PARALLEL AND HETEROGENEOUS CATALYTIC ELECTRODE REACTIONS
4111
On the Distinction between the Kinetics of Parallel and Heterogeneous Catalytic Electrode Reactions by J. D. E. McIntyre Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey
(Received March $0,1969)
The kinetic behavior of a heterogeneous catalytic electrode reaction can be described in terms of a ‘Lpseudoparallel” reaction scheme which is analogous in form to that for simple parallel reactions. Such kinetic schemes are currently employed in studies of the complex mechanism of the oxygen electrode reaction. Criteria are developed for distinguishing between these two types of reaction mechanism with particular reference to use of a rotating ring-disk electrode system. In a recent communication,’ Damjanovic, Genshaw, and Bockris have discussed the use of the rotating disk electrode (RDE) with a concentric ring electrode for distinguishing between intermediates and products evolved in parallel electrochemical reactions. It is the purpose of this article to show that the diagnostic plots for a heterogeneous catalytic electrode reaction (HCER) have features in common with those for simple parallel reactions and to provide criteria for establishing the actual reaction mechanism.
Theory I. Parallel Electrochemical Reactions. An electrode process consisting of parallel electrochemical reactions which yield the same final product can be represented schematically
o + nle ki R
(1%)
R+Qe-%S
(Ib)
o + n3e ks_ s
(IC)
Since the flux of R (moles per square centimeter per second) a t the disk surface is
JR(0) =
iZ
the absolute value of the ring current is given by
I,
=I
nlFNAJR(0)
(2)
where F is the Faraday constant and A is the geometric surface area of the disk. Hence
I,
=
N[Il
-
(n1/nz)IzJ
(3)
where I1 and IZ are the partial disk currents corresponding to reaction steps I a and b. The total disk current is Id
11
f IZ
+ 13
(4)
From eq 3 and 4
+ (w‘ndII1 - ( n h > ( I r / N ) +
(5) We now define the quantity, x, to represent the ratio of the rates of consumption of 0 via reactions I a and c Id
where kl, kz, k3, and nl, n2,n3 represent the formal heterogeneous electrochemical rate constants and the number of electrons transferred in the individual steps, respectively. The partial current densities corresponding to the rates of these steps on the disk electrode will be denoted as $1, iz, and is and are assumed to be uniform over the entire disk surface (cf. ref 3). Cathodic currents and fluxes directed away from the electrode surface (x = 0) are taken as positive. The ring-electrode current corresponds to the diffusion-limited rate of reoxidation to 0 of that fraction of the intermediate species, R, which escapes from the disk and is transported to the ring by convective diffusion. The collection efficiency, N, of the ring electrode is a function of the geometry of the ring and disk but is independent of the angular velocity of the assembly.s-s A general relation for the ratio of the disk current to the ring current for reaction I is obtained as follows.
-nlF nzF il
-
= [1
13
(1) A. Damjanovic, M. A. Genshaw, and J. O’M. Bockris, J . Chem. Phys., 45, 4057 (1966). (2) For consistency with previoud treatmentsat4 of the kinetics of the HCER, i t has been necessary to alter the numbering of the steps in the reaction scheme used by Damjanovic, et al.1 The parameters na and ka in their communication are identical with nz and kz in reaction scheme I above. (3) J. D. E. McIntyre, J . Phvs. Chem., 71, 1196 (1967). (4) J. D. E. McIntyre, ibid., 73,4102 (1969). (5) Yu. E. Ivanov and V. G. Levich, Dokl. Akad. Nauk SSSR, 126, 1029 (1959). (6) V. G. Levich, “Physicochemical Hydrodynamics,” PrentioeHall, Inc., Englewood Cliffs, N. J., 1962, p 329. (7) A. C. Riddiford, “Advances in Electrochemistry and Electrochemical Engineering,” Vol. 4, P. Delahay, Ed., Interscience P u b lishers, New York, N. Y., 1966, p 108. (8) W. J. Albery and S. Bruckenstein, Trans. Faraday SOC.,62, 120 (1966).
Volume 78,Number l d
December 1969
41 12
J. D. E. M C I N T Y R ~
From eq 5 and 6
(7) The steady-state flux of a species i at an R D E is
J%(O)= L%U’/”C,(O)
pR1S 0 I
- C,O]
(8) where C,(O) and Cto are the surface and bulk solution concentrations of this species. The Levich constant, L,, is given by9J0
Li
=
0.62048D,”/”~-’/~
1
+ 0.298O(Di/~)~//” + 0.14514(D,/v)”/”
(i = 0, R , , . .) (9) where D,is the diffusion coefficient of species i in the supporting electrolyte with kinematic viscosity, V. The product, L,ol/’, is equal to the ratio of D,to the diffusion-boundary layer thickness, 6,. The partial current density, iz,is given by
iz
= nzFk&(O)
+ qS
Reaction IIc represents a surface-catalyzed chemical regenerative reaction of order p (with respect to R1) and with a heterogeneous chemical rate constant, kh, which is independent of potential. The kinetic behavior of the HCER has been analyzed in detail in recent communications. 3,4 It is particularly relevant to studies of the kinetics and mechanism of oxygen reduction on solid metal electrodes which act as catalysts for the heterogeneous chemical decomposition of hydrogen peroxide (cf. references cited in ref 3 and 4). On defining the steady-state regeneration rate fraction of reaction I1 as
(10)
With CR’ = 0, we have from eq 1,2,8,and 10 11
=
[1
+
(~Z/LRW’/’) 1 ( 1 r / ~ )
(11)
Hence from eq 7 and 11, the ratio of the disk current to the ring current is
Ir 5 = N‘[(23).nl
.)”([I‘
+ I] + N
as
+
nl
3 +)“(]1 nl
L ~ w ~ (12) / ~
Equation 12 is a more general form of the diagnostic relation originally derived by Damjanovic, Genshaw, and B0ckris.l These authors have analyzed reaction scheme I and deduced the form of plots of Id/Ir vs. for several possible combinations of the rate constants kl, kz, and k3. For this scheme, the rates of the parallel electrochemical reactions are coupled through the variation of Co(O), which is a function of both potential and mass transport rate. However, the quantity, x, which expresses their relative velocity, is independent of mass transport; x is potential dependent, if the Tafel parameters, bo1 and bo3, of reaction steps I a and c are different. This analysis has been to provide diagnostic criteria for the elucidation of the mechanism of the electrochemical reduction of oxygen on noble metal electrodes and to establish the role of the intermediate electrolysis product, hydrogen peroxide. To determine the actual mechanism for this complex electrode process, it is particularly important to distinguish between the results of the diagnostic analysis for parallel electrochemical reactions1 and those for a heterogeneous catalytic electrode reaction. I I . Heterogeneous Catalytic Electrode Reactions. The HCER is represented schematically as T h e Journal of Physical Chemistry
it was shown4that reactions IIa and c can be combined
where nl*, the equivalent number of Faradays of charge required to reduce each mole of 0 supplied to the electrode by either mass transport or regeneration, is given by nl*
=
( L ) n , P--8
On rearranging, reaction I11 can be resolved into two “pseudo-parallel” reactions corresponding to the limiting cases obtained when kh = 0,-8 = 0, and kh = m , 0 = 1, respectively. The kinetic behavior of the HCER can then be expressed in terms of a “parallel” reaction scheme which is equivalent in over-all stoichiometry and kinetics to reaction I1 but which is identical in form with the simple parallel reaction scheme 114
0
+ nle
--t
R1
ova)
(9) Reference 6,p 69. (10) J. Nemman, J. Phys. Chem., 70, 1327 (1966). (11) A. Damjanovic, M. A. Genshaw, and J. O’M. Bockris, ibid., 70, 3761 (1966);ibid., 71, 3722 (1967). (12) A. Damjanovic, M. A. Genshaw, and J. O’M. Bockris, J. Electrochem. SOC.,114, 466, 1107 (1967). (13) M. A. Genhsaw, A. Damjanovic, and J. O’M. Boclrris, J. Electroanal. Chem., 15, 163,173 (1967). (14) The species Rz and S may be chemically identical. Here they are denoted as separate entities for mathematical convenience.
41 13
PARALLEL AND HETEROGENEOUS CATALYTIC ELECTRODE REACTIONS The total rate of consumption of 0 is thus
(27)
It must be emphasized that the ratio of the velocities of reactions IVa and c is not simply dependent on the potential, as is the case for reactions I a and c, but rather is a complex function of potential, mass transport rate, and chemical kinetics. The rates of consumption of 0 via the two parallel paths are equal to the velocities of the reactions IVaandc. Now VIVa
+
= J~l(0)
JR@)
(15)
and
Examination of eq 24 and 26 reveals that the rates of consumption of 0 via the equivalent parallel reactions IVa and c are not simply equal to i,l,hlF and [ ( p - l)/’p]ioat/n1F, respectively. When k h > 0,the rate of reaction IVa and the absolute flux of 0 to the surface are less than their normal value, inl/nlF. The total current density is greater than normal, however, since more charge is consumed in reducing 0 by reaction IVc than by reaction IVa. To find the quantity, x, it is most convenient to express Co(0) as4
The quantity, x, for the HCER is therefore given by
x = - VIVO
(17)
VIVa
(29)
Now for reaction I1 J S ( 0 ) = pkh[CRi(O)
1”
(18)
Hence from eq 16 and 18 VIVc
= (p
- l)kh[CRi(O)]”
(19)
For the HCER, the surface concentration of 0 is4
From eq 17, the ratio of the rates of consumption of
0 is then given by the simple relation The normal component of il (the value of il when 0) is
When
kh
> 0, we define ioat =
i1
- inl
From eq 20,21, and 22
Therefore
The current density,
il,
i~ = n i F v I v a From eq 22 and 24
is
+ [p/(p - l ) l n ~ F v ~ v ~
kh
=
Since the equivalent parallel reaction scheme IV for the HCER is of the same form as that for the “simple parallel” reactions I, the form of plots of I d / I r us. w-’/’ for the HCER can be determined from the diagnostic expression, eq 12 and the value of x given by eq 32. Figure 1 illustrates’s the form of the steady-state current-voltage (i-E) curve of the HCER depicted in reaction scheme I1 for the case p = 2, disk angular velocities of 10, 200, and 1000 radians sec-’ and the parametric values: nl = n2 = 2, k,l = cm sec-1, k,2 = cm sec-’, b , ~= b , ~= 0.118315 V, EcloEczO= - l.OV, p = 2, k h = lo4om4mol-’ sec-l, Coo = mol cm-*, DO = DRI = D R = ~ cm2sec-I, v = cm2 sec-l, and N = 0.4, where k , ~ k, , ~and Eclo, Eczoare the formal standard rate constants and potentials of reactions IIa and IIb, respectively. The solid curves represent the behavior of the HCER with kh = lo4 om4 mol-’ sec-l, the dashed curves show the current corresponding to the sum of the reaction velocities, VIVa and VIVb, while the dotted curves illustrate the (16) All illustrations were generated automatically with a General Electric 646 computer.
Volume 73, Number 18 December 1960
J. D. E. MCINTYRE
4114 8.0
-
,
5.0
-
4.0
-
I
first current wave of Figure 1. The solid curves correspond to the HCER, the single dotted line tB the normal case. In this potential region, kz = 0, and since there is no catalytic current for the normal case, 2 = 0 as well. The normal ratio I d / I r is independent of both potential and mass transport and has a constant value equal to 1/N. The variation of I d / ' I r for the HCER is revealed
N I
E
.--E
22.0 200
3.0
-
2.0
-
20.0
18.0
1.0
................."...............**
16.0
-
14.0
IO L
H
0.0. 0.4
\
I
-0.4
0.0
-1.2
-0.8
0
-1.6
-2.0
12.0
CI
10.0
8.0
Figure 1. Steady-state current-voltage curves for heterogeneous catalytic ( p = 2, k h = 104 cm4 mol-' sec-l) and normal ( k h = 0) electrode reactions on an RDE for disk angular velocities w = 10, 200, and 1000 radians sec-1: HCER: , i.) ----, 72lFVIVa $- nZFvIVb; normal : . . . . . . . i.
6.0
-
4.0 2.0 I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 (
20.0
1
A
1
I
I
0.8 0.9
1.0
I / w " ~ (rod sec-l)-l/'
Figure 3. Variation of Id/Ir with u - ' I 2 for an HCER ( p = 2, k h = lo4 cm4 mol-' sec-1) and a normal electrode reaction at potentials in the lcd region of the first wave: -, HCER; normal case.
.......
-0.15,
A
60.0
10.0
c
/A 50.0
40.0
L
0.0 0.0
'0 C
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l/wi/2 (rod s e c - 1 ) - ' / 2
Figure 2. Variation of I d / I r with w - ' / ~ for an HCER ( p = 2, kh = l o 4 om4mol-' sec-l) and a normal electrode reaction at potentials, E - Eelo(volts) in the first current wave: -, HCER; . . . . . ., normal case.
normal form when k h = 0. The values of the rate constants are such that the i-E curves exhibit two welldefined limiting curreht regions. The detailed characteristics of the curves have been discussed previously.' I n Figure 2, the ratio Id/Ir is plotted as a function of for a series of potentials on the rising section of the The Journal of Physical Chemistry
30.0
20.0
........
10.0
0.0
0.1
..............a.............
I
I
I
I
0.2
0.3
0.4
0.5
I/ W I / ~
I
I
0.6 0.7 (rad s e d ) - ' / *
I
I
0.8
0.9
Figure 4. Variation of Id/Ir with w-l/2 for an HCER ( p = 2, kh = lo4 om4 mol-' sec-1) and a normal electrode reaction a t potentials, E
-,HCER;
- Eo*' (volts), in the second current wave:
. . . . . . . normal case.
1.0
PARALLEL AND HETEROGENEOUS CATALYTIC ELECTRODE REACTIONS by examination of eq 12 and 32. As the disk potential is scanned cathodically through the first wave with w held constant, the value of nl* increases monotonically from nl, a t the foot of the wave, to its maximum value nlCl* a t the limiting current density (lcd) plateau. Simultaneously l d / I r increases from the value 1/N, when x = 0, to the value (l/N) [1/(2 - nlc1*/nl)l. A constant potential plot of I d / I r vs. w-l/’has an inter* n~ cept equal to 1/N on the w-’/’ = 0 axis since n ~ + a n d $ + O a s w + QJ.A s w + O , ~ + l , a n d J R l ( 0 ) + 0; hence, I r -+ 0 and I d / I r + . I n the lcd region where x is independent of potential, the constant potential curves coalesce to form a single curve with I d / I r = (1/N) [1/(2 - nl,l*/n1)]. Figure 3 shows the variation of I d / I r with disk angular velocity in this region. Finally, Figure 4 illustrates the form of plots of I d / I r vs. w-l/’ for potentials in the rising part of the second current wave. I n this region, kz > 0 and R1 is partially consumed by the consecutive electrochemical reaction IIb. The slopes of the plots for both the normal and catalytic electrode reactions increase as the potential is made more negative. At fixed potential, the ratio I d / I r is greater for the HCER since R1 is consumed by both the electrochemical reduction process I I b and the chemical regenerative reaction IIc. The curvature of the solid plots for the HCER reflects the variation in nl* with w. For potentials near the lcd region of the second wave, nl* --t nl and x --+ 0; the solid curves for the HCER tend to merge with the dotted lines for the normal case. I n the lcd region, R1 is totally consumed on the disk and I r = 0. QJ
Discussion Although the rate constant, k h , of the reaction IIc, by which S is formed is independent of potential, the rate of consumption of R1 by this reaction varies with potential in the rising sections of both the first and second current waves since the surface concentration, C,,(O), is potential dependent. As a result, the plots of I d / I r vs. w-”’ for the HCER exhibit features characteristic of several of the “diagnostic” forms deduced’ for the simple parallel reaction scheme (cf. Figure 1, ref 1). For the rising section of the first wave of the HCER,
4115
x > 0 but
k2 = 0. I n the medium-to-high rotational speed range (w-l/‘ = 0.03-0.2 sec-’/’), the constant potential curves for the HCER closely resemble the family of lines for reaction I with kz > 0 and x = 0. At low rotational speeds, the plots for the HCER are similar to those for reaction I when kl > 0, kz > 0, k3 > 0, and x > 0. I n the limit a t very high rotational speeds, the HCER plots become tangential to the line, I d / I r = 1/N, corresponding to the behavior of reaction I when kl > 0, k2 = kl = 0, and x = 0. For the rising section of the second current wave of the HCER, kl > 0, k2 > 0, and x > 0. At low rotational speeds, the constant potential plots of I d / I r vs. w-”’ for the HCER resemble those for reaction I when kl > 0, k2 > 0, kl > 0, and x > 0; the latter have intercepts greater than 1/N on the w - l j ’ = 0 axis. At high rotational speeds, x + 0 as w --+ ; the constant-potential plots for the HCER then resemble those for reaction I when kl > 0, ICz > 0, k ) = 0, and x = 0. I n the restricted range of rotational speeds commonly employed in kinetic studies with the ring-disk electrode system, the plotsof I d / I r V S . w-”’ for an HCER bear a close resemblance to those for simple parallel electrochemical reactions. To gain a detailed understanding of the actual reaction mechanism and the catalytic role of the electrode substrate, it is essential to distinguish between these two kinetic schemes. The characteristic curvature of HCER plots can only be revealed properly by obtaining data over a wide range of disk angular velocities, sufficient to alter the diffusion layer thickness, 60, and the value of the ratio, kh/L0W1/’, by a t least afactor of 10. A rotational speed range of 100-10,000 rpm or more is required. HCER plots exhibit the greatest curvature for potentials in the lcd region of the first wave. A simpler procedure, which does not require the use of a ring-disk assembly, is to examine the form of the Levich plots of ilcl vs. w’/’ for curvature and/or nonzero intercepts on the wl/’ = 0 axis (cf. ref 3). A comparison of the kinetics of the HCER and those of an ECE mechanism with a rate-controlling chemical step which is noncatalytic will be presented in a separate article. Application of these kinetic analyses to the interpretation of the kinetics and mechanism of the oxygen electrode reaction will be discussed in forthcoming communications. QJ
Volume 73, Number 12 December 1969
