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natural abundance, caution should be used in applying this procedure. LITERATURE CITED (1) Bremner, J. M. In "Methods of Soil Analysis", Black, C. A,, Ed.; American Society of Agronomy: Madison, Wis., 1965; p 1179-1237. (2) Bremner, J. M.; Keeney, D. R. Soil Sci. SOC.Am. Roc. 1966, 30, 577-02. (3) Stanford, G.; Carter, J. N.; Simpson, E. C., Jr.; Schwaninger, D. E. J . Assoc. Off. Anal. Chem. 1973, 5 6 , 1365-68. (4) O'Deen, W. A.; Porter, L. K. Anal. Chem. 1979, 51, 586-89. (5) Porter, L. K.; O'Deen, W. A. Anal. Chem. 1977, 4 9 , 514-16.
(6) Fawcett, J. K.; Scott, J E. J . Clin. Pathol. 1960, 13, 156-59
RECEIVED for review October 29, 1979. Accepted February 8, 1980. Contribution of USDA, Science and Education Administration, Agricultural Research, Fort Collins, Colo., in cooperation with Colorado State University Experiment Station and published as Scientific Series No. 2487. Trade names used in text are included for the reader's convenience and do not constitute any preferential endorsement of products named over similar products available on the market.
Evaluation of Electrochemical Cell Impedance Parameters Dleter Britz Chemistry Department, Aarhus University, 8000 Aarhus C, Denmark
There are times when an electrochemical experiment requires reasonably accurate knowledge of double-layer capacitance c d l and/or uncompensated series resistance R,. An example is the application of iR compensation-one needs to know how much there is to compensate (if R, is constant and a n appropriate technique for this is employed). The simplest expedient is t o increase the level of compensation until the potentiostat/cell system is no longer stable. In a recent review ( I ) , i t was pointed out that this can be a rather inaccurate procedure, as many systems may be unstable a t rather less t h a n 100% compensation and, conversely, some are stable even beyond 100%. T h e alternative of carrying out highly accurate measurements using a bridge ( 2 ) or by impedance phase-angle and -magnitude, either by hand ( 3 )or by means of suitable (and often expensive and otherwise perhaps not needed) instruments may be too costly and time-consuming a n d unnecessarily accurate. For measurements requiring about 1 LTC accuracy, several simple and fast techniques suggest themselves. If only R, is wanted one may simply measure the ac cell current at some sufficiently high frequency, where Cdl plays only a small part (normal ac polarography ( 4 ) ) . Similarly, c d l may be measured at low frequencies. This often works but equally often is rather uncertain because of poor convergence of the impedance 2 to R , at high, and Cd at low, frequencies, and to the equipment failure at high audio frequencies; it also necessitates accurate measurement or control ( 5 ) of the ac signal amplitude, which is not always very convenient. We have used, for some years now, the following technique: assume that the electrochemical cell is a pure RC series element, as in Figure 1, frequency independent. Let Z1and Z 2 be t h e absolute magnitudes of the impedances of this cell at frequencies fl and fi, respectively. These are given by the equations
These are 2 simultaneous equations with 2 unknowns, R , and Cdl, for which we obtain, after some simple algebra:
R, =
(
Z I 2 f l 2- z,2f22 )li2 fI2 -
fZ2
(3)
0003-2700/80/0352-1166$01 .OO/O
(4) or cdl
=
1 4 ~ * f ~ ' -( z RU ,~ 2)
If we then measure the magnitude of the impedance a t a t least 2 frequencies, we can calculate R, and Cd. Our method consists of measuring Z ( f )a t a number (>2) of frequencies, and systematically computing R, for all possible pairs of frequencies (except, by choice, nearest neighbors) and taking the mean of all R, values. A simple computer program is used for these computations. An additional refinement is used: having obtained the mean R,, the program computes Cd, using Equation 5, for all frequencies. A least-squares straight line is then fitted to the set of (f,C d l ) points and the value of R, iteratively adjusted to minimize the slope of the line The search is stopped when the adjustment to R, causing b to change sign, is less in magnitude than a given level e (we or of b < lo4 pF/Hz. In practice, this use e = 6R,/R, = adjustment turns out usually to be rather small, of the order of 0.1 R. We generally take measurements at about 6 to 8 frequencies, and skip frequency neighbor pairs in the computations, giving a total of 10-21 values for R,, all printed out as a check. The measurement of impedance magnitude is done as follows (here for a dropping mercury electrode): a potentiostat is connected, via a switch as shown in Figure 2, alternatively to the cell, or to a dummy cell, consisting of a standard capacitor Cs and a resistance decade Rs in series. Cs is unfortunately necessary to prevent large dc loads on the potentiostat when the dc cell potential is not close to zero. At the end of each mercury drop, the ac current (signal from current follower, amplifier, rectifier, and low-pass filter) is read by a digital voltmeter. The applied ac voltage is about 10 mV rms but need not be accurately known. Cs must be larger than Cd]. Rs is adjusted until the digital current reading is the same for both cell and measuring dummy cell. T h e frequency is read accurately by a digital meter, and Z(f) is computed (by the program) given f, Cs, and Rs. In the case of the dropping mercury electrode, there is a timing problem-one wants the measurement a t a definite time after drop birth. In our present setup, a minicomputer C 1980 American Chemical Society
-R,
frequency a t which measurements are most accurate, and we try to cluster our frequencies around it, perhaps, in a second measurement set. We have used this method for some time now and have built the switch in Figure 2 into our potentiostat. The only accurate components required are the capacitor Cs (an:y calibrated good-quality capacitor will do, or a decade) and a resistance decade box. A series of measurements takes about 30 min and reproducibility has been in the same range as the scatter reported above. T h e foregoing assumes t h a t there is (1) no frequency dispersion and (2) no Faradaic admittance. If frequency dispersion is found, due to distributed cell parameters, there is, in any case, no simple value of R, and one should perhaps then try to eliminate the dispersion by making the cell/electrode system as symmetrical as possible ( I ) . Faradaic components can often be removed simply by measuring a t a different potential-R, is usually (except with passivating layers) potential-independent; if C d l is the wanted quantity, a more complicated procedure such as that of Canagaratna and Karunathilaka (6),which could also be adapted for use with absolute values of impedances, could be used.
cdl
Figure 1. Equivalent cell circuit
Ac input
4P
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Anal. Chem. 1980, 52, 1167-1168
RE
.
Figure 2. Measuring setup: P, potentiostat with current-follower; CE,
counter electrode; RE, reference electrode; WE, working electrode; S, 3-pole, 2-position switch; C, calibrated Capacitor (or decade); R, resistance decade. Drop-knocking and timing circuitry for use with DME not shown
ACKNOWLEDGMENT
I thank my colleague Lars Kryger for generously allowing me time on his minicomputer during much of this work.
controls this timing, knocking off a drop, and obtaining a digital sample (average of 20 over one 5 0 - H ~cycle) of the rectified and filtered ac current signal a t a certain time; however, analog sample-and-hold plus timing circuitry does just as well. A typical run, a t 7 frequencies ranging from 800 to 6000 Hz, in 0.5 M Na2S04a t 0.00 V vs. Ag/AgCl electrode, produced 15 R, values ranging from 68.2 to 70.1 R , with mean 69.3 R, standard deviation 0.8%. The iteration procedure produced a new value of 69.2 R and the resulting C d l values a t the 7 frequencies showed a standard deviation of 0.8%. T h e program also outputs the frequency at which 45" phase-shift obtains, or where R, = 1/(27rJCdI). This is that
LITERATURE CITED (1) Britz, D. J . Electroanal. Chem. 1978, 88, 309-352. (2) Armstrong, R. D.; Race, W. P.; Thirsk, H. R. Electrochim. Acta 1968, 13, 215-239. (3) Britz, D.; Bauer, H. H. J . Sci. Instrum. 1967, 4 4 , 843-846. (4) Breyer, B.; Bauer. H. H. "Alternating Current Polarography and Tensammetry"; Interscience: New York, 1963. (5) Britz. D.; Jackson, J. S.; Bauer. H. H. Chem. Instrum. 1971, 3 , 229-234. __. _. (6) Canagaratna, S. G.;Karunathilaka, S. A. G. R . J . Nectroanal. Chem. 1973, 48, 183-188.
RECEIVED for review December 10, 1979. Accepted March 3, 1980.
Determination of Subnanogram and Nanogram Amounts of Fluoride by Fluoride and Calomel Reference Electrodes Dan Deutsch* and Shahla Zarini Department of Dental Research, Dental School, Hebrew University
of Jerusalem, Jerusalem, Israel AnalaR reagent. A digital pH meter/millivolt meter MBK model A4031 is used to measure the electropotential. Standards. Fluoride standard solutions 0.0, 0.1, 0.5, 1.0, 2.0, and 10.0 ppm F- in 1 M acetate buffer IpH 5.3) are prepared according to the method described by Hallsworth, Weatherell, and Deutsch, 1976(2). Fluoride Determination. To determine the fluoride concentration of the sample solution, a 1-pL drop is placed at the center of the inverted fluoride electrode. The calomel reference electrode is then lowered t o 0.5 mm from the fluoride electrode surface, confining the sample solution drop between this surface and the surface of the ceramic frit junction. The electropotential of the solution is read after 2 min. This potential reflects the fluoride concentration of the solution. To avoid contamination by fluoride adsorbed to the surface of the electrode from a previous determination, the surfaces of both electrodes are rinsed between successive fluoride determinations with distilled water, to reach a potential 50 mV more positive than that obtained for the buffer blank (0.0 ppm F). The
In recent years extremely sensitive methods for determining subnanogram amounts of fluoride with the fluoride electrode have been developed (1-3). These methods involve the use of very small sample volumes placed between the ordinary fluoride electrode and a reference system specially constructed for this purpose. The present method dispenses with the need t o construct a reference electrode by employing the commercially available calomel electrode commonly used for p H measurements. EXPERIMENTAL Fluoride Electrode Assembly. The fluoride electrode assembly (Figure 1) consists of an Orion fluoride electrode, model 96-09 or 94-09, clamped in an inverted position, and a reference calomel electrode radiometer type K401, which is mounted above. The lower end of the calomel electrode is flat and contains at the center a ceramic frit junction, 0.75 mm in diameter. The reference filling solution consists of saturated KC1 solution prepared from 0003-2700/80/0352-1167$01.00/0
C
1980 American Chemical Society
