Fundamental and Second Harmonic Alternating Current Polarography of Electrode Processes with Coupled First-Order Preceding Chemical Reactions: Experimental Study of the Cadmium-Nitrilotriacetate System Kathryn R. Bullock' and Donald E. Smith2 D e p a r t m e n t of Chemistry, Northwestern University, Evanston, 111. 60207
Fundamental and second harmonic ac polarographic measurement results are reported for the reduction of the cadmium-nitrilotriacetate complex (CdNTA) at the mercury-aqueous solution interface. One purpose of the study was to obtain data concerning the fidelity of ac polarographic rate laws derived for systems with coupled preceding first-order chemical reactions, particularly for the second harmonic response where careful tests of the theory have not been reported. The CdNTA polarogram has been shown to consist of two waves under appropriate conditions: the first involving kinetic influence of the CdNTA dissociation preceding the heterogeneous charge transfer step. Measurements were performed at pH's of 4.95, 4.28, and 3.05 over which range the chemical kinetic influence varies from slight to almost solely rate determining in the dc polarographic context. Data were obtained using a digital Fourier transform approach to ac polarography in the harmonic multiplex mode. Agreement of data with theoretical predictions is excellent. Kinetic and thermodynamic parameters for the CdNTA dissociation obtained from the ac polarographic observables are consistent with previous electrochemical and N MR spectroscopic investigations.
The electrochemical behavior of the cadmium-nitrilotriacetate (CdNTA) system has been widely investigated (1-17). Koryta and coworkers were the first to pursue such studies in detail (1-8). Their efforts encompassed the polarographic determination of the stability and dissociation rate constants of the CdNTA complex over a wide range of pH values, ionic strengths, ligand concentrations and buffer systems. Other studies concerned with the kineticPresent address, T h e Gates R u b b e r Co., 999 South Broadway. Denver, Colo. 80217 T o whom correspondence should b e addressed. (1) J Koryta and I Kossler, Collect. Czech. Chem. Commun., 15, 241 (1950). (2) J. Koryta, Proc. 7st. Intern. Polarogr. Congr.. 1, 798 (1951) (3) J. Koryta, Z.Phys. Chem. (Leipzig), 157, July 1958, (Sonderheft). (4) J. Koryta, Collect. Czech. Chem. Commun.. 24, 2903 (1959) (5) J. Koryta, Collect. Czech. Chem. Commun., 24, 3057 (1959). ( 6 ) J. Cizek. J. Koryta. and J. Koutecky, Collect. Czech. Chem. Commun., 24, 663 (1959). (7) J. Koryta, Z. Elektrochem., 64, 196 (1960). (8) C. V. D'Alkaine and J . Koryta. Collect. Czech. Chem. Commun., 34, 2138 (1969). (9) K. Morinaga and T. Morinaga. Nippon Kagaku Zasshi, 79, 200 (1958). (10) P. Papoff, J. Amer. Chem. SOC.,81, 3254 (1959). (11) N. Tanaka, K. Ebata, T. Takahari, and T Kumagai, Bull. Chem. SOC.Jap., 35, 1836 (1962). (12) N. Elenkova. God. Khim. Techno/. Ins?.,8 , 113 (1961) (13) N. Elenkova, God. Khim. Techno/. Inst., 8 , 1 (1961). (14) N Elenkova, C. R. Acad. Bulg. Sci., 18, 347 (1965). (15) C . Fischer, Collect. Czech. Chem. Commun.. 27, 1119 (1962) (16) Y Masuda. Mem. Fac. Sci., Kyushu Univ., Ser. C, 6, 9 (1967). (17) M . S Shuman and I . Shain, Anal. Chem., 41, 1818 (1969).
mechanistic aspects of CdNTA electrochemistry have been performed by Morinaga and Morinaga (9), Papoff (IO),Tanaka, Ebata, Takahari, and Kumagai (11), Elenkova (12-14), Fischer (1.9, Masuda (16), and Shuman and Shain (17 ) using either dc polarography, oscillopolarography, chronopotentiometry, and/or cyclic voltammetry. From these investigations, a reasonably clear picture of the electrode reaction mechanism a t the mercury-aqueous solution interface has evolved. In particular, under conditions of concern in this report, the accepted mechanism is one originally proposed by Koryta ( 5 ) , and recently substantiated in a detailed cyclic voltammetric and dc polarographic study by Shuman and Shain (17). According to this mechanism, the first polarographic wave is due to the electrode reaction
which is preceded by the homogeneous chemical reaction
Cd(NTA)-
+ H+
k
Cd2+
k--2
+ HNTA'-
(2)
The pH dependence of the current of the first wave is a result of the latter reaction. Rabenstein and Kula (18) have obtained proton NMR spectroscopic data which further substantiated Reaction 2. The values of k2 and k 2 which they reported are in close agreement with those of Shuman and Shain (17). A second polarographic wave arises from the electrode reaction:
This reaction is characterized by a very small k,z-value so that it is essentially irreversible under polarographic conditions. The homogeneous preceding chemical reaction
(In,
Cd(NTA)-
k k-1
Cd2+
+ NTA3-
(4)
also has been postulated, but the rate constants are too small to be measured by cyclic voltammetry or polarography ( 1 7 ) . This insight into the electrochemical behavior of the CdNTA complex provided by the studies just quoted, together with environmental concerns (19), appears to have stimulated the successful development of polarographic methods for assaying NTA in natural waters (20-22). The basic approach in each report involves forming the (18) D. L. Rabenstein and R . J. Kula, J. Amer. Chem. Soc., 91, 2492 (1969). (19) S. S. Epstein. Inf. J. fnviron. Srud., 2, 291 (1972). (20) R . L. Daniel and R . B. LeBlanc, Anal. Chem., 31, 1221 (1959). (21) J. Wernet and K. Wahl. fresenius' Z. Anal. Chem., 251, 373 (1970). (22) J. Asplund and E. Wanninen. Anal. Lett., 4, 267 (1971). A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 8, J U L Y 1974
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CdNTA complex and then studying the polarographic cadmium wave to determine the quantity of NTA present. The theory for the fundamental (23, 24) and second harmonic (25, 26) ac polarographic response of systems with first-order preceding chemical reactions has been developed in its most exact form for experiments with a dropping mercury electrode by McCord and Smith. The calculations were based on a rigorous solution of the expanding plane boundary value problem (23). The predictions of these equations have been discussed in detail (24, 26). Matsuda and Tamamushi (27) have used the fundamental harmonic equations in a successful study of the decomposition of the Cd-EDTA complex. In a more recent paper, Hawkridge and Bauer (28) characterized the preceding chemical reaction in the Cu(II)/LiN03 system by means of fundamental harmonic measurements and the McCord-Smith theory. To our knowledge, these remain the only quantitative applications and tests of the theory for the fundamental harmonic response, with the mechanism in question. Furthermore, no experimental studies of the second harmonic ac polarography with such systems, involving quantitative theory-experiment comparisons, have come to our attention. Because experimental contributions in this area have been relatively scarce, we undertook a study of the fundamental and second harmonic ac polarographic behavior of the CdNTA system. This system was chosen because it is well-characterized, because of the number of careful studies described above whose results therefore provided a basis for evaluation of our work. Conditions utilized by Shuman and Shain (17) were selected for our investigation, so their data provide the primary basis for comparison with previous reports. The primary objectives of this experimental effort were to provide for the fundamental harmonic response additional evidence and for the second harmonic response initial evidence concerning the fidelity of the relevant theoretical rate laws. Results of this investigation are reported here. EXPERIMENTAL Apparatus. The polarographic data were obtained using the "harmonic multiplex mode" (29, 30) with the aid of a computerized (Raytheon 704) instrument which provided the following capabilities: a) dc, fundamental harmonic ac, and second harmonic ac polarograms are acquired simultaneously with the aid of on-line Fourier transform data processing (30); bj raw waveform data are obtained over a 0.1-sec interval a t a precisely-timed point in the life of a mercury drop (30); c) polarograms are subject to digital filtering using the interactive Fourier transform smoothing technique (31); d j non-faradaic effects are made negligible a t the frequencies of interest by positive-feedback iR drop compensation (32-34) and subtractive compensation of the double-layer charging current (32, 34); e) real-time analog readout of the ac polarographic waves is provided by a storage oscilloscope display; f ) polarograms are acquired by incremental scanning of the dc potential, with 3-mV increments employed in the present study; gj digital printout of polarographic current amplitudes and phase angles is provided following completion of the dc potential sweep. Full details of the electronic apparatus have been given (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)
T. G . McCord and D. E . Smith, Anal. Chem., 40, 1959 (1968) T. G . McCord and D. E. Smith, Anal. Chem.. 41, 116 (1969) T. G . McCord and D. E . Smith Anal. Chem., 40, 1967 (1968) T. G . McCord and D. E . Smith, J. Electroana/. Chem., 26, 61 (1970), K . Matsuda and R . Tamamushi, BuK Chem. SOC.Jap., 41, 1563 (1968). F. N . Hawkridge and H H . Bauer, Anal. Chem., 44,364 (1972) D. E . Glover and D. E . Smith Anal. Chem., 44, 1140 (1972). D. E . Glover and D. E . S m i t h , Anal. Chem., 45, 1869 (1973). J. W. Hayes, D . E . Glover, D E . S m i t h , and M W. Overton, Anal. Chem., 45, 277 (1973). E R . Brown, T. G McCord, D. E . S m i t h , and D. D. DeFord. Ana/. Chem., 38. 1119 (1966) E R . Brown, D. E . Smith, and G. L. Booman, Anal. Chem., 40, 1411 (1968). E. R. Brown, H. L. Hung, T. G . McCord, D. E . S m i t h , and G . L. Booman, Anal. Chem., 40, 1424 (1968)
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VOL. 46, NO. 8, JULY 1974
previously (30, 35). A computer program written by McCord (36) was modified for the Control Data Corporation Model 6400 computer. It provided the basis for calculating the theoretical response. A Beckman Expandomatic pH meter with a Model 39301 glass electrode was used to determine the p H of the solution. The meter was standardized using buffers of pH 5 and 7 thermostated a t 25 "C. The polarographic cell employed a dropping mercury working electrode (DME), a platinum wire auxiliary electrode, and a saturated calomel reference electrode, Reagents. A stock solution of 1M C d ( N 0 3 ) ~was prepared by dissolving CdC03 in dilute nitric acid, boiling the solution to drive off the carbon dioxide, and diluting to volume. The NTA was recrystallized four times from a hot 1mM Cd(N03)Z solution. NTA is insoluble in water a t room temperature. Therefore, it was necessary to dissolve the NTA in hot water, add sufficient NaOH to bring the solution to a pH near 7, cool and finish titrating to pH 7, and dilute to volume. The NaOH solution was prepared by diluting an aliquot from a saturated NaOH solution to the proper volume. The NTA was an Eastman product, KNOJ and NaOH were Baker Analyzed Reagents, and all other chemicals used were Allied Chemical B and A reagent Grade. The polarographic solutions used in this study were 1.0 X 10- 3M in Cd(NO&, 4.0 X 10- zM in disodium nitrilotriacetate, 1.OM in KN03, and 0.1M in sodium acetate. Enough glacial acetic acid was added to achieve the desired pH. On the basis of evidence from previous studies (5, 13,the resulting solution's buffer capacity is sufficient to assure that the preceding homogeneous reaction (Reaction 2) is pseudo first-order. Procedure. Full details of data acquisition and processing procedures have been given previously (30, 35). Waveform acquisition was performed at the 3.5-sec point in the life of the DME (mechanically-controlled; the instantaneous drop area a t 3.5 sec was 0.023 cm2). Frequencies employed were in the range between 39 and 448 Hz. Specific values are indicated below. A stream of prepurified nitrogen (Matheson) was bubbled through a solution identical to the supporting electrolyte before it entered the Polarographic cells. The cells were degassed for 15 minutes prior to measurement, and were kept under a nitrogen atmosphere throughout the experiment. The temperature of the water-jacketed cells and bubbler was held a t 25.0 f 0.2 "C.
RESULTS AND DISCUSSION
AC polarograms of the CdNTA system exhibited two waves, as expected. The first (most positive) wave showed clear evidence of kinetic contributions of the preceding chemical reaction at all pH's studied, while no evidence of heterogeneous charge transfer kinetic effects was apparent. The second (most negative) wave exhibited characteristic behavior for an electrochemically irreversible process. The qualitative evidence definitely was consistent with the mechanism indicated above (Reactions 1, 2, 3),. The absence of charge transfer kinetic effects a t the first wave greatly simplified evaluation of the pseudo firstorder dissociation constant, kf(kf = kz[H+]) and the equilibrium constant, K = k f / k b (b= kz-[HNTA2-]). Work' ~ 3 9 , and ratios of first ing curves such as cot @ us. ~ 1 (24, and second wave heights us. frequency proved useful in obtaining first estimates for the preceding reaction's rate and equilibrium constants. These were subsequently refined to improve the theory-experiment agreement regarding wave shape, particularly in the case of the second harmonic wave. Heterogeneous charge transfer kinetic parameters for Reaction 3 (second wave) were obtained on the basis of ac polarographic wave shapes ( ( ~ 2 ) and the second wave's separation from the first wave ( k s ) . Rationale and further details regarding these procedures are given below. The values of the diffusion coefficients of were assumed t o be equal. Cd2+ (Do) and CdNTA- (Dy) Shuman and Shain (17) measured the value of Do as 3.6 (35) D. E. Glover, P h . D . Dissertation, Northwestern University, Evanston, Ill., 1973. (36) T G McCord, P h . D Dissertation, Northwestern University, Evanston, ill., 1970 (37) D. E. Smith, in "Electroanalytical Chemistry." Vol. 1. A . J. Bard. Ed , M . Dekker, Inc.. New Y o r k , N . Y . 1966.
“I
24 I
A , a
B
I
0
21 46
-056
-062
-068
-074
-080
-386
-092
-098
I ~
I 2
I
~
c0 -5 0
Ee
j /
-356
-362
i do’Is1
Ed
(volli!
Figure 1. Fundamental harmonic ac polarographic results with C d N T A system at pH 4.28 0 = experimental points for total fundamental harmonic faradaic current using 20-mV peak-to-peak sine wave at frequencies of 39.0 Hz ( A ) , 105 Hz ( E ) , 228 Hz (C),and 448 Hz (D). System: 1.00 X 10-3M Cd2+, 4.0 X 10-2M NTA. 1.OM K N O J , and 0.10M acetate buffer, pH 4.28. - = theoretical profile for n1 = n2 = 2, T = 298 O K , A € = 10.0 mV, Do = DR = 3.6 X c m 2 sec-’, kSl = m , k S 2 ’ = 1 . 5 X lo-’ c m sec-’, a 2 = 0.63, k f = 9.5 s e c - l , K = 7.0 X 10-3, CO* = 1.00 X lO-3M.Expanding plane model
x 10-6 cm2 sec-1, and this value was used in the theoretical calculations presented here. It should be mentioned that the mercury drop area cannot be determined with great precision, since the formula relating the drop area to the measured capillary characteristics neglects the inactive area a t the orifice of the capillary. This introduces a small error in the theoretical current magnitudes. In order to avoid this problem, the factor required to equalize the theoretical and experimental current levels a t the peak of the fundamental harmonic ac wave a t one frequency has been determined and all theoretical currents have been multiplied by this factor. This normalization procedure does not affect the determination of the kinetic parameters, however, since the working curves used in this determination involve current ratios, where any error in the drop area cancels out. The shape and location of the wave are also independent of the drop area. However, one should recognize that theoryexperiment correlations shown below which involve absolute current magnitudes are influenced slightly by this normalization process. In reporting rate and thermodynamic parameters obtained from our measurements, uncertainties given define the acceptable range of values outside of which degradation of the theory-experiment agreement is quite evident upon casual inspection of graphical presentations. Large numbers of replicate measurements were not performed, so valid quotation of average or standard deviations based on direct observation is not possible. However, on the basis of our knowledge of current measurement uncertainty, we estimate that average deviations would be 5-10 times smaller than the uncertainty ranges indicated below, if apparent parameters from individual measurements were taken as the intermediate value in the “ac-
A
B
i
t
‘e< ’
“C8li
1
Figure 2. Second harmonic ac polarographic results with CdNTA system at pH 4.28 0 = experimental points for total second harmonic faradaic current using 20-mV peak-to-peak sine wave at frequencies of 39 Hz ( A ) and 105 Hz ( B ) . = same as Figure 1
-
ceptable range.” The uncertainties indicated for the Shuman-Shain results used for comparison are based on our estimate of the uncertainty (range of reasonable values) associated with the interpolations which we had to perform. The difference in the basis for estimating parameter ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 8, J U L Y 1974
1071
30
r
/
,
/
,
/
/
/
20
IO
0
IO
20
30
40
w”2Irec
o ‘
0
IO
30
20
SO
40
w
‘’E
(
rec
60
70
EO
-‘I*)
Figure 3. Ratio of peak currents of first/second waves as function of frequency at pH 4.28 with CdNTA system 0 = experimental points for total fundamental harmonic faradaic current: conditions as in Figure l. 0 = experimental points for total second harmonic faradaic current: conditions as in Figure 1. - = theoretical profile for fundamental harmonic current. - = theoretical profile for second harmonic current
--
uncertainty should be recognized when comparing our results with those of Shuman and Shain. Results at pH 4.28. Figures 1 and 2 illustrate the correlation between typical fundamental and second harmonic ac polarograms of the CdNTA system at a pH of 4.28 and the predicted theoretical curves. Two waves are present. The first wave is reversible and is produced by the reduction of uncomplexed cadmium, ion. It is kinetically controlled by the rate of decomposition of the CdNTA complex and diffusion. Because of the reversibility of the electrode reaction, k,l and a1 could not be measured. An essentially infinite k,l value (1 x IOzo cm sec-l) and an a2 = 0.5 were used in the computer program to express the reversibility of the electrode reaction. The pseudo-firstorder rate constant of the dissociation reaction, k f , yielding the best theory-experiment fit (Figure 1) was determined to be 9.5 f 1.0 sec-l and the equilibrium constant of the chemical reaction, K, was found to be (7.0 f 1.0) X 10- 3 a t this pH. The second ac polarographic wave is irreversible and is due to the direct reduction of the CdNTA complex. The kinetic parameters used to obtain the theoretical curve were ks2 = (1.5 0.3) x lo-’ cm sec-l and a2 = 0.63 f 0.03. In principle, the shape and magnitude of the second wave may be used to further verify the kinetic parameters determined from the first wave because, while the mechanism of the electrode reaction which produces the first wave may be expressed as
*
Y
ki
S
O
k,,a , n
* R
kb
the mechanism of the electrode reaction which generates the second wave may be written
Thus the kinetic parameters kf and k+, are included in the theory for the second wave. However, under the conditions of this experiment, kf is small relative to k b and thus the reaction favors species Y. It is for this reason, together with the slow electrode reaction, that a negligible effect of the preceding reaction on the second wave is observed. It can be seen from Figures 1 and 2 that the height of the predicted and observed second wave decreases relative to the height of the first wave in a uniform manner as the frequency increases. In Figure 1 a t 39 Hz, the second wave is larger than the first wave; a t 228 Hz, the two waves are 1072
ANALYTICAL CHEMISTRY, VOL. 46, NO. 8, J U L Y 1974
50
€0
70
EO
-v= 1
Figure 4. Results for fundamental harmonic phase angle cotangent at current peak of first wave at pH 4.28 with CdNTA system 0 = experimental points: conditions as in Figure 1. profile of cot $
- = theoretical
nearly equal in height; at 448 Hz, the first wave is the larger of the two. Actually, the heights of both waves increase slightly with increasing frequency, but the height of the first wave changes with frequency more than that of the second wave because the latter is nearly irreversible (38). This trend suggested the use of the fundamental and/or second harmonic working curves shown in Figure 3. In that graph, the ratio of the peak current of the first wave to that of the second wave is plotted us. the square root of the angular frequency for both harmonics. As with the polarograms, excellent theory-experiment agreement is afforded by the rate parameters given above. The shapes of the second-harmonic ac polarograms are particularly interesting from the standpoint of verifying the mechanism of the reaction. A nearly reversible electrode process, as the Cd(II) ion exhibits in the absence of NTA, gives rise to a second harmonic ac polarogram with two nearly equal peaks (37) [slight deviations from equality result from spherical diffusion effects (39)]. However, in the present case, the second harmonic polarogram of the first wave is reduced to a peak with only a shoulder on the anodic side due to the kinetic effect of the preceding chemical reaction (26). The irreversible nature of the electrode process in the second wave generates a peak with a shoulder on the cathodic side (38). Both waves are much smaller than with a comparible reversible cadmium system. Another sensitive test of the mechanism of the reaction is shown in Figure 4, where the cotangent of the fundamental harmonic phase angle a t the current peak of the first wave is plotted as a function of the square root of the angular frequency. In a reversible process, the cotangent of the phase angle has a value of unity over the entire frequency spectrum (37). A “hump” in the curve in Figure 4 a t low frequencies is indicative of either a preceding or a following chemical reaction (37). The limiting slope at 0 Hz is a function of the rate of the chemical reaction (24). At higher frequencies, the effect of the homogeneous chemical reaction is outrun and the value of cot 4 approaches the limiting value of unity. As is seen when comparable curves a t other pH values are examined, the cot 4 us. a 1 1 2 plot also is very responsive to changes in the kinetic parameters of the preceding chemical reaction. For example, changing the value of kp from 9.5 to 15 sec-l will change the value of cot 4 a t 448 Hz from 1.5 to 2.5. Thus, to obtain the close agreement between theory and experiment in Figure 4, magnitudes of the kinetic parameters of the preceding chemical reaction must be selected rather precisely. (38) D. (39) T.
E.Smith and T. G. McCord, Anal. Chem., 40,474(1968) G. McCord, E. R. Brown, and D. E. Smith, Anal. Chern.,
1615 (1966).
38,
Table I. Potential Separation of the Current Peaks of the First and Second Waves at pH 4.2& Frequency, HZ
448 228 105 39
Fundamental ac polarogram, mV
2nd harmonic ac polarograms, mV
Theory
Experiment
Theory
198 198 192 186
203 191 194 205
180 174 174 168
Experiment
185
... 181 165
Conditions are the same as in Figure 1. Experimental values are precise to +6 mV. Theoretical values based on k S 2and a*-values in Table 111.
Whereas a2 can be calculated from the shape of the second wave, the magnitude and shape of such irreversible waves are ks-independent (28). However, the position of the second wave relative to the first wave on the potential axis is a measure of the value of the heterogeneous rate constant, k s 2 , for the direct reduction of the CdNTA complex. This method of measuring ksz is not terribly accurate, but it is the best method available when the wave is highly irreversible. Table I contains a list of the potential separation between the current peaks of the two waves a t various frequencies for both the fundamental and second harmonic ac polarograms. From these data, the value of ks2 for the second wave a t a p H of 4.28 was determined. The ks2-va1ue obtained (quoted above) is essentially identical to the (1.3 f 0.3) X lo-' cm sec-l value obtained by interpolating the Shuman and Shain results. Similar excellent agreement applies to the 0.63 a-value obtained. Interpolation of the data of Shuman and Shain from a plot of kf us. pH gave a value of kf = 25 f 3 seca t a pH of 4.28. The value reported here is 9.5 A 1.0 sec-l. In other words, the values differ by a factor of about 2.6. This difference may be attributed to any of a number of factors such as differences in systematic error level attending the ac and cyclic voltammetric procedures, failure to precisely duplicate solution conditions in the two studies, etc., all of which would be merely speculative. Regardless, the disparity is not unduly serious considering the marked difference in experimental techniques and the fact that this represents the largest disparity between our results and the Shuman-Shain data. All other kinetic and thermodynamic parameters reported here a t pH 4.28 and the other pH's agree with the results of Shuman and Shain within the experimental error attending the ac polarographic measurements and interpolation of the Shuman-Shain data. Results at pH 3.05. At pH 3.05, only one wave appears in the dc polarogram. A plot of log (&-i)/ius. potential is linear with a slope of 32.2 mV, indicating that the electrode process is a reversible two-electron transfer on the dc time scale. However, in the fundamental and second harmonic ac polarograms, a very small second wave can be detected. The basic shape of the second harmonic polarograms is the same as that of those measured a t pH 4.28. The first wave has a shoulder on the positive side while a very small shoulder appears on the negative side of the second wave. Kinetic effects of the preceding chemical reaction on the first wave are easily detected. As theory predicts for more facile preceding chemical reactions (24, 3 3 , the hump in the cot 4 - u1/2 profile is lower and more broad than a t pH 4.28. At high frequencies, the cot 4 values again approach unity. The rate parameters giving the best theory-experiment agreement at this pH were determined to be kf = (3.5 f 1.0) x lo2 sec-l and K = (4.9 f 1.0) x These values are virtually identical to those interpolated from the data of Shuman and Shain. From the shape of the second wave, the value of a2 was
Table 11. Potential Separation of the Current Peaks of the First and Second Waves at pH 3.050 Fundamental ac polarograms, mV
2nd Harmonic ac polarograms, mV
Frequency,
Hz
Theory
Experiment
Theory
448 228 105 79 39
156 150 144 141 129
148 145 138 142 139
138 132 129 123 117
Experiment
119 125 151 113 104
a Conditions are the same as in Figure 5. Experimental values are precise to 1 1 0 mV. Theoretical values based on k., and otvalues in Table 111.
50
4o
:
r
/
I
1
/
A
/
/ /.
3
20
A,
0
o
p
I O
01 0
10
20
30
w
40
50
60
iSec-7')
Figure 5. Results for ratio of c a t h o d i c / a n o d i c second h a r m o n i c p e a k currents for CdNTA system first wave at p H 3.05 a n d 4.28
0 = experimental points for pH 3.05, other conditions same as in Figure 1. 0 = experimental points for conditions same as in Figure 1 (pH 4.28). - = same as in Figure 1, except ks2 = 1.6 X cm sec-', kl = 3.5 X l o 2 sec-', K = 4.9 X l o T 2 .- - - - - = same as in Figure 1 (solid curve)
again found to be 0.63. From potential differences between the first and second wave current peaks (Table 11), it was concluded that ksz = (1.6 A 0.5) x 10-6 cm sec-l. Another useful test of the values of the kinetic parameters is shown in Figure 5, for pH 3.05 and 4.28, in which the ratio of the peak current to the shoulder current for the first wave of the second-harmonic ac polarogram is . curve is very sensitive plotted as a function of ~ 1 1 2 This to changes in the rate and thermodynamic constants .of the preceding chemical reaction. For example, it is easy to distinguish between a kp of 2.0 x lo2 and a kf of 3.5 x IO2 using this type of working curve. Since this curve involves a ratio of currents, i t is also free from error in measurement of drop area. Results at pH of 4.95. A further test of the theory was made by obtaining data a t a pH of 4.95. This is the opposite extreme from the conditions a t pH 3.05, since the first wave is now so small as to be almost undetectable, while the second wave is now large. Since the chemical rate parameters are determined largely from the first wave, this is perhaps the most difficult test of the ac polarographic technique. Shuman and Shain did not report a value of kf at this pH, although they report a value of K of 3.5 x They measured kf to a pH of 4.57 and an extrapolation of their data (Table 111 of Reference 17-mean value of three results a t pH 3.99 employed in extrapolation) yields kf = 3.4 f 1 sec- a t pH 4.95. Although substantial uncertainty attends this extrapolation because of the scatter in the Shuman-Shain data (at pH 3.99), the foregoing values of K and kf do produce theoretical ac polarographic responses which agree quite well with our experimental results for the first wave a t pH 4.95. The correlations were somewhat less satisfactory than those obtained in the A N A L Y T I C A L CHEMISTRY, VOL. 46, N O . 8, J U L Y 1 9 7 4
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Table 111. Compilation of Kinetic and Thermodynamic Parameters Obtained in This Work and from the Data of Shuman and Shain pH 3.05
Parametera
K r (sec-1)
K hSz(cm sec-l) 012
This work S and S This work S and S This work S and S This work S and Sb
(3.5 -I 1.0) x 102 (3.5
+ 0.5)
*
X
(4.9 1.0) x (4.9 i 0.4) X (1.6 i 0 . 5 ) X (4.0 i 1.0) X 0.63 i 0.3 0.63
lo2
10-2 10-2
10-6 10-6
pH 4 . 2 8
pH 4 . 9 5
9.5 i 1.0 25 i 3 (7.0 1.0) x 10-3 (8.0 i 0.3) x (1.5 f 0.3) X 10-7 (1.3 + 0.3) x 10-7 0.63 + 0.3
3.4 i 0.8 3.4 i 1.2 (3.5 i 0.4) x 10-3 ( 3 . 5 + 0.1) x 10-3
*
0.63
... ... ...
...
*
Parameters evaluated and uncertainties estimated as indicated in text. Values taken directly from Shuman and Shain text where uncertainties were not indicated.
more ideal pH range (-4.3), but this is to be expected considering the more demanding operating conditions which existed. Regardless, it was readily concluded that K and kf values which were essentially identical to those deduced from the Shuman-Shain work gave the best theoryexperiment fit with our data.
CONCLUSIONS The results presented above are quite satisfactory concerning theory-experiment correlation and agreement between rate ‘parameters deduced in this study and precedents in the literature. Specifically, it is reasonable to observe that: a) The experimental results are in excellent agreement with the theoretical predictions for both the fundamental and second harmonic ac polarographic techniques. Any discrepancies between theory and experiment are well within the limits of experimental error. b) The results of the two ac methods are in mutual agreement. c) The values of the rate parameters obtained in this study are nearly identical with those obtained by Shuman and Shain ( 1 7 ) using cyclic voltammetry. As they have shown, a change in hydrogen ion concentration of about two orders-of-magnitude will produce a change in the rate constant of the decomposition reaction preceding charge transfer of two orders-of-magnitude and a change in the equilibrium constant of about one order-of-magnitude. The results also agree with those reported by Rabenstein and Kula (18) using proton NMR spectroscopy. Table 111 provides a compilation of our results and those of Shuman and Shain a t the three pH’s investigated in this work. In evaluating the data and the foregoing remarks, it is important to recognize that the ac polarographic measurements were performed under far-from-ideal conditions with regard to current amplitude. In all instances, the second polarographic wave was characterized by the rather minimal response associated with an electrochemically irreversible process (such waves were essentially undetected in many early studies) (38). Relative to the reversible response, the first polarographic wave also was severely attenuated due to the kinetic effects of the preceding chemical reaction. Indeed, a t two pH’s employed the first wave was comparable to (pH 4.28) or smaller than (pH 4.95) the second irreversible wave. Despite these factors, the data acquisition and processing techniques employed (29-35) revealed the desired observables with sufficient clarity to produce chemical kinetic data whose precision
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was a t least comparable to that provided by competing chemical and electrochemical relaxation methods, and often with greater tolerance to the choice of experimental conditions (e.g., dc polarography appears to be inapplicable a t pH 3.05, and the extreme K-values a t all pH’s employed would have precluded most chemical relaxation methods at these pH’s). Furthermore, the data presented here make evident the possibility of performing certain ac polarographic measurements under even more demanding conditions, if necessary. The shape of the second harmonic polarogram is a particularly good test of the mechanism of the reaction. The ratio of the height of the cathodic peak to the height of the anodic peak ,in the first wave of the second harmonic polarogram is especially sensitive to the rate of the preceding chemical reaction. In either polarographic technique, the ratio of the peak currents of the first and second waves may be used to determine the rate and equilibrium constants of the preceding chemical reaction. In fundamental ac polarography, a plot.of the cotangent of the phase angle as a function of frequency varies in a predictable manner with changes in the kinetic and thermodynamic status of the preceding reaction. When the measurement of both fundamental and second harmonic ac polarograms is practical, the results of the two techniques can provide a useful cross-check. The development of harmonic multiplexing (29, 30), in which the dc, fundamental, and second harmonic polarograms are measured simultaneously, has greatly facilitated this marriage of the two methods. With some systems, of course, use of both techniques may not be possible. For example, if difficulties with large residual charging currents are encountered, the second harmonic ac polarogram may be the better of the two methods. The important thing to note is that either technique may be used to obtain accurate kinetic data and to elucidate the mechanism of the reaction. The choice, if one is to be made, must rest on the experimental problems a t hand.
ACKNOWLEDGMENT The authors are indebted to D. E. Glover, S. C. Creason, and J. W. Hayes for helpful suggestions and discussion. Received for review October 15, 1973. Accepted February 25, 1974. This work was supported by National Science Foundation Grant GP-28748X. K.R.B. was an N M Fellow, 1969-72 and an Electrochemical Society Summer Fellow, 1969.
