e!ectiveness factor in terms of the DamkoK hler dimensionless number, Da , results in (Cassiere & Car berry, 1973; Carberry, 1976): 1 " . 1#Da
(4)
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Finally, the global reaction rate R may be expressed in terms of the e!ectiveness factor (Cassiere & Carberry, 1973; Carberry, 1976): R"R "k C . (5) For an electrochemical process, the ideal rate constant k "k a must be considered, being k the charge trans fer constant for reduction reaction at the cathode (rotating cylinder), and a"A/< , the speci"c area, while A is 0 the rotating cylinder area (immersed in the solution) and < the electrolytic solution volume. C "C is the con0 centration in bulk of the oxidized species O (Cu(II)). Performing these substitutions in Eq. (5), the global cathodic process rate results in 1 k a R" C " C . (6) 1#Da - 1/k a#1/k a K For an electrochemical reactor with rotating cylinder electrode in simple batch, the mass balance for cathodic process indicates that dC ! - "R dt
(7)
or dC - "!k aC . dt
Fig. 2. Rotating cylinder electrode scheme.
These two extreme cases suggest that as for the 0.5 value, the best equation to represent the results is that corresponding to (8), since the electrochemical process is under mixed control.
(7)
Integrating the foregoing expression and considering that for t"0, C "C (0) and for t"t, C "C (t), the oxidized species concentration as a function of time is C (t)"C (0) exp![k a t] or
(8)
1 C (t)"C (0) exp! t . (8) 1/k a#1/k a K In Eq. (8), it is possible to visualize the contribution of the intrinsic reaction (charge transfer) and the mass transport in the global cathodic process that is being carried out within a RCE in batch operation; therefore, we have two extreme cases: E for relatively high DamkoK hler values, the e!ectiveness coe$cient P1/Da , indicating that the process re mains under mass transport control. In this case, Eq. (8) is transformed C (t)"C (0) exp![k a t]. K
(9)
E for relatively small DamkoK hler values, the e!ectiveness coe$cient P1, indicating the control by intrinsic reaction (charge transfer). Similarly, Eq. (8) is transformed into: C (t)"C (0) exp![k a t].
(10)
4. Experimental The solutions used in this work were prepared of analytic grade reactants and deionised water Milli}Q2+. The Cu(II) concentration was 15.7 mM in 1.0 M H SO . Fig. 2 shows the diagram of a device built up for laboratory studies, consisting of a 500 cm glass reactor with a temperature bath and 8.5 cm inner diameter. A 304-type stainless-steel cylinder with a 2.1 cm diameter and a length of 6.0 cm was used as a cathode. As anodes were used four 13 cm long, 1.8 cm wide and 0.3 cm thick graphite bars which were attached to the reactor walls and connected between each other. 115 V and 70 W Caframo2+ electric motor of variable velocity was used to rotate the inner cylinder. The reference electrode was an SSE Tacussel2+. The potentiostat utilized was PAR2+ Model 273. For performing impedance tests, FRA signal generator, 1260 Solartron2+, was employed. 4.1. Determination of electrolysis potential for Cu(II) to Cu(0) reduction process The working electrode was previously coated with copper in order to simulate the real deposition process. Voltamperogram of the Cu(II) solution in the cathodic region (!0.43}!1.1) V vs. SSE, was determined at 20 mV/s sweep rate with an immobile electrode in order
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to identify the potential range in which the mass transfer controls the global electrochemical process. 4.2. Evaluation of the intrinsic reaction rate constant (charge transfer rate constant) To evaluate the intrinsic reaction rate constant, the copper-coated RCE was used as a working electrode. To calculate the charge transfer rate constant of the Cu(II)/Cu(0) reaction, we employed the electrochemical impedance spectroscopy technique near the equilibrium potential (without bulk mass transport phenomena). Furthermore, as we know that the charge transfer varies exponentially with the overpotential (Bard & Faulkner, 1980; Pletcher, 1991; Walsh, 1993), we assessed the value of the charge transfer rate constant at the corresponding electrolysis potential. The impedance spectrum was obtained applying a direct current potential of !5 mV from the equilibrium potential (!0.43 V vs. SSE). A sinusoidal signal was used with amplitude of 5 mV within a frequency interval of 0.18}1888 Hz. The experiments were made at room temperature with static electrodes. Data were obtained at 10-decade intervals with the help of `Zplota software of Solartron equipment. The analysis of data was carried out by an adjustment of non-linear regression algorithm created by Bouckamp (1993). 4.3. Experimental evaluation of the specixcs mass transfer coezcients The experimental determination of the speci"c mass transfer coe$cients k a was carried out by electrolysis K experiments at controlled potential. These experiments, made in a potential range with a limiting current plateau, were performed with the purpose of making the bulk mass transport controls the global process at these conditions of potential. The electrolysis potential was of !0.90 V vs. SSE. The studies were made for Reynolds numbers within 34;10(Re(1.15;10 interval, in turbulent #ow (Gabe, 1974).
5. Analysis of results and discussion 5.1. Determination of electrolysis potential for Cu(II)/Cu(0) reduction process Fig. 3 shows the voltamperogram made at 20 mV/s sweep rate with an immobile electrode; the Cu(II) ionreduction process starts at !0.43 V vs. SSE and reaches a peak at approximately !0.85 V vs. SSE; such fast decay of the current is observed because the Cu(II) deposition is performed at a copper metal surface. Here, it was possible to identify a region of potential between
Fig. 3. Voltamperogram of the cathodic region for Cu(II)/Cu(0) process in a copper-coated cylinder [Cu(II)]"15.7 mM, [H SO ]"1 M, A"11.87 cm, sweep rate 20 mV/s, ¹"298 K.
!0.85 and !1.00 V vs. SSE in which the reduction process is controlled by the mass transport determined by the arrival of Cu(II) species toward the interface. Hence, the electrolysis potential was selected at !0.90 V vs. SSE. 5.2. Determination of the intrinsic reaction rate constant (charge transfer rate constant) As previously mentioned, the technique employed to assess the intrinsic reaction rate constant at the equilibrium was that of electrochemical impedance spectroscopy. However, the charge transfer constant at the electrolysis potential was assessed considering its exponential variation with the overpotential (Bard & Faulkner, 1980; Pletcher, 1991; Walsh, 1993). Fig. 4(a) shows the Nyquist diagram of impedance spectra obtained within a frequency range from 0.18 to 1888 Hz, for the Cu(II)/Cu(0) deposition process onto metallic copper. In this spectra two loops are observed, one less de"ned at high frequencies between 376 and 1888 Hz (see Fig. 4(b)) and the other, better de"ned at low frequencies between 376 and 0.18 Hz (see Fig. 4(a)). The out-of-"t point at 60 Hz is assumed as an electronic problem in the laboratory electric net. The adjustment shown in Fig. 4 was obtained with a non-linear regression using the equivalent circuit model shown in Fig. 5 that represents the deposition process of a metallic ion (McDonald, 1987; Reid & Allan, 1987; Wiart, 1990; Proud & Muller, 1992; Miranda 1999). In this model C is the double-layer capacitance, R is the charge transfer resistance, Q is the constant phase element, R is the charge transfer resistance and Z is the U corresponding Warburg impedance. It is appropriate to notice that continuous line in Fig. 4 represents the adjustment obtained to the Nyquist experimental
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Fig. 4. Nyquist diagram for the Cu(II)/Cu(0) process in a copper-coated cylinder [Cu(II)]"15.7 mM, [H SO ]"1 M, A"11.87 cm, amplitude of 5 mV, frequency sweep from 0.18 to 1888 Hz, E"!0.435 V/SSE, ¹"298 K.
Table 1 Equivalent circuit impedance results
Fig. 5. Equivalent circuit used to "t the experimental data of the impedance spectra for the Cu(II)/Cu(0) process.
diagram. Table 1 shows the values obtained for the equivalent circuit elements. To check that no frequency in the range has been unfairly biased in the data treatment, the percentage relative error as a function of frequency, has been calculated. The errors established using a Kramers}Kronig's error model were between !0.20 and 0.35% (McDonald, 1993; Morris & Smyrl, 1990). In studies related to the reduction kinetics of Cu(II) in acid media it has been found that the Cu(II)/Cu(0) mechanism occurs in two steps (Reid & Allan, 1987). In accordance with this, R corresponds to the charge transfer resistance of Cu(I)/Cu(0) and R to the Cu(II)/Cu(I) which is limited by a Warburg di!usion component, Z . U This impedance element is associated with the surface di!usion of the Cu (II) ions around the deposited nucleus (Miranda, 1999). So, the Cu(II)/Cu(I) reduction process is the lowest step because R is ap proximately 13 times greater than R (see Table 1). The phase constant element Q corresponds to
R () Q
R ()
1Z (u) R () U
C (F)
1> (u sL) n
1.60
0.77
0.93
281.60
0.012
10.47
0.68
the pseudocapacitance of the deposited nucleus. The high value of capacitance, C, is due to the large electrode area. Furthermore, in order to calculate the copper reduction process rate constant for the step that governs the intrinsic process, it is necessary to obtain the exchange current density, J . This calculation is performed consid ering the following equation in the open-circuit potential (Bard & Faulkner, 1980; McDonald, 1987; Reid & Allan, 1987): R¹ J " , zFR AR
(11)
where R "R is the charge transfer resistance with the AR value of 10.47 for the step governing the Cu(II)#1e\PCu(I) intrinsic process, A, the geometric area of cylinder electrode (11.875 cm), ¹, temperature (298 K), R, the universal gas constant (8.314 J mol\ 3K\), z, the number of exchangeable electrons during electrochemical reaction, F, Faraday constant (96 485 C mol\) and J , the exchange current density (A cm\). The exchange current density J , calculated using Eq. (11) was 2.06;10\ A cm\, therefore, the rate constant for the Cu(II)#1e\PCu(I) process should be calculated (Bard & Faulkner, 1980; McDonald, 1987; Reid
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& Allan, 1987) considering J kH" . (12) zFC? C\? - 0 For a small-amplitude ac process, in which one species of the couple Cu(II)/Cu(I) is in rapid equilibrium with a unit activity solid, Eq. (12) may be stated as (Reid & Allan, 1987) J (13) kH" . zFC? This equation should be valid for the Cu(II)/Cu(0) redox couple since the cuprous ion is in rapid equilibrium with the copper metal interface. Taking "0.5 from Reid and Allan (1987) in our work, kH results to be 5.39;10\ cm s\, while this value was reported by Reid and Allan (1987) as 1.6;10\ cm s\. This di!erence arises from the fact that Reid and coworkers had used a copper disk, while we have used a copper-coated stainless-steel cylinder electrode. It is worth mentioning that at the electrolysis potential it is possible to use the following expression (Bard & Faulkner, 1980; Pletcher, 1991; Walsh, 1993)
k "kH exp!
zF(E!E ) C , R¹
(14)
in order to assess only the intrinsic reaction rate constant (charge transfer) without the in#uence of bulk mass transfer, where the charge transfer rate constant is known to vary exponentially with the overpotential (E!E ). In C this work, the overpotential E!E "!0.47 V vs. SSE, C thus the value obtained for k using Eq. (14) is
Table 2 Speci"c mass transfer coe$cients via Cu(II) electrolysis, in the RCE, at a constant electrolysis potential of !0.90 V/SSE and di!erent Reynolds. A"29.68 cm, < "230 cm, a"0.129 cm\, at 298 K 0 Re
k a (1/s);10 K
34 636 53 108 68 117 98 135 115 447
6.39 8.19 15.60 17.10 18.40
5.09;10\ cm s\. With the value of a"0.129 cm\, k a is 6.56;10\ s\. This value will be used to evaluate the e!ectiveness factors in Eq. (4). 5.3. Determination of the specixc mass transfer coezcients in the RCE The speci"c mass transfer coe$cients k a were deterK mined at the RCE via electrolysis at a constant potential of !0.9 V/SSE. Fig. 6 shows a logarithmic decay of the Cu(II) concentration vs. time. Eq. (9) makes possible the assessment of these transport parameters at di!erent Reynolds values, whereas bulk mass transport governs at this electrolysis potential, !0.9 V vs. SSE. Table 2 shows that both parameters, Re and k a increase proporK tionally due to the fact that mass transport resistance diminishes with the increase of the rotation rate. Fig. 6 shows that as the Reynolds increases, the logarithmic concentration decay is faster, and naturally the k a values increase (Table 2). In addition, it can be seen K
Fig. 6. Logarithmic decay of the Cu(II) concentration versus electrolysis time in the RCE at di!erent Reynolds. A"29.68 cm, < "230 cm, 0 a"0.129 cm\, at a constant electrolysis potential of !0.90 V/SSE, ¹"298 K.
J. L. Nava-M. de Oca et al. / Chemical Engineering Science 56 (2001) 2695}2702 Table 3 E!ectiveness factors for the Cu(II)/Cu(0) process in the RCE at di!erent Reynolds. k a"6.56;10\ s\, A"29.68 cm, < "230 cm, 0 a"0.129 cm\, at 298 K Re
Da
34 636 53 108 68 117 98 135 115 447
10.30 8.02 4.21 3.84 3.57
0.08 0.11 0.19 0.21 0.22
that the slopes are not very distant for the Reynolds of 98 135 and 115 447; one might think that the transport of reactants is e$cient and that there is a beginning of competence between the charge transfer and the mass transfer at such hydrodynamic conditions. However, the aforementioned is just a hypothesis, since what really occurs is that rotating cylinder walls start showing vortices giving rise to a diminution in the speci"c area (a"A/< ) during the electrolysis. This happens even at 0 Reynolds within turbulent #ow. The justi"cation of the above fact is also analyzed through the e!ectiveness factors (see Table 3) as discussed in the following section. 6. RCE modeling in simple batch using e4ectiveness factors The Damkohler number, Da , was evaluated by Eq. (3) using the value for the intrinsic reaction rate constant (charge transfer rate constant) determined in this work k a"6.56;10\ s\, corresponding to the electrolysis potential of !0.90 V vs. SSE, and the k a values for K each Re. Moreover, the e!ectiveness factors, , were calculated by Eq. (4). The values of Da and are shown in Table 3. It is seen that as Re increases, Da , diminishes. This permits to visualize the way in which the hydrodynamics favors the mass transport. In addition, with the increase of Re, increases too and this phenomenon results from the decrease in the bulk mass transport resistance. On the other hand, the analysis of catalytic e!ectiveness factors 0.08((0.22 established that the mass transport controls the global electrochemical process. The aforementioned allows us to think about an extreme situation: C (t)"C (0) exp![k at]. (9) K This expression is typically used for RCE in simple batch (Gabe & Walsh, 1984; Genders & Weimberg, 1992; Walsh, 1993). So, through the e!ectiveness factors we have corroborated that this expression is reasonably valid for the Cu(II)/Cu(0) process.
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Nevertheless, through these factors, it is possible to determine rotating velocities at which vortices and other hydrodynamic phenomena, unfavorable for reactant transport, appear. For example, at Re of 98 135 and 115 447 (see Fig. 6) it would be expected that +0.50 as mentioned in Section 5.3, but the values obtained were 0.21 and 0.22 (see Table 3). So, there is not competition between the charge transfer and bulk mass transport. The real phenomenon occurs due to the appearance of vortices which diminish the e!ective area, a, on the rotating cylinder walls during the electrolysis. It is important to mention that for other electrochemical reactions whose charge transfer rate does not occur so fast, the mass transport is probably not the only global process-limiting step.
7. Conclusions A straightforward analysis of the RCE through e!ectiveness factors using the dimensionless DamkoK hler number for an electrochemical interface process, Cu(II)/Cu(0), shows the competition between the intrinsic reaction (charge transfer) and bulk mass transport phenomena. E!ectiveness factors indicate that the mass transfer controls this process as it was expected considering the potential imposed during electrolysis. The technique employed to assess the intrinsic reaction rate constant at the equilibrium was the electrochemical impedance spectroscopy. It was performed within an RCE taking into account that the charge transfer constant at electrolysis potential was assessed considering its exponential variation with the overpotential. The speci"c mass transfer coe$cients were obtained by electrolysis at controlled potential in order to calculate a mean value. Furthermore, through e!ectiveness factors, it is possible to determine the Reynolds range at which some vortices and other hydrodynamic phenomena, unfavorable for the current #ow through the cell, appear. It is recommended to analyze the e!ectiveness factors for other electrochemical reactions, since some of them are slow and the mass transport would not be the only step to limit the global process. In addition, it is recommendable to use a methodology analog to the one developed in this paper for other types of electrochemical reactors.
Notation a A CL CL C -
speci"c area ("A/< ), cm\ 0 rotating cylinder electrode area, cm surface concentration, mol cm\ concentration in bulk, mol cm\ oxidized species concentration in mol cm\
bulk,
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C (0) -
initial oxidized species concentration in bulk, mol cm\ C (t) oxidized species concentration at t time in bulk, mol cm\ C reduced species concentration, mol cm\ 0 C double-layer capacitance, Faraday d rotating cylinder diameter, cm Da dimensionless DamkoK hler number ("k /k a) K E potential, V E equilibrium potential, V f rotation frequency, rev s\ F Faraday constant, 96485 c mol\ J current density, A cm\ J exchange current density, A cm\ k reaction rate constant for nth order kH charge transfer constant in redox equilibrium, cm s\ k charge transfer constant at a cathodic over potential, cm s\ k mass transfer constant, cm s\ K n reaction order r rotating cylinder radius, cm R real reaction rate Re dimensionless Reynolds number, (";d/) R charge transfer resistance, R solution resistance, Q R ideal reaction rate R , R charge transfer resistance of the processes 1 and 2, respectively, R universal gas constant, 8.314 J mol\ K\ ¹ temperature, K ; peripheral velocity, ;"2rf, cm s\ < electrolytic solution volume, cm 0 z number of electrons transferred during electrochemical reaction Z Warburg impedance, u U Greek letters
dimensionless charge transfer coe$cient di!usion layer thickness, cm kinematic viscosity, cm s\ dimensionless isothermal e!ectiveness factor, (1/(1#Da ))
Acknowledgements J. L. N and E. S. H. are grateful to CONACyT for the obtained grants. The authors acknowledge the "nancial aid of CONACyT (Proyect No.400200-5-32539-T).
References Bard, A. J., & Faulkner, L. R. (1980). Electrochemical methods fundamentals and applications. New York, USA: Wiley. Bouckamp, B. A. (1993). Equivalent circuit, Ver. 4.51. University of Twente, Netherlands. Carberry, J. J. (1976). Chemical and catalytic reaction engineering. New York, USA: Mc Graw Hill. Cassiere, G., & Carberry, J. J. (1973). Chemical Engineering Education, 1, 22}26. Gabe, D. R. (1974). Journal of Applied Electrochemistry, 14, 91}108. Gabe, D. R., & Walsh, F. C. (1984a). Journal of Applied Electrochemistry, 14, 555}564. Gabe, D. R., & Walsh, F. C. (1984b). Journal of Applied Electrochemistry, 14, 565}572. Gabe, D. R., & Walsh, F. C. (1985). Journal of Applied Electrochemistry, 15, 807}824. Gabe, D. R., Wilcox, G. D., Gonzalez-GarcmH a, J., & Walsh, F. C. (1998). Journal of Applied Electrochemistry, 28, 759}780. Genders, J. D., & Weimberg, N. L. (1992). Electrochemistry for a cleaner environment. New York: The Electrosynthesis Company Inc. Fogler, H. S. (1992). Elements of chemical reaction engineering. Englewood Cli!s, NJ, USA: Prentice-Hall. McDonald, J. R. (1987). Impedance spectroscopy emphasizing solid materials. New York, USA: Wiley. McDonald, J. R. (1993). Electrochimica Acta, 38, 1883}1890. Miranda, M. H. (1999). Te& cnicas electroqun& micas para discriminar las diferentes etapas del proceso de electrocristalizacio& n de plata en disoluciones acuosas de etilamina y amoniaco. Tesis doctoral. Universidad AutoH noma Metropolitana-I. MeH xico. Morris, R., & Smyrl, W. (1990). Journal of Electrochemical Society, 137(2), 406}411. Pletcher, D. (1991). A xrst course in electrode processes. Romsey: The Electrochemical Consultancy. Pletcher, D., & Walsh, F. C. (1993). Industrial electrochemistry. Great Britain: Blackie. Proud, W. G., & Muller, C. (1992). Journal of Applied Electrochemistry, 22, 405}413. Reid, J. D., & Allan, P. D. (1987). Journal of Electrochemical Society, 6, 1389}1394. Walsh, F. C. (1993). A xrst course in electrochemical engineering. Romsey: The Electrochemical Consultancy. Wiart, R. (1990). Electrochimica Acta, 35, 1587}1593.
E!ectiveness factors in an electrochemical reactor with rotating cylinder electrode for the acid-cupric/copper cathode interface process JoseH Luis Nava-M. de Oca, E. Sosa, Carlos Ponce de LeoH n, M. T. Oropeza* Departament of Chemistry, Universidad Auto& noma Metropolitana-Iztapalapa Av. Michoaca& n y la Purn& sima, A.P. 55-534, C.P. 09340, Me& xico D.F., Me& xico, USA Received 2 March 2000; received in revised form 9 October 2000; accepted 17 October 2000
Abstract An expression for the global electrochemical deposition rate of copper in a rotating cylinder electrode (RCE) was developed through the e!ectiveness factor. The e!ectiveness factor of an electrochemical interface process, calculated using the DamkoK hler number, leads to identify the competition between intrinsic reaction (charge transfer) and bulk mass transport. The charge transfer constant was evaluated by electrochemical impedance spectroscopy and the speci"c mass transfer coe$cients were determined by electrolysis at controlled potential. Catalytic e!ectiveness factors were found between 0.08 and 0.22 establishing that the mass transport of Cu (II) ion controls the global electrochemical process at an imposed electrolysis potential of !0.9 V vs. SSE in turbulent #ow. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Rotating cylinder electrode; Electrolysis; Mass transfer; E!ectiveness factor
1. Introduction It is well known that in the chemical reaction engineering, the di!usion and intrinsic reaction are the main phenomena that determine the global reaction rate within a heterogeneous catalytic reactor. For this reason, it is interesting to study an electrochemical transformation as a heterogeneous catalytic process since the bulk mass transport and charge transfer (intrinsic reaction) participate in a global electrochemical transformation. There are di!erent factors in the electrochemical reactor operation that determine its behavior, such as polarization of the working electrode necessary to assure quantitative transformation and the relationship between mass transfer phenomena and the rotation speed of the polarized electrode (Gabe & Walsh, 1984a,b, 1985; Gabe, Wilcox, Gonzalez-Garcia, & Walsh, 1998). In particular, the reactor with rotating cylinder electrode (RCE) is one of the most studied models regarding the mass transfer phenomena and hydrodynamics. In the literature, besides dimensionless correlation concern* Corresponding author. Tel.: #52-5-724-4670; fax: #52-5-7244666. E-mail address: [email protected] (M. T. Oropeza).
ing mass transfer coe$cient variation and hydrodynamic regimen at a RCE (Gabe, 1974; Gabe & Walsh, 1984, 1985; Gabe & Weimberg, 1998), some authors have used an equation that predicts the concentration of a reactant species versus time within a simple batch type reactor (Gabe & Walsh, 1984; Genders & Weimberg, 1992; Walsh, 1993). In all the above papers, it was assumed that the mass transfer, determined by the limiting current, governs the global electrochemical process. The foregoing approach suggests a detailed analysis of the global electrochemical process kinetics from the heterogeneous catalysis viewpoint, since the operating principles of catalytic reactors can be helpful in the understanding of the operation of an RCE. The purpose of this paper is to show the competition between the charge transfer and bulk mass transport at an imposed electrolysis potential using the e!ectiveness factors that group together the contributions of each phenomenon involved. It is important to mention that in order to obtain a better estimate of each transfer coe$cient, this study analyzes the charge transfer (intrinsic reaction within double layer) by electrochemical impedance spectroscopy and the bulk mass transport by copper ion depletion in solution during electrolysis.
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 1 4 - 5
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Fig. 1. Concentration pro"le as a function of the distance under steady-state conditions for an electrochemical process.
2. Interface model of copper deposition process Since the Cu(II)/Cu(0) system is widely utilized as a reaction model for the characterization of RCE processes (Gabe & Walsh, 1984a,b, 1985; Pletcher & Walsh, 1993) it was selected to be used in an acid aqueous environment. The concentration pro"le, as a distance function (Fig. 1), helps the identi"cation of bulk mass transfer phenomena. In the Helmholtz plane (HP), the reaction can take place in two consecutive steps (Reid & Allan, 1987): Cu(II)#1e\Cu(I), Cu(I)#1e\Cu(0). However, if we impose a potential corresponding to the limiting current plateau (electrolysis potential), the Cu(II) ion transport from the bulk to the Nernst di!usion layer, can limit the global electrochemical process. The aforementioned implies that in order to reach HP, the Cu(II) species early needs to cross a -thick "lm (Nernst di!usion layer).
3. E4ectiveness factor in rotating cylinder reactor Two central parts in heterogeneous transformation process engineering are the catalytic reaction and the di!usion of species involved in the chemical system. In the electrodeposition of a metallic species both of them are present. The "rst one as a need for enough surface energy to perform the desired transformation (electronic transfer reaction) and the second, as a resistance that an electroactive species has to overcome in order to reach the electrode surface. One of these phenomena may control the global rate, depending on the potential imposed in the electrode processes (Gabe, 1974; Gabe & Walsh, 1984a,b, 1985).
In this work, we "rst identify the catalytic surface where an electroactive species is transformed and which is going to be called the electrocatalyst. Then the interface di!usion and reaction at isothermal conditions will be discussed as contributions to the global electrochemical transformation rate within the RCE. The heat transport phenomena are not mentioned, however, it does not mean they are undervalued. Finally, an expression for global cathodic process rate is determined using general e!ectiveness factor assumptions. E!ectiveness factor, , of an interfacial chemical process is de"ned by the relationship between the rate of an interface process R (real rate), when the concentration of an electroactive species at the surface is C and the rate of an ideal process, R , when the surface concentration is equal to that of the species in `bulka C (Cassiere & Carberry, 1973): R " (1) R considering "rst a order reaction, the equation is transformed to
k C C " " , k C C where k is the intrinsic reaction rate constant. In steady-state conditions, the electroactive species arriving by a mass transport mechanism disappears through the intrinsic reaction (Cassiere & Carberry, 1973): k a(C !C)"k C, (2) K where the surface concentration (the unobservable, C) in terms of bulk concentration (the observable, C ) is (Car berry, 1976): 1 C" C . (2) 1#k /k a K In this equation, the DamkoK hler dimensionless number appears in an explicit form, Da , which is the ratio of the intrinsic reaction rate constant with regard to the speci"c mass transfer constant: k (3) Da " . k a K It is important to mention that in steady-state conditions, mass transport and intrinsic-reaction rate are equal, but it does not mean that mass and charge transfer coe$cients have the same value. Furthermore, the global reaction rate is equal to the rate of the lowest step in the mechanism (Carberry, 1976; Fogler, 1992). The e!ectiveness factor in terms of the DamkoK hler dimensionless number, Da , results in (Cassiere & Car berry, 1973; Carberry, 1976): 1 " . 1#Da
(4)
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Finally, the global reaction rate R may be expressed in terms of the e!ectiveness factor (Cassiere & Carberry, 1973; Carberry, 1976): R"R "k C . (5) For an electrochemical process, the ideal rate constant k "k a must be considered, being k the charge trans fer constant for reduction reaction at the cathode (rotating cylinder), and a"A/< , the speci"c area, while A is 0 the rotating cylinder area (immersed in the solution) and < the electrolytic solution volume. C "C is the con0 centration in bulk of the oxidized species O (Cu(II)). Performing these substitutions in Eq. (5), the global cathodic process rate results in 1 k a R" C " C . (6) 1#Da - 1/k a#1/k a K For an electrochemical reactor with rotating cylinder electrode in simple batch, the mass balance for cathodic process indicates that dC ! - "R dt
(7)
or dC - "!k aC . dt
Fig. 2. Rotating cylinder electrode scheme.
These two extreme cases suggest that as for the 0.5 value, the best equation to represent the results is that corresponding to (8), since the electrochemical process is under mixed control.
(7)
Integrating the foregoing expression and considering that for t"0, C "C (0) and for t"t, C "C (t), the oxidized species concentration as a function of time is C (t)"C (0) exp![k a t] or
(8)
1 C (t)"C (0) exp! t . (8) 1/k a#1/k a K In Eq. (8), it is possible to visualize the contribution of the intrinsic reaction (charge transfer) and the mass transport in the global cathodic process that is being carried out within a RCE in batch operation; therefore, we have two extreme cases: E for relatively high DamkoK hler values, the e!ectiveness coe$cient P1/Da , indicating that the process re mains under mass transport control. In this case, Eq. (8) is transformed C (t)"C (0) exp![k a t]. K
(9)
E for relatively small DamkoK hler values, the e!ectiveness coe$cient P1, indicating the control by intrinsic reaction (charge transfer). Similarly, Eq. (8) is transformed into: C (t)"C (0) exp![k a t].
(10)
4. Experimental The solutions used in this work were prepared of analytic grade reactants and deionised water Milli}Q2+. The Cu(II) concentration was 15.7 mM in 1.0 M H SO . Fig. 2 shows the diagram of a device built up for laboratory studies, consisting of a 500 cm glass reactor with a temperature bath and 8.5 cm inner diameter. A 304-type stainless-steel cylinder with a 2.1 cm diameter and a length of 6.0 cm was used as a cathode. As anodes were used four 13 cm long, 1.8 cm wide and 0.3 cm thick graphite bars which were attached to the reactor walls and connected between each other. 115 V and 70 W Caframo2+ electric motor of variable velocity was used to rotate the inner cylinder. The reference electrode was an SSE Tacussel2+. The potentiostat utilized was PAR2+ Model 273. For performing impedance tests, FRA signal generator, 1260 Solartron2+, was employed. 4.1. Determination of electrolysis potential for Cu(II) to Cu(0) reduction process The working electrode was previously coated with copper in order to simulate the real deposition process. Voltamperogram of the Cu(II) solution in the cathodic region (!0.43}!1.1) V vs. SSE, was determined at 20 mV/s sweep rate with an immobile electrode in order
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to identify the potential range in which the mass transfer controls the global electrochemical process. 4.2. Evaluation of the intrinsic reaction rate constant (charge transfer rate constant) To evaluate the intrinsic reaction rate constant, the copper-coated RCE was used as a working electrode. To calculate the charge transfer rate constant of the Cu(II)/Cu(0) reaction, we employed the electrochemical impedance spectroscopy technique near the equilibrium potential (without bulk mass transport phenomena). Furthermore, as we know that the charge transfer varies exponentially with the overpotential (Bard & Faulkner, 1980; Pletcher, 1991; Walsh, 1993), we assessed the value of the charge transfer rate constant at the corresponding electrolysis potential. The impedance spectrum was obtained applying a direct current potential of !5 mV from the equilibrium potential (!0.43 V vs. SSE). A sinusoidal signal was used with amplitude of 5 mV within a frequency interval of 0.18}1888 Hz. The experiments were made at room temperature with static electrodes. Data were obtained at 10-decade intervals with the help of `Zplota software of Solartron equipment. The analysis of data was carried out by an adjustment of non-linear regression algorithm created by Bouckamp (1993). 4.3. Experimental evaluation of the specixcs mass transfer coezcients The experimental determination of the speci"c mass transfer coe$cients k a was carried out by electrolysis K experiments at controlled potential. These experiments, made in a potential range with a limiting current plateau, were performed with the purpose of making the bulk mass transport controls the global process at these conditions of potential. The electrolysis potential was of !0.90 V vs. SSE. The studies were made for Reynolds numbers within 34;10(Re(1.15;10 interval, in turbulent #ow (Gabe, 1974).
5. Analysis of results and discussion 5.1. Determination of electrolysis potential for Cu(II)/Cu(0) reduction process Fig. 3 shows the voltamperogram made at 20 mV/s sweep rate with an immobile electrode; the Cu(II) ionreduction process starts at !0.43 V vs. SSE and reaches a peak at approximately !0.85 V vs. SSE; such fast decay of the current is observed because the Cu(II) deposition is performed at a copper metal surface. Here, it was possible to identify a region of potential between
Fig. 3. Voltamperogram of the cathodic region for Cu(II)/Cu(0) process in a copper-coated cylinder [Cu(II)]"15.7 mM, [H SO ]"1 M, A"11.87 cm, sweep rate 20 mV/s, ¹"298 K.
!0.85 and !1.00 V vs. SSE in which the reduction process is controlled by the mass transport determined by the arrival of Cu(II) species toward the interface. Hence, the electrolysis potential was selected at !0.90 V vs. SSE. 5.2. Determination of the intrinsic reaction rate constant (charge transfer rate constant) As previously mentioned, the technique employed to assess the intrinsic reaction rate constant at the equilibrium was that of electrochemical impedance spectroscopy. However, the charge transfer constant at the electrolysis potential was assessed considering its exponential variation with the overpotential (Bard & Faulkner, 1980; Pletcher, 1991; Walsh, 1993). Fig. 4(a) shows the Nyquist diagram of impedance spectra obtained within a frequency range from 0.18 to 1888 Hz, for the Cu(II)/Cu(0) deposition process onto metallic copper. In this spectra two loops are observed, one less de"ned at high frequencies between 376 and 1888 Hz (see Fig. 4(b)) and the other, better de"ned at low frequencies between 376 and 0.18 Hz (see Fig. 4(a)). The out-of-"t point at 60 Hz is assumed as an electronic problem in the laboratory electric net. The adjustment shown in Fig. 4 was obtained with a non-linear regression using the equivalent circuit model shown in Fig. 5 that represents the deposition process of a metallic ion (McDonald, 1987; Reid & Allan, 1987; Wiart, 1990; Proud & Muller, 1992; Miranda 1999). In this model C is the double-layer capacitance, R is the charge transfer resistance, Q is the constant phase element, R is the charge transfer resistance and Z is the U corresponding Warburg impedance. It is appropriate to notice that continuous line in Fig. 4 represents the adjustment obtained to the Nyquist experimental
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Fig. 4. Nyquist diagram for the Cu(II)/Cu(0) process in a copper-coated cylinder [Cu(II)]"15.7 mM, [H SO ]"1 M, A"11.87 cm, amplitude of 5 mV, frequency sweep from 0.18 to 1888 Hz, E"!0.435 V/SSE, ¹"298 K.
Table 1 Equivalent circuit impedance results
Fig. 5. Equivalent circuit used to "t the experimental data of the impedance spectra for the Cu(II)/Cu(0) process.
diagram. Table 1 shows the values obtained for the equivalent circuit elements. To check that no frequency in the range has been unfairly biased in the data treatment, the percentage relative error as a function of frequency, has been calculated. The errors established using a Kramers}Kronig's error model were between !0.20 and 0.35% (McDonald, 1993; Morris & Smyrl, 1990). In studies related to the reduction kinetics of Cu(II) in acid media it has been found that the Cu(II)/Cu(0) mechanism occurs in two steps (Reid & Allan, 1987). In accordance with this, R corresponds to the charge transfer resistance of Cu(I)/Cu(0) and R to the Cu(II)/Cu(I) which is limited by a Warburg di!usion component, Z . U This impedance element is associated with the surface di!usion of the Cu (II) ions around the deposited nucleus (Miranda, 1999). So, the Cu(II)/Cu(I) reduction process is the lowest step because R is ap proximately 13 times greater than R (see Table 1). The phase constant element Q corresponds to
R () Q
R ()
1Z (u) R () U
C (F)
1> (u sL) n
1.60
0.77
0.93
281.60
0.012
10.47
0.68
the pseudocapacitance of the deposited nucleus. The high value of capacitance, C, is due to the large electrode area. Furthermore, in order to calculate the copper reduction process rate constant for the step that governs the intrinsic process, it is necessary to obtain the exchange current density, J . This calculation is performed consid ering the following equation in the open-circuit potential (Bard & Faulkner, 1980; McDonald, 1987; Reid & Allan, 1987): R¹ J " , zFR AR
(11)
where R "R is the charge transfer resistance with the AR value of 10.47 for the step governing the Cu(II)#1e\PCu(I) intrinsic process, A, the geometric area of cylinder electrode (11.875 cm), ¹, temperature (298 K), R, the universal gas constant (8.314 J mol\ 3K\), z, the number of exchangeable electrons during electrochemical reaction, F, Faraday constant (96 485 C mol\) and J , the exchange current density (A cm\). The exchange current density J , calculated using Eq. (11) was 2.06;10\ A cm\, therefore, the rate constant for the Cu(II)#1e\PCu(I) process should be calculated (Bard & Faulkner, 1980; McDonald, 1987; Reid
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& Allan, 1987) considering J kH" . (12) zFC? C\? - 0 For a small-amplitude ac process, in which one species of the couple Cu(II)/Cu(I) is in rapid equilibrium with a unit activity solid, Eq. (12) may be stated as (Reid & Allan, 1987) J (13) kH" . zFC? This equation should be valid for the Cu(II)/Cu(0) redox couple since the cuprous ion is in rapid equilibrium with the copper metal interface. Taking "0.5 from Reid and Allan (1987) in our work, kH results to be 5.39;10\ cm s\, while this value was reported by Reid and Allan (1987) as 1.6;10\ cm s\. This di!erence arises from the fact that Reid and coworkers had used a copper disk, while we have used a copper-coated stainless-steel cylinder electrode. It is worth mentioning that at the electrolysis potential it is possible to use the following expression (Bard & Faulkner, 1980; Pletcher, 1991; Walsh, 1993)
k "kH exp!
zF(E!E ) C , R¹
(14)
in order to assess only the intrinsic reaction rate constant (charge transfer) without the in#uence of bulk mass transfer, where the charge transfer rate constant is known to vary exponentially with the overpotential (E!E ). In C this work, the overpotential E!E "!0.47 V vs. SSE, C thus the value obtained for k using Eq. (14) is
Table 2 Speci"c mass transfer coe$cients via Cu(II) electrolysis, in the RCE, at a constant electrolysis potential of !0.90 V/SSE and di!erent Reynolds. A"29.68 cm, < "230 cm, a"0.129 cm\, at 298 K 0 Re
k a (1/s);10 K
34 636 53 108 68 117 98 135 115 447
6.39 8.19 15.60 17.10 18.40
5.09;10\ cm s\. With the value of a"0.129 cm\, k a is 6.56;10\ s\. This value will be used to evaluate the e!ectiveness factors in Eq. (4). 5.3. Determination of the specixc mass transfer coezcients in the RCE The speci"c mass transfer coe$cients k a were deterK mined at the RCE via electrolysis at a constant potential of !0.9 V/SSE. Fig. 6 shows a logarithmic decay of the Cu(II) concentration vs. time. Eq. (9) makes possible the assessment of these transport parameters at di!erent Reynolds values, whereas bulk mass transport governs at this electrolysis potential, !0.9 V vs. SSE. Table 2 shows that both parameters, Re and k a increase proporK tionally due to the fact that mass transport resistance diminishes with the increase of the rotation rate. Fig. 6 shows that as the Reynolds increases, the logarithmic concentration decay is faster, and naturally the k a values increase (Table 2). In addition, it can be seen K
Fig. 6. Logarithmic decay of the Cu(II) concentration versus electrolysis time in the RCE at di!erent Reynolds. A"29.68 cm, < "230 cm, 0 a"0.129 cm\, at a constant electrolysis potential of !0.90 V/SSE, ¹"298 K.
J. L. Nava-M. de Oca et al. / Chemical Engineering Science 56 (2001) 2695}2702 Table 3 E!ectiveness factors for the Cu(II)/Cu(0) process in the RCE at di!erent Reynolds. k a"6.56;10\ s\, A"29.68 cm, < "230 cm, 0 a"0.129 cm\, at 298 K Re
Da
34 636 53 108 68 117 98 135 115 447
10.30 8.02 4.21 3.84 3.57
0.08 0.11 0.19 0.21 0.22
that the slopes are not very distant for the Reynolds of 98 135 and 115 447; one might think that the transport of reactants is e$cient and that there is a beginning of competence between the charge transfer and the mass transfer at such hydrodynamic conditions. However, the aforementioned is just a hypothesis, since what really occurs is that rotating cylinder walls start showing vortices giving rise to a diminution in the speci"c area (a"A/< ) during the electrolysis. This happens even at 0 Reynolds within turbulent #ow. The justi"cation of the above fact is also analyzed through the e!ectiveness factors (see Table 3) as discussed in the following section. 6. RCE modeling in simple batch using e4ectiveness factors The Damkohler number, Da , was evaluated by Eq. (3) using the value for the intrinsic reaction rate constant (charge transfer rate constant) determined in this work k a"6.56;10\ s\, corresponding to the electrolysis potential of !0.90 V vs. SSE, and the k a values for K each Re. Moreover, the e!ectiveness factors, , were calculated by Eq. (4). The values of Da and are shown in Table 3. It is seen that as Re increases, Da , diminishes. This permits to visualize the way in which the hydrodynamics favors the mass transport. In addition, with the increase of Re, increases too and this phenomenon results from the decrease in the bulk mass transport resistance. On the other hand, the analysis of catalytic e!ectiveness factors 0.08((0.22 established that the mass transport controls the global electrochemical process. The aforementioned allows us to think about an extreme situation: C (t)"C (0) exp![k at]. (9) K This expression is typically used for RCE in simple batch (Gabe & Walsh, 1984; Genders & Weimberg, 1992; Walsh, 1993). So, through the e!ectiveness factors we have corroborated that this expression is reasonably valid for the Cu(II)/Cu(0) process.
2701
Nevertheless, through these factors, it is possible to determine rotating velocities at which vortices and other hydrodynamic phenomena, unfavorable for reactant transport, appear. For example, at Re of 98 135 and 115 447 (see Fig. 6) it would be expected that +0.50 as mentioned in Section 5.3, but the values obtained were 0.21 and 0.22 (see Table 3). So, there is not competition between the charge transfer and bulk mass transport. The real phenomenon occurs due to the appearance of vortices which diminish the e!ective area, a, on the rotating cylinder walls during the electrolysis. It is important to mention that for other electrochemical reactions whose charge transfer rate does not occur so fast, the mass transport is probably not the only global process-limiting step.
7. Conclusions A straightforward analysis of the RCE through e!ectiveness factors using the dimensionless DamkoK hler number for an electrochemical interface process, Cu(II)/Cu(0), shows the competition between the intrinsic reaction (charge transfer) and bulk mass transport phenomena. E!ectiveness factors indicate that the mass transfer controls this process as it was expected considering the potential imposed during electrolysis. The technique employed to assess the intrinsic reaction rate constant at the equilibrium was the electrochemical impedance spectroscopy. It was performed within an RCE taking into account that the charge transfer constant at electrolysis potential was assessed considering its exponential variation with the overpotential. The speci"c mass transfer coe$cients were obtained by electrolysis at controlled potential in order to calculate a mean value. Furthermore, through e!ectiveness factors, it is possible to determine the Reynolds range at which some vortices and other hydrodynamic phenomena, unfavorable for the current #ow through the cell, appear. It is recommended to analyze the e!ectiveness factors for other electrochemical reactions, since some of them are slow and the mass transport would not be the only step to limit the global process. In addition, it is recommendable to use a methodology analog to the one developed in this paper for other types of electrochemical reactors.
Notation a A CL CL C -
speci"c area ("A/< ), cm\ 0 rotating cylinder electrode area, cm surface concentration, mol cm\ concentration in bulk, mol cm\ oxidized species concentration in mol cm\
bulk,
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C (0) -
initial oxidized species concentration in bulk, mol cm\ C (t) oxidized species concentration at t time in bulk, mol cm\ C reduced species concentration, mol cm\ 0 C double-layer capacitance, Faraday d rotating cylinder diameter, cm Da dimensionless DamkoK hler number ("k /k a) K E potential, V E equilibrium potential, V f rotation frequency, rev s\ F Faraday constant, 96485 c mol\ J current density, A cm\ J exchange current density, A cm\ k reaction rate constant for nth order kH charge transfer constant in redox equilibrium, cm s\ k charge transfer constant at a cathodic over potential, cm s\ k mass transfer constant, cm s\ K n reaction order r rotating cylinder radius, cm R real reaction rate Re dimensionless Reynolds number, (";d/) R charge transfer resistance, R solution resistance, Q R ideal reaction rate R , R charge transfer resistance of the processes 1 and 2, respectively, R universal gas constant, 8.314 J mol\ K\ ¹ temperature, K ; peripheral velocity, ;"2rf, cm s\ < electrolytic solution volume, cm 0 z number of electrons transferred during electrochemical reaction Z Warburg impedance, u U Greek letters
dimensionless charge transfer coe$cient di!usion layer thickness, cm kinematic viscosity, cm s\ dimensionless isothermal e!ectiveness factor, (1/(1#Da ))
Acknowledgements J. L. N and E. S. H. are grateful to CONACyT for the obtained grants. The authors acknowledge the "nancial aid of CONACyT (Proyect No.400200-5-32539-T).
References Bard, A. J., & Faulkner, L. R. (1980). Electrochemical methods fundamentals and applications. New York, USA: Wiley. Bouckamp, B. A. (1993). Equivalent circuit, Ver. 4.51. University of Twente, Netherlands. Carberry, J. J. (1976). Chemical and catalytic reaction engineering. New York, USA: Mc Graw Hill. Cassiere, G., & Carberry, J. J. (1973). Chemical Engineering Education, 1, 22}26. Gabe, D. R. (1974). Journal of Applied Electrochemistry, 14, 91}108. Gabe, D. R., & Walsh, F. C. (1984a). Journal of Applied Electrochemistry, 14, 555}564. Gabe, D. R., & Walsh, F. C. (1984b). Journal of Applied Electrochemistry, 14, 565}572. Gabe, D. R., & Walsh, F. C. (1985). Journal of Applied Electrochemistry, 15, 807}824. Gabe, D. R., Wilcox, G. D., Gonzalez-GarcmH a, J., & Walsh, F. C. (1998). Journal of Applied Electrochemistry, 28, 759}780. Genders, J. D., & Weimberg, N. L. (1992). Electrochemistry for a cleaner environment. New York: The Electrosynthesis Company Inc. Fogler, H. S. (1992). Elements of chemical reaction engineering. Englewood Cli!s, NJ, USA: Prentice-Hall. McDonald, J. R. (1987). Impedance spectroscopy emphasizing solid materials. New York, USA: Wiley. McDonald, J. R. (1993). Electrochimica Acta, 38, 1883}1890. Miranda, M. H. (1999). Te& cnicas electroqun& micas para discriminar las diferentes etapas del proceso de electrocristalizacio& n de plata en disoluciones acuosas de etilamina y amoniaco. Tesis doctoral. Universidad AutoH noma Metropolitana-I. MeH xico. Morris, R., & Smyrl, W. (1990). Journal of Electrochemical Society, 137(2), 406}411. Pletcher, D. (1991). A xrst course in electrode processes. Romsey: The Electrochemical Consultancy. Pletcher, D., & Walsh, F. C. (1993). Industrial electrochemistry. Great Britain: Blackie. Proud, W. G., & Muller, C. (1992). Journal of Applied Electrochemistry, 22, 405}413. Reid, J. D., & Allan, P. D. (1987). Journal of Electrochemical Society, 6, 1389}1394. Walsh, F. C. (1993). A xrst course in electrochemical engineering. Romsey: The Electrochemical Consultancy. Wiart, R. (1990). Electrochimica Acta, 35, 1587}1593.