(3) Delahay, P., Berzins, T., J . Am. Chem. SOC.75, 2486 (1953).
(4) Delahay, P., Trachtenberg, I., Ibid., 79,2355 (1957); 80, 2094 (1958). (5) Flaschka, H., Butcher, J., Speights, R., ?Manta 8, 400 (1961). (6) Froneaus, S., Dissertation Lund (Sweden), 1948. (, 7,) Frumkin. A. S . , Trans. Faradau SOC. 5 5 , 156 (1959). ’ (8) Kolthoff, I. M., Okinaka, V., J . Am. Chem. SOC.81, 2296 (1959).
(9) Koryta, J., Collection Czech. Chem. Commun. 18, 206 (1956). (10) Lefebore, J., J . Chzm. Phys. 54, 601 (1957). (11) Loshkarev, M. A., Kryvkova, A. A., Zh. Fiz. Khim. 23, 209 (1949); 30, 2236 i1956’3. (12) Meites, ’L., J . Am. Chem. SOC.71, 2269 ---- 11949) \----,.
(13) Murray, R. W., Kodama, 31 ., ASAL. CHEY.37, 1759 (1965). (14) Reilley, C. N., Stumm, W., “Ad-
sorption in Polarography” in “Progress in Polarography,” Volume I, P. Zuman
and I. M. Kolthoff., eds.., Interscience. New York, 1962. (15) Reinmuth, W. H., AKAL.CHEM.33, 485 (1961). (16) Schmid, R. W., Reilley, C. N., J . Am. Chem. SOC.80,2087 (1958). (17) Tikhonov, A. S., Bel’skaya, V. P.,
Obschchei Khim., Akad. Nauk S.S.S.R. 2, 1211 (1953).
RECEIVEDfor review June 24, 1965. Accepted October 1, 1965. Work supported in part by Advanced Research Projects -4gency under Contract SD-100.
Derivative Vo Itu mmetry with Irreversible Systems Application to Spherical Electrodes S. P. PERONE and C. V. EVINS Department of Chemistry, Purdue University, lafayetfe, Ind. Theory for derivative voltammetry has been extended to irreversible systems at spherical electrodes. Expressions are presented for first-, second-, and third-derivative currentvoltage behavior, and tabulated functions for calculation of theoretical curves at both planar and spherical electrodes are included. The theory was tested by application to the reduction at the hanging mercury drop electrode of nickel(l1) in chloride medium. The simplicity with which the kinetic parameter ano could be evaluated was demonstrated with several systems including nickel(l1) in chloride, nitrate, and thiocyanate media, iodate at pH 7.2, and cobalt(l1) in a chloride medium. The analytical applicability was demonstrated with nickel(l1) over the range of 5.0 X to 4.0 X
1 0 -?M.
P
Ivith derivative voltammetry (7, 9) has demonstrated the enhanced analytical sensitivity obtainable with reversible systems. The work presented here shows that the increased sensitivity can be extended to irreversible systems. Furthermore, it s h o w that derivative voltammetry is an extremely convenient tool for evaluating heterogeneous kinetic parameters. In an earlier communication ( 8 ) , xhich presents the theory for derivative voltammetry a t planar electrodes, it is suggested that the kinetic parameter, ann, can be determined with unique simplicity with derivative voltammetry. For example, the ratio of peak heights of successive derivatives is directly related to ana, and the only experimental parameter which must be known t o evaluate the proportionality constant REVIOUS
is the scan rate. -4 considerablv abbreviated curve-fitting procedure can be used also xith derivative voltammetry, simply relating the separation of peak and/or zero potentials in a single curve to ano. Here, the theory has been extended to spherical electrodes, and experimental correlations were carried out with nickel(I1) in a chloride medium. First-, second-, and third-derivative voltnmmetric curves were obtained a t the hanging drop mercury electrode; ano was evaluated with several different types of measurements; a comparison of experimental curves with theoretical predictions for spherical electrodes was made; and an analytical study with sensitivity to 4 x 10-7V was carried out. Other irreversible systems were investigated also, but only for the evaluation of an,. EXPERIMENTAL
WORK
Instrumentation. The instrumentation used in this work has been described in detail previously (9). Basically, i t consisted of a general purpose instrument based on the operational features of analog computer amplifiers. X three-electrode system was utilized so that the potential between the working and reference electrodes could be accurately controlled while the current passed between the working electrode and a counter electrode. For the most part, voltammetric curves were recorded with an Esterline Angus Speed Servo, Model S6O1SJ variable range, ’/((-second recorder in conjunction l\-ith a low-pass filter network. In every case, the filtering was adjusted so that the signal of interest was unattenuated. Curves obtained a t scan rates exceeding 300 mv./second were recorded by photographing the trace on a Tektronix Model 536 oscilloscope with Types D and T
plug-in units. The camera attachment was a Dumont S o . 2620 Polaroid camera, using Type 47/Speed 3000 film. The operational amplifier differentiator units were calibrated by observing their output for a triangular wave input of appropriate frequency. The response of each was linear over the entire frequency range covered in this work; but each unit had a systematic error of 1 2 to 8% depending on the accuracy of the input capacitor. Cells and Electrodes. The cell design has been described previously (9). A hanging mercury drop electrode was used as the working electrode. Each electrode consisted of three drops of mercury collected from a conventional dropping mercury electrode capillary. The area and radius of the electrode were obtained from calculations based on the weight of 50 drops. The area was 0.0582 sq. cni., and the radius was 0.0681 cm. A fresh electrode was used for each run. The reference electrode was a large saturated calomel electrode. The counter electrode consisted of a graphite rod immersed in a 1.OJI KC1 solution. The entire cell assembly was immersed in a constant temperature water bath (Sargent Thermonitor, E. H. Sargent and Co., Chicago, Ill.). Temperature was maintained a t 25.0’ =k 0.1’ C. The cell assembly was mounted on a ring stand which rested on a thick pile of folded paper; this mounting reduced vibration considerably. Reagents. All chemicals used were reagent grade. Stock nickel(I1) nitrate solutions were standardized by an EDTA titration procedure (4). Water used to make up solutions was purified by distillation and passage through a mixed bed cation-anion exchange resin. All solutions were deaerated by passing high purity nitrogen through a gas-washing bottle containing the inert electrolyte solution and then through the sample solution for approximately 15 minutes. VOL. 37. NO. 13. DECEMBER 1965
1643
RESULTS AND DISCUSSION
Theory. Expressions for first-, second-, and third-derivative currentvoltage behavior for irreversible systems a t a planar stationary electrode have been presented previously (8). These relationships were obtained by simple differentiation of the expressions presented by Kicholson and Shain (6) and derived by Reinmuth (IO) for stationary electrode polarography with irreversible systems. To extend this theory to spherical electrodes, the expression presented by Reinmuth (IO) for stationary electrode polarography
with irreversible systems a t a spherical electrode was used. The relationship given by Reinmuth is
i
nFACo*
=
d6D0X
where
P=-
diz T
di
(3)
and Table 1.
Values of Functions xl(bt) and qh(bt)
E’, volts
xi(bt) 0.0035 0,0075 0,0161 0.0233 0.0336 0 0480 0 0673 0 0923 0 1221 0.1536 0.1794 0,1866 0.18823 0,1816 0.1669 0.1128 0,0388 0.0000 -0,0295 -0.0802
0.160 0.140 0.120 0.110 n.100 0 090 0 080 0 070 0 060 0,050 0.040 0.035 0.03155“ 0,025 0,020 0,010 0.000 - 0.00534 -0 010 -0.014
+i(bt) 0.0000 0,0001 0.0005 0,0011 n . on24 0 0052 0 0108 0 0219 0 0434 0.0821 0.1461 0.1892 0.2184 0.2931 0,3481 0.4376 0.4598 0.4355 0.4004 0.3345
All peak and zero potentials are underlined. a
Table 11.
0.160 n. _ 140 ~ 0.120 0.110 0,100 0.090 0.080
o om
0.060 0,0565” 0,050 0,040 0.035 0.03155 0.025 0.020 0.010 0.00215 0.000 -0.005 -0.010 -0.014
xdbt) .
0.0035 n 0074 0.0155 0.0222 0.0312 0.0429 0.0568 0.0711 0.0810 0 0819
0.0778
0.0497 0.0231 0.0000 -0.0534 -0.0981 -0.1740 -0.1960 -0.1942 -0,1778 -0,1478 -0,1182
ddbt) 0.0000 0,0002 0.0011 0.0022 0.0047 0 I0099 0.0200 0,0394 0.0739 0.0907 0.1290 0.2022 0.2395 0.2617 0.2846 0.2756 0.1617 0.0015 -0.0552 -0.1680 -0.2233 -0.5534
a All peak and zero potentials are underlined.
1644
E’
- Eo)+
= m,(E
(RT/F) In (?rDob)1/2/k, (4) The other symbols have their usual significance ( 1 ) . Differentiating Equation 1, one obtains
di - = Z‘= dt
_di.
dE
dE -= dt
nFACo*Do1’2b3’2 >(
dE where dT
=
-v.
Analogous expressions are obtained for the second and third derivatives:
Values of Functions XB(bt) and 6dbf)
E’, volts
ANALYTICAL CHEMISTRY
Exactly analogous expressions can be obtained for the second- and thirdderivative current-voltage behavior a t spherical electrodes.
where the functions X,(bt) are the series solutions for the planar case only, and have been defined previously (8). The series expressions for xl,(bt) and the spherical correction terms, &(bt), were evaluated with the IBM 7094 computer and results are presented in Tables I, 11, and 111. Values are given for the function in the potential region from the foot of the wave up to about 14/(m, mv. beyond the conventional peak. The presentation of only limited data beyond the conventional peak is due to the failure of the series to converge for larger negative values of E’. However, the theoretical data available are sufficient for experimental correlations, because all important features of the derivative curves occur before E’ = -0.01, Experimental Correlations. The advantages of the derivative voltammetric technique have been discussed previously (9). Enhanced sensitivity, increased scan rate dependence, minimization of charging current interference, and analytical resolution of a minor constituent in the presence of a more easily reduced major component are some of the more important advantages. With regard to irreversible
Table
111.
Values of the Functions x&t) and 43(bt)
E‘, volts
xa(bt)
+a(bt)
0.160
0.0034 0.0072 0.0103 0.0145 0.0200 0.0266 0.0333 0.0375 0.0377 0.0339 0.0132 0.0000 -0.0350 -0.1137 - 0 1990 -0.2309 - 0.2287 -0.1394 0.0000 0.0410 0.1239 0.1788
0.0001 0.0005 0.0010 0.0021 0.0043 0.0090 0.0183 0,0358 0.0400 0.0664 0.1134 0.1329 0.1692 0.1978 0.1236 -0.0595 -0.1218 - 0.4588 -0.6070 -0.5987 - 0.5246 - 0.3799
0.140
0 .i30 Equations 5 , 6, and 7 were evaluated for many different values of p . The difference between the spherical and planar solution was in each case a linear function of p for all values of p below about 0.01. Thus, a spherical correction term, @“(bt),could be calculated. The spherical correction term would be useful for values of p below 0.01, and it has been demonstrated previously (6) that this would include most reasonable applications. Thus, for first-derivative voltammetry at a spherical electrode
0 120 0.110 0.100 0.090
0.080 0.0781“ 0.070 0.060 0.0565 0.050 0.040 0.030 0.0219 0.020 0,010 0.00215 -0.000 -0.005 -0.010
I
a All peak and zero potentials are underlined.
processes, two extremely simple methods for determining the kinetic parameter, an,, at a planar electrode have been presented (8). The first is basically an abbreviated curve-fitting method which can be carried out conveniently with derivative curves by simply noting the separation of characteristic peaks and/ or zero potentials. Thus, for a firstderivative curve
A more accurate measurement of ano can be obtained by taking the ratio of successive derivative peak heights:
Table IV. Determination of an, for Nickel(l1) in 0.1M KCI
Method
0.80 0.80 0.77 0.74 0.74
(14.800)an,
a
(16 - 9 4 ~an, )
where E D Pis~the potential a t which the first peak occurs in the first-derivative curve, and E Z is the potential !%-here the derivative current goes through zero. Similar data can be obtained from other derivative curves; for example, for a third-derivative curve it is possible to utilize the difference between the first two potentials a t which the signal goes through zero :
Figure 1. First- and second-derivative current-voltage curves for nickel
(11) 5.00 X 1 O-4M nickel(l1) in 0.1 M KCI; scan rate, 2 9 0 mv./sec.; VI.
initial potential,
- 0.350
volt
S.C.E.
Upper t r a c e second-derivative scale, 67 jtA./sec.2/large div.; - 0.1 45 volt/large div. Lower trace: first-derivative scale, 10.0 wa./sec./large scale, -0.1 45 volt/large div.
curve. Vertical horizontal scale, curve. Vertical div.; horizontal
ana
( 4 5 . 8 5 ~ ) ~ n(15) , where i, is the current at the conventional voltammetric peak; d& (bt) is the series solution for the conventional curve, previously evaluated by Sicholson and Shain (6); I,’ and I,” are the derivative currents a t the first peak in each of the first- and second-derivative curves, respectively; I”’p2and I)’,? are the derivative currents of the second (inverted) peaks in each of the secondand third-derivative curves, respectively. The relationship between derivative current-voltage behavior at a plane stationary electrode and the heterogeneous rate constant, k,, has been discussed previously (8). The rate constant can be evaluated from the zero potential of the first-derivative curve; k, (and an,) can be evaluated also from the intercept and slope of a plot of ln(I,‘/v) os. ( E D p - E O ) . However, this last approach is not useful a t spherical electrodes because spherical diffusion is important at slow scans, and instrumental limitations make it difficult to carry out the experiment over a wide range of rapid scan rates. Correlation with Spherical Electrodes. The preceding correlations discussed are limited in application to planar electrodes. These same relationships cannot be applied strictly when spherical electrodes are used and when spherical diffusion is important. Xot only are peak currents altered, but peak and zero potentials are shifted. Moreover, referring to Tables I, 11, and 111, the amount of spherical correction is different at each point in the curve. For any given potential, however, the spherical contribution can be made negligible by reducing the value of p ; this can be done simply by using an appropriately rapid scan rate. For the evperiniental work reported here, kinetic parameters mere computed from data obtained at spherical electrodes, but utilizing sufficiently rapid scan rates to reduce the spherical contribution at all potentials of significance to less than 1.5% in the \Torst case. On the other hand, experiments not for the purpose of evaluating
b
First-derivative curve. Third-derivative curve.
Table V. Determination of ano for Various Irreversible Systems
System Ni(II), O.lJ1 KNO, Xi(I1))0.1M KCNS IO,-. DH 7.2 a
an a
(obs) 0.81 1.12 0.28 0.57
ana
(reported) 0 . 3 1 (6)” 1.33 ( 2 ) b 0.26 ( 3 ) c 0 . 8 2 (2)d
0.2X K S 0 3 , 0.005c;C gelatin.
0.5.11 KCXS, no gelatin. c
d
pH 7.2 phosphate buffer, no gelatin. 1.0M KC1, no gelatin.
kinetic parameters were carried out a t slow scan rates, where the spherical contribution was significant. Determination of an,. Before further correlations between theory and experimental result. could be made with the nickel(I1) chloride system, it wab necessary to establish evperimentally the value ot an, for the system. Thiq was done in t w o ways: taking the ratio of the various derivative peak., and measuring the separation of the various characterizing potentials. The data are summarized in Table IT. Experimental curves typical of thoze from which the data in Table IV were computed are shown in Figure 1. In addition, the ratio I p t / i pwas obtained with more accurate potentiometric recording and the corresponding an, value (0.80) is considered the most reliable. It was used in subsequent calculationi. There is good agreement throughout Table IV and also with previous report. of an, for nickel(l1) in chloride media. For example, Delahay and Mattax ( 2 ) obtained a value of 0.70 using chionopotentiometric measurements on nickel(I1) in chloride media. To demonstrate further the utility of derivatire voltammetry for the determination of an,, several other irreversible systems were investigated. These systems were nickel(I1) in 0.1V nitrate, nickel(I1) in 0.1M KCSS, iodate in pH 7 . 2 , 0.l.U phosphate buffer, and cobalt(I1) in 0 . l X KC1. Only the peak ratio method was used in each case, taking the first-derivative and conventional peaks and using Equation VOL. 37,
NO. 13, DECEMBER
1965
1645
I
,
Lwtc
I
P
zoo
I
-2.w’
1
-I.-
,
I
1
I
b
1
- 1.10
VOLTS
VI.
I
- 1.10
I
1
S.C.E.
Figure 2. Comparison of experimental to theoretical first-derivative current-voltage behavior with spherical diffusion 5.00 X 10-4M nickel(l1) in 0.1M KCI; scan rote, 31.5 mv./sec.; diffusion coefficient, 6.9 X
10-6 cm.21sec.
13. The scan rate used in these determinations was 300 mv./second. The data are summarized in Table V. Correlation with previously reported data vas not good for every case in Table V. However, in those cases where conditions were closest to those of previous studies, correlation was good. For example, with iodate and with nickel(I1) in thiocyanate, conditions were nearest to those of the previous work, and correlation with previous data was best. With nickel(I1) in nitrate, gelatin was used in the previous work and probably resulted in the lower value obtained ( 2 ) . With cobalt(I1) in chloride, the electrolyte concentration used in previous work was 10-fold higher than used here. Attempts to use the higher electrolyte concentration for the work reported resulted in the reduction wave being shifted too cathodic for accurate measurements.
Comparison of Experimental to Theoretical First-Derivative Curves. -4 comparison between theoretical and experimental first-derivative voltammetric curves was made under conditions where the spherical contribution was important. This comparison served several purposes. One of these was to demonstrate that the shape of the entire voltammetric curve was consistent with the an. value obtained. Another objective was to evaluate the adequacy of the instrumentation. Finally, this comparison was a check on how nearly the experimental system approached ideality. Data were obtained with a 5.00 x 10-4M nickel(I1) solution in 0.1M KCl. A scan rate of 31.5 mv./second was used, and the correlation is shown in Figure 2. The diffusion coefficient used in the calculations was that used by Kivalo, Oldham, and Laitinen (5). The experimental curve agrees well with the predicted curve, the maximum deviation a t any point being 2%. Analytical Study. The dependence of the first-derivative peak height on concentration was determined experimentally using nickel(I1) in 0.131 chloride, varying the concentration to 4.0 X lO-7M. from 5.0 X Correlation was made with the first peak occurring at -1.09 volt us. S.C.E. Table VI presents the analytical data, showing the expected linear dependence of peak height on concentration. Figure 3 shows the nature of the first-derivative current-voltage curve obtained a t 1.0 x 10-6M nickel(I1). Analytical data could not be obtained with the conventional technique a t concentrations more dilute than 10-5M.
Analysis of Nickel(l1) in 0.1M KCI by First-Derivative Technique” Peak Concn., GO*, rurrent, ZP‘, Rel. std. moles/l. fia./sec. I , /co * dev., 70
Table VI.
x x x 1.00 x 3.00 X 1.00 x 4.00 x 5.00 1 .oo 4.00
a
1646
0
2.61 X 5.22 x 10-3 10.9 5.12 x 10-7 5.12 x 10-3 1.5 2 . 0 7 x 10-7 5 . 1 7 x 10-3 1.7 5.22 X 10-8 5.22 x 10-3 1.6 1.56 X 5 . 2 8 x 10-3 3.0 10-6 5.28 x 10-9 5.28 x 10-3 4.1 10-7 2 . 3 3 x 10-9 5 . 8 3 x 10-3 4.9 Data represent average of 5 to 9 replicate runs at each concentration,
10-4 10-4 10-5 10-5
ANALYTICAL CHEMISTRY
- 0.80
- 1.00
-1.20
V O L T S vs. S.C.E.
Figure 3. Experimental first-derivative curve for dilute nickel(1l) solution 1-00X 1 0 3 4 nickel(l1) in 0.1 M KCl; scan rate, 31.5 mv./sec.; dashed line represents experimental blank LITERATURE CITED
(1) Delahay, Paul, “New Instrumental Methods in Electrochemistry,” second ed., p. 119, Interscience, New York, 1954. (2) Delahay, P., Mattax, C. C., J . Am. Chem. SOC.76, 874 (1954). (3) DeMars, R. D., Shain, I., Zbid., 81, 2654 (1959). (4) Flaschka, H. A., “EDTA Titrations,” p. 79, Pergamon Press, N. Y., 1959. (5) Kivalo, P., Oldham, K. B., Laitinen, H. A., J . Am. Chem. SOC.75, 4148 (1953). \ - - - - I .
(6) Nicholson, R. S., Shain, Irving, ANAL. CHEM.36, 706 (1964). ( 7 ) Perone, S. P., Birk, J. R., Zbid., 37, 9 (1965). (8) Perone, S. P., Evins, C. V., Zbid., p. 1061. (SjPerone, S. P., Mueller, T. R., Zbid., p.
(1tj Reinmuth, W. H., (1961).
Zbid., 33, 1793
RECEIVEDfor review August 16, 1965. Accepted September 29, 1965. Division of Analytical Chemistry, 150th Meeting, ACS, Atlantic City, y. J., September 1965. Work supported in part by Public Health Service Research Grant No. CA07773-02 from the National Cancer Institute.
