dilutions and extrapolating to obtain the thermodynamic dissociation constants. While their values for the first and last mentioned acids were in good agreement with other data, the value for phenylacetic acid was lower than previously reported (28, 29) values by 0.006. Recently, however, Smolyakov and Primanchuk (30) have reported a value from conductance measurements slightly lower than the result of King and Prue (27). Further studies are required of the possible effect of small errors of the glass electrodes on the extrapolation to obtain pK. The results presented in Table I1 for the emf of cell I1 suggest a method for checking the performance of glass electrodes in cells without liquid junction without the necessity of using the hydrogen electrode. The difference between the emf given (28) J. F. J. Dippy and F. R. Williams, J . Chem. SOC.,1934, p 161. (29) G. H. Jeffery and A. I. Vogel, Ibid., p 166. (30) V. S. Srnolyakov and M. P. Prirnanchuk, R u m J . Phys. Chem., 40, 493 (1966).
in Table 11, column 5, for any two buffer solutions is the theoretical emf change for a perfect glass electrode transferred between two chloride-containing buffer solutions, each containing a silver-silver chloride reference electrode. Any deviation from this value amounts to an error of the glass electrode, which can be attributed to one of the solutions if there is good reason to believe that there is no error in the other. It would be advisable to choose the highest buffer molality given in Table 11. This method is considered superior to methods involving cells with liquid junction. A somewhat similar method but using high ionic strength solutions has been suggested recently by Light and Fletcher (31). RECEIVED for review Oct. 23, 1967. Accepted Nov. 24, 1967. One of us (A.E.B.) thanks the Science Research Council for the award of a Research Studentship. (31) T. S . Light and K. S. Fletcher, ANAL.CHEM., 39, 70 (1967).
Derivative Chronopotentiometry of Multicomponent Systems P. E. Sturrock, W. D. Anstine, and R. H. Gibson1 School of Chemistry, Georgia Institute of Technology, Atlanta, Ga. 30332
The theory and technique of derivative chronopotentiometry are extended to systems containing two or more electroactive species. Theoretical equations, applicable to reversible electrode processes, are derived and experimentally verified. As in most voltammetric methods, the sensitivity is limited by the double-layer charging current. The feasibility of standard addition procedures is shown, even in cases where the initial solution contains such a low concentration that a significant portion of the current is used in charging the double layer.
IN PREVIOUS PAPERS (1-3) the theory and instrumentation for derivative chronopotentiometry, as well as applications to systems of one electroactive species, have been reported. In this paper the relationships between the minimum of the dE/dt function and the transition times are derived for systems containing more than one electroactive species. Delahay and Mamantov ( 4 ) reported the relationship between concentrations and transition times for two consecutive electrode processes. Reilley, Everett, and Johns ( 5 ) extended the relationship to multiple consecutive electrode processes. The potential-time relationships in such systems have not been reported previously but follow readily from application of the response function additivity principle of Murray and Reilley (6). Present address, Department of Chemistry, University of North Carolina at Charlotte, Charlotte, N. C . 28205. (1) P. E. Sturrock, J. Electroanal. Chem., 8, 425 (1964). (2) D. G. Peters and S . L. Burden, ANAL.CHEM., 38,530 (1966). (3) P. E. Sturrock, Gregg Privett, and A. R. Tarpley, J. Electro-
anal. Chem., 14, 303 (1967). (4) Paul Delahay and Gleb Mamantov, ANAL.CHEM.,27, 478 (1955). ( 5 ) C . N. Reilley, G. W. Everett, and R. H. Johns, Zbid., 27, 483 (1955). (6) R. W. Murray and C. N. Reilley, J. Electroanal. Chem., 3, 182 (1962).
Using this principle, the bulk concentration of the jth electroactive species, Co,j*,is given by Equation 1.
During thejth step of the chronopotentiogram-Le.,
c
for
j-1
3
Tm
m=l
> t >m = lTm
[t"2
-
(5
m-1 Tm)'/z]
(2)
where Co,jand C,,5 are the surface concentrations of the oxidized and reduced forms of the jth electroactive component. The initial bulk concentration of the reduced form, C,,5* is assumed to be zero. Solving Equations 1 and 2 for Co,jr and C r , jand substituting into the Nernst equation gives the potential time relationship for the jth step of the chronopotentiogram.
Equation 3 is valid, provided the jth couple is reversible. However, reversibility of preceding electrochemical processes is not necessary. Differentiating Equation 3,
dE -=dt
-RT 2n5Ft1l2
X
(4)
VOL. 40, NO. 3, MARCH 1968
505
For the case, j = 1, Equation 4 reduces to that for a singlecomponent system (2). Following from the IUPAC sign convention, the electrode potential for a reduction process becomes more negative with the passage of current and thus the dE/dt function is negative. Therefore, the derivative of the point of minimum slope of a reduction chronopotentiogram is a maximum. However, a practice often used in chronopotentiometry is to plot - E us. t and in an analogous manner -dE/dt is plotted us. t or E for a derivative chronopotentiogram. The maximum of the dE/dt function then appears as a minimum in the plotted curve. This minimum is evaluated by taking the second derivative and equating it to zero. nj
(g
rm)
m=l
6)
m,n =
a
+ (a2--4.5a RT/F + +1X
1 2a + (a2 - a + 1)UZ - 1
1 p 2
1
+
a
+ (a2 - a +
-2
1
(5) where I
I
I
I
1
Figure 1. Chronopotentiograms for 9.4 x 10-sF Cd(NO& in the presence of 1.0 X 10-2F Pb(N03),
A digital computer was programmed to solve the right side of Equation 5 for specified values of a and to print out the results in tabular form. All of the terms on the left side of Equation 5 are either known or obtained by experiment. Then the corresponding value of a can be read from the table. Equations 5 and 6 are also applicable to single-component, multistep cases. EXPERIMENTAL
The instrument used for these studies incorporates a Philbrick SP 456 amplifier as a galvanostat and P45AU amplifiers for all other functions. The basic circuit is similar to that previously reported ( I ) . Readout was obtained with a Tektronix 561A oscilloscope equipped with a 3A72 dual-trace vertical amplifier and a 3B4 time-base unit. Photographs of the oscilloscopic traces were taken with a Tektronix C-12 camera using Polaroid 3000 film. Cell. The test electrode was either a hanging mercury drop (Metrohm E-410) or a dropping mercury electrode of conventional design fitted with a solenoid drop detacher. A platinum wire counterelectrode was used for all experiments. The reference electrode was either a Beckman 39170 fiberjunction saturated calomel or a locally fabricated one using an agar-agar salt bridge. The cell was a spoutless 100-ml Berzelius beaker with the electrodes and nitrogen bubbler inserted through a rubber stopper. Chemicals. All chemicals were Baker Analyzed and were used without further purification. Procedure. Conventional and derivative chronopotentiograms were performed on solutions 1.O x 10-T in Pb(NO&, 1.OF in KN03, 1.0 X 1OP2Fin "OB, and ranging from 1.9 x to 1.1 X 10-2F in Cd(NO&. In some experiments the lead concentration was decreased to 5.0 X 1O-V and the solution was then made 5.0 X 10-3F in Cu(NO&. A volume of 25.0 ml of the lead (or lead and copper) solution with supporting electrolyte was placed in the cell and deoxygenated. A second solution was then added to the cell in increments, the cell deoxygenated after each addition, and the chronopotentiogram performed. The solution which was added contained identical concentrations of lead and supporting electrolyte as did the original solution in the cell, and in Instrumentation.
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ANALYTICAL CHEMISTRY
Horizontal axis, 10 mseconds for each major division a. Conventional chronopotentiogram. Vertical axis, each major division equals 0.2 volt b. Derivative chronopotentiogram. Vertical axis, each major division equals 1.05 volts per second c. Derivative chronopotentiogram. Vertical axis, each major division equals 10.2 volts per second d. Continuation of trace c showing derivative chronopotentiogram for cadmium
addition contained cadmium nitrate. Thus the concentrations of lead and supporting electrolyte were held constant while the cadmium concentration was increased throughout a series of chronopotentiograms. Several runs were made first using the hanging drop electrode of known surface area and then repeated using the dropping mercury electrode with the mercury height and timing circuit delay adjusted so as to obtain the same transition time for the lead as was obtained with the hanging drop. The only noticeable difference was that a greater reproducibility was obtained using the DME and so this electrode was used for the majority of the remaining experiments. The dual-trace amplifier of the oscilloscope was usually operated in the alternate mode. Thus derivative and conventional chronopotentiograms were recorded on successive drops of the DME. Usually at least five chronopotentiograms of each type were photographed on one film for each cadmium concentration. Commonly these repeated traces were so reproducible that the photograph appeared to be of only one trace of each type. No depletion effect was noted between successive drops, since the chronopotentiograms of the first few drops were not recorded. Furthermore, depletion effects should be much less than in polarography because polarization occurs for less than 0.1 second out of a drop time of approximately 5 seconds. Current densities ranging from 2 X to 4 X ampere per sq cm were used. The lower current densities were used for the lower cadmium concentrations so as to minimize the effect of the double-layer charging. For the two-component systems, the lead transition time was determined by conventional graphical techniques and
also calculated from derivative measurements using the equation of Peters and Burden (2). The agreement between the two methods was usually within 2 % . The worst discrepancy was 5 % , which occurred at the highest current density. In the three-component systems the derivative technique was not satisfactory for the copper transition because of the irreversible nature of the copper wave. Since the nature of this irreversibility was not relevant to the purpose of this paper, it was not investigated further. RESULTS AND DISCUSSION
Verification of Theoretical Equations. A representative oscilloscopic trace is shown in Figure 1. The minimum of the dE/df trace for the cadmium wave can be easily read from the photograph. However, only the order of magnitude of the transition time can be obtained from the conventional chronopotentiogram for cadmium. A summary of the results is presented in Table I. Both the (&d/D1.b)1’2 and r2values were calculated from rearrangements of Equation 6 for the particular system.
where ri is the transition time for the lead reduction obtained by the derivative technique. Although somewhat obscured by scatter, the ( D C d / D P b ) ’ ’ 2 values in Table I show a gradual increase as the cadmium concentration decreases, followed by a sudden increase as the cadmium concentration is decreased below 3 X 10-4F and r2 becomes less than 3 mseconds. An approximate calculation indicates that at this point the charging current exceeds 1 % of the faradaic current for the combined reduction of cadmium and lead. The sudden increase in the apparent ( D C d / D P b ) 1 / 2 values is therefore attributable to the increasing portion of the current needed for charging the double layer. The gradual increase in the calculated ( D C d / D P b ) 1 ’ 2 values prior to the increase due to charging current was at first thought to be due to experimental error. The average of all values with cadmium concentrations from 3 X lop4 to 1.1 X 10-?F is 0.969 with a standard deviation of 0.023. Such a deviation is well within the range of scatter of conventional chronopotentiometric data for single-component systems and each (DCd/Dpb)’’’ value is calculated from four experimental parameters (two concentrations and two derivatives). However, several previous workers (2, 5 , 7) have reported data indicating an apparent increase of the diffusion coefficient as concentration is decreased. The increasing ( D C d / p b ) ” * values are therefore thought to be due to an apparent gradual increase of Ded as the cadmium concentration decreases, Regardless of the validity of the above point, the relative values is satisfactory evidence constancy of the ( D C d / & b ) l ’ of the validity of the theoretical equations. Table I1 contains the results of a series of experiments on the Cu-Pb-Cd system. In this case, where the concentrations of copper and lead are equal, Equation 6 may be rearranged to give
(7) D. C. Noonan, Ph.D. dissertation, Columbia University, 1967.
Table I. Summary of Results [Cd2+]
[Pb2+] 9.41 X 9.41 X 9.41 X 9.41 X 9.41 x 9.41 X 9.41 X 9.41 X 9.41 X 9.41 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 x 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X 9.95 X
l W 3 1.09 X lW3 1.01 X 9.25 X 8.35 X 10-3 7.39 x 6.36 X 5.26 X 4.09 X 2.93 X 1.47 X 1.37 X IOW2 1.27 X 1.16 X 1.05 x 9.30 X 8.01 X 6.63 X 5.15 X 3.56 X 2.75 X 2.54 X 2.33 X 10-2 2.10 x 1.86 X 1.85 X 1.60 X 1.33 X 1.03 X 9.42 X 7.12 X 3.70 X 1.89 X
io, amp/ sq cm
3.8 X 3.8 X 3.8 x 3.8 X 10-3 3 . 8 x 3.8 X 3.8 X 3.8 X 3.8 X 3.8 X 2.2 x 10-3 2 . 2 X 2.2 X 2.2 x 2.2 x 2.2 X 2.2 X 2.2 X 2.2 X IO-’ 2 . 2 X 2.2 X 2.2 X 10-4 2 . 2 x 2.2 X 2.2 X 2.2 X 2.2 X 2.2 X 2.2 X 2.2 X 2.2 X 2.2 X
10-2 10-2
10-2
10-2
10-2 10-2
10-2 10-2
rl,
72,
msec
msec
14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 49.1 49.1 49.1 49.1 49.1 47.6 49.1 49.1 49.1 47.6 49.1 49.1 49.1
51.6 44.9 40.1 35.5 30.4 25.0 18.8 14.0 8.90 4.43 14.2 12.8 11.7 10.5 9.19 7.87 6.39 4.99 3.34 2.87 2.69 2.43 2.18 1.98 1.94 1.80 1.59 1.40 1.26 1.16 0.96 0.87
(Dcdl Dpb)liZ
0.971 0.945 0.948 0.956 0.961 0.958 0.944 0.952 0.925 0.942 1.014 0.994 0.990 0.993 0.988 0.989 0.977 0.989 0.967 1.043 1.056 1.042 1.036 1.064 1.087 1.130 1.201 1.362 1.394 1.634 2.625 4.646
Table 11. Experiments on Cu-Pb-Cd System
1.37 x 1.27 x 1.16 x 1.05 X 9.30 x 8.01 x 6.63 x
10-3 10-3 10-3 10-4
10-4
10-4 10-4 10-4 1.85 X 10-4
x 3.56 x 5.15
9.42
x
10-6
0.988 0.933 0.948 0.930 0.925 0.929 0.904 0.899 0.894 0.955 1.206
15.2 13.2 12.0 10.8 9.45 8.13 6.50 4.99 3.40 1.88 1.20
[Cu2+] = 5.00 X 10-3M, [Pb2+] = 4.98 X 10-3M, r1 = 13.1 msec, r2 = 39.4 msec, io = 2.1 X lo-* ampere/sq cm.
These data are significant, not only because of the extension to a three-component system, but also because the copper chronopotentiogram is irreversible. Thus the Peters and Burden equation could not be used to calculate the transition time for the copper wave. The total transition time for the combined copper and lead was therefore obtained from the conventional chronopotentiograms. Double-Layer Charging. For a single-component system, a Bard-type correction (8) for the double-layer charging current has proved to be feasible (3). However, attempts to devise a similar technique for the multicomponent case failed because the correction term in the Bard equation is a com(8) A. J. Bard, ANAL.CHEM., 35,340 (1963). VOL. 40, NO. 3, MARCH 1968
e
507
posite of the double-layer charging and the current for the preceding electrode processes. The flux, and therefore the current, for the latter decreases rapidly after the end of the prior transition. An instrumental technique for charging current compensation has been reported by Shults et al. (9), but it seems doubtful to the present authors that such a technique would prove to be feasible for transitions shorter than a millisecond. In the present work the ratios of cadmium to lead concentrations which could be studied were limited by the nature of the test electrode. As indicated above, the dropping mercury electrode was selected because of the greater reproducibility obtainable with it and the rapidity of obtaining repeat experiments. However, it was then necessary to limit total transition times to about 50-msec, and even times of this length appear somewhat questionable. The necessary current densities then led to the charging current limit discussed above. The use of shielded planar electrodes would have allowed far longer transitions and lower current densities with a concurrent lowering of the charging current and increased sensitivity. Application to Analysis. The straightforward application of derivative chronopotentiometry to the analysis of multicomponent systems is theoretically possible by means of Equation 10, which is obtained by combination of Equations 1 and 6.
However, the accumulation of errors from each of the factors would lead to an intolerably high error in the calculated concentration, even as in the application of the IlkoviE equation to polarographic analysis. Thus practical analysis appears to be most feasible by use of either titration or standard addition procedures. A standard addition technique is especially feasible because the a factor is essentially independent of dilution. Thus it is only necessary to make a series of standard additions, calculate a by means of Equation 5 (and the computer-prepared table), and then plot (a - 1) versus the amount added. Any convenient units may be used for the amount added. Figure 2 shows such a plot. Since the horizontal axis represents the amount of Cd(1I) added, the intercept should be zero if no Cd(I1) was present in the original solution and negative if Cd(I1) was originally present. I n this case the intercept (9) W. D. Shults, F. E. Haga, T. R. Mueller, and H. C. Jones, ANAL.CHEM., 37, 1415 (1965).
508
ANALYTICAL CHEMISTRY
Figure 2. Standard addition of Cd(I1) to Pb(I1) See text
indicates that 5.3 pmoles of cadmium were present in the sample, which actually contained 5.0 pmoles of cadmium in the presence of 590 pmoles of lead. Double-layer charging is responsible for the curvature at the low cadmium concentrations in Figure 2. However, the standard additions were continued until the linear region was reached at higher concentrations. Then back extrapolation to the (a - 1) = 0 axis gave the cadmium initially present. '~ of The slope of Figure 2 is [ ( D c ~ / D P L ,X) ~ (micromoles lead)-'] and can be used to calculate approximately the amount of lead in the sample. Additional investigations using derivative chronopotentiometry are under way in this laboratory. These studies include: titration techniques for multicomponent systems (both with and without prebias of the electrode), extension of the technique to microsecond transition times, and application to systems in which homogeneous chemical reactions are coupled to the charge transfer step. The results of these studies will be published in the near future. RECEIVED for review September 11,1967. Accepted December 14, 1967. Presented in part before the 153rd Meeting, American Chemical Society, Miami Beach, Fla., April 1967. Work supported in part by a NSF Faculty Summer Research Participation Grant to R. H. G. and an NSF Institutional Grant to the Georgia Institute of Technology.
