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The hanging meniscus rotating disk (HMRD) Part 3. Application to a charge transfer preceded by a chemical reaction in equilibrium
Journal of Electroanalytical Chemistry 418 (19%) 159- 165
EISEVIER
The hanging meniscus rotating disk ( HMRD) Part 3. Application to a charge transfer preceded by a chemical reaction in equilibrium H.M. Villullas INFIQC.
Dep.
de F&co
Quimicn,
Fowltad
de Cirncias
Quhiras,
*, M. L6pez Teijelo llnioersidad
Nacioml
de C&d&i,
Suc.16,
C.C.61,
5016
Chdobo,
Argentina
Received 2 November 1995; revised 10 May 1996
Abstract The hanging meniscus rotating disk (HMRD) is a configuration in which a cylinder of the electrode material is mounted without an insulating shroud. Despite the differences in the hydrodynamic boundaries introduced because of the meniscus of electrolyte that rises above the general surface level, the hydrodynamic behavior of the HMRD is similar to that of conventional rotating disks and it can also be used to study the kinetics of simple charge transfer reactions. In this paper, the current-rotation rate equations for a chemical reaction in equilibrium that precedes the charge transfer reaction at the HMRD are developed. The applicability of the HMRD to the study of the kinetics of chemical reactions in equilibrium in the bulk electrolyte is discussed. Keywords: Hanging meniscus rotating disk; Preceding chemical reaction; Charge transfer reaction; Current-rotation rate equations; Kinetics
and then raised to develop a hanging meniscus.Measure-
1. Introduction
ments for the ferro/ferricyanide The conventional rotating disk electrode @DE) [I], widely used in research work in electrochemistry [2-41, is usually constructed by press-fitting the electrode material into an insulating mantle such as Teflon or another plastic. However, this procedure is not always possible and/or convenient. An extreme example is found for metal single crystals, where severe damage could be introduced into the crystal structure when the electrode is encased. In contrast, in the hanging meniscus rotating disk (HMRD) the elec-
trode material is used without an insulating mantle, making easierthe use of single crystals as rotating electrodes[5,6]. Moreover, the absenceof an insulating shroud introduces potential advantagesnot only for systemswhere creating a sealat the electrode/mantle interface may be difficult, but also for those that require special treatments to the electrode surface (annealing, electropolishing, etc.) before mounting. The HMRD usesan electrode of cylindrical form which is simply lowered until contact with the electrolyte is made
* Corresponding author. Fax: + 54 51 334174. 0022-0728/96/$15.00 PII
SOO22-0728(96)04765-
Reff = R, - K( v/o)“*
Copyright 0 1996 Elsevier Science S.A. All rights reserved 1
redox couple
on Au have
shown that, despite the differences in the hydrodynamic boundaries,the behavior of the HMRD is similar to that of a conventional RDE [S,7]. Linear plots of limiting current I, vs. square root of rotation rate w were obtained, showing the sameslope as that expected for a conventional RDE and a small negative intercept [5]. The HMRD has been used with Au [6,8,9] and Pt [lo-l 5] single-crystal electrodes, as well as in corrosion studies[ 16,171,although the experimental study of its hydrodynamic behavior is quite recent [7]. The approach to describe the hydrodynamic behavior of the HMRD is based on the main featuresof this configuration, i.e. the absenceof an insulating mantle and the meniscusof electrolyte that rises above the general electrolyte level. Since the meniscusfree surface imposes a clear limit to mass transport, when the liquid in the hydrodynamic layer reaches the disk edge, it has to change direction and flow down along the meniscus surface. This reversal of flow produces a reduction of the effective radius of the electrode Ref, for the convective diffusion current, according to [7,18]
(1)
H.M.
160
Villullas,
M. Ldpcz Te~jelo/Jourml
oj’Elrctroanulyticrtl
where R, is the geometric radius and K is a proportionality constant [IS]. When the reduction of the effective radius promoted by the change in flow direction at the disk edge is considered, the effective electrode area (A,,, = rrR&) has to be introduced into the Levich equation. From this [7,18], dropping second-order terms: I,(HMRD)
= 0.620nFD2/3v-1/6~~w’/2 XA[l
-2KR,‘(~/w)l’~]
(2)
where A is the geometric area and all the other symbols have their usual meaning 119,201. The first term in Eq. (2) is exactly the standard Levich equation, and the second term, which does not depend on rotation speed and accounts for the negative intercept, is given by intercept = ( - 0.620nFD2/3~-‘/6c,*A)2KR;
‘v”~
(3)
where the quantity in parentheses is exactly the slope of the standard Levich equation. It was demonstrated that Eq. (2) describes the hydrodynamic behavior of the HMRD and allows the slope value and the negative intercept to be explained [7]. The thickness of the flow reversal region was found to be a function of rotation speed, meniscus height and electrode radius. The value of the proportionality constant K depends on meniscus height h, as follows: K= k( h, - h,)
(4)
where h, corresponds to the value of the critical meniscus height above which h, should be set to prevent meniscus climb on the cylinder side [7]. It should be noted that at and below h,, lateral wetting is expected to occur and Eq. (2) is no longer valid [7]. The values of k and h, depend on specific properties of the electrodelmeniscus interface
171. The HMRD was also applied to the study of the kinetics of simple charge transfer reactions under mixed control [21]. By employing the usual methodology to extract quantitative kinetic information for charge transfer reactions based on the method of Frumkin and Tedoradse [22] for the RDE, i.e. plotting l/Z vs w-‘j2 for first-order kinetics, nonlinear plots are obtained for the HMRD configuration. However, it was demonstrated [21] that the general equation for first-order kinetics l/Z = (1/L!,)
+ (l/Z,)
Chemimy
418 (1996)
159-165
electrode and liquid, depend strongly on h, [23]. Furthermore, it was also demonstrated that, under rotation, the excess pressure exerted by the centrifugal force on the liquid interface also produces a variation of meniscus shape and local contact angle [23]. Thus, since the shape of the boundary is unknown and depends on h, and w, solving the hydrodynamic equations and the convective diffusion equation for the HMRD appears to be a very difficult task. The aim of this paper is to show that the use of the HMRD can be extended to other types of system. The current-rotation rate equations for a chemical reaction in equilibrium that precedes the charge transfer reaction on the HMRD are developed using a semi-empirical approach similar to that employed in previous papers [7,21]. Results of calculations are used to analyze the applicability of the HMRD to the study of the kinetics of chemical reactions in equilibrium in the bulk electrolyte.
2. The basic theory 2.1. The equations for the RDE Since the description of the RDE by Levich 111, a number of extensions to the fundamental theory of RDE hydrodynamics have been published [2-41, including the particular case of a charge transfer reaction preceded by a chemical reaction in equilibrium [24,25]. The model kinetic system k, A,*A, k2 A, fne-+
(6) A,
(7)
considered by Koutecky and Levich [24,25] involves the rapid interconversion of A I and A, that keeps the system in equilibrium throughout the bulk of the solution. Near the electrode surface, this equilibrium is perturbed by the reduction of A,. It is assumed that the latter process occurs under complete mass transport control at potentials where A, is not reduced and that D = D, = D,. The limiting current for the reduction of A, is [24,25]
(5)
is valid for the HMRD, as well as the equivalent equations for reaction orders other than one [211. Consequently, the kinetic current I,,, obtained on the HMRD at any given potential and for any h, set above h,, is identical to that obtained using a conventional RDE, even in the potential region close to equilibrium [20,21]. The influence of h, and rotation rate on the shape of the free surface of the meniscus and its relation to lateral wetting and meniscus stability have been studied recent!y [23]. It was shown that the shape of the free surface of the meniscus, as well as the local contact angle between
(8)
Z,=(nFADc,*)/
where c,* is the overall bulk concentration of the solution, 6, is the thickness of the hydrodynamic boundary layer (the ‘Nemst layer’) and S, is the thickness of the kinetic boundary layer. Introducing the definitions of 6, and 6, [24,25], Eq. (8) can be rearranged to give the well-known expression [X41 [ = &,‘/2 L
- B’1w’/2]
L
(9)
H.M.
Villullm,
of Electroanalytical
M. LOpc,- Teijrlo/Jourd
where B’ = 0.620nFAD2/3v-‘/6c*
0
(10)
and B”=0.620D”6v-“6(k,/k,)(k,
+k2)-“*
(11)
Thus, the kinetic parameters for the chemical reaction in equilibrium that precedes the charge transfer can be obtained from measurements of ZL as a function of w [2-41. According to Eq. (91, B’ and B” are given by the intercept and slope respectively of a plot of ZJO”~ vs. I,. If D is known, the observed intercept may be compared with the calculated value of B’ (EIq. (10)) and, if the equilibrium constant is also known, the rate constants k, and k, can be evaluated from Eq. (11). The RDE has been used to study the kinetics of a number of preceding chemical reactions [4], such as the dissociation and recombination of weak acids [26-291. For acetic acid, more accurate measurements were carried out using two synchronously rotating electrodes connected by a bridge [30]. The rate constants for the dissociation and recombination of trinitromethane in ACN were also measured using an RDE [31]. Hale [32] and Chernenko and coworkers [33,34] have investigated, for galvanostatic conditions, the effect of the chemical reaction kinetics on the establishment of the steady-state, while Albery [35] has considered the effect of the electric field in the diffusion part of the double layer and of migration on the measured values of the dissociation constants. 2.2. The equations for the new hydrodynamic
boundaries
Despite the fact that the HMRD behaves similar to the conventional RDE, the hydrodynamic boundaries are obviously different. A clear limitation to mass transport at the disk edge is imposed by the omission of the insulating mantle and the presence of a hanging meniscus of electrolyte which produces a region of flow direction reversal [7,18]. Therefore, the effective radius of the electrode will be the reduced value given by Eq. (1) and, in consequence, Eq. (2) will hold for the HMRD at all accessible meniscus heights above h,. Introduction of the effective electrode area A,rf instead of the geometric area of the disk A into Eq. (8) yields (ignoring second-order terms) Z,(HMRD)
=nFADc;(l
-2KR,‘\/v/o)/
Chemistry
418 (1996)
159-165
161
It is clear from comparison of Eq. (9) and Eq. (13) that the equation that describes the behavior of the HMRD differs from that of the RDE only in the correction term B’Z, which accounts for the reduction of effective radius observed at the HMRD because of the inversion of flow direction at the edge and is identical to the second term in Eq. (2).
3. Calculation
of kinetic parameters
3.1. Deviations causedby the Bow reversal region It is well known that a deviation of a plot of I, vs. w”* from a straight line that intersects the origin is an indication of some kinetic step involved in the electron transfer reaction taking place on an RDE [2-4,191. Similar behavior should be expected for the HMRD, despite the fact that the reversal of flow direction at the edge of the disk will cause a shift of the curves towards smaller currents for increasing values of h,, producing nonzero intercepts [5,7,18,21]. Curves calculated using Eq. (12) for several values of K, which is directly proportional to h, through Eq. (4), are depicted in Fig. 1. The curves correspondingto the RDE behavior with and without the kinetic step involved in the charge transfer reaction have also been included. While plots of Z,/W’/~ vs. I, are linear for the RDE, nonlinear plots should be expected for the HMRD because of the correction term B’Z in Eq. (13). This correction term, which is a function of h, and the particular properties of the system according to Eq. (3) and Eq. (4), is independent of w [7]. It is clear from Eq. (13) that a significant curvature should be exhibited by a plot of
RDE
(without
d,)
I
1.76
&,+a,> i
i 6)
J3q. (12) can be rearranged to give Z,(HMRD)
= B’w “* - B”Z,( HMRD)
0”
- B’Z
(13)
where Z=2KR,]v’/*
(14)
Fig. 1. Calculated I, vs. w’/’ curves for the RDE and HMRD configurations: n=l; c,‘=O.lM; R,=O.l5cm; D=1X10~5cm2~~‘; Y= O.Olcm’ SC’; k, /k2 = 5.7X 104; k, + k, = IX lO”cms-I. The values of K for the HMRD are indicated in the figure.
162
H.M.
Villullas.
of Electroanulytical
M. Lbpez Teijelo/Journal
Chemistry
418 (1996)
159-165
3.0 RDE (without
--------
d,) 2.5
,
RDE (with
\\
dJ
o.“oo
.li
,
a hm /cm
O.iS
0.20
G/ d Fig. 2. Calculated I, /w “* vs. I, curves for the RDE and HMRD configurations. The values of K for the HMRD are indicated in the figure. Other parameters as for Fig. I.
I&B”2
vs. I, for the HMRD, causedby the variation of the ratio B’Z/W’/~, and that the HMRD curve should approach the linear behavior at high values of I,. Additionally, curvature should be a function of the value of B’Z, i.e. it should become more pronounced for greater values of K (i.e. higher values of h,). Curves calculated using data taken from Fig. 1 for different values of K are presentedin Fig. 2. The deviation from linearity shown in Fig. 2 may appear as a limitation to the applicability of the HMRD to this type of study. However, it will be shown below that such a limitation may be overcome if measurementsof I, as a function of w are taken at different values of h,. For the HMRD set at any value of h, above h,, the dependenceof the limiting current on w is given by Eq. (12). Introducing K as given by Eq. (4) and rearranging yields (15)
Xcm, = 2 R, ‘( v/w)
“2
0.14
Fig. 3. Calculated values of I, for the HMRD plotted vs. h, at different rotation rates as indicated in the figure. Parameters as for Fig. 1.
At constant rotation rate, GcW,and XcW, will have fixed values and, consequently, the measuredvalue of I, will be determined only by the value of h, at which the HMRD is set. Thus, according to Eq. (151, a plot of I, vs. h, should be linear for any given value of w. I, values taken from calculated curves of Fig. 1 are plotted vs. h,, for different rotation rates in Fig. 3. Since Xcw, can be calculated using Es. (161, G(w, could be evaluated for each rotation rate from the slopesin Fig. 3, provided that the parameter k for the system is available. Unfortunately, this is not likely to be the case, since the value of k, as well as the critical meniscusheight h, (Eq. (4)), is related to the particular properties of the electrode material and electrolyte used [7,23]. Nevertheless, evaluation of both k and h,, should always be possible from the sameset of data. 3.2. Calculation
of parameters
k and h,
From Eqs. (15)~(171, it is clear that the intercepts and slopesof plots IL as a function of h, should be expected to change for the different values of o. This is, indeed,
(16)
and Gcw, = nFDAc,*/
It should be noted, from comparisonof Eq. (8) and Eq. (17), that Gcw, representsthe limiting current that would be measuredon an RDE of geometric radius R,. Thus, if it
were possible to evaluate Cc,,,, from data taken on the HMRD, this would be equivalent to correcting each measured value of I, for the HMRD, taking into account the reduction of effective electrode areaproduced by the reversal of flow direction at the edge.
Fig. 4. Quotient lintercept/slopel, w”‘. Parameters as for Fig. 1.
Q(,,
calculated
for curves of Fig. 3 vs.
H.M.
\\
\\
Villullus,
M. L&m
Teijelo /Journul
of Electronnalyticul
418 (1996)
159-165
\
\
0.1 1 1.5
.
RDE HMRD
\ \
2.b Cf,,
2.3 /
mA
Fig. 5. Calculated values of G~wj/w”’ vs. Gcw,. The calculated curve I, /co’/2 vs. I, for the RDE as shown in Fig. 2 is also included. Parameters as for Fig. 1.
observed in Fig. 3. The quotient lintercept/slopel, for each one of these curves (Eq. (15)) is
Q,,,,,,
Eq. (16) into Eq. (18) yields
Q,,, = w”~R,/(~~v”~)
+ h,
(19)
Then, if Q,,, is calculated for each value of w of Fig. 3 and plotted vs. w112 (Fig. 4), the value of k can be obtained from the slope. In addition, it is interesting to note that although k is the only parameter required to carry out the calculation of the rate constants for the previous chemical reaction, k, and k,, the value of the critical meniscus height h, is also obtained, since it is given by the intercept in Fig. 4. 3.3. Analysis of data for calculation of kinetic parameters As stated above, the quantity G,,, can be calculated for each value of rotation rate from the slopes of plots of I, vs. h, (Fig. 3) provided that the value of k has been determined for the system (Fig. 4). Consequently, since GcW,, according to Eq. (8) and Eq. (17), represents the limiting current that would be measured on an RDE of geometric radius R,, a plot of G(w)W2 vs. Y($ should be linear and coincident with the plot of IL/w vs. I, for the RDE, as shown in Fig. 5, yielding exactly the same values of B’ and B” from which the diffusion coefficient and the kinetic parameters can be obtained as discussed above [2,4]. Alternatively, B’ and the parameters for the previous chemical reaction may also be obtained from a plot of vs. l/o ‘I* . From Eq. (17) and the definitions of W(u) S,, and 6, [24,25]
It is quite evident that, according to Eq. (13), a plot of (I, + B’Z)/w “2 vs. I, for the HMRD, should be linear and lie on the same straight line obtained when plotting Z,/W”2 vs. I, for the RDE. The appropriate correction term B’Z that should be added to each measured value of I, may be obtained for the particular h, set for the experiments, by calculating Z according to Eq. (4) and Eq. (14). The parameters k and h, are required and can be evaluated from the slope and intercept of a plot of Q,,, vs. w’12 as in Fig. 4. A plot of (IL + B’Z)/w’/* vs. I,, for values taken from calculated curves of Fig. 1, is shown in Fig. 6, where data points for the RDE are included for comparison. It is worth noting that although the calculated points for the HMRD at different h, values lay on the same straight line as those calculated for the RDE, and consequently yield the same values of B’ and B”, as should be expected, data points for the HMRD are shifted with respect to the corresponding point for the RDE. As shown in Fig. 1, limiting currents at the HMRD are smaller than those corresponding to an RDE of the same geometric area caused by the inversion of flow direction at the disk edge [5,7,18,21]. 3.5. Limiting currents for unequal diffusion coeflcients In the treatment given for the HMRD in the preceding sections, we have kept, for the sake of simplicity, the assumption that D = D, = D, as in the original derivation by Koutecky and Levich for the conventional RDE [24,25]. The more general case of unequal diffusion coefficients has been solved by Dogonadze [36], who showed that this assumption does not have a significant effect on the method
c ‘; o.57 \ ?7 3 3
zJ
I:;L
1.0 IL
1 -= Gw
1 B'W'/2
(k, +
163
3.4. Limiting current values at the HMRD ---
‘R
Introducing
Chemistry
+k2)
-"'Wkd
nFAc,* D1/2
(20)
/
1.5 mui
2.0
2
Fig. 6. Calculated (I, + B’Z)/W’/~ vs. I, curves for the RDE ( n ) and HMRD configurations. The values of K for the HMRD are (0 ) 0.44, (0 ) 1.32; (0) 2.20; other parameters as for Fig. 1.
164
H.M.
Villullas.
M. L&r:
Teijelo/Journal
c~fElrctrounalyricai
of calculation. The limiting current for the RDE when D, # D, turns out to be [36]
(21) where D=
Chemistry
418
(1996)
159-165
obtained using the KDE. From the ments, the parameters k and h, system can also be obtained. The presented above is also valid for the
same set of measurethat characterize the method of calculation general case D, # D,
Acknowledgements k,D2
+ k2D,
(22)
k, +k* 8, = 1.61( D/v)“‘(
v/w)*‘*
1
8, = k,
d
o+o 1
(24) 2
References [I]
It is quite obvious that Eq. (8) and Eq. (21), as well as S,, and 6, [24,25] for both cases, have a similar mathematical form. Therefore, introducing Eq. (23) and Eq. (24) into Eq. (21) yields an expression for the RDE which is identical to Eq. (9), whereas, for the HMRD, introducing the effective electrode area Aeff instead of the geometric area of the disk A into Eq. (21) and combining with EZq. (23) and Eq. (24) yields a relation completely equivalent to E$. (13). The parameters B’ and B” for the D, # D, case are “(jc 0*
B”=0.620D-“3v-“6(k,D,/k,D,)(k,/D,
CONI-
(23)
k2
B’ = 0.620i~FAD*‘~v-
This research has been supported by CONICET, COR and SECYT (UNC).
121 [3] [4] [5]
[6]
[7]
(25) +k,/Dz)-“2
(26) Despite the fact that, in order to get the kinetic data for the chemical reaction, more lengthy computations will be required when the parameters B’ and B” are given by Eq. (25) and Eq. (26), it is quite clear that the case D, # D, does not differ, in principle, from the simpler case of equal diffusion coefficients [24,25,36]. Thus, the method presented above for calculation of B’ and B” for a chemical reaction in equilibrium that precedes the charge transfer reaction on the HMRD is also valid for the general case D,#D2.
[8] 191 [lo] [ll] [12] [13] [14] [15] [16] [17]
4. Conclusions The HMRD can be used to study the kinetics of a chemical reaction in equilibrium in the bulk electrolyte that precedes the charge transfer. Plots of IJo”~ vs. I, are not linear. Measurements of limiting current as a function of rotation rate should be carried out for several meniscus heights. The reduction of effective area promoted by the change in flow direction at the edge of the disk can be taken into account and, in consequence, corrected limiting currents corresponding to the whole geometric area of the disk can be calculated. The kinetic parameters should be independent of meniscus height and identical to those
[18] [ 191 /20] [21] [22] [23] [24] 1251 [26]
V.G. Levich, Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1962. A.C. Riddiford, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 4, Interscience, New York, 1966. R.N. Adams, Electrochemistry at Solid Electrodes, Marcel Dekker, New York, 1969. F. Opekar and P. Beran, J. Electroanal. Chem., 69 (1976) 1. B.D. Cahan and H.M. Villullas, ECS Meet., Chicago, 1988, Abstr. 707; B.D. Cahan and H.M. Villullas, J. Electroanal. Chem., 307 (1991) 263. B.D. Cahan, H.M. Villullas and E.B. Yeager, ECS Meet., Atlanta, 1988, Abstr. 532; B.D. Cahan, H.M. Villullas and E.B. Yeager, J. Electroanal. Chem., 306 (1991) 213. H.M. Villullas and M. L6pez Teijelo, 43rd ISE Meet., Cordoba, Argentina, 1992, Abstr. 6-014; H.M. Villullas and M. Lopez Teijelo, J. Electroanal. Chem., 384 (1995) 25. T. Abe, Y. Miki and K. Itaya, Bull. Chem. Sot. Jpn., 67 (1994) 2075. R.R. Adzic, J. Wang and B.M. Ocko. Electrochim. Acta, 40 (1995) 83. F. El Kadiri, R. Faure and R. Durand, J. Electroanal. Chem.. 301 (1991) 177. F.H. Feddrix, E.B. Yeager and B.D. Cahan, J. Electroanal. Chem., 330 (1992) 419. H. Kita, S. Ye and Y. Gao, J. Electroanal. Chem., 334 (1992) 351. N. Markovic, R.R. Adzic, B.D. Cahan and E.B. Yeager, J. Electroanal. Chem., 377 (1994) 249. H. Kita, H.W. Lei and Y. Gao, J. Electroanal. Chem.. 379 (1994) 407. T. Abe, G.M. Swain, K. Sashikata and K. itaya, J. Electroanal. Chem., 382 (1995) 73. M.M. Laz, R.M. Souto, S. Gonzalez, R.C. Salvarezza and A.J. Arvia, Electrochim. Acta, 37 (1992) 655. R.M. Souto, M. Perez Sanchez, M. Barrera, S. Gonzalez, R.C. SaIvarezza and A.J. Arvia, Electrochim. Acta, 37 (1992) 1437. B.D. Cahan, Proc. Symp. on Modelling of Batteries and Fuel Cells, ECS Meet., Phoenix, 1991. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. E. Gileadi, Electrode Kinetics, VCH, New York, 1993. H.M. Villullas and M. Lopez Teijelo, J. Electroanal. Chem.. 385 (1995) 39. A. Frumkin and G. Tedoradse, Z. Elektrochem., 62 (1958) 251. H.M. Villullas and M. L6pez Teijelo, J. Appl. Electrochem., 26 (1996) 353. J. Koutecky and V.G. Levich, Dokl. Akad. Nauk SSSR, 117 (1941) 441. J. Koutecky and V.G. Levich, Zh. Fiz. Khim., 32 (1958) 1565. W. Vielstich and D. Jahn, in I.S. Langmuir (Ed.), Advances in Polarography, Vol. 1, Pergamon. New York, 1960.
NM.
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TeiJelo / Journul
of Elrctroundyticd
[27] W. Vielstich and D. Jahn, Z. Elektrochem., 64 (1960) 43. [281 M. von Stackelberg, W. Vielstich and D. Jahn, An. R. Sot. Esp. Fis. Quim. Ser. B:, 56 (1960) 475. [29] W. Vielstich, Z. Anal. Chem., 173 (1960) 84. [30] W.J. Albery and R.P. Bell, Proc. Chem. Sot., (1963) 169. [31] V.A. Kokorekina, L.C. Feoktistov, V.Y. Filinovskii and S.A. Sevelev, Elektrokhimiya, 6 (1971) 1196.
Chemistry
418 (19961 159-165
[32] J.M. Hale, J. Electroanal. Chem., 8 (1964) 332. [33] V.I. Chemenko, Elektrokhimiya, 7 (1971) 820. [34] V.I. Chemenko and K.I. Litovchenco, Elektrokhimiya, 1410. [35] W.J. Albery, Trans. Faraday Sot., 61 (1965) 2063. [36] R.R. Dogonadze. Zh. Fiz. Khim., 32 (1958) 2437.
165
7 (1971)
EISEVIER
The hanging meniscus rotating disk ( HMRD) Part 3. Application to a charge transfer preceded by a chemical reaction in equilibrium H.M. Villullas INFIQC.
Dep.
de F&co
Quimicn,
Fowltad
de Cirncias
Quhiras,
*, M. L6pez Teijelo llnioersidad
Nacioml
de C&d&i,
Suc.16,
C.C.61,
5016
Chdobo,
Argentina
Received 2 November 1995; revised 10 May 1996
Abstract The hanging meniscus rotating disk (HMRD) is a configuration in which a cylinder of the electrode material is mounted without an insulating shroud. Despite the differences in the hydrodynamic boundaries introduced because of the meniscus of electrolyte that rises above the general surface level, the hydrodynamic behavior of the HMRD is similar to that of conventional rotating disks and it can also be used to study the kinetics of simple charge transfer reactions. In this paper, the current-rotation rate equations for a chemical reaction in equilibrium that precedes the charge transfer reaction at the HMRD are developed. The applicability of the HMRD to the study of the kinetics of chemical reactions in equilibrium in the bulk electrolyte is discussed. Keywords: Hanging meniscus rotating disk; Preceding chemical reaction; Charge transfer reaction; Current-rotation rate equations; Kinetics
and then raised to develop a hanging meniscus.Measure-
1. Introduction
ments for the ferro/ferricyanide The conventional rotating disk electrode @DE) [I], widely used in research work in electrochemistry [2-41, is usually constructed by press-fitting the electrode material into an insulating mantle such as Teflon or another plastic. However, this procedure is not always possible and/or convenient. An extreme example is found for metal single crystals, where severe damage could be introduced into the crystal structure when the electrode is encased. In contrast, in the hanging meniscus rotating disk (HMRD) the elec-
trode material is used without an insulating mantle, making easierthe use of single crystals as rotating electrodes[5,6]. Moreover, the absenceof an insulating shroud introduces potential advantagesnot only for systemswhere creating a sealat the electrode/mantle interface may be difficult, but also for those that require special treatments to the electrode surface (annealing, electropolishing, etc.) before mounting. The HMRD usesan electrode of cylindrical form which is simply lowered until contact with the electrolyte is made
* Corresponding author. Fax: + 54 51 334174. 0022-0728/96/$15.00 PII
SOO22-0728(96)04765-
Reff = R, - K( v/o)“*
Copyright 0 1996 Elsevier Science S.A. All rights reserved 1
redox couple
on Au have
shown that, despite the differences in the hydrodynamic boundaries,the behavior of the HMRD is similar to that of a conventional RDE [S,7]. Linear plots of limiting current I, vs. square root of rotation rate w were obtained, showing the sameslope as that expected for a conventional RDE and a small negative intercept [5]. The HMRD has been used with Au [6,8,9] and Pt [lo-l 5] single-crystal electrodes, as well as in corrosion studies[ 16,171,although the experimental study of its hydrodynamic behavior is quite recent [7]. The approach to describe the hydrodynamic behavior of the HMRD is based on the main featuresof this configuration, i.e. the absenceof an insulating mantle and the meniscusof electrolyte that rises above the general electrolyte level. Since the meniscusfree surface imposes a clear limit to mass transport, when the liquid in the hydrodynamic layer reaches the disk edge, it has to change direction and flow down along the meniscus surface. This reversal of flow produces a reduction of the effective radius of the electrode Ref, for the convective diffusion current, according to [7,18]
(1)
H.M.
160
Villullas,
M. Ldpcz Te~jelo/Jourml
oj’Elrctroanulyticrtl
where R, is the geometric radius and K is a proportionality constant [IS]. When the reduction of the effective radius promoted by the change in flow direction at the disk edge is considered, the effective electrode area (A,,, = rrR&) has to be introduced into the Levich equation. From this [7,18], dropping second-order terms: I,(HMRD)
= 0.620nFD2/3v-1/6~~w’/2 XA[l
-2KR,‘(~/w)l’~]
(2)
where A is the geometric area and all the other symbols have their usual meaning 119,201. The first term in Eq. (2) is exactly the standard Levich equation, and the second term, which does not depend on rotation speed and accounts for the negative intercept, is given by intercept = ( - 0.620nFD2/3~-‘/6c,*A)2KR;
‘v”~
(3)
where the quantity in parentheses is exactly the slope of the standard Levich equation. It was demonstrated that Eq. (2) describes the hydrodynamic behavior of the HMRD and allows the slope value and the negative intercept to be explained [7]. The thickness of the flow reversal region was found to be a function of rotation speed, meniscus height and electrode radius. The value of the proportionality constant K depends on meniscus height h, as follows: K= k( h, - h,)
(4)
where h, corresponds to the value of the critical meniscus height above which h, should be set to prevent meniscus climb on the cylinder side [7]. It should be noted that at and below h,, lateral wetting is expected to occur and Eq. (2) is no longer valid [7]. The values of k and h, depend on specific properties of the electrodelmeniscus interface
171. The HMRD was also applied to the study of the kinetics of simple charge transfer reactions under mixed control [21]. By employing the usual methodology to extract quantitative kinetic information for charge transfer reactions based on the method of Frumkin and Tedoradse [22] for the RDE, i.e. plotting l/Z vs w-‘j2 for first-order kinetics, nonlinear plots are obtained for the HMRD configuration. However, it was demonstrated [21] that the general equation for first-order kinetics l/Z = (1/L!,)
+ (l/Z,)
Chemimy
418 (1996)
159-165
electrode and liquid, depend strongly on h, [23]. Furthermore, it was also demonstrated that, under rotation, the excess pressure exerted by the centrifugal force on the liquid interface also produces a variation of meniscus shape and local contact angle [23]. Thus, since the shape of the boundary is unknown and depends on h, and w, solving the hydrodynamic equations and the convective diffusion equation for the HMRD appears to be a very difficult task. The aim of this paper is to show that the use of the HMRD can be extended to other types of system. The current-rotation rate equations for a chemical reaction in equilibrium that precedes the charge transfer reaction on the HMRD are developed using a semi-empirical approach similar to that employed in previous papers [7,21]. Results of calculations are used to analyze the applicability of the HMRD to the study of the kinetics of chemical reactions in equilibrium in the bulk electrolyte.
2. The basic theory 2.1. The equations for the RDE Since the description of the RDE by Levich 111, a number of extensions to the fundamental theory of RDE hydrodynamics have been published [2-41, including the particular case of a charge transfer reaction preceded by a chemical reaction in equilibrium [24,25]. The model kinetic system k, A,*A, k2 A, fne-+
(6) A,
(7)
considered by Koutecky and Levich [24,25] involves the rapid interconversion of A I and A, that keeps the system in equilibrium throughout the bulk of the solution. Near the electrode surface, this equilibrium is perturbed by the reduction of A,. It is assumed that the latter process occurs under complete mass transport control at potentials where A, is not reduced and that D = D, = D,. The limiting current for the reduction of A, is [24,25]
(5)
is valid for the HMRD, as well as the equivalent equations for reaction orders other than one [211. Consequently, the kinetic current I,,, obtained on the HMRD at any given potential and for any h, set above h,, is identical to that obtained using a conventional RDE, even in the potential region close to equilibrium [20,21]. The influence of h, and rotation rate on the shape of the free surface of the meniscus and its relation to lateral wetting and meniscus stability have been studied recent!y [23]. It was shown that the shape of the free surface of the meniscus, as well as the local contact angle between
(8)
Z,=(nFADc,*)/
where c,* is the overall bulk concentration of the solution, 6, is the thickness of the hydrodynamic boundary layer (the ‘Nemst layer’) and S, is the thickness of the kinetic boundary layer. Introducing the definitions of 6, and 6, [24,25], Eq. (8) can be rearranged to give the well-known expression [X41 [ = &,‘/2 L
- B’1w’/2]
L
(9)
H.M.
Villullm,
of Electroanalytical
M. LOpc,- Teijrlo/Jourd
where B’ = 0.620nFAD2/3v-‘/6c*
0
(10)
and B”=0.620D”6v-“6(k,/k,)(k,
+k2)-“*
(11)
Thus, the kinetic parameters for the chemical reaction in equilibrium that precedes the charge transfer can be obtained from measurements of ZL as a function of w [2-41. According to Eq. (91, B’ and B” are given by the intercept and slope respectively of a plot of ZJO”~ vs. I,. If D is known, the observed intercept may be compared with the calculated value of B’ (EIq. (10)) and, if the equilibrium constant is also known, the rate constants k, and k, can be evaluated from Eq. (11). The RDE has been used to study the kinetics of a number of preceding chemical reactions [4], such as the dissociation and recombination of weak acids [26-291. For acetic acid, more accurate measurements were carried out using two synchronously rotating electrodes connected by a bridge [30]. The rate constants for the dissociation and recombination of trinitromethane in ACN were also measured using an RDE [31]. Hale [32] and Chernenko and coworkers [33,34] have investigated, for galvanostatic conditions, the effect of the chemical reaction kinetics on the establishment of the steady-state, while Albery [35] has considered the effect of the electric field in the diffusion part of the double layer and of migration on the measured values of the dissociation constants. 2.2. The equations for the new hydrodynamic
boundaries
Despite the fact that the HMRD behaves similar to the conventional RDE, the hydrodynamic boundaries are obviously different. A clear limitation to mass transport at the disk edge is imposed by the omission of the insulating mantle and the presence of a hanging meniscus of electrolyte which produces a region of flow direction reversal [7,18]. Therefore, the effective radius of the electrode will be the reduced value given by Eq. (1) and, in consequence, Eq. (2) will hold for the HMRD at all accessible meniscus heights above h,. Introduction of the effective electrode area A,rf instead of the geometric area of the disk A into Eq. (8) yields (ignoring second-order terms) Z,(HMRD)
=nFADc;(l
-2KR,‘\/v/o)/
Chemistry
418 (1996)
159-165
161
It is clear from comparison of Eq. (9) and Eq. (13) that the equation that describes the behavior of the HMRD differs from that of the RDE only in the correction term B’Z, which accounts for the reduction of effective radius observed at the HMRD because of the inversion of flow direction at the edge and is identical to the second term in Eq. (2).
3. Calculation
of kinetic parameters
3.1. Deviations causedby the Bow reversal region It is well known that a deviation of a plot of I, vs. w”* from a straight line that intersects the origin is an indication of some kinetic step involved in the electron transfer reaction taking place on an RDE [2-4,191. Similar behavior should be expected for the HMRD, despite the fact that the reversal of flow direction at the edge of the disk will cause a shift of the curves towards smaller currents for increasing values of h,, producing nonzero intercepts [5,7,18,21]. Curves calculated using Eq. (12) for several values of K, which is directly proportional to h, through Eq. (4), are depicted in Fig. 1. The curves correspondingto the RDE behavior with and without the kinetic step involved in the charge transfer reaction have also been included. While plots of Z,/W’/~ vs. I, are linear for the RDE, nonlinear plots should be expected for the HMRD because of the correction term B’Z in Eq. (13). This correction term, which is a function of h, and the particular properties of the system according to Eq. (3) and Eq. (4), is independent of w [7]. It is clear from Eq. (13) that a significant curvature should be exhibited by a plot of
RDE
(without
d,)
I
1.76
&,+a,> i
i 6)
J3q. (12) can be rearranged to give Z,(HMRD)
= B’w “* - B”Z,( HMRD)
0”
- B’Z
(13)
where Z=2KR,]v’/*
(14)
Fig. 1. Calculated I, vs. w’/’ curves for the RDE and HMRD configurations: n=l; c,‘=O.lM; R,=O.l5cm; D=1X10~5cm2~~‘; Y= O.Olcm’ SC’; k, /k2 = 5.7X 104; k, + k, = IX lO”cms-I. The values of K for the HMRD are indicated in the figure.
162
H.M.
Villullas.
of Electroanulytical
M. Lbpez Teijelo/Journal
Chemistry
418 (1996)
159-165
3.0 RDE (without
--------
d,) 2.5
,
RDE (with
\\
dJ
o.“oo
.li
,
a hm /cm
O.iS
0.20
G/ d Fig. 2. Calculated I, /w “* vs. I, curves for the RDE and HMRD configurations. The values of K for the HMRD are indicated in the figure. Other parameters as for Fig. I.
I&B”2
vs. I, for the HMRD, causedby the variation of the ratio B’Z/W’/~, and that the HMRD curve should approach the linear behavior at high values of I,. Additionally, curvature should be a function of the value of B’Z, i.e. it should become more pronounced for greater values of K (i.e. higher values of h,). Curves calculated using data taken from Fig. 1 for different values of K are presentedin Fig. 2. The deviation from linearity shown in Fig. 2 may appear as a limitation to the applicability of the HMRD to this type of study. However, it will be shown below that such a limitation may be overcome if measurementsof I, as a function of w are taken at different values of h,. For the HMRD set at any value of h, above h,, the dependenceof the limiting current on w is given by Eq. (12). Introducing K as given by Eq. (4) and rearranging yields (15)
Xcm, = 2 R, ‘( v/w)
“2
0.14
Fig. 3. Calculated values of I, for the HMRD plotted vs. h, at different rotation rates as indicated in the figure. Parameters as for Fig. 1.
At constant rotation rate, GcW,and XcW, will have fixed values and, consequently, the measuredvalue of I, will be determined only by the value of h, at which the HMRD is set. Thus, according to Eq. (151, a plot of I, vs. h, should be linear for any given value of w. I, values taken from calculated curves of Fig. 1 are plotted vs. h,, for different rotation rates in Fig. 3. Since Xcw, can be calculated using Es. (161, G(w, could be evaluated for each rotation rate from the slopesin Fig. 3, provided that the parameter k for the system is available. Unfortunately, this is not likely to be the case, since the value of k, as well as the critical meniscusheight h, (Eq. (4)), is related to the particular properties of the electrode material and electrolyte used [7,23]. Nevertheless, evaluation of both k and h,, should always be possible from the sameset of data. 3.2. Calculation
of parameters
k and h,
From Eqs. (15)~(171, it is clear that the intercepts and slopesof plots IL as a function of h, should be expected to change for the different values of o. This is, indeed,
(16)
and Gcw, = nFDAc,*/
It should be noted, from comparisonof Eq. (8) and Eq. (17), that Gcw, representsthe limiting current that would be measuredon an RDE of geometric radius R,. Thus, if it
were possible to evaluate Cc,,,, from data taken on the HMRD, this would be equivalent to correcting each measured value of I, for the HMRD, taking into account the reduction of effective electrode areaproduced by the reversal of flow direction at the edge.
Fig. 4. Quotient lintercept/slopel, w”‘. Parameters as for Fig. 1.
Q(,,
calculated
for curves of Fig. 3 vs.
H.M.
\\
\\
Villullus,
M. L&m
Teijelo /Journul
of Electronnalyticul
418 (1996)
159-165
\
\
0.1 1 1.5
.
RDE HMRD
\ \
2.b Cf,,
2.3 /
mA
Fig. 5. Calculated values of G~wj/w”’ vs. Gcw,. The calculated curve I, /co’/2 vs. I, for the RDE as shown in Fig. 2 is also included. Parameters as for Fig. 1.
observed in Fig. 3. The quotient lintercept/slopel, for each one of these curves (Eq. (15)) is
Q,,,,,,
Eq. (16) into Eq. (18) yields
Q,,, = w”~R,/(~~v”~)
+ h,
(19)
Then, if Q,,, is calculated for each value of w of Fig. 3 and plotted vs. w112 (Fig. 4), the value of k can be obtained from the slope. In addition, it is interesting to note that although k is the only parameter required to carry out the calculation of the rate constants for the previous chemical reaction, k, and k,, the value of the critical meniscus height h, is also obtained, since it is given by the intercept in Fig. 4. 3.3. Analysis of data for calculation of kinetic parameters As stated above, the quantity G,,, can be calculated for each value of rotation rate from the slopes of plots of I, vs. h, (Fig. 3) provided that the value of k has been determined for the system (Fig. 4). Consequently, since GcW,, according to Eq. (8) and Eq. (17), represents the limiting current that would be measured on an RDE of geometric radius R,, a plot of G(w)W2 vs. Y($ should be linear and coincident with the plot of IL/w vs. I, for the RDE, as shown in Fig. 5, yielding exactly the same values of B’ and B” from which the diffusion coefficient and the kinetic parameters can be obtained as discussed above [2,4]. Alternatively, B’ and the parameters for the previous chemical reaction may also be obtained from a plot of vs. l/o ‘I* . From Eq. (17) and the definitions of W(u) S,, and 6, [24,25]
It is quite evident that, according to Eq. (13), a plot of (I, + B’Z)/w “2 vs. I, for the HMRD, should be linear and lie on the same straight line obtained when plotting Z,/W”2 vs. I, for the RDE. The appropriate correction term B’Z that should be added to each measured value of I, may be obtained for the particular h, set for the experiments, by calculating Z according to Eq. (4) and Eq. (14). The parameters k and h, are required and can be evaluated from the slope and intercept of a plot of Q,,, vs. w’12 as in Fig. 4. A plot of (IL + B’Z)/w’/* vs. I,, for values taken from calculated curves of Fig. 1, is shown in Fig. 6, where data points for the RDE are included for comparison. It is worth noting that although the calculated points for the HMRD at different h, values lay on the same straight line as those calculated for the RDE, and consequently yield the same values of B’ and B”, as should be expected, data points for the HMRD are shifted with respect to the corresponding point for the RDE. As shown in Fig. 1, limiting currents at the HMRD are smaller than those corresponding to an RDE of the same geometric area caused by the inversion of flow direction at the disk edge [5,7,18,21]. 3.5. Limiting currents for unequal diffusion coeflcients In the treatment given for the HMRD in the preceding sections, we have kept, for the sake of simplicity, the assumption that D = D, = D, as in the original derivation by Koutecky and Levich for the conventional RDE [24,25]. The more general case of unequal diffusion coefficients has been solved by Dogonadze [36], who showed that this assumption does not have a significant effect on the method
c ‘; o.57 \ ?7 3 3
zJ
I:;L
1.0 IL
1 -= Gw
1 B'W'/2
(k, +
163
3.4. Limiting current values at the HMRD ---
‘R
Introducing
Chemistry
+k2)
-"'Wkd
nFAc,* D1/2
(20)
/
1.5 mui
2.0
2
Fig. 6. Calculated (I, + B’Z)/W’/~ vs. I, curves for the RDE ( n ) and HMRD configurations. The values of K for the HMRD are (0 ) 0.44, (0 ) 1.32; (0) 2.20; other parameters as for Fig. 1.
164
H.M.
Villullas.
M. L&r:
Teijelo/Journal
c~fElrctrounalyricai
of calculation. The limiting current for the RDE when D, # D, turns out to be [36]
(21) where D=
Chemistry
418
(1996)
159-165
obtained using the KDE. From the ments, the parameters k and h, system can also be obtained. The presented above is also valid for the
same set of measurethat characterize the method of calculation general case D, # D,
Acknowledgements k,D2
+ k2D,
(22)
k, +k* 8, = 1.61( D/v)“‘(
v/w)*‘*
1
8, = k,
d
o+o 1
(24) 2
References [I]
It is quite obvious that Eq. (8) and Eq. (21), as well as S,, and 6, [24,25] for both cases, have a similar mathematical form. Therefore, introducing Eq. (23) and Eq. (24) into Eq. (21) yields an expression for the RDE which is identical to Eq. (9), whereas, for the HMRD, introducing the effective electrode area Aeff instead of the geometric area of the disk A into Eq. (21) and combining with EZq. (23) and Eq. (24) yields a relation completely equivalent to E$. (13). The parameters B’ and B” for the D, # D, case are “(jc 0*
B”=0.620D-“3v-“6(k,D,/k,D,)(k,/D,
CONI-
(23)
k2
B’ = 0.620i~FAD*‘~v-
This research has been supported by CONICET, COR and SECYT (UNC).
121 [3] [4] [5]
[6]
[7]
(25) +k,/Dz)-“2
(26) Despite the fact that, in order to get the kinetic data for the chemical reaction, more lengthy computations will be required when the parameters B’ and B” are given by Eq. (25) and Eq. (26), it is quite clear that the case D, # D, does not differ, in principle, from the simpler case of equal diffusion coefficients [24,25,36]. Thus, the method presented above for calculation of B’ and B” for a chemical reaction in equilibrium that precedes the charge transfer reaction on the HMRD is also valid for the general case D,#D2.
[8] 191 [lo] [ll] [12] [13] [14] [15] [16] [17]
4. Conclusions The HMRD can be used to study the kinetics of a chemical reaction in equilibrium in the bulk electrolyte that precedes the charge transfer. Plots of IJo”~ vs. I, are not linear. Measurements of limiting current as a function of rotation rate should be carried out for several meniscus heights. The reduction of effective area promoted by the change in flow direction at the edge of the disk can be taken into account and, in consequence, corrected limiting currents corresponding to the whole geometric area of the disk can be calculated. The kinetic parameters should be independent of meniscus height and identical to those
[18] [ 191 /20] [21] [22] [23] [24] 1251 [26]
V.G. Levich, Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1962. A.C. Riddiford, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 4, Interscience, New York, 1966. R.N. Adams, Electrochemistry at Solid Electrodes, Marcel Dekker, New York, 1969. F. Opekar and P. Beran, J. Electroanal. Chem., 69 (1976) 1. B.D. Cahan and H.M. Villullas, ECS Meet., Chicago, 1988, Abstr. 707; B.D. Cahan and H.M. Villullas, J. Electroanal. Chem., 307 (1991) 263. B.D. Cahan, H.M. Villullas and E.B. Yeager, ECS Meet., Atlanta, 1988, Abstr. 532; B.D. Cahan, H.M. Villullas and E.B. Yeager, J. Electroanal. Chem., 306 (1991) 213. H.M. Villullas and M. L6pez Teijelo, 43rd ISE Meet., Cordoba, Argentina, 1992, Abstr. 6-014; H.M. Villullas and M. Lopez Teijelo, J. Electroanal. Chem., 384 (1995) 25. T. Abe, Y. Miki and K. Itaya, Bull. Chem. Sot. Jpn., 67 (1994) 2075. R.R. Adzic, J. Wang and B.M. Ocko. Electrochim. Acta, 40 (1995) 83. F. El Kadiri, R. Faure and R. Durand, J. Electroanal. Chem.. 301 (1991) 177. F.H. Feddrix, E.B. Yeager and B.D. Cahan, J. Electroanal. Chem., 330 (1992) 419. H. Kita, S. Ye and Y. Gao, J. Electroanal. Chem., 334 (1992) 351. N. Markovic, R.R. Adzic, B.D. Cahan and E.B. Yeager, J. Electroanal. Chem., 377 (1994) 249. H. Kita, H.W. Lei and Y. Gao, J. Electroanal. Chem.. 379 (1994) 407. T. Abe, G.M. Swain, K. Sashikata and K. itaya, J. Electroanal. Chem., 382 (1995) 73. M.M. Laz, R.M. Souto, S. Gonzalez, R.C. Salvarezza and A.J. Arvia, Electrochim. Acta, 37 (1992) 655. R.M. Souto, M. Perez Sanchez, M. Barrera, S. Gonzalez, R.C. SaIvarezza and A.J. Arvia, Electrochim. Acta, 37 (1992) 1437. B.D. Cahan, Proc. Symp. on Modelling of Batteries and Fuel Cells, ECS Meet., Phoenix, 1991. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. E. Gileadi, Electrode Kinetics, VCH, New York, 1993. H.M. Villullas and M. Lopez Teijelo, J. Electroanal. Chem.. 385 (1995) 39. A. Frumkin and G. Tedoradse, Z. Elektrochem., 62 (1958) 251. H.M. Villullas and M. L6pez Teijelo, J. Appl. Electrochem., 26 (1996) 353. J. Koutecky and V.G. Levich, Dokl. Akad. Nauk SSSR, 117 (1941) 441. J. Koutecky and V.G. Levich, Zh. Fiz. Khim., 32 (1958) 1565. W. Vielstich and D. Jahn, in I.S. Langmuir (Ed.), Advances in Polarography, Vol. 1, Pergamon. New York, 1960.
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of Elrctroundyticd
[27] W. Vielstich and D. Jahn, Z. Elektrochem., 64 (1960) 43. [281 M. von Stackelberg, W. Vielstich and D. Jahn, An. R. Sot. Esp. Fis. Quim. Ser. B:, 56 (1960) 475. [29] W. Vielstich, Z. Anal. Chem., 173 (1960) 84. [30] W.J. Albery and R.P. Bell, Proc. Chem. Sot., (1963) 169. [31] V.A. Kokorekina, L.C. Feoktistov, V.Y. Filinovskii and S.A. Sevelev, Elektrokhimiya, 6 (1971) 1196.
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[32] J.M. Hale, J. Electroanal. Chem., 8 (1964) 332. [33] V.I. Chemenko, Elektrokhimiya, 7 (1971) 820. [34] V.I. Chemenko and K.I. Litovchenco, Elektrokhimiya, 1410. [35] W.J. Albery, Trans. Faraday Sot., 61 (1965) 2063. [36] R.R. Dogonadze. Zh. Fiz. Khim., 32 (1958) 2437.
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7 (1971)