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by initially subtracting 50 000 000 from each, however, the resultant array values of 0 and 2 lead to an eq 2 numerator of 4-2=2
The correct answer thus requires but one digit in the calculation.
ACKNOWLEDGMENT The authors are appreciative of and were greatly aided by the critiques and helpful comments on an initial version of this paper by two anonymous referees. LITERATURE CITED (1) Graham, M. M.; Lewellen, T. K.; Griep, R. J.; Rudd, T. G. J . Nud. Med. 1981, 22, 193. (2) Bork, A. Am. J . Phys. 1980, 4 8 , 421-422. (3) Brown, H. A. Am. J . Phys. 1980, 48, 905.
(4) Hewlett-Packard Co. “HP-97 Owner’s Handbook and Programming Guide”; Rev. F 9/78; Hewlett-Packard: Cupertino CA, 1976; p 83.
’
Queens Hospital Medical Center, Oahu, HI. *Department of Justice, Rlverside, CA. 3Chevron Research Co., Richmond, CA.
Philip M. Wanek* Richard E. Whipple’ Terry E. Fickied Patrick M. Grant3 Medical Radioisotope Group CNC-3 Los Alamos National Laboratory Los Alamos, New Mexico 87545 RECENED for review June 4,1981. Resubmitted April 26,1982. Accepted May 7, 1982. This work was partially performed under the auspices of the United States Department of Energy.
Comments on Curve-Fitting Methods Sir: We have been using the method of Deming (1) (as described by Wentworth (2))to obtain “best-fit” parameters from data for a number of years. We had no reason to question its appropriateness until we ran across a paper by Christian et al. (3), which compares “usual” least-squares parameters for a given set of data to ones obtained by “proper” minimization of residuals. Our subsequent application of the Deming method to the data in the article, in fact, produced exactly the parameters and residuals listed as “usual” values, and therefore, by implication, “improper” ones. The result of our looking into the matter may be of general interest to those of us who do curve fitting. Christian et al. (3) define the weighted sum of squares of residuals in terms of the weights of the individual points, rather than the individual measured variables. The result is a functional dependence of the weighting factors on the parameters to be determined. (To an experimentalist, this seems absurd, since the “goodness” of a measurement surely does not depend on the values of arbitrary parameters forced onto that measurement; nonetheless, mathematically, at least, it appears defensible.) A derivation of the so-called “normal equations” from this starting point, with explicit recognition of the functional dependence of weights on parameters in the minimization step leads to an extra term on the right-hand side of those equations. (Christian et al. (3) used a different procedure to incorporate the dependence into their analysis.) Wentworth’s equation (31), for example, becomes
Table I. Parameters Quoted by Christian et al. ( 3 ) to Include Standard Deviations dependent variable A, A, A,
least-squares result A , = (0.646 i 0.377)A2 t (0.0908 i 0.107)A3 A , = (0.419 * 0.279)A, + (0.169 i 0.077)A3 A , = (0.914 i 0.844)A1 + (2.630 i 0.816)A2 Proper Minimization Gives A , = (1.550 f 0.585)A2 t (-0.166 i 0.167)A3
S
7.89 6.06 12.96
5.67
where WaLis the partial derivative of W,with respect to a, and = 1/Li. Using these modified normal equations, we reproduce exactly the parameters Christian et al. label as the result of “proper minimization” (see Table I). The essential point to be made here is that the standard deviations in the parameters, omitted in ref 3 but here included in Table I, are quite large fractions of the parameters themselves, indicating that the latter are not particularly well-defined by the data. It is this characteristic of the example chosen by Christian et al. that leads to different results from those obtained with the method in Wentworth’s papers. The extra term introduced in the “alternative” derivation contains (Fio)2,which represents the square of the residual of the ith point. That is, if the fit were perfect, this term would disappear, and the two derivations would in this idealized situation yield identical normal equations. Thus, for data which conform well to the correlating function there is no practical difference in the two approaches. (For example, when both sets of normal equations are used to fit vapor pressure data for water to the Antoine equation, the best-fit parameters are identical to the sixth digit, while the standard deviations occur in the fourth.) We conclude that whenever there is a sound theoretical basis for the form of the correlating function used (or the appropriateness of an empirical functional relationship between dependent and independent variables has been established), there is no disadvantage whatever in using the Deming method of curve fitting.
wi
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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
