J. D. E. MCINTYRE
4102 plex for reaction 2, and used activated complex theory22 to calculate the A factors for the forward (2) and reverse (-2) reactions for comparison with experiment. n-Butane was used as a model for n-butyl radicaL23 I n the activated complex the four bending and rocking frequencies were lowered to '/lS4 of their value in butane. This is the effect of lowering the force constants for these vibrations by a factor of 2. Two of the three torsions were adjusted. The torsion about the partial double bond was increased by 1.4 of the difference between its frequency in n-butane (102 cm-') and its frequency about a double bond (578 cm-l). The torsion about the partial single bond was lowered by l/3. The frequencies and entropy of activation are shown in Table V.
This simplified description of frequency changes can nevertheless provide realistic calculated values of the A factors. The calculated value A S * = 1.70 eu leads to an A factor for reaction -2 of 1013.60sec-l a t 298"K, in good agreement with the experimental value sec-'. Using AS2O = -33.2 eu, this leads to an A factor for the association reaction 2 of about 1011*17 cc mol-1 sec-1, which is fortuitously exactly equal to the experimental value. Acknowledgments. The authors gratefully acknowledge the financial support of the Faculty Improvement Committee of Colorado State University, and the Research Corporation, Burlingame, California. We thank Mr. D. R. Lawson, who prepared the azoisopropane.
The Kinetics of Heterogeneous Catalytic Electrode Reactions. 11. Charge-Transfer Kinetics by J. D. E. McIntyre Bell Telephone Laboratories, Incorporated, Murray Hill, New JeTSey
(Received March $0,1969)
The kinetic behavior of an electrochemical reaction coupled with a heterogeneous chemical regenerative reaction is analyzed. The steady-state current-voltage curve for this electrode process contains a catalytic current component over the complete potential range in which reduction of the electroactive species occurs. Methods of determining the kinetic parameters of the charge-transfer reaction, per se, are presented with particular reference to use of the rotating disk electrode. This analysis provides a basis for the elucidation of the kinetics and mechanism of an oxygen electrode reaction where hydrogen peroxide is formed as an intermediate reduction product which undergoes a surface-catalyzed decomposition.
The kinetic behavior of an electrochemical reaction which is coupled with a surface-catalyzed chemical regenerative reaction was analyzed in a preceding communication' with particular reference to the kinetics of the regenerative process. The course of this heterogeneous catalytic electrode reaction (HCER) can be represented schematically as
sumed in the regenerative cycle after each successive occurrence of the electrolysis reaction, it was shown that the over-all electrode process can be represented in the simple form
IC1
O + n e F R k- 1
-+
PR ?-To
qs
(1%) Ob)
On defining the regeneration rate fraction2 to repre-
sent that fraction of the electrolysis product, R, conThe Journal of Physical Chemistry
The equivalent number of Faradays of charge, n*, required to reduce each mole of 0 supplied to the electrode by either mass transport or regeneration is thus given by eq 2. (1) J. D. E. McIntyre, J, Phys. Chem., 71, 1196 (1967). (2) The symbols and sign conventions used in reaction scheme I and eq 1 are defined in ref 1.
HETEROGENEOUS CATALYTIC ELECTRODE R~ACTIONS
n* =
(+-) n
I < - n* - 1, however, 8 and n* are functions of potential in the region of mixed mass transport, chemical kinetic, and potential control. I n this region, n* varies continuously with potential, increasing monotonically from the value n at the foot of the current wave to the value n*1, a t the Icd plateau, while the catalytic component of the current density increases from zero to its maximum value ( i l c - id). For an irreversible charge-transfer reaction
i - = klCo(0) nF
=
k, = kl0 exp( -
(17)
= IC-1"
where E," is the formal standard potential. For an irreversible electrode reaction, the current density is
i
=
nFh, exp[-2.3(E
- E,")/~,]CO(O) (18)
From eq 9 and 18 we obtain two equivalent forms of the steady-state i-E curve for an irreversible HCER a t an
RDE
E = E,"
+ b,log
E
+ b, log(-) Loo1/2 + b,log(j
n*
=
E,"
-
):
(19b)
These relations may be compared to that for the normal case (kh = 0)
a. First-Order Catalytic Reaction. When p
For the case p = 1
= 1,
n* = n*1,for all E and eq 19 can be written as
b, log
Whenp = 2
+ (kl/~ow'/2)] b = LRW'/'[~+ (kl/Lo~'/2)] a
3,") (3,")
k ~ "exp
where the symbols klo,b,, E , and F have their usual significance. General analytic expressions for the surface concentrations of 0 and R can be obtained, for the case of the RDE, by solution of eq 6, 10, and 12. We find
where
I . Steady-State Current-Voltage Curves. The formal standard electrochemical rate constant, k,, of the electrolysis reaction I a is defined as
= kh[2
c = -klCoO
(154
+
+
t+
- 1)
(21)
For this case, a plot of log (ilc/i- 1) vs. E is linear and has a slope equal to the reciprocal of the Tafel parameter, b,. The half-wave potential, El/%, where i = ilC/2, is given by
(154
(16)
It is apparent from eq 16 that the ratio i/& is only equal to the ratio n*/n in the Icd region, where LOW^/^ >> 1. The Journal of Phvical Chemistry
b, log
(15b)
From eq 5 and 9 it is apparent that at potentials in the rising part of the current wave, the primary effect of the regenerative process is to enhance Co(0) and, hence, the current, in comparison with the normal case (kn = 0). The forms of the relations involving n* are particularly useful for conceptual purposes. For generation of steady-state i-E curves with a computer, eq 13, 14, and 15 are employed. From eq 8, 9, and 12, the ratio of the current density for the catalytic case to that for the normal case, a t a fixed potential, is
_i -- 1 (k1/L0w1/3 i, 1 (n/n*)(kl/Loo1'2>
p)+
Since n*1, > n, the effect of the coupled catalytic reaction is to shift El/, cathodically from its normal value when k h = 0. This shift reflects the extension of the range of potential control by the action of the chemical regenerative process. b. Higher-Order Catalytic Reactions. When p > 1, n* is a complicated function of both potential and mass transport rate for intermediate values of w and cannot be simply evaluated from the steady-state currentvoltage curves. The effects of the variation of n* with potential can be revealed by recasting eq 19 in the form
4105
HETEROGENEOUS CATALYTIC ELECTRODE REACTIONS 1000
6.0 100 5.0
-
IO
4.0
YI
N
s
.--
I
3.0 y2
0.I 2.0
.01
6
E-E:
E-E:
(volt
(Vait
Figure 1. Steady-state current-voltage curves for heterogeneous catalytic ( p = 2, k h = 104 cm4mol-' sec-l) and normal ( k h = 0) electrode reactions on an RDE for the disk angular velocities w = 10, 100, and 1000 radians sec-l: -, HCER; . . . .I . ., normal case; 0, half-wave potentials.
The last term on the right side varies only slightly with potential in the range of interest, since the shape of the i-E curve for the HCER closely resembles that for the normal case (kh = 0). To a good approximation, a plot of log (ilC/i- 1) vs. E simply represents a lateral displacement (to more negative potentials) of a graph of log [ (nil,/n*I,i>- (n/n*) ] vs. E, coupled with a slight decrease in slope (ca. 2%). The Tafel parameter b, can thus be determined with reasonable accuracy from the reciprocal slope of a plot of log ( i l , / i - 1) os. E. An alternative method of determining b, is discussed in a following section. Figure 1 illustrates9 the form of the steady-state i-E curves a t disk angular velocities of 10, 100, and 1000 radians sec-' for the case p = 2 and the parametric values: n = 2, DO = DR = cm2 sec-', v = low2 cm2sec-', Coo = mol cm-a, IC, = loe3 cm sec-', CY = 0.5, b, = 0.118315 V, and T = 25.0". The dotted curves indicate the normal form when kh = 0; the solid curves represent the behavior of a catalytic electrode reaction with kh = lo4 cm4 mol-' sec-'. Half-wave potentials are indicated by open circles. The extension of the range of potential control and the enhancement of the current due to increasing mass transport rate and the action of the catalytic reaction are apparent. As the electrode is polarized cathodically, the i-E curve for the HCER appears to grow out of the curve for the normal case, reflecting the increase in n* with increasing CR(O). The catalytic component of the current density attains its maximum value a t the Zcd plateau. From Figure 1 it is apparent that the relative enhancement of the current is greater a t low rotational speeds,
Figure 2. Exact and approximate plots for determination of the Tafel parameter, bo, of an HCER with p = 2, k h = IO4 em4 mol-' sec-1, and w = 10, 100, and 1000 radians sec-1: -, exact: Y1 = i d / i - n / n * ; . . * * * * . , approximate: Yz = il& - 1.
in accord with the variation of 0 and nl,* with w. The kinetic behavior of this lcd region has already been discussed in detail (cf. ref 1). I n Figure 2, graphs of Y1 = (&/i - n/n*) and Yz = ( i l o / i - 1) are plotted on a logarithmic scale as functions of E - E," for the HCER discussed above. From this figure, it is apparent that "approximate" plots of log Y2 vs. E (dotted lines), which can readily be constructed from experimental data, are very nearly linear and almost parallel to the theoretical "exact" plots of log Y1 vs. E (solid lines). It should be noted that the exact plots are independent of kh and CO"(cf. eq 19). II. Evaluation of Kinetic Parameters. The method of evaluating the heterogeneous rate constant, kh, of the chemical regenerative reaction was discussed in a preceding communication. To determine kinetic parameters characteristic of the charge-transfer reaction Ia, per se, the effects of electrode polarization due to the rate of mass transfer and depolarization due to the regenerative reaction must be taken into account. The steady-state RDE method is particularly well suited to this application. By measuring the current density as a function of rotational speed a t a series of potentials in the rising part of the steady-state i-E curve and extrapolating the results to infinite rotational speed, the effects of mass transfer and chemical kinetic control on the rate of the electrolysis reaction can be completely removed. The definition of n* in eq 2 enables a kinetic treatment to be developed which is similar to those em(9) All illustrations in this article were automatically generated with a General Electric 645 computer.
Volume 75,Number 18 December 1969
J. D. E. MCINTYRE
4106 ployed for RDE studies of other less-complicated types of electrode processes.'O-'~ For an irreversible discharge reaction, we obtain, from eq 9 and 12, the relation 1 -- - 1 i nFklCoo
1 n*FLoCoo
($)
10.0
(24)
At constant electrode potential, IC1 is fixed and eq 24 represents a curve with a local slope of l/n*FLoCO0 and intercept, on the l/wl/' = 0 axis, of
1
i,
-
12.0
-
N
1 nFICICOo
which corresponds to an infinite mass-transfer rate constant. Since lime = 1 0.0 I 0.0
-0
I
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(rad s e c - ' ) - " * Figure 3. Variation of l/i with 1/u1/'for an H C E R ( p = 2, k h = l o 4 cm4 mol-1 sec-l) and a normal electrode reaction a t the potentials, E E,' (volts), indicated: HCER; , normal case. I/W''~
and n* =
(ee)
-
n
it is evident for the case p = 2, the slope of the curve doubles between the limits w = 0 and w = O D . The limiting slope for all p at the l/a"* = 0 axis is l/nFLoCoo. The value of the intercept, l/i,, is completely independent of parameters pertaining to mass transport and the kinetics of the regenerative process; the variation of i, with E is solely due to charge-transfer polarization. From eq 17 and 25 i, = nFlc,Co" exp[-2.3(E
- E,")/b,]
(27)
Hence log i, = log nFk,Co"
-
(E
- E,")/b,
(28)
A plot of log i, us. E is analogous to a conventional Tafel plot and has a slope equal to the reciprocal of the Tafel parameter, b,. By extrapolating this plot to the standard potential, E,", the value of the standard electrochemical rate constant, IC,, of reaction l a can be determined. The standard exchange current density is defined by
i,"
= nFlc,
=
nFk, exp[-2.3(Er
- E,")/b,]Co"
(30)
and can be evaluated by extrapolation of the plot of log i, us. E to the hypothetical reversible potential, Er.14 Figure 3 illustrates the form of the l/i us. l/wl/' plots for an HCER with p = 2 and k h = lo4cm4mol-' sec-I. The dotted lines indicate the kinetic behavior for the normal case ( k h = 0); the solid curves refer to the catalytic The Journal of Physical C h m h t r y
reaction. It is important to note the changes in curvature which occur as the electrode is polarized more cathodically. For potentials near the foot of the current wave (upper curves) the curvature in the plot of l/i is gradual as the l/al/z= 0 axis is approached. At more negative potentials (lower curves), the curvature becomes increasingly sharper. In the limit, the solid curves become tangential to the dotted lines since 9 -+ 0 as w -*. O D . To avoid serious errors in the extrapolation of experimental plots, it is necessary to obtain data a t high disk rotational speeds (10,000 rpm or greater). Incorrect extrapolation will yield Tafel slopes, b,, which are too high and standard rate constants, k,, which are too low. III. Consecutive Electrochemical Reactions. The kinetic behavior of two consecutive electrochemical reactions which are coupled with a heterogeneous chemical regenerative reaction is now considered. The reaction scheme is (111s) (IIIb)
(29)
The apparent exchange current density under the actual experimental conditions is defined by
io
--,
I , . , , .
(IIIC) L
-1
(10) A. N.Frumkin and G. Tedoradse, Z.Elektrochem.,62,251 (1958). (11) D.Jahn and W. Vielstich, J. Electrochem. SOC., 109, 849 (1962). (12) Z.Galus and R. N. Adams, J. Phys. Chem., 67, 866 (1963). (13) A. C. Riddiford, "Advances in Electrochemistry and Electrochemical Engineering," Vol. 4, P. Delahay, Ed., Interscience Publishers, New York, N. Y.,1906,p 96. (14) I n experimental studies of the HCER, i t is usually expedient to make CRO = 0 to avoid a continuous decomposition of R via reaction Ib. The hypothetical values of Er and io correspond to the case CRO = Coo.
HETEROGENEOUS CATALYTIC ELECTRODE REACTIONS
4107
All reactions are assumed to be irreversible. The components of the total current density which correspond to the raters of the individual charge-transfer steps I I I a and I I I b are denoted by il and i z . The flux boundary condition for reaction I11 is JO(0)
+ JRi(0) +
JRs(0)
(p
- 1)hh[CRi(O)IP= 0
(p
and the regeneration rate fraction now becomes
Le.
It follows that (33) We define, as before, the quantity n*l to represent the number of Faradays of charge which are required to reduce (via steps IIIa and IIIc) each mole of 0 supplied to the electrode surface by either mass transport or regeneration. Thus
nl* =
- l)]$ -
(p
i.e.
where i d l is the normal diffusion-limited current density of reaction I I I a when kh = 0. When the bulk solution concentration of R1is zero, the surface concentration of R1 is
(36) General analytic expressions for the surface concentration of 0 and R1can be obtained by a procedure similar to that used for reaction I, for generating steady-state i-E curves with a computer. The total current density of reaction I11 is
+ iz
(37)
The steady-state i-E curve for reaction 111a t an RDE is then given by
2}
(39)
- 11
(40)
- l)]; -
Whenp = 2
[-(-
bCzlog n2 2n1 - 1); nl n*l
il
(38)
wherek,l, ICs2, Eelo,and ECz0 are the formal standard rate constants and potentials of reactions I I I a and IIIb, and bcl and bcz are the corresponding cathodic Tafel slopes. When the values of kl, kz, and k h are such that the steady-state i-E curve exhibits two well-defined limiting current regions, the kinetic parameters for the successive steps can be found. For potentials more positive than the limiting current region of the first wave, izis negligibly small and the second term on the right side of eq 36 can be neglected. The method of determining the kinetic parameters for the first step is then identical with that discussed previously for reaction I. For potentials in the second wave, the variation of i 2 with potential is given by
(34)
Qn denoting i l O las the lcd for reaction I I I a and k*i,,as the corresponding value of n*l, the surface concentration of 0 is given by
i=
$}
(31)
Figure 4 illustrates the form of the steady-state i-E curves for consecutive electrochemical reactions a t disk rotational speeds of 10,100, and 1000 radians sec-l, with nl = nz = 2, ks2 = loF3cm sec-l, kBz= cm = 1.0 V.16 sec-1, bCl = bcz = 0.118315 V, and Eclo- ECZ0 Other parameters have the same values as for Figure 1. The solid curves represent the kinetic behavior of an HCER with p = 2 and k h = lo4 cm4 mol-’ sec-l; the dotted lines illustrate the normal behavior when k h = 0. Dashed curves show the variation of the component, il, with potential. Two well-defined current waves are evident in Figure 4. The kinetic behavior of the first wave is identical with that discussed for reaction I. At the foot of the second wave, il = i l o l and n*1 = n*1,,. At more negative potentials, however, CR~(O)falls due to the increased rate of consumption of R1 via step IIIb. As a (15) Note particularly in this example that EczO is more positive than Eol0,as is the case for the reactions O2 2H+ 2e & HZOz E o = 0.713 V
+ + 2Hf -+ 2e & 2H20
+
HzOz
E o = 1.77 V
Volume 73, Number 18 December 1968
4108
J. D. E. MCINTYRE
3'0 2.0 1.0
1I F-== i
1
100
'.--
.......... ............**. - - - __ __ ..................................:n2Fkz(Lo/L~,)C0" 1
(2
1
(5)
- n * l / n ~ ) n 2 F L ~ C ~ o (43) nl as w -+. w , the reciprocal of the intercept (2
0.4
0.5
07
0.6
0.8
0.9
1.0
I / w " ~ ( r a d sec-l)-"'
(voit )
(id2
0.3
Since n*1+ of the curve represented by eq 43 on the l/wl/' = 0 axis for the HCER (solid curves) and those for the normal case (dotted lines) for three potentials in the rising part of the second wave. The solid curves lie above the corresponding dotted lines, since i 2 for the catalytic elec-
Figure 7. Variation of l/i with l/t.o'/' for the second waves of an HCER ( p = 2, k h = lo4 cm4mol-' sec-I) and normal consecutive electrochemical reactions at the potentials, E ECzo(volts), indicated: -, HCER; , normal.
. . .. . .
trode process is less than that for the normal case, in contrast to the behavior of i l in the first wave (cf. Figure 3). The separation of curve and line decreases as the electrode potential is made more negative, reflecting the decrease of 0 and n*1 with increasing CR~(O). It should be noted that at constant potential both lim ( 2 - n*l/nl) = 1 -0
(44%)
and lim (2 -0
- n*l/nl) =
1
(44b)
The latter result arises since for fixed potentials in the second wave, n*l passes through a maximum as w is inThis variation in n * ~is made creased from 0 to more evident by observing how a constant-potential line cuts through the second waves of the steady-state i-E curves in Figure 4. At low rotational speeds the intersection occurs near the lcd plateau, where n*l -t nl since CB,(O) + 0. At high rotational speeds the line crosses the i-E curve near the foot of the second wave, where n*l -t n*l,,. I t will be recalled that as w -+. O D , n*1,, nl. The dotted lines in Figure 7 represent approximate experimental plots of l / ( i - i l 0 J us. l/wl/'. These plots have the same intercepts on the l/wl" = 0 axis as the exact plots and enable values of izm to be determined as a function of potential. Q),
--f
Discussion The preceding kinetic analysis is particularly relevant to studies of the kinetics and mechanism of oxygen reVolume 75, Number 18 December 1969
J. D. E. MCINTYRE
4110 duction on solid electrodes which act as catalysts for the heterogeneous chemical decomposition of the intermediate electrolysis product, hydrogen peroxide. Determination of all the kinetic parameters pertinent to this complex electrode process is essential for a detailed understanding of how the mechanism is affected by the pH of the electrolyte and the electrocatalytic properties of the electrode substrate material. Steady-state forced-flow techniques such as the R D E method are particularly useful since they often permit kinetic parameters to be determined in a potential region where the electrode is not covered (or is only partially covered) by a surface oxide film. It is important to distinguish between the kinetic behavior of the HCER discussed here and that of an electrode process in which 0 is reduced to the final product S and the intermediate R1 by simple parallel reactions which are not coupled through an intermediate. The latter case has recently been analyzed by Damjanovic, Genshaw, and Bockris" for studies of the kinetics of oxygen electrode reactions using an R D E with a concentric ring electrode.18-20 The ratio of the rates of consumption of 0 via the two parallel paths is denoted by x. For simple parallel reactions, the value of x is dependent on potential but not on mass transport rate. For the HCER, however, the quantity, x, is dependent on potential, mass transport rate, and chemical kinetics. The analysis of the form of the diagnostic plots of Idisk/ Iring vs. w-"' for the HCER is considerably more com-
The Journal of Physical Chemistry
plex than that for the simple parallel reactions although the plots for these two types of mechanism have several features in common. The distinction between the kinetics of heterogeneous catalytic and parallel electrode reactions is discussed in detail elsewhere.21 A close similarity also exists between the kinetic behavior of an HCER and that of an ECE mechanism
Jlkdnae x-s in which the rate of reduction of the intermediate, RI, to S is controlled by a heterogeneous chemical decomposition step. Methods of distinguishiag between these two reaction types will be presented in a forthcoming communication.
Acknowledgment. It is a pleasure to acknowledge helpful discussions with Dr. P. C. Milner and assistance in computer programming by W. F. Peck, Jr. (17) A. Damjanovic, M. A. Genshaw, and J. O'M. Bockris, J . Chem. Phys., 45, 4057 (1966). (18) A. Damjanovic, M. A. Genshaw, and J. O'M. Bockris, J . Phys. Chem., 70, 3761 (1966); 71, 3722 (1967). (19) A. Damjanovic, M. A. Genshaw, and J. O'M. Bookris, J. Electrochem. SOC.,114,466, 1107 (1967). (20) NI. A. Genshaw, A. Damjanovic, and J. O'M. Bockris, J. Electroanal. Chem., 15, 163, 173 (1967). (21) J. D. E. McIntyre, J . Phys. Chem., 73, 4111 (1969).
