Determination of Electron Transfer Parameters via the Semi-integral J. H. Carney and H. C. Miller Department of Chemistry, The University of Alabama, University, Ala. 35486
The semi-integral evaluation of surface concentrations has been utilized for the direct determination of electron transfer rate parameters of Cd(ll) in 0.5M Na,S04, 1.OM HCIO4, 1.OM KN03, and 1.OM KCI, Zn(ll) in 1.OM KNO3, and Co( II I) tris(ethy1enediamine) in 0.1M ethylenediamine/l .OM NaC104. The semi-integral is particularly useful for dealing with uncompensated solution resistance since the form of the potential-time function need not be known a priori. However, the current must be separated into faradaic and nonfaradaic components. The shortest time at which data could be taken under the experimental conditions was about 30 psec; thus the maximum rate constant that could be determined is 0.8 cm sec-'. The systems Cd(1l) in KNO3 and KCI and Co(lll) tris(ethy1enediamine) had rate constants greater than this maximum.
played on a Tekronix 564-B storage oscilloscope, equipped with a type 3A9 vertical amplifier. Data were photographed with a Tektronix C12 Polaroid back camera. The syringe type hanging mercury drop electrode was a Metrohm E410,supplied by Brinkmann. The counter electrode was a 2.0 cm2 platinum flag, and the reference was a saturated calomel electrode with ionic contact made through a luggin capillary. Current-time curves obtained in potential step experiments were noticeably affected by changes in relative electrode positions. Thus, it was necessary to measure the uncompensated resistance of the cell for each experiment. Stock solutions of all chemicals were made by determinant weighings from reagent chemicals. All solutions were prepared with water distilled from borosilicate glass. Data analysis was carried out on an IBM 360-50 computer.
RESULTS AND DISCUSSION The basic equation relating current and potential for electron transfer controlled processes is ( 5 )
Oldham has demonstrated that for electrochemical processes represented by
OX
+ ne-
red
1)
where both ox and red are soluble and which do not involve rate-limiting steps prior to or following electron transfer, the concentrations of the species a t the electrode surface are given by Co,(O, t ) = Coxbulk- ( l / n F A D o x ' / 2d) - " 2 [ i ( t ) ] / d t - 1 1 2 (2a)
C r e d ( O , t ) = CredbUlk+ ( l / n F A D , e d ' ' 2 ) d - " 2 [ i ( t ) ] / d t - 1 ' 2 (2b)
Semi-infinite linear diffusion is assumed to be the only means of transport for both species. Coxbulk and Credb"lk are the bulk concentrations of the respective species, i ( t ) the current, Do, and D r e d the diffusion coefficients of the respective component, A, the electrode area, and d - 1 / 2 / dt-1'2 the semi-integral operator (1-4). As has been pointed out, knowledge of surface concentrations would allow direct determination of electron transfer rate constants from current-time curves which are generated by arbitrary, but known, voltage perturbations (1). We wish to illustrate the application of the semi-integral technique to potential step chronoamperometry for determination of kinetic parameters.
where hO is the standard rate constant and a the transfer coefficient. Thus, it is necessary to determine C,,(O,t) and c , e d ( o , t ) in order to find kO and a . An exact equation can be written which eliminates the dependence on the two concentration variables. However, it has been used only in an abbreviated form as the first term of an expansion in an infinite sum (6). Since the two concentration variables may be determined by the semi-integral method uia Equation 2, it is easy to obtain the rate constant and transfer coefficient by direct use of Equation 3. The semi-integral has been evaluated by use of the Riemann-Liouville definition of the operator ( I ) . The current-time curve is approximated by a series of linear segments of the form of Equation 4.
Here, Ail is the current a t the beginning of the hth segment, B k the slope of that segment, and t k It 5 til + 1. The current a t time t in the nth time segment may then be represented by an appropriate summation of the individual segments.
EXPERIMENTAL Potential step experiments were carried out with a Wenking Model 68 FR0.5 fast rise potentiostat, supplied by Brinkmann Instruments. Input pulses to the potentiostat were generated by a Tektronix 161 pulse generator. The current-time curves were dis(1) K . B. Oldham and J. Spanier, J. Electroanai Chem.. 26, 331 (1970). (2) K. B. Oldham, A n a l . Chem.. 41, 1904 (1969). (3) M . Grenness and K. B. Oldham, Anal. Chem.. 44, 1121 (1972). (4) K. 8. Oldham, A n a l . Chem.. 44, 196 (1972).
The terms in braces arise from the way that the summation is taken over all time segments and is a correction for (5) H. A . Laitinen, "Chemical Analysis," McGraw-Hill, New York, N.Y., 1960, p 307. (6) P. Delahay, "New Instrumental Methods in Electrochemistry," Inter-
science, New York, N . Y . , 1954, p 74.
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Table I. Potential Step Chronoamperometry t.
i.
psec
mA
Cox(OJ).n mM
15 25 95 195 295 395 465
5.44 3.85 1. a i 1.25 1.01 0.871 0.803
2.434 2.467 2.497 2.499 2.500 2.501 2.501
CO,(O,t), mM
*
3.123 2.966 2.739 2.666 2.635 2.61 7 2.608
*
The current at zero time is 16 mA. The current at zero time is 6 mA.
the h - 1 current segment. The validity may be checked by writing down the terms for two specific values of k . Equation 5 can be rearranged to give Equation 6.
the semi-integral method for data analysis; that the value of C,,(O,t) a t any time t is affected by all values of current prior to t, and the calculation of C,,(O,t) is thus adversely affected by all errors in approximating the current. This is a particular problem in potential step experiments for which the non-faradaic current introduces a large uncertainty for the points a t and near zero time. The magnitude and extent of such an effect can be estimated from columns three and four of Table I for which different initial currents were employed. The values of C,,(O,t), determined for the case in which i(l = 0 ) = 16 mA, settle down quickly to the correct value of 2.5mM. For the case in which the initial current employed was 6 mA. the calculated value of the surface concentration only slowly approaches the correct number. In both cases, the currents a t a given time were the same except a t zero time. As pointed out by one reviewer, a more realistic approximation to the theoretical current-time curve described by Equation 10 is
~ ( t= ) A, Except for the case in which h = 1, the term in braces is equal to zero, leaving
i(t)
=
"
AI +
Bk-i)(t
-
th)]
The semi-integral can thus be approximated for diffusion to a plane electrode by
(2/3)
[(Bi,
- Bk-i)(t - tkj3 1' 1
(8)
Bo
0
(9)
b=1
=
The current-time approximation of Equation 7 is also suitable for determination of surface concentration values by consideration of chronopotentiometric theory. Reilley and Ashley have derived a general equation which describes the variation of surface concentration in a chronopotentiometric experiment for the case in which the current imposed is a piecewise continuous function of the form of equation (7). The result of this treatment is a piecewise continuous surface concentration-time function equivalent to the results of the semi-integral of the current. One test of the ability of the algorithm of Equation 9 to approximate the semi-integral is afforded by the theoretical current-time behavior for a potential step experiment (8).The behavior is given by Equation 10.
where t = ( & x / & e d ) 1 ' 2 and 6' = exp [ ( n F / R T )( E - Eo)]. For a reversible couple, if the potential is stepped to the standard potential of the couple, C,,(O,t) = 1/2Coxbu1k, with Do, = Dred.The results are shown in Table I. There is a wide discrepancy between the calculated C,,(O:t) and the expected values near t = 0, which is caused by the fact that though it is necessary to put in some finite number for the current a t zero time, the theoretical current is infinite. This point illustrates an important weakness of (7) J. W. Ashley and C. N. Reilley, J. Electroanai Chem., 7, 253 (1964). (8) P. Delahay. "New Instrumental Methods in Electrochemistry," Interscience, New York, N.Y.. 1954, p 5 3 .
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A N A L Y T I C A L C H E M I S T R Y , VOL.
(11)
n-1
=
Ai
[Bk(th+l
-
th)-"']
(12)
/ =1
(7
h=l
B,(t - t,)-"'
where
A, [(Bb -
+
Equation 11 should in fact allow an exact fit to the theoretical current-time data. However, we would not expect it to provide any better approximation to experimental data which is affected by uncompensated resistance and double layer charging than would linear segments. The validity of Equation 8 for determining surface concentrations can also be illustrated for an electron transfer rate limited process. Theoretical current-time data were calculated for a kO of 0.5 cm/sec and an a of 0.30, using the equation given by Delahay (6).
i(t)
= nFAhCOxb"lkexp(p2t)erfc@t1")
(13)
where p = hDox-1i2
+
hD,,d-1'2 e x p [ ( n F / R T ) ( E - E ' ) ] (14)
h = h o exp[(-anF/RT)(E
- E')]
(15)
The theoretical currents generated by Equation 13 were employed in Equation 8 to determine the semi-integral and thus the surface concentrations C,,(O,t) and Cred(O,t). These concentrations, along with i ( t ) , and the potential a t which the current-time curve was calculated were then employed in Equation 3 to determine K O and a . Figure 1 shows a plot of In ( h ) as defined by Equation 15 us. ( E - EO).The method seems most accurate in the region near EO with deviations from the expected straight line a t potentials more anodic than EO by 15 mV and more cathodic by 45 mV. The calculated value for kO was 0.51 cm sec-1 with an a of 0.29. Rate constants as large as 5.0 cm sec-1 could be successfully determined from synthetic data by the semi-integral method. However, for the larger values of hO, data must be taken a t times less than 1psec. The effects of kinetic control cannot be detected if the value of the actual faradaic current, i, is greater than about 95% of the value of the diffusion-controlled current, id, a t the same potential. This limit of 95% was found by experiment, and is probably set by the method of approximation to the semi-integral. Delahay's Equation 4-11 allows one to estimate the maximum time a t which currents
45, NO. 13, N O V E M B E R 1973
demonstrate significant kinetic control a t the standard potential (6). For the purposes here, Delahay’s term X is given by
where t,,, is the maximum time a t which data may be taken; and X is approximately 3.0 for the case in which i / i d must be less than or equal to 0.95. For the conditions given in Figure 1, significant data must be taken a t times less than 90 psec. An experimental test of the method for determining kinetic parameters was made with the Cd(II)/Cd(Hg) system in 0.5M Na2S04. The rate constant is small enough so that the electron transfer reaction is at least partially rate-determining a t times for which double-layer charging is almost complete. However, in order to correctly determine the semi-integral, the faradaic current for times near zero must be known; and therefore the current a t any time must be separated into faradaic and nonfaradaic components. An RCdl time constant for double-layer charging in the absence of any faradaic current was determined via a potential step experiment. The uncompensated resistance R was determined in the same experiment. The charging current, calculated from the equation
-1 0
---
-
Eapplied
-k
-0.5-
1 C
0.0-
+o&
1’0
Equations 17 and 18 are consistent with assignment of a positive sign to cathodic currents. Results for a 5.OmM solution of Cd(1I) in 0.5M Na2S04 are shown in Figure 2. For currents taken between 30 and 400 psec, the points fall on a straight line as predicted. At times less than 30 psec, the method of analysis breaks down because of the uncertainties involved in separation of the charging and faradaic currents. Use of positive feedback in the potentiostat for lowering the uncompensated resistance as described by Lauer and Osteryoung would improve the linearity at short times (9). Points taken a t times much greater than 400 psec show the effect of nearly complete diffusion control of the overall process. It may be noted that because of the uncompensated resistance there is a variation in rate constant for a given potential step. This arises because E e f f e c t i v e becomes more cathodic as the current decreases according to Equation 12. The value of the standard rate constant determined from the intercept at EO is 0.10 f 0.02 cm sec-I with a n CY of 0.22. These values of the parameters may be compared to 0.085 cm sec-1 for the standard rate constant with an CY of 0.30 determined by chronocoulometry with resistance compensation (9). The reduction of 1.79mM Cd(I1) in 1.OM HC104 was also investigated. The value of kO a t -0.546 V was 0.14 f 0.02 cm sec-1 with a n CY of 0.30. The value of kO determined by Biegler and Laitinen uia faradaic impedance is 0.35 cm sec-1 ( 1 0 ) . (9) G. Lauer and R. A. Osteryoung, Ana/. Chem., 38, 1106 (1966). (10) T. Biegler and H. A . Laitinen, Anal. Chem., 37, 572 (1965).
-io
-60
-40
Figure 1. Determination of kinetic parameters by potential step chronoamperometry using theoretical current-time curves The values of parameters employed are n = 2, D,,,= Dred = 1.0 X cm2 sec-’, CoxbUlk = 1.0 X 10-3M, k o = 0.5 cm sec-’. 01 = 0.3
-2.5
-20
L
(18)
tiR
A
E-E’ (mv)
was then subtracted from the total current, it, obtained in the presence of electroactive material. The potential effective in driving the electron transfer, E e f f e c t l v e , was determined by correcting for the ohmic drop across the uncompensated resistance R.
E effective -
-
0
-lo10
-10
-20
-30
-40
E-Eo (mv)
Figure 2. Determination of kinetic parameters by potential step chronoamperometry for 5mM Cd( I I ) in 0.5M Na2S04. Eapplied is given below 0 , -0.730 V; A , -0.710 V; 0 , -0.690 V ; 0, -0.670 V; D O X= 0.8 X cm2 sec-’, and E o = -0.606V cm2 s e c - l , Dred = 2.0 X
-35 1
0
-10
-x)
-30 -40 E-E’ (mv)
-50
-Eo
I
Figure 3. Determination of kinetic parameters by potential step chronoamperometry for 5.0mM Zn(ll) in 1.OM K N 0 3 Do, = 0.67 X -1.011 v
cm2 sec-’, Dred = 1.57 X
cm2 sec-’, Eo =
The system Zn(I1) in 1.OM K N 0 3 was investigated in the same manner as for Cd(I1). Results are shown in Figure 3. The standard rate constant for this system was
ANALYTICAL CHEMISTRY, VOL. 45, NO. 13, NOVEMBER 1973
2177
0.061 f 0.001 cm sec-1 with a transfer coefficient of 0.27. Note that the rate constant in this case does not change for a given potential step since the current and therefore the effective potential were almost constant during the time in which currents were measured. Each point on the plot represents one experiment. Attempts were also made to determine kinetic parameters for reduction of Cd(I1) in 1.OM KN03, Cd(I1) in 1.OM KC1, and Co(II1) tris(ethy1enediamine) in 0.1M ethylenediamine/l.OM NaC104. All three systems proved to have rate constants too large to be measured under the conditions of our experiments. As noted above, the shortest time a t which meaningful 'data could be taken was 30 psec. For times of this order, the largest rate constant which could be determined, that is distinguished from diffusion control, is 0.8 cm sec-l. This lower limit of 0.8 cm
sec-1 may be compared to the range of 0.6 to 5.0 cm sec-1 found by ac impedance methods for Cd(I1) in nitrate and chloride media (11). Laitinen and Randles reported a rate constant of 0.13 cm sec-1 for Co(II1) tris(ethy1enediamine) (12); however, Sluyters-Rehback and Sluyters demonstrated that the reduction of the similar system Co(II1) in diethylenetriamine is diffusion controlled though complicated by adsorption of the cobalt complex (13). Received for review August 24, 1972. Accepted May 18, 1973. This work was partially supported by the University of Alabama Research Grants Committee. (11) N. Tanakaand R. Tamamushi, Electrochim. Acta, 9,963 (1964). (12) H . A. Laitinen and J. E. 6 . Randles, Trans. Faraday SOC.. 51, 54 (1955). (13) M. Sluyters-Rehbach and J. H. Sluyters, J. Phys. Chem., 75, 2209 (1971).
Experimental Study of the Phase-Selective Anodic Stripping Analysis of Micromolar Cadmium(l1) at the Micrometer Hanging Mercury Drop Electrode in 0.1M Potassium Chloride E. D. Moorhead and P. H. Davis Departments of Chemical Engineering and Chemistry, University of Kentucky, Lexington, Ky. 40506
Conflicting reports in the literature pertaining to the observed magnitude of the ac phase-selective current obtained in the anodic stripping of Cd from the HMDE prompted an experimental reassessment of the stripping behavior of this metal using a micrometer-type HMDE and 0.1M KCI as base electrolyte. Reproducibility of the stripping analysis was indicated by a 1.20% average deviation in the measured peak height obtained for ten independent runs at the micromolar Cd(ll) level. The functional dependence of peak height on signal frequency, applied ac voltage, and cadmium concentration conformed to theory developed previously for ac polarography. However, observed in-phase peak currents obtained during the stripping step were substantially smaller than those reported by previous authors for the same experiment. The influence of stripping scan rate on the CdCd(Hg) system was examined, and peak heights were found to be strongly dependent on this parameter. The importance of maintaining well-controlled conditions during pre-electrolysis was indicated from brief studies of peak height dependence on solution volume and stirring rate.
Anodic stripping voltammetry (ASV) has been utilized frequently for trace metal analysis a t various electrodes with most of the practical applications to date having been made with the hanging mercury drop electrode [HMDE]. The technique's extraordinary measurement sensitivity hinges on a preconcentration step whereby the trace material in a typical 10-ml to 100-ml sample solution is first electrochemically deposited into a mercury cm3 volume. Applicamicroelectrode of about 5 x 2178
tion of a positive-going voltage sweep to the resulting dilute amalgam then results in re-oxidation or "stripping" of the dissolved metals a t their characteristic potentials. In most of the ASV work reported heretofore, measurement of the collected trace material has involved a rapid, linear reverse scan of electrode potential with simultaneous recording of the direct current peak associated with each of the collected species. As described in several reviews (1-3), such analyses have employed either the Kemula electrode configuration (2), designated here as HMDE(K), where the electrode consists of a stationary drop extruded from a micrometer-driven capillary, or the platinum contact configuration [HMDE(Pt)], in which one or more mercury drops obtained from a dropping mercury electrode are suspended from a Hg-plated Pt contact sealed in glass, the flush Pt ,surface having been etched back ca. 0.1 mm to improve drop stability and shielding of the Pt from the solution ( 4 ) . For either electrode configuration, dc voltammetric stripping results in unsymmetric current peaks which "tail." This theoretically predictable feature of the dc process offers no major disadvantage a t moderate to high trace concentrations, but a t ultra-trace levels with low signal/noise ratios, such drawn-out peaks seriously impair both precision and resolution. To circumvent this prob(1) I . Shain in "Treatise on Analytical Chemistry," Part I , Vol. 4, I . M . Kolthoff and P. J. Elving, Ed., Interscience, New York, N.Y., 1963, Chap. 50. ( 2 ) W. Kernula and Z. Kublik in "Advances in Analytical Chemistry and instrumentation," Vol. 2 , C. N. Reiiley, Ed., interscience, New York. N.Y., 1963, Chap. 3. (3) E. Berendrecht in "Electroanalytical Chemistry-A Series of Advances," Voi. 2, A. J. Bard, Ed., Marcel Dekker, New York, N.Y., 1967, p 53. (4) A. M. Hartley, A. G. Heibert, and J. A. Cox, J . Electroanai Chem., 17, 81 (1968).
A N A L Y T I C A L C H E M I S T R Y , V O L . 45, N O . 13, N O V E M B E R 1973
