1879
Anal. Chem. 1982, 5 4 , 1879-1880
ACKNOWLEDGMENT Thanks are due J. A. Meyer for writing the computer programs that reproduced the results of ref 3. LITER,ATURECITED (1) Demlng, W. E. "Statistical Adjustment of Data"; Wiley: New York, 1943. (2) Wentworth, W. E. J . Chorn. Educ. 1965, 42, 96 and 162.
(3) Christian, S.
D.;Lane, E. H.; Garland, F. J . Chern. Educ. 1074, 51,
476 7 . V.
Edwin F. Meyer Chemistry Department DePaul University Chicago, Illinois 60614
RECEIVED for review March 8,1982. Accepted June 8, 1982.
Temperature-Dependent Determination of the Standard Heterogeneous Rate Constant with Cyclic Voltammetry Sir: Due to its relative simplicity, cyclic voltammetry has become almost a standard technique in the study of electrode kinetics and in the determination of the standard heterogeneous rate constant k , ~ (1'--5). , The determination of ks,h with cyclic voltammetry is fully outlined by a method developed by Nicholson (6),a t a not mentioned temperature which was presumedly 25 "C (7). As is generally known the knowledge of the rate constant at one temperature can yield only the free activation energy (AG*) where 2,is the heterogeneous frequency and the other symbols have their usual meaning. AG*contains contributions from both the solvent and the molecular reorganization energy necessary for an electron transfer to occur (8). However a better insight in the nature of the activation process is possible if the enthalpy (A") and entropy (AS*) of activation are known (9). For example, a nonadiabatic contribution to the activation process is reflected in the value of AS* (10). A classical way for the determination of AH* and AS* is to construct an Ahrrenius plot (Le., R In ks,h vs. 1/T), so measurement of k,,h as a function of the temperature is required. I t has been shown (6) that under certain restrictive conditions cyclic voltammograms can be interpreted on the basis of only one kinetic parameter \k defined by eq 2. Here, Do
P = ks,h/ d?rDonFv/ R T
(2)
is the diffusion coefficient of the depolarizer, u is the scan rate, and the other symbols have their usual meaning. \k is obtained from a AE,-\k working curve in which AE, is the anodiccathodic peak potential difference, an easily measured parameter. In order to carry out temperature-dependent measurements of ks,h, a,-*working curves have to be available a t various temperatures. For an infinite switching time (A) Matsuda (11)derived the following expression:
a,= (ZRT/nF)C(P,a)
(3)
The function C(\k,cu)can be evaluated numerically. For constant \k and a eq 3 becomes = c0nstant-T
(4)
(~~vT)[AE,TI.,~
(5)
so [ ~ ~ , 2 9 8 1 ~ ,=,
where AE,'
is the peak potential difference a t temperature 0003-2700/82/0354-1879$01.25/0
T. Equation 5 enables us to use the AE,298-Pcurve for any other temperature. For finite values of X eq 3 is no longer valid and there exist in this case no expression for the AE,-P relation. Consequently the temperature dependence of [AE,],,mis now unknown. It has been shown for a reversible electron transfer that the deviation from eq 3 is small if the switching potential is at least 65/n mV beyond the half-wave potential ( I ) . Thus for a quasi-reversible electron transfer the temperature dependence of [AE,],,a remains unclear. However intuitively we expect that the deviation from eq 5 will now also be small. To check whether this assumption is valid, we computed numerically AEp-P points in the temperature range of 303-253 K and we compare these with the points calculated from the data a t 298 K by means of eq 5. COMPUTATIONAL SECTION For the numerical calculation of the current-voltage curves the explicit-finite-difference method as described by Feldberg (12) was used. The laws of Fick for semiinfinite linear diffusion and the Butler-Volmer rate law are incorporated. This design allows the use of the program for other purposes like simulating follow-up reactions or multielectron transfers. The program relied entirely upon the highly efficient "Continuous System Modeling Program" (CSMP 111) and was solved on a IBM 370/158 machine. The resolution in AEp was 1 mV. The charge-transfer coefficient was set equal to 0.5 while Do= D,. In all calculations the switching potential was chosen so that IEx - Ellzl > 150/n mV. The procedure of generating AE,-\k points is simple then: for various values of \k the corresponding cyclic voltammogram are calculated and from these the AE, values are easily obtained. RESULTS AND DISCUSSION As the temperature range of 303-253 K is especially suitable for those low boiling solvents as acetone, dichloromethane, and acetonitrile which are nowadays frequently used as solvents in electrochemical studies of coordination compounds we have calculated AE, for 303 K, 298 K, 283 K, and 253 K (see Table I and Figure 1). The nine chosen values are mainly restricted to the useful1 working range of the AE,-\k curve i.e., 0.3 < \k < 4.0. The figure shows the very good agreement between Nicholson's curve and our calculated points a t 298 K which are within 1 mV; however our points tend to be somewhat lower. As can be seen from the table the agreement between the simulated Ups and via eq 5, calculated Upb, is 0 1982 American Chemlcal Society
1880
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
Table I. Anodic-Cathodic Peak Potential Difference, AE, (in mV), a vs. the Kinetic Parameter 9 T = 303 K
*
A
E
~ A~~
*
, b
T = 283 K
AEpa
AE,~
*
T = 253 K
aEpa
A~ , b
0.103 0.264 0.367 0.528 0.793 1.050 2.000 3.000 4.215
a
213 21 1 0.100 202 201 185 185 0.094 138 0.255 138 132 132 121 121 0.24 1 120 0.355 120 114 114 0.336 104 105 105 0.511 104 98 99 91 90 0.483 90 92 85 0.766 87 79 77 0.724 83 84 1.015 78 80 72 71 0.960 72 1.931 73 68 69 62 61 1.826 69 68 2.902 64 65 2.744 59 58 66 66 61 62 4.073 56 55 3.851 A E , ~= from the numerical calculated cyclic voltammograms. A E , ~= from Nicholson's curve at 298 K and eq 5.
cell potential is that one does not have to be concerned about the temperature dependence of the potential of the reference electrode, for hE, is independent from the reference potential. Thus there is no need to keep the reference electrode at a fixed temperature. From the preceding discussion it is clear that Nicholson's original working curve, valid a t 298 K, can also be used for the calculation of k,, at other temperatures because expression 5 is valid with a maximum deviation of 2-3 % , provided IEx - E,lzl > 150/n mV. So the cyclic voltammetric technique is now extended for temperature-dependent measurements of ks,h.
AExn.rnV
1
3
-
a-
Figure 1. Peak potential difference A€,, X n vs. \k: (curve A) solid line is constructed from the data at 298 K taken from ref 2, triangles are the numerically calculated points at 298 K; (curve B) open circles are the numerically calculated points at 253 K, the solid line represents a smooth curve draw through these points.
ACKNOWLEDGMENT We thank Professor J. J. Steggerda for the critical reading of the manuscript.
within 2-3%. This is comparable with the deviation which is introduced by setting a equal to 0.5 (there is 5% variation of AE, for a changing from 0.3 to 0.7 if \k > 0.5 (6)). This is an important result because one is saved from the need to construct hE,-\k working curves at every needed temperature. In application of eq 2 a t various temperatures a difficulty arises: the value of the diffusion coefficient should be known a t every temperature. To solve this problem one has two possibilities: (a) Iri a first approximation we calculate Do with the aid of the Stokes-Einstein relation
Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 3 6 , 706-723. Sharp, M. J. Nectroanal. Chem. 1978, 88, 193-203. Klingler, R. J.; Kochi, J. K. J. Am. Chem. SOC. 1960, 702, 4790-4798. Amatore, C.; Pinson, J.; Savgant, J. M. J. Nectroanal. Chem. 1980, 707, 59-74. Ahlberg, E.; Parker, V. D. J. Electroanal. Chem. 1981, 727, 57-71. Nicholson, R. S. Anal. Chem. 1965, 3 7 , 1351-1355. Bard, A. J.; Faulkner, L. R. "Electrochemical Methods", 1st ed.; Wiley: New York, 1981; p 231. Marcus, R. A. J. Phys. Chem. 1965, 4 3 , 679-701. Weaver, M. J. J. Phys. Chem. 1976, 8 0 , 2645-2651. Waisman, E.; Worry, G.; Marcus, R. A. J. Electroanal. Chem. 1977, 82, 9-28. Matsuda, H.; Ayabe, Y. Z . Elektrochem. 1955, 59, 494-503. Feidberg, S. W. "Electroanalytical Chemistry"; Marcel Dekker: New York, 1969; Vol. 111, pp 199-296.
Do= k T / 6 a r q
(6)
where r is the radius of the particle and the viscosity of the medium and k is the Boltzmann constant. Note that the viscosity is also temperature dependent so one has to take care to insert the correct value in eq 6. (b) A second more time-consuming but much more scrupulous method is to determine at every temperature the limiting current (id) from, for example, a normal pulse polarogram. From id it is possible to calculate Do and to use it further in expression 2. An advantage both of the cyclic voltammetric method and of measuring ks,hrather than the rate constant a t a constant
LITERATURE CITED
J. E. J. Schmitz J. G. M. van d e r Linden* Department of Inorganic Chemistry University of Nijmegen Toernooiveld, 6525 ED Nijmegen The Netherlands RECEIVED for review March 18,1982. Accepted May 3,1982. This research was supported by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organisation of Pure Research (ZWO).
