J. Phys. Chem. 1983, 87, 3911-3918
3911
-145 kcal/mol as a lower limit to the heat of formation of tetrafluorocyclopropane. However, if the uncertainties in the Arrhenius parameters are considered, higher and lower values of Emincan fit the experimental results yielding values of AHf0(c-C3H2F4) between -130 and -150 kcal/mol. Considering that the activation energy for the decomposition of tetrafluorocyclopropane is 48.5 f 2 kcal/mol and that E , for the addition of CF, to CH2=CF2 can be estimated as 10 f 2 kcal/m01,~,the heat of reaction of tetrafluorocyclopropane = 38.5 f 4 kcal/mol. With this value and AHfo(CF2) = -43.5 f 2.0 kcal/moP3 and AHfo(CH2=CF2) = 71.5 f 5 kcal/moP2 as estimated by
group additivity, AHfO(c-C3H2F4)= -153.5 f 11 kcal/mol is obtained. This value is 13.5 kcal/mol lower than that calculated from the chemical activation results in this work, although they are both within the error limits. However, it must be noted that the best agreement is obtained with the lower set of Arrhenius parameters. Unfortunately, the thermochemistry is so uncertain that it is difficult at present to make a decision about the best values. Also, if AHf0(CH2)were lower than 102 kcal/mol, a better agreement could be obtained.
(32) Benson, S. W.; O’Neal, H. E. “Kinetic Data on Gas Phase Unimolecular Reactions”; National Bureau of Standards: Washington, DC, 1970. (33) Stull, D. R., Prophet, H., Eds. “JANAF Thermochemical Tables”, 2nd ed.; U.S. Department of Commerce: Washington, DC, 1971.
partial financial support through Programa de Investigacidn Fisicoquimica. Registry No. CH,, 2465-56-7; C2F4,116-14-3; 1,1,2,2-tetrafluorocyclopropane, 3899-71-6.
Acknowledgment. We express our appreciation to Dr. D. W. Setser for providing most of the programs used in the calculations and to the CONICET (R. Argentina) for
Square Wave Voltammetry and Other Pulse Techniques for the Determination of Kinetic Parameters. The Reduction of Zinc( I I ) at Mercury Electrodes John J. O’Dea, Janet Osteryoung,’ and Robert A. Osteryoung Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214 (Received: December 14, 1982)
Square wave voltammetry was used to determine kinetic parameters for the couple Zn(II)/Zn(Hg). Rapid-scan square wave voltammograms for the reduction of Zn(I1) were analyzed numerically to yield estimates and confidence intervals for reversible half-wave potential (ET1,*), transfer coefficient ( a ) ,and electron-transfer rate constant (he,). Simplex optimization was used to minimize the difference between the experimental voltammogram and that calculated by using the boundary condition for a quasi-reversible reaction. As expected, the kinetic parameters which characterize this reaction were found to be independent of the step height and frequency of the excitation wave form. The utility of rapid-scan square wave voltammetry for kinetic studies at trace concentration levels (3.2 pM) was also demonstrated. The values of Er1/2,a , and hec at 25 “C in 1 M NaN03 (without double-layer correction) were found to be 0.998 f 0.003 V vs. SCE, 0.23 f 0.01, and (4.6 f 0.3) X cm/s, respectively.
Modern theories of electrode reactions predict relationships of kinetic parameters which can be tested experimentally.’ This has heightened interest in obtaining accurate values for these parameters. The range of values reported in the literature for nominally identical systems2B confirms the complexity and difficulty of this straightforward-seeming task. There are three different considerations which enter into the quality of the values of parameters derived from experiment. First, the experiment must be free of chemical artifact; that is, the rate determined experimentally must be characteristic of the electrochemical system as described. For example, reaction rates often depend strongly on trace quantities of surfactants which may be introduced inadvertently. This problem area is highly specific to the laboratory and the experimental system and will be addressed only inciden(1) Hush, N. S., Ed. “Reactions of Molecules at Electrodes”; WileyInterscience, New York; 1971. (2) Tamamushi, R. “Kinetic Parameters of Electrode Reactions of Metallic Compounds”; Butterworths: London, 1975. (3) Meites, L.; Zuman, P., Eds. “Handbook Series in Organic Electrochemistry”; CRC Press: Boca Raton, FL, 1976; Vol. 1, and succeeding volumes. 0022-3654/83/2087-39 1 1$01.50/0
tally here. Second, the mathematical description used to relate the rate of the reaction to the kinetic parameters must correspond to the actual experimental situation. Common problems in this regard include ignoring deviations from conditions of semiinfinite linear diffusion, lack of accurate potential control in transient techniques, and the intrinsic problem of subtracting “background” currents. Third, the data and the mathematical desccription must be brought together in some way to yield values of the kinetic parameters. Published procedures for doing this are uniformly inadequate in lacking a well-defined method for calculating the uncertainty associated with the derived values of parameters. In the following we describe the application of pulse voltammetric techniques, especially square wave voltammetry, for the determination of kinetic parameters. Square wave voltammetry as employed here is a largeamplitude technique carried out at fast scan rates. Thus, it is quite different from the steay-state technique employed by Tamamushi and M a t s ~ d a .Although ~ it can be (4) Tamamushi, R.; Matsuda, K. J. Electroanal. Chem. 1977, 80,
201-8.
0 1983 American Chemical Society
3912
The Journal of Physical Chemistry, Vol. 87, No.
20, 1983
implemented by using analog instruments,5 ideally it is carried out under computer control. From the experimental point of view rapid-scanning square wave voltammetry has the following advantages. First, because scan rates are large, data can be acquired rapidly and the small extent of reaction minimizes problems which occur due to reactions of the product (e.g., film formation). Second, the differential current measurement tends to subtract out background currents and to cancel terms describing slight deviations from simple theory (e.g., nonlinear diffusion). Third, like all pulse techniques, square wave voltammetry discriminates well against charging currents. Finally, the precision and sensitivity of the current measurement are sufficiently good that the technique can be used to do physical-chemical studies at micromolar concentration levels. This makes it practical to use concentration as an experimental variable over a significant concentration range. We now turn to the question of obtaining values of parameters from data. The power and availability of digital computers has made calculation of detailed theoretical models for complicated electrode reactions commonplace. Numerical approaches to solution of diffusion problems in the electrochemical context were developed systematically first for cyclic voltammetry because it is experimentally convenient, but its mathematical description is quite complex.6 The method employed, step-functional integration of the appropriate integral equation, has been described in detail.7 Similar convolution techniques have been employed to transform the experimental currentpotential response to one more tractable for a n a l y s i ~ . ~A- ~ second general approach has employed ac voltammetry and the mathematics of the fast Fourier transform.1° This approach, although powerful, is quite expensive and complex as well, and has not been widely employed. Both experimental implementation and data analysis employ a small-amplitude perturbation, which in turn limits sensitivity. A third experimental approach is based on chronoamperometry and mathematically equivalent techniques such as normal pulse voltammetry. The latter technique is experimentally convenient and has simple and well-known mathematical descripti0ns.l' Pulse techniques employing more than one step, however, have quite complicated mathematical descriptions for all but the reversible case. In particular, the double-potential step case has received considerable attention in a variety of ~ontexts.'~-'~ However, voltammetric wave forms composed of a sequence of potential steps are ideally suited tQ mathematical modeling based on the techniques of numerical integration popularized by N i c h o l ~ o n .This ~ approach is elegant in its generality, can be made arbitrarily precise, and is extremely efficient. It has been used by us recently to calculate theoretical responses for rapid-scan square wave (5) Yarnitsky, Ch.; Osteryoung, R. A.; Osteryoung, Janet. Anal. Chem. 1980.52. 1174-8. ( 6 ) Nicholson, R. S.; Shain, I. A n d . Chem. 1964,36, 706-23. (7) Nicholson, R. S.; Olmstead, M. L. In "Electrochemistry: Calculations, Simulation and Instrumentation"; Mattson, J. S., Mark, H. B., MacDonald, H .C., Eds.; Marcel Dekker: New York, 1972; Vol. 2. (8) Imbeaux, J. C . ; SavBant, J. M. J . Electroanal. Chem. 1973, 44, 169-87. (9) Oldham, K. B.; Spanier, J. J. Electroanal. Chem. 1970, 331-41. (10) Schwall, R. J.; Bond, A. M.; Smith, D. G. Anal. Chem. 1977,49, 1805-12. (11) Christie, J. H.; Parry, E. P.; Osteryoung, R. A. Electrochim. Acta 1966, 1I , 1525-9. (12) Aoki, K.; Osteryoung, Janet; Osteryoung, R. A. J . ElectroanaL Chem. 1980, 210, 1-18. (13) LovriE, M.; Osteryoung, Janet. Electrochim Acta 1982,27,963-8. (14) Evans. D. H.: Kellv. M. J. Anal. Chem. 1982. 54. 1727-9. (15) Hanafey, M. K.; Slo'tt, R. L.; Ridgway, T. H.; Reilly, C. N. Anal. Chem. 1978, 50, 116-37.
O'Dea et al.
voltammetry for the first-order kinetic cases.16 Of course, the problem facing the experimenter is the converse of this: given the experimental voltammograms and the boundary conditions, how does one determine the values of the kinetic parameters and their associated uncertainties? Working curves15 are used for this purpose by plotting data on the same graph as the curves calculated by using different values of the parameters and picking by eye the calculated curve which "best" fits the data. Convolutiona yields the well-known s-shaped voltammogram which is then analyzed by using the equation for a quasi-reversible reaction together with an independently determined value of the reversible half-wave potential.17 Constraining one of the parameters in this fashion is unnecessary and may be unwise. Neither approach provides an unbiased estimate of the precision of the derived quantities. In this work we have used a general method applicable to any voltammetric wave form composed of a sequence of potential steps. The method is illustrated for the case of slow heterogeneous charge-transfer kinetics. The voltammetric technique emphasized is square wave voltammetry, and the experimental system is the reduction of Zn(I1) at a stationary mercury electrode. The numerical treatment of data yields the reversible half-wave potential, transfer coefficient, electron-transfer rate constant, and their respective confidence intervals from pulse voltammograms. The boundary conditions assumed for the calculation are as follows: semiinfinite planar diffusion, only oxidized species initially present, and electron-transfer governed by the Erdy-Gruz and Volmer equation. For the case of slow electron transfer the integral equations describing the surface concentrations, Co(O,t)and C,(Ot), of the oxidized and reduced forms under conditions of planar semiinfinite diffusion are
Cp,(O,t) =
( ~ O ' ! ~ C ~ * / [ I T ( D R T ) ~ / ~ ] ) ~' ~u)~'/' ()U du ) / ( ~
(3)
where $ ( t )is a dimensionless current given by
i(t) = ~ ( t ) n F A D o 1 / 2 C O * / ( 9 r ) 1 / 2
(4)
and T is a characteristic pulse width of the wave form. Co* is the bulk concentration of the oxidized form and Do and DR are diffusion coefficients of the oxidized and reduced forms. For convenience we define K
6
= kec/Do('-a)/2DRa/2
(5)
= exp[(nF/RT)(E(t) - Erli2)]=
[co(o,t)/c,(o,t)l(00/0~)"~ (6) where ke, is the heterogeneous charge-transfer rate constant referred to the formal potential, Ee, and E ( t ) is specified by the experiment. Then, the Erdy-Gruz and Volmer equation can be written
i ( t ) = IzFAK€-a[Do1/2Co(0,t) - t D ~ ' / ~ c ~ ( o , t () 7] ) Combination of eq 2 and 3 with eq 7 yields a new integral equation (16) O'Dea, J. J.; Osteryoung, Janet; Osteryoung, R. A. Anal. Chem. 1981, 53, 695-701. (17) SavBant, J. M.; Tessier, D. J . ElectroanaL Chem. 1975, 65, 57-66.
The Journal of Physical Chemistty, Vol. 87,No. 20, 1983 3913
Square Wave Voltammetry for Kinetic Measurements
*(t) = K E P ( ( T T ) ' / ~-
[(l+ ~ ) / s ' / ~0 ] J ~ [ $ ( -u u) /) (' /t ~du] ] (8)
By the method of Nicholson and O l m ~ t e a d we , ~ now replace the integral in eq 8 with a finite sum and $ ( t )by b(m),the estimate of $(t). Further algebraic manipulations eventually yield the explicit solution for b(m).
b(m) = (r11/2/[2(1+
t)]
-
'E1b(i)s;j/(I i=l
+
(a1)1/2/[2Kt-"(l
+
E)])
(9)
Here 1 is the number of subintervals per period of the wave form, b(i) is the estimate of $ ( t )a t t = h / l , S!J = j 1 / 2- (j - 1)1/2,and j = m - i + 1. Thus, to the extent that the boundary conditions describe acurately the experimental system, for all potentials and times
(10) +c = nFADo1/2Co*/(a7)1/2. The constant c allows iexptl
= aJ/(ff,K,E1,2)
where a the intercept to be nonzero and thus ensures that arbitrary offset in the zero of current has no effect on the derived results. Hence, for any wave form a plot of i vs. J/ will be linear with a correlation coefficient, R, of unity for the correct estimates of a, K , and Erlj2.In practice some deviation from unity, R = 1 - R, is always observed due to noise in the experimentally measured currents. Our goal then is to fiid the set of parameters, a, K , and ErlI2,which produce the least deviation from perfect correlation with the experimental data and to evaluate the influence that each parameter has on that deviation. The problem is formulated in terms of a three-dimensional minimization wherein a , ZPl12,and log K T 1 l 2 are the dimensions and the quantity to be minimized is the deviation from perfect correlation, 8.Because of its simplicity and reliability,ls the modified simplex method of Nelder and Meadlg is employed to make this optimization. (The simplex technique has also been used in a somewhat different way for fitting electrochemical data by Birke et a1.20) Given the initial estimates, the modified simplex algorithm simultaneously varies the coordinates until a predetermined criterion for convergence on the best estimates (those which yield the minimum value of R ) is satisfied. For this work the algorithm was stopped when the coordinates of the vertices of the final simplex agreed to within 0.01 % of each other in each individual dimension. The reversible half-wave potential and a are obtained directly from the coordinates of the optimum vertex. The standard rate constant (ke, in eq 5) can be calculated from the third coordinate of the optimum vertex by using assumed or measured values of Do, DR,7 , and CY.If the surface area of the electrode is known, bulk concentrations (Co* in eq 4) can also be found since a, the slope of the plot of i vs. $, is fixed by the regression. Intuitively a good fit between theory and experiment over the accessible range of potentials and times means that the data are free of systematic error and that the model is appropriate. The estimates of the kinetic parameters derived from the curve-fitting process are, however, difficult to interpret without some estimate of confidence intervals associated with the values obtained. (18) Ryan, B. P.; Barr, R. L.; Todd, H. D. Anal. Chem. 1980, 52, 1460-7. (19) Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308-13. (20) Birke, R. L.; Kim, M.-H.; Strassfeld, M. Anal. Chem. 1981, 53, 852-6.
Because of the complicated dependence of goodness of fit on the values of kinetic parameters and 7 , the quality of the fit (Le., R ) is not simply related to the uncertainty in the derived parameters. We have developed an empirical but well-defined procedure for estimating confidence intervals which makes use of information furnished by the simplex calculation. After the best set of parameters is obtained from one voltammogram, values of each parameter are found, holding the other two at the optimum value, which give a value R = 3Rmin,where Rminis the value for R when all the parameters have their optimum values. The two values of each parameter corresponding to R = 3Rmin are located by using the secant algorithm.21 The resulting six values define three confidence intervals associated with the values of kinetic parameters which give the best fit. The width of these intervals depends both on the quality of the original data and on the sensitivity of the data collected to the parameter in question. A poor signal-tonoise ratio in the original data will decrease the correlation of a plot of i vs. $ and consequently increase the width of the interval defined by tripling R . More subtle than this is the widening of the confidence interval due to decreasing sensitivity of the data to the parameter. For example, uncertainties in a calculated from nearly reversible voltammograms will be larger than the uncertainties in CY calculated from quasi-reversible voltammograms. In the latter case small changes in a have drastic effects upon the wave or peak shape; in the former case they do not. The intrinsic noise of the experimental data set therefore propagates more successfully to a in the nearly reversible case and makes us less certain of its true value. Similarly, the value of ET1 is well-defined in nearly reversible voltammograms hut becomes progressively less well-defined for less reversible voltammograms. We have applied this scheme for numerical analysis to many types of voltammetric wave forms including normal pulse, differenced normal pulse, differential normal pulse, differential pulse, reverse pulse, cyclic staircase, and square wave, all with equal success. Although this approach is not constrained in any way to the analysis of square wave voltammograms, we emphasize square wave voltammetry here because of its many experimental virtues and potential widespread use.22 Experimental Section All calculations, experimental control, and data collection were carried out on a DEC PDP8/e laboratory minicomputer equipped with RLOl disk drive, floating point processor, real time clock, and point plot display. The point plot display is particularly useful in showing immediately the results of an experiment and in showing the intermediate results of the curve-fitting procedure superimposed on the experimental data as the calculation takes place. This display allows the user to observe any systematic deviations of the data from the model and to appreciate how changes in kinetic parameters affect the peak or wave shape. ADC's and DAC's were used to interface the PDP8/e to a homemade potentiostat described elsewhere.23 An EG&G PARC Model 303 static mercury drop electrode (SMDE) equipped with a temperature-controlled cell (25 " C ) was used as the working electrode. The large drop size (5.514 mg) was selected for all current measurements. Assuming spherical geometry, this corresponds to an area (21) Atkinson, K. E. "An Introduction to Numerical Analysis"; Wiley: New York, 1978. (22) Borman, S. A. Anal. Chem. 1982, 54, 698A-705A. (23) O'Dea, John J. Ph.D. Dissertation, Colorado State University, Fort Collins, CO, 1979.
3914
The Journal of Physical Chemistry, Vol. 87,
No. 20, 1983
O’Dea et
al.
,. a
_I
z
a H
tZ
4-
/ / ”
W
t0 [L
c
Ei
1
01
-050
-¶50
PDTENTIRL
/ui
TIME Figure 1. Wave form and nomenclature for rapid-scan square wave. See Experimental Section.
of 2.65 X cm2with a drop radius of 5 X lo-* cm. The reference electrode was an EG&G PARC saturated calomel electrode. All solutions were purged for 20 min with argon passed over BASF catalyst to remove residual oxygen before analysis. A t trace concentrations background currents obtained in the supporting electrolyte alone were subtracted and voltammograms were ensemble averaged ( N = 25) to suppress random noise. All chemicals were reagent grade and were used without further purification. Water was purified by passing distilled water through a Millipore MilliQ purification system. The square wave wave form is defined by Figure 1. There AE,is the step height of the staircase on which is superimposed the square wave of amplitude (peak-to-peak) 2E,, and frequency 1/21. Note that in previous discussions of square wave voltammetry 7 has been used to symbolize the square wave period; in the present case 7 is defined as the half-period of the symmetrical square wave. With this definition, 7 is readily identified as the characteristic time of the experiment. The delay time, t d , generally 2 s, is chosen to be long enough for the drop newly dispensed by the SMDE to stabilize mechanically. Unless otherwise specified AE, is 5 mV and E,, is 25 mV. The step height provides good potential resolution for the voltammogram, adequate numbers of points per voltammogram, and scan rates which are acceptable even at the lowest frequencies (4 s per scan of 400-mV range). This square wave amplitude gives the maximum ratio of peak height to peak width for a reversible system for n = 2 and has been adopted for most work in this laboratory. Note that, although the magnitude of the current is a strong function of the value of E,,, that dependence contains almost no kinetic information. The current is measured at points 1 and 2, that is, at nominal times 7 and 27 in each cycle, and the resulting voltammogram is &(E,)- i2(E,)vs. E = Ei + E,, + jm,, where Ei is the initial potential and j the number of the square wave. The actual time of current measurement is determined by the timing cycle of the computer and is an aperture of 0.04 ps at time T - 7.5 ps. Although from the experimental point of view the square wave is applied on top of the staircase, as shown in Figure 1, the index potential for the voltammogram is chosen a t the midpoint of each square wave cycle. This ensures that for a reversible system the experimental peak potential occurs at E , = Er1,2. Because the current measurement is differential, many experimental artifacts are avoided. The difference eliminates much of the background current. It also eliminates
- 1050 VS.
- 1150 SCE
- 1250J
rMV1
Figure 2. Spurious peaks in background currents; 0.2 M NaNO,, A€, = 1 mV, E,, = 25 mV. f (Hz) for peaks from left to right: 26.0, 25.5, 25.0, 24.5.
00
,. a
70
60
-
50 r
Z W
40-
U
3
50
U 20 10 0
.IO -850
-950
- 1050
POTENTIRL VS-
-
SCE
1 I50 tMV1
- 1250
Figure 3. Experimental (0)and theoretical (-) square wave voltammograms for reduction of 3.2-pM Zn(I1) in 1.0 M NaNO, at 200 Hz ( 7 = 2.5 ms). Triangles represent residuals. A€, = 5 mV; E,, = 25 mV.
effects of spherical diffusion which would otherwise be apparent at the lower frequencies employed. (For a reversible system, nonlinear diffusion has no effect on the shape or position of the net current voltammogram, even in the steady-state limit; 24 any effects in the kinetic case are negligible, as the results below demonstrate.) Spurious background peaks are occasionally seen when square wave voltammetry is employed at the SMDE. These were first observed at the dropping mercury electrode and are attributed to a fortuitous mechanical resonance of the gradually increasing drop mass and the excitation wave form.25 Because the drop mass is constant at the SMDE, we were at first surprised to see spurious peaks whose position is a strong function of frequency as seen in Figure 2. Apparently the monotonic change in the mean surface tension of the drop during the scan is sufficient to sweep the system through mechanical resonance and produce a spurious peak. A t trace concentrations these features can interfere with the qualitative interpretation of voltammograms by suggesting electrochemical reactions when in fact there are none. As in the case of the dropping mercury electrode these peaks can always be eliminated by changing the square wave frequency. Apparently diffusional mass transport is not disrupted by the resonance as simple background subtraction yields normally shaped peaks. (24) Aoki, K.; O‘Dea, J. J.; Kataoka, M.; Osteryoung, cJanet,unpublished results. (25) Feder, A. L.; Yarnitsky, C.; ODea, J. J.; Osteryoung, Janet. Anal. Chem. 1981,53, 1383-6.
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983 3915
Square Wave Voltammetry for Kinetic Measurements
TABLE I : Values of Kinetic Parameters and Their Confidence Intervals for the Reduction of Zn(II)u ~~
J
SO
~
origin a1
smoothed
c z w
+0.078 0.171- 0.070
II:
3
F
1.32 X 17.1
SINC
lo-’
4.71 x 10-3 28.8
-101
-
-050
’
‘ Square wave voltammetry, data of Figure 3. Calculated by using D o = 7.8 X cm2/sand D R = 1.89 X C ~ ~ / S Reference . ~ ~ 11. Reference 36. e Defined as the ratio of the peak height t o the rms deviation of the experimental points from the model.
-
-350 POTENTIRL
- 1050 VS-
-
SCE
1150 (MV>
- 1250I
Figure 5. Experimental data (0) of Figure 2 with six curves (-) calculated from the original (nonsmoothed) confiience intervals of Table
I. lee swOW-
5
78
-
ee5e
-
ie
O.O1
t
0.ooi
0.1s
e
--e. 101 05 0.20
0.25
0.30 , -
= -0.9984
Results and Discussion Figure 3 shows a particularly noisy square wave voltammogram for reduction of 3.2-pm Zn(I1) in 1.0 M NaNO3 Although not representative, it is useful for illustrating the method by which the kinetic parameters and their associated confidence intervals are obtained. The lack of trend in the residuals of Figure 3 suggests that the model accounts for the potential dependence of the current. Table I shows the calculated kinetic parameters and their associated confidence intervals for this data set. The charge-transfer coefficient and the apparent heterogeneous electron-transfer rate constant (calculated from K ) are in good agreement with literature values measured by different techniques at much higher concentrations. Figure 4 shows the relationship of the coordinate CY of the simplex to the quantity being optimized, R , and the confidence limits for CY as defined by the intersection of the curves with a threshold 3 times as high as the optimum value of R. Corresponding plots for the coordinates E l j Z and log ~ 7 are similar. As expected, all of the curves possess varying degrees of asymmetry due to the inherent nonlinearity of the problem. The width of these confidence intervals is not appreciably affected by a small uncertainty in the location of the minimum due to the finite size of the final simplex, which is 100 times smaller than the intervals. A family of six curves can be calculated from the limits of confidence, using optimum values for two parameters and the upper and lower limits for the third. Superimposed on the experimental data, these form an envelope of uncertainty as seen in Figure 5. With the exception of a few outlying points, all of the experimental data fall
-
1.85 VS. SCE
-1. 15 [VOLTS)
- 1. z5I
= Figure 6. As Figure 3,but 0.5 M NaNO,, 25 scan average. -0.9992 f 0.0007 v; CY = 0.212 0.033,-0.027; 1% KT’” = -0.256 0.079, -0.067.
RLPHR
Figure 4. Plot of R vs. CY for data shown in Figure 2. V, log K (s-’”) = 0.171.
- e . as
POTENTIRL
- - E d
+
~
+
within this envelope. This suggests that these intervals, based upon a predefined critrion, are of reasonable width and that the best estimates of the coordinates do indeed fall within them. Additional digital filtering of the data set (three passes through a seven-point, second-order, moving polynomial fit) improves the signal-to-noise ratio and reduces the width of the confidence intervals about 40% without affecting the best estimates (Table I). Heavier filtering, of course, distorts the peak shape and results in systematic errors in the calculation. A quantitative evaluation of this distortion, while possible, is beyond the scope of this work. It should be emphasized that the voltammogram of Figure 3 is exceptionally noisy. A more characteristic example in the same concentration range is shown in Figure 6. Even this does not convey intuitively a good feeling for the signal-to-noise ratio or, in analytical terms, the detection limit, for the instantaneous current measurement employed here for convenience in comparing / ~ experiment with theory is very poor in its ability to discriminate against noise. Analog filtering by gated integration of the current would decrease the noise significantly.26 Apparently over the range of times used here the current i,, = Q(t,)/t,, where Q(t,) is the integral of current over the time period 7 - t , to r , has the same functional dependence as i ( T ) , provided t , / r < We have not examined this point exhaustively. (26) Brumleve, T. R.; O’Dea, J. J.; Osteryoung, R. A.; Osteryoung, Janet. Anal. Chem. 1981,53, 702-6. (27) Andreu, R.; Sluyters-Rehbach, M.; Remijnse, A .G.; Sluyters, J. H.J.Electroanal. Chem. 1982, 134, 101-15.
3918
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983
O’Dea et ai.
35
0.211
POTENTIRL
VS-
SCE
Figure 7. Square wave current as a function of frequency, 1 mM Zn(1I) in 1.05 M NaNO,. E , = 5 mV, E sw = 25 mV. Experimental (0)and theoretical (-) currents with frequencies in ascending order of curves at -1100 mV; 10, 25, 50, 100, 200, 300,400, 500, 700, and 1000 Hz. ____ -333
10
15 20 25 SQRTCFREQUENCYJ
30
I
35
Flgure 9, a and confidence intervals vs. f”* for the data of Figure 7.
1.301
,-==- t
q ,
5
[MVl
T
T
-337
-336
T 1. 15
5
i
-3saL
5
10
15
20
25
30
3
Figure 10. 7.
K
10
15 20 SORT CFREQUENCYI
25
30
I
15
and confidence intervals vs. f ’ / * for the data of Figure
5QRT (FREQUENCY1
Flgure 8. 7.
and confidence intervals vs. f ’ / * for the data of Figure
The ability to follow quantitatively electrode kinetics a t trace concentrations via square wave voltammetry is expected to open up new areas of investigation both in the study of fundamental electrode process and in analytical applications. Further exploration of the utility of square wave voltammetry for kinetic studies was carried out by using data collected a t millimolar concentrations. The reduction of Zn(I1) in 1.0 M NaN03 was studied over a two-decade range of square wave frequency. Almost the entire spectrum of electrochemical behavior is observed, from nearly reversible to nearly irreversible peak shapes. Figure 7 shows a comparison of experimental and theoretical currents produced by the numerical analysis. Best estimates and confidence intervals for the kinetic parameters as a function of the square root of frequency are seen in Figures 8-10. The constancy of the best estimates is remarkable in view of the dramatic changes in peak shape seen in Figure 7 and demonstrates the success of the model (eq 7) for this system. The systematic trends in the widths of the confidence intervals can be explained as follows. In the case of a (Figure 9), the width incrases only at low frequencies and nearly reversible peak shapes. This is because a has little effect upon the measured current under these conditions and hence is less well-defined. For suf~ for 1 a~ reversible ficiently large values of log ~ 7 (i.e., system), the calculation of confidence intervals warns that the model is inappropriate by failing catastrophically. Similarly, the reversible half-wave potential (Figure 8) for the process is less and less well-defined as the peak shape becomes more and more irreversible. The systematic trend
toward more negative potentials is probably due to increasing uncompensated IR drops in the cell at higher frequencies. The width of the confidence interval for the rate constant (Figure 10) passes through a minimum at about 200 Hz. This parameter is least well quantified at both ends of the plot because of its insensitivity to peak shape at low frequencies and its lack of definition relative to Bl12 at high frequencies. As one might guess it is best measured at the frequency where the peak is most drawn out and flattened by the effects of slow electron transfer. For totally irreversible systems, the quality of the fit of the quasi-reversible model to the data becomes independent of the choice of ETll2or log K T ~ but I ~ , rather depends on the combination (anFPll2/RT)log ~ 7 ’ 1 ~Thus, . in such cases the model for a totally irreversible system should be used. The success of eq 7 in explaining the shape of square wave voltammograms for Zn(I1) reduction over a large range of potentials and frequencies conflicts with reports that a itself is a function of potential in this system. Matsuda and TamamushiZ8claim a potential dependence of the form a = 0.27 + b(nF/RT)(E - E r l j 2 ) where the coefficient b is 0.013 f 0.01. Matsuda and Tamamushi cite this as experimental confirmation of their prediction that b lies in the range 0.0083-0.0103 based on Marcus theory. On the basis of our data for the reduction of Zn(I1) a t 200 Hz we estimate that the coefficient b cannot be larger than 0 f 0.0005. This estimate is based upon a fit in which b is employed as a fourth independent (28) Matsuda, K.;Tamamushi,
831-9.
R. J.Electroanal. Chem. 1979, 100,
The Journal of Physical Chemistty, Vol. 87, No. 20, 1983 3917
Square Wave Voltammetry for Kinetic Measurements
TABLE XI: Values of Kinetic Parameters and Their Confidence Intervals as a Function of Step Height for t h e Reduction of Zn(II)a no. of points
Erm, v
5
81
-0.cJg75+0.0025 -0.0024
10
41
-0.9983+ 0.0035 -0.0035
25
17
50
9
AE, mV
log
01
K (S-"')
0.1 50:
0.234::::
:1
;:
0.160$:::22!
1.0 mM Zn(I1) in 1.0 M NaNO,; f = 200 Hz,E,, = 25 mV, E , = -0.85 V. TABLE 111: Values of Kinetic Parameters and Their Confidence Intervals for Reduction of Zn(I1) by Various Voltammetric Techniques -.
normal pulseQ differential pulseb differential normal pulseb differenced normal pulseb square waveb staircase Data of Figure 12.
-0.0020 - 1 . 0 1 2 5 + 0.0021 -0.0030 -1'0121+0.0030 -1.0134 - 0 ~ 0 0 3 0 + 0.0030 -1.012g-0.0026 +0.0026 -1.0124-0.0016 + 0.0016 - 1.0 146 -0'0024 + 0.0023
0.253;::;;; 0.234;::::; 0.240,:::;; o.257 -0.037 +0.045 o.199 -0.040 f 0.044
-0.546 -0'11' +0.133 -0.532 -0.024 + 0.026 - 0.56 3 -0'089 +0.113 -0.520 -0.051 +0.056 - 0.536 + 0.037 -0.538 -0.066 A 0.079 - o.523-o.051 +0.055
Data of Figure 11.
dimension in simplex space. Larger values of b produce distortions in the peak shape of the square wave voltammograms which are simply not observed experimentally. Thus, we find no evidence for a potential dependence of a in this system. It has been suggested that the reduction of Zn(I1) to Zn(Hg) involves two, slow, one-electron transfers separated by a chemical step.n The question of the power of the data presented here to distinguish between this alternative model and the simple kinetic model of eq 7 is beyond the scope of this work. We will discuss alternative models for reduction of Zn(1I) in a forthcoming publication. It is worth noting, however, that the variance of the data from the best fit (see, e.g., Figure 6) to the simple kinetic model of eq 7 does not differ markedly from the variance of the background current. Only a few current measurements are required to characterize the electrode process. Table I1 shows that the values of the parameters are only a weak function of the number of pulses employed in-the experiment. Even voltammograms consisting of just nine net currents contain kinetic information identical with that obtained from far more detailed scans. Perhaps this is to be expected since only five points are required to determine the parameters and additional data serve only to define the system further. The selection of a large (i.e., 25 vs. 5 mV) step height for the excitation wave form produces voltammograms which are less pleasing aesthetically but can save considerable amounts of experimental and computational time. This may be particularly useful in applications where square wave voltammetry is used for electrochemical characterization in real time as in liquid c h r ~ m a t o g r a p h yor~ flow ~ injection analysis. The same experimental system and programs for data analysis were used for seven different pulse voltammetric techniques. The same characteristic times and pulse am(29) Samuelsson, R.; O'Dea, J. J.; Osteryoung, Janet. Anal. Chem. 1980, 52, 2215-6.
-0.131 0.239 + 0.124 -0.013 0.237 +0.014
1
20
-2 1 -0.85
-0.35
POTENTIRL
-
-1.05
v s - scc
I. 15 ~VOLTS)
- 1.25
Flgure 11. Voltammograms for reduction of 1.0 mM Zn(I1) in 1.0 M NaNO,, delay time = 1 s, pulse height (2€,,) = 50 mV, pulse width (T) = 50 ms. Differential pulse (0),differential normal pulse ( O ) , differenced normal pulse (A),square wave (O), staircase (X). 40 35
38 25 20
! I 15
2 I
10
-
5
zE 0 2
-5
-10
- 15
-201 -25
-30 -35 -0.85
- a . 95
POTENTIRL
-1.05
VS-
-1.15
SCE
1
-1.25
[VOLTS)
Figure 12. As Figure 11 but normal (upper curve) and reverse (lower curve) pulse voltammetry.
plitudes were used throughout. Results for Zn(I1) including experimental points and the optimum theoretical
3918
J. Phvs. Chem. 1983, 87,3918-3925
curves are shown in Figure 11 (differential differential normal pulse,12,26differenced normal staircase,31and square wave voltammetry) and in Fi lure 12 (normal" and r e v e r ~ epulse ~ ~ , voltammetry). ~~ The values of a , and log K T ~ /displayed ~ , in Table 111, all fall within the calculated confidence intervals. With the possible exception of reverse pulse voltammetry the quality of the fit is about the same in each case. Note the greater sensitivity of the voltammetric wave (Figure ll),and, among the differential or derivative techniques (Figure lo), the greater sensitivity of square wave voltammetry. Of these techniques only normal and, of course, differenced normal pulse voltammetry have well-established, although questionable, methods of data analysis, as described above. In the limit of small step height and high frequency, staircase voltammetry approaches linear scan voltammetry, and so these voltammograms could, with some resulting errors, be analyzed as such. The data analysis employed here depends on the calculation of currents over time intervals short compared with the pulse time, but the calculations must be done over the entire electrolysis time. Of these techdiques reverse pulse and differential pulse voltammetry are the only ones with extended electrolysis times. In reverse pulse, only one such calculation needs to be performed, but for differential pulse the calculation must be done for each experimental point. With our system as described the entire curve-fitting procedure typically takes minutes for square wave, tens of minutes for reverse pulse, and hours for differential pulse voltammetry. Thus, the latter is not competitive as a technique
for kinetic investigations. More detailed consideration of techniques other than square wave voltammetry is beyond the scope of this work. The response in differenced normal, differential normal, and differential pulse voltammetry for quasi-reversible systems is discussed in detail by LovriE et al.34 We conclude from this work that square wave voltammetry is an excellent technique for determining kinetic parameters. Its inherent reliance on computer-based systems for variation of experimental parameters and for numerical calculation of the response expected from various models may be turned to advantage by using objective techniques for curve fitting which yield best values of the parameters of the model along with realistic estimates of their confidence intervals. The techniques used here also apply without modification to all pulse voltammetric wave forms. As in other cases, the modified simplex algorithm must be used with judgment, for the optimum values found may represent saddle points or local maxima. The confidence intervals lack rigorous statistical definition but are well-defined mathematically and seem to provide realistic estimates of the uncertainty in the values of the derived parameters.
(30) Anderson, J . E.; Bond, A. M. Anal. Chem. 1981, 53, 504-8. (31) Christie, J. H.; Lingane, P. J. J. Electroanal. Chem. 1965, 10, 176-82. (32) Osteryoung, Janet; Kirowa-Eisner, Emilia. Anal. Chem. 1980,52, 62-6. (33) Oldham, K. B.; Parry, E. P. Anal. Chem. 1970, 42, 229-33.
(34) LovriE, Milivoy; O'Dea, J. J.; Osteryoung, Janet. Anal. Chem. 1983,55, 704-8. (35) Baranski, A.; Fitale, S.; Galus, Z. J.Electroanal. Chem. 1975,60, 175-81. (36) Dillard, J . W.; O'Dea, J. J.; Osteryoung, R. A. Anal. Chem. 1979, 51: 115-9.
Acknowledgment. This material was presented in part at the Middle Atlantic Regional American Chemical Society Meeting, Newark, DE, April 21-24,1982. This work was supported in part by the National Science Foundation under Grants CHE 7917543 and CHE 8305748. Registry No. Zn, 7440-66-6; NaNO,, 7631-99-4.
Thermal-Induced and Photoinduced Atropisomerization of Picket-Fence Porphyrins, Metalloporphyrins, and Diacids: A Means for Examining Porphyrin Solution Properties Ruth A. Freltag and David G. Whltten' Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27514 (Received: January 3, 1983)
An investigation of the thermal-induced and photoinduced atropisomerization of the free base, diacid, and metal complexes of tetrakis(0-pivalamidopheny1)porphyrin (H2PF,TPiv),tetrakis(0-hexadecanamidopheny1)porphyrin (H,PF,THA), and tetrakis(0-propionamidopheny1)porphyrin(HzPF,TPro) was carried out. The isomerization process involves rotation of a single phenyl ring through a coplanar transition state, which requires substantial distortion from planarity of the porphyrin core. Relative thermal isomerization rates and photoisomerization efficiencies were found to depend on the size of the ortho substituent as well as on the atom(s) coordinated to the central core. The nature of the latter dependence suggests that porphyrin deformability in solution varies with central substituent and increases according to the following order: Zn(I1) complex < Pd(I1) and Cu(I1) complexes < free base < diacid < Ni(I1) complex. This order is consistent with published reports of nonplanar core distortions observed by X-ray crystallography for tetraphenylporphyrin and its metal complexes. The nonplanar conformation characteristic of porphyrin diacids provides an explanation for their atropisomerization rates as well as for spectral shifts observed for the four isomers upon protonation.
Restricted rotation of phenyl rings resulting in the existence of room-temperature stable atropisomers has been demonstrated for a variety of compounds. Early inves0022-3654/83/2087-39 18$01.50/0
tigations focused on biphenyls, most notably optically active ortho-substituted biphenyls which undergo thermally induced or photoinduced racemi~ation.l-~More 0 1983 American Chemical
Society
