Theory of Chronopotentiometry with Current Reversal for Measuring Heterogeneous Electron Transfer Rate Constants Floyd H. Beyerlein a n d Richard S. Nicholson Chemistry Department, Michigan State University, East Lansing, Mich. 48823 Theory of chronopotentiometry with current reversal is used to show that a system which behaves reversibly for some given current density can be made to exhibit kinetic behavior at higher current densities, as indicated by a separation of cathodic and anodic quarterwave potentials. An equation is derived which relates this difference of quarter-wave potentials to the standard rate constant for electron transfer. The method appears to provide a rapid and simple way to evaluate electrode kinetics.
CHRONOPOTENTIOMETRY with current reversal provides a useful qualitative estimate of reversibility of an electrode process. It should be possible with this technique to obtain quantitative estimates as well. Consider a redox system characterized by its heterogeneous rate constant, k,. For some given current density, k , will be large enough that electrochemical equilibrium will be established during the entire chronopotentiometric experiment. Under these conditions the system will appear Nernstian, and, for example, the difference of quarterwave potentials during the forward and reverse parts of the experiment will be nearly zero. However, with this same system if current density is progressively increased, a point will be reached where it is impossible for electrochemical equilibrium to be maintained. In this case the difference of quarter-wave potentials will be different from zero, and together with the current density at which this happens, should be a measure of ks. The purpose of this paper is to present equations with which exactly such correlations can be made.
-300
- 200
t
c
1
I
I
0.2
I
1
1 .o
0.6
\I ,
I .4
t/q Figure 1. Theoretical chronopotentiograms showing effect of kinetic parameter, il., when equls 0.5 and R equals -1 (Y
THEORY
Theory of chronopotentiometry with current reversal and electron transfer described by the electrochemical absolute rate equation is given by Anderson and Macero ( I ) ; their results provide the starting point of our treatment. In the following discussion we assume that only the oxidized form of depolarizer is present at the beginning of the experiment. Hence we consider reduction to occur initially with current density, iy, and label the first transition time, rY. After current reversal at ry, the current density is i,, and the second (oxidation) transition time measured from t = r Ywill be labeled, r7. Under these conditions, and with slightly different notation, Equation 4 of Reference ( I ) becomes O < t I r y
IJ exp[ag(E)] ry
5 t I rr
IJ Rexp[ag(E)]
=
1
= 1
- y1/2- Y
1/2
expk(El1
(1)
+ (1 - R) ( y - 1)lj2 - y 1 / 2+
[(I - R) (Y -
W 2- ~ ~ / 2 l e x ~ k ( E(2) ll
(1) L. B. Anderson and D. J. Macero, ANAL. CHEM., 37, 322
(1965).
286
ANALYTICAL CHEMISTRY
The following definitions apply to Equations 1 and 2 =
iy(DR)a/2/nFksCo*(Do)OL/2
(3)
Y = t1Y .
(4)
g(E) = (nF/RT) ( E - E d
(5)
R
=
illif
(6)
There Ellz is the conventional polarographic half-wave potential, k , the standard heterogeneous rate constant, a the transfer coefficient, DOand DRdiffusion coefficients of the oxidized and reduced forms respectively of depolarizer, CO*initial bulk concentration of depolarizer, and remaining terms have their usual meaning. Calculation of Potential-Time Curves. Because of the nonlinear form of Equations 1 and 2, except for some limiting cases discussed below, these equations cannot be solved explicitly for g(E), and therefore to calculate &E) as a function of y would require numerical solution of Equations 1 and 2. If the only object is to construct theoretical potential-time curves, however, it is simpler to consider y as the dependent variable. In this case Equations 1 and 2 can be solved explicitly for y
Tf
5t 5
7 7
=
yl/z
+ { K (E)' + [(l - R)'
-K(E)
(1
- 11 [(l - R)'
- R)2 - 1
+ K(E)']}'/' (8)
where
K ( E ) = (1 - R+exp[ag(E)1)/(1
+ expMEll}
Equations 7 and 8 can be used to construct theoretical potential-time curves without performing extensive numerical calculations. Potential-time curves calculated in this manner are shown in Figure 1. In general curves like those of Figure 1 depend on the three parameters +, a, and R . Effects of these parameters on potential-time curves are discussed in following sections. Effect of R. The parameter R affects only the anodic portion of potential-time curves (see Equations 7 and 8). In general R affects the shape of the potential-time curve slightly, but the major effect is in terms of T,. Often R would be - 1, but from an experimental point of view sometimes it is convenient to select R so that T and rr will be of similar magnitude. The exact relationship between T,-, r r ,and R is (2) 7,/7/
= 1/[(R
-
1)'
- 11
(Y
+,
g(E) = ln[(l - yl/z)/yl/z]
5t5
g(E)
=
- R) (Y - 1>1'21} (11)
We find that potential-time curves are described by Equations 10 and 11 within a few millivolts whenever +is less than 0.01. For Equation 10 Eli2 occurs [g(E) = 01 when y = 0.25 (quarter-wave potential). For Equation 11 the value of y at which Eli2occurs is given by y g ( E )=0 =
c
-1
+ { l + [(l - R)'
- 11 [4(1 - R)' 2[(1 - R)' - 11
I
0.5
I
I
0.7
l
I
0.9
Figure 2. Dependence of AEon charge transfer coefficient, a
+
A second limiting case arises when is sufficiently large that the processes for oxidation and reduction can be treated separately as the totally irreversible case. Equations 7 and 8 then reduce to the following familiar expressions ( I ) o ln[[(l - R ) ( y
- l)l/'
- y1/']/R) (14)
We find that potential-time curves are described within a few millivolts by Equations 13 and 14 whenever is greater than 2.5. Generally a affects potential-time curves in the expected < 0.01, curves are independent of (Y manner. Thus, for (Equations 10 and 11). For > 2.5, the effect of a is given by Equations 13 and 14. For values of between these limits, a affects both the symmetry of potential-time curves, and their position on the potential axis. This latter behavior is illustrated in Figure 2 where A E ( = E, - El) is plotted cs. CY for two values of Because for the mechanism being considered here CY is typically about 0.5, and rarely outside the range 0.3-0.7, these data of Figure 2 show that for reasonable values of a, the parameter AE tends to be independent of a, the dependence becoming less as decreases. The explanation of this fact is that as CY varies both E, and E, shift in the same direction, and these shifts tend to cancel in terms of AE. Nevertheless, for extreme values of a near 0 or 1, AE is markedly dependent on a . The reason for this effect is that, for example, as a approaches 1, E , tends to be independent of a (see Equation 13), whereas E, tends to vary exponentially with a (see Equation 14).
+ l]]"' ]'
+
+
+
+.
(12)
Thus, for example when R equals - 1, from Equation 12 El/' occurs at y = 1,0716,or relative to r r ,when y-1 = 0 . 2 1 5 ~( I~) . Potentials corresponding to these two times (0.25 and yo(^) - 0 ) hereafter are referred to as E, and E,, respectively. (2) R. W. Murray and C. N. Reilley, J . Electroanal. Chem., 3, 182 ( 1962).
I
+
+ (1 - R ) ( y - l)lj2 yl/2]/[y1/2- (1
0.3
(10)
71.
ln([l
1
CY
(1 - &(E) =
O5tI71
I
0.1
(9)
Except where indicated subsequent discussions are limited to R = -1. Effects of $ and (Y.Both $ and affect the potential-time curves of Figure 1, and in general these effects cannot be separated. However, some limiting cases exist where the relationship between g(E), R, +, and a can be stated explicitly. For example for small the redox system always is in equilibrium, and Equations 7 and 8 reduce to the following well known relationships which are independent of kinetic parameters ( I )
71
I
+
VOL. 40, NO. 2, FEBRUARY 1968
287
AE as Measure of k,. In the preceding discussion it was concluded that for small values of $ (a quantitative discussion of the limits for $ is given below) and reasonable values of a, AE is independent of a. This fact is important for two reasons. First, when AE is independent of a, AE is determined uniquely by $, and therefore AE is a simple measure of $, and hence k, (see Equation 3.) Second, for the special case of a equal 0.5, Equations 7 and 8 can be solved explicitly for g(E). This means that an equation can be derived for AE which always is valid when a equals 0.5, and which, depending on $, may be exact for any between 0.3 and 0.7. These expressions for g(E) when (Y equals 0.5 take the following form (Y
O_
