Anal. Chem. 1903. 55. 1988-1990
1988
Flgure 4. (A) Concematbn profile of sample of 1 mM Fe"'(oralate), in 0.25 M sodium oxalate. pH 4.5. Flow rate was 0.98 mL min-'. The posilwn of the vertical mrs represents time (10 s. 13.5 s. and 17.0 s. respeclhrely) and the width of lhe bars represents the duration of tb cyclic voltammograms which are shown in Figure 4 (insert). Cordbbns of the cyclic voltammograms (insert)are 3 V s-', 100 mV div-'. and p A div-'. E , = zkO.0 V.
undiluted sample. For ow experiment Dd = i*/ip = k where i* is the steady-state current obtained when the constant concentration of the depolarizer is supplied to the electrode. The simplest way to obtain this value is to inject a large (5Ml pL) sample of the iron(II1) oxalate to the background electrolyte so that a steady-state current value is reached. By this procedure the Y axis of Figure 4 has been calibrated in the units of concentration. In the next experiment a 70-pL sample of 1 mM Fe111(oxalate)3 was again injected and single-cyclic voltammograms at 3 V s-l were recorded at times indicated by the vertical bars in Figure 4. These voltammograms are shown in Figure 4 (insert, 2 pA/div and 100 mV/div). The width of the bars in Figure 4 represents the duration of the one cvcle in the correswndine cvclic voltammomam - in F i-m e 4b (insert). The combination of FIA and cyclic voltammetry is attractive because of the flexibility of the former and the diagnostic power of the latter. In planning these experiments the following strategy is recommended. The choice of the FIA manifold must be made to suit the chemistry to be performed. Ample examples of various manifold configurations can be found in the basic FIA literature (8). Although the flow rate and the size of the working electrode can now be well reproduced, the geometry of the detector is less well defined and, therefore, the prediction of the absolute values of the currents is not possible at present. Fortunately, this can be circumvented, by using a well-behaved redox couple to calibrate the system. In the first step, we need to determine the maximum sweep rate a t which the couple behaves reversibly. This will establish the minimum limit of the time window of the experiment. Next, we inject a large (500 pL) sample of this I
_
standard redox couple solution, in order to establish the magnitude of current when sample dispersion equals one (cf. Figure 2), and to calibrate the Y axis in concentration units (cf. Figure 4). It follows from FIA theory and experimental evidence that it is reasonable to expect that a diluted solution of the compound of interest will always have the same dispersion in the same flow geometry. Next, by varying the sample volume, flow rate, and scan rate, we can find the CV, HV, and transition region for given experimental conditions. Finally, the injection of the sample under thus found conditions will allow a direct comparison with the electrochemical behavior of the chosen standard redox couple or to perform assays of an unknown species. T o summarize, the above observations and preliminary results demonstrate the compatibility and ease with which FIA and voltammetric techniques can be combined. It is important to realize that the experiment summarized in Figure 4 represents yet another variation on a theme of gradient FIA techniques ( I ) . The flow-through detector described here has a very small effective volume and, therefore, allows the experimental exploration of a large variety of new approaches based on combination of FIA gradient techniques and voltammetric techniques for both research and technical applications. LITERATURE CITED (1) Leach. R. A,: R i l i a s . J.: Hank. J. M. Anal.
1889-1673.
aWrm. 1983. 55.
(2) Ririi5ka. J: Hansen. E. H. Anal. CMm. Acta 1983. 145. 1-15. (3) Janata. J.: RZIEkka. J. Anal. chim. Acta 1982. 139. 105-115. (4) Current Separalbns. R. E. Shoup. Ed.. Bioanapical systems Inc.. Purdue lndu~lrlalResearch Park. 1205 Kent A".. West Lafayelle. IN 47906. (5) ROiEka. J.: Pansen. E.H.: Ramsing. A. U. Anal. l%h. Acta 1981. 134. 55-71. (8) Bard, A. J.: Faukner. L. "EIBCh~~hemlcaI Mamods": Wiley: New York. 1980. (7) Bard. A. J. "Encyclopedia a1 Ebcbochemishy of me Elements": Marcel Dekksr: New Yon. 1982 Volume IX. Pan A. ( 8 ) RuiiE*a. J.: Hansen. E. H. "Flow i n b e b n Analysis": Wiley: New YWk. 1981.
'Permanent address: Chemirhy DepRmenl A. The Technical Universw 01 Denmark. 2800 Lyngby. Denmark.
Niels Thogersen' J j i i Janata* Jaromir Riii2ka' Department of Bioengineering University of Utah Salt Lake City, Utah 84112
RECEIVED for review April 25, 1983. Accepted July 5, 1983. The support from the Danish Council for Science and Industrial Research to N.T. is greatly appreciated. Part of this work was supported by the NIGMS, Grant Number 22952.
Precision of the Double Known Addition Method in Ion-Selective Electrode Potentiometry Sir: One of the most important problems in analytical potentiometry is the precision in the determination of sample concentration. It is well-known that 1 mV error in the emf reading causes 4% error in the calculated sample concentration if a perfect, i.e., error free calibration line is used and the slope of the calibration line is about 59 mV/decade. Comparable alternative analytical methods, e.g., UV-visible spectrophotometry, typically allow precision and accuracy better than about 1%. There have been therefore many efforts to keep the standard deviation of the emf readings below ca.
0.1 mV and to eliminate possible sources of systematic error. One approach to achieve these goals is to use the calibration method with careful thermostatization, electrical shielding, matching of standards to samples, etc. Another approach is to use addition or subtraction methods or potentiometric titrations. In the latter cases the analyst has to balance the usually contradicting goals of high precision, high accuracy, and low cost of labor. In an earlier study (1-4) we investigated the effect of the precision of individual emf readings on the precision of the
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 12, OCTOBER 1983
sample concentration calculated from the emf readings. Appropriate equations were derived for the relative standard deviation of the sample concentration or, if no explicit solution could be found, we used the Monte Carlo simulation method to obtain quantitative results. This study covered practically all important methods of analytical potentiometry from the calibration method to t#itrations. One of these, the double known addition method (DKAM) has been the subject of a recent paper in this journal ( 5 ) . The authors, who were apparently unaware of the above mentioned publications, used the Monte Carlo simulation method to derive quantitative conclusions about the effect of volumetric and emf errors on the precision of the calculated sample concentration. There are two important differences between this latter (5) and the earlier (1, 4 ) study that deserve attention. One is in the mathematical apparatus used while the other is in the conclusions.
1989
where f denotes the left-hand side of eq 4. By insertion of these values into eq 1 one obtains
DERIVATION OF THE STATISTICAL ERROR Both approaches are based initially on propagation of error statistics. The sample concentration, c,, is calculated from the three emf readings Eo,E,, and Ez,in the sample, after the first addition and after the second addition, respectively, and from the two concentration increments Acl and Ac2. The variance of the calculated c, can then be expressed as
where u denotes the standard deviation of the indexed quantity. uE, and uaC,can be estimated by experiments. T o obtain the five derivatives on the right-hand side of eq 1,it would be desirable to express c, as a function of the EI's and the Acjs. An explicit equation relating c, to these quantities in the form (2) c, = dEo, E l , E29 A% Ac2) where g denotes any suitable function, does not exist, however. Instead one obtains only an implicit equation (3) f(c,, Eo, El, E2, ACl, 4 ) = 0 which cannot be solvecl for c, by rearrangement. The actual equation is C, Ac, Ac, C, + Ac, (E, - E,) In -- (E, - E,) In =o
+
C,
+
CX
14) Since this equation cannot be solved explicitly for c,, it has been assumed in ref 5 that the derivatives dc,/dE, and dc,/dAc, cannot be expressed explicitly either, hence eq 1 cannot be used. ucxwas estimated therefore from simulated experiments with simulated random errors of known variance in the El's and Ac,'s. To obtain ucxfor a single set of the variables 500 parallel experiments were simulated. For each of these, c, had to be calculated by a numerical method. Then sc,, the empirical standard deviation of the 500 c, values was calculated, which was itself only an estimate for uc,and was only reproducible within h3%. For any other set of variables the whole process had to be repeated. This tedious and approximative calculation can be avoided if it is realized that the derivatives dc,/8El and dc,/dAc, can be explicitly calculated from eq 4. A theorem of differential calculus (6) asserts that these derivatives can be determined by the following formulas
ac, = --af/aEl -
aEL
af/ac,
(5)
and
ac,
-=
CVAC]
af/aAc, -~
af/ac,
(6)
L
-
'
,,(,,.,)] i./.,(,,..,..)I c,
c,
ACl
1 ++ - 1
' I-2 -9
c, c,
c, c,
(7)
C,
Here uE, = uE1= uE2 = UE was assumed, and S* is S/ln 10. Equation 5 of ref 1was obtained in the same way but errors of Acl and Ac2 were neglected compared to the effect of emf errors. Equation 7 is an exact and explicit solution of the precision problem of the DKAM. With this equation RSDcx the relative standard deviation of c, can be calculated for any given set of c,, Ac,, Acz, BE, S, uAcland uAC2. It was noticed in the course of Monte Carlo simulations (5) that RSDcI did not depend separately from c,, A q , and Ac, but only from the two ratios Acl/c, and Acz/Acl (or equivalently from Acl/cx and Acz/c,). From eq 7 it is clearly seen why this result was found when uAcl/Acl = aAc,/Ac2was assumed. Clearly, eq 7 can be used even if RSD,, is not constant. The dependence of RSDcron UE could only be vaguely described on the basis of Monte Carlo simulations ( 5 ) ,while eq 7 gives the correct answer. Summing up, eq 7 is a simple, straightforward, and general solution to the precision problem of DKAM.
PRACTICAL LIMITATIONS OF THE DKAM Although different mathematical methods have been used, both studies on DKAM give about the same numerical results. Thus both can conclude that the DKAM has a surprising error magnifying effect if used within the experimentally reasonable ranges of the parameters. In a typical example aE is 0.1 mV and uac/Ac is 0.5%, still the RSDczis 5 % . Compared to an RSDczof 0.4% caused by uE = 0.1 mV when c, is read from a calibration line, this result is obviously disappointing. As was shown earlier ( I ) , this serious loss of precision is caused by the use of small volume increments of the standard solu-
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Anal. Chem. 1983, 55, 1990-1992
tions, which is in turn dictated by the effort to avoid systematic error resulting from a possible change in the sample matrix. The question is then under what conditions is the serious loss of precision a reasonable bargain for the gain in accuracy. The question can be asked also in a more general way: what are the conditions for the DKAM to be preferred to other potentiometric techniques? The sufficient and necessary condition for the DKAM to give valid results is that the modified Nernst equation
E
= E’
+ S log c
does hold during the DKAM measurement. E’ and S must be independent of c but they may change from sample to sample either because the sample matrix causes such changes or because E’ and S are drifting. If E’ and S were not changing from sample to sample, one would use the calibration method. If only E’ changed, the single addition method would be used. There remains one situation when the DKAM could be reasonably used: if S changes from sample to sample (E’ may also change or may remain constant). If the change in S is caused by variations in the sample matrix, we expect from our experience that eq 8 will not hold for such samples, Le., the calibration line of the cell will be nonlinear. This observation limits the range of usefulness of the DKAM to situations when S is drifting with time. This problem can, however, often be overcome by better methods than the DKAM, e.g., frequent redetermination of S or elimination of the drift by a suitable adjustment buffer or by using the analate addition method. If these methods cannot be used for some reason or do not help, only then does the use of the inherently imprecise DKAM seem to be justified. It should also be recognized that the DKAM does not allow for drift that is too fast either, since S has to remain practically constant during one DKAM measurement.
The main limitation in the use of the DKAM was seen to be its error magnifying effect. This may be occasionally counteracted by a rather low uB It has been observed ( 4 , 5 )that the emf differences measured in the course of standard additions (not only double but also single or multiple additions) have a much smaller variance than one would expect on the basis of repeated single readings with a change of corresponding standard solutions. This gain in Q may a t least partly compensate for the error magnifying effect of the DKAM. It should also be noted that if large volume additions are allowed, i.e., they do not cause disturbing matrix effects, then the precision of the DKAM can be considerably improved (1).
In light of the foregoing discussion one might envisage some specific situations when DKAM may prove to be the method of choice; however, in most typical potentiometric applications it appears to be less appealing then other, suitably chosen methods.
LITERATURE CITED HoNai, G.: Pungor, E. Anal. Chlm. Acta 1980, 113, 287. HoNai, G.;Pungor, E. Anal. Chlm. Acta 1980, 113, 295. HoNai, G.;Pungor, E. Anal. Chlm. Acta 1980, 116, 87. Horvai, G. “Critical Study of Potentiornetrlc Measuring Techniques”; Candidate’s Thesis, Budapest, 1979. Efstathiou, C. E.; Hadjiioannou, T. P. Anal. Chem. 1982, 54, 1525. Harbarth, K.; Riedrich, T. “Differentialrechnung fur Funktionen mit mehreren Variablen”; Teubner: Leipzig, 1978. Brand, M. J. D.; Rechnitz, G. A. Anal. Chem. 1970, 42, 1172.
Gyorgy Horvai Erno Pungor* Institute for General and Analytical Chemistry Technical University Budapest, Gell6rt t6r 4,H-1111 Hungary
RECEIVED for review February 14,1983. Accepted July 1,1983.
Analysis at the Micromolar Level by Cyclic Voltammetry Sir: The main limitation to the effectiveness of voltammetric electroanalysis a t low concentrations arises from the need to distinguish between the faradaic signal and interfering “background currents. The latter are particularly severe when the electrode potential and/or area are changing with time, because the background currents then contain a capacitative component. Accordingly, sensitive electroanalytical techniques generally rely on measurements made on stationary (or very slowly growing) electrodes at constant (or very slightly variable) potential. While such techniques do efficiently separate the nonfaradaic interference from the faradaic signal, they suffer the disadvantage of requiring many separate measurements to construct a voltammogram, so that experiments of long duration are needed for a single analysis. Cyclic voltammetry is a technique that has proved extremely useful in qualitative studies of electrode reactions, but it has rarely been used for quantitative analysis. Chief among the reasons for this is the low signal-to-noiseratio. The capacitative current may exceed the faradaic current at concentrations of lo“ M or less, making accurate quantification impossible in this important concentration range. The faradaic sensitivity increases with the applied sweep rate, but the capacitative “background signal” increases faster. The name “cyclic voltammetry” has been applied to a number of related techniques. Here we mean the imposition of a single isosceles triangular potential wave form on an electrode and the recording of the resultant current as a function of potential. The potential excursion is selected to
start a t a value at which the analyte of interest is not reduced (oxidized), to traverse the region of the reduction (oxidation) wave to a reversal potential in the concentration-polarized range, and then to return to the initial potential. A typical cycle might consist of a ramp that changes the electrode potential to a value more negative by about 15RT/nF followed by a similar positive-going ramp; at a sweep rate of order 100 mV sF1, the entire experiment occupies a few seconds only. The cyclic voltammogram consists of two branches, as illustrated in Figure ICfor a reduction process. One branch, the forward branch I‘,is generated by the negative-going sweep; the backward branch i- arises during the positive-going return sweep. Figure la,b, respectively, shows typical faradaic and nonfaradaic contributions to the branches of the cyclic voltammogram. Whereas Figure l a could be used for chemical analysis, Figure ICis useless for this purpose because it contains a nonfaradaic component of unknown magnitude. In principle, voltammogram l a could be reconstructed from IC by subtracting a replica of curve b, determined from a separate experiment on supporting electrolyte alone. Such experiments are time-consuming and the subtractive correction is not always reliable. In this paper we advocate an alternative, and simpler, method of correcting for background interference: adding the
i” and i branches of cyclic voltammogram ICto produce the single wave-shaped curve shown in Figure Id. Because the
0003-2700/83/0355-1990$01.50/00 1983 American Chemical Society
