is fixed while the time required to traverse a given potential interval is set by the scan rate. The more rapid the scan rate, the larger the potential region in which the current obeys the simple exponential relation. However, in any case this relation gives the limiting behavior a t the foot ofthe wave. Thus the behavior of the system on variation of scan rate is qualitatively that shown in Figure 1. The region in which the current obeys the simple exponential relation can be defined quantitatively by the following procedure. The maximum flux which can be obtained is given by Equation 1, with the bulk concentration replacing the instantaneous concentration. If this maximum flux is used as a boundary condition in the solution of the Fick's law equation, the resulting surface Concentration is the minimum which could be observed. If it is assumed that currents can be measured to an accuracy of I%, then the simple exponential relation is obeyed when the minimum surface concentration calculated by this procedure differs from the initial concentration by less than 1%or when
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POTENTIAL
Figure I . Current-potential behavior at the foot of a stationary electrode polarogram as function of potential scan rate 1 to 4. Increasing scan rate
0.341 aec.-l/a (4) assuming a drop time of 5 seconds for the dropping electrode. I n order for the simple exponential relation to be obeyed in stationary electrode polarography a t the potential corresponding
to the conventional polarographic Ex/,, the scan rate would have to be 60 volts per second, assuming an = 0.5 (where a is the charge transfer coefficient) and a temperature of 25" C. This corresponds to a 30-C.P.S. triangular wave of 1volt amplitude, a simply attainable experimental value. The fact that the current is independent of scan rate a t fixed potential a t the foot of irreversible waves allow rapid and simple distinction between these and reversible systeme In cases of the latter type the current i s proportional to the square root of scan rate over the entire polarogram t6), while in irreversible cases square root dependence is achieved only near and beyond the peak potential ( I ) . The analogous criterion of reversibility in conventional polarography, dependence of current on mercury height, has been employed ( 3 ) . The basic principles underlying the two cases are identical. Clearly the increase of current in Nernstian systems with the square root of scan rate can continue only until the current approaches the limit set by the rate of the charge-transfer process.
SIR: Much interest has attended the determination of trace amounts of metals by voltammetric methods. With conventional techniques the limiting factor appears to be the capacitive current associated with the charging of the electrical double layer. Attempts to circumvent this problem have been made in a number of ways, but electrolytic concentration in the
electrode phase prior to determination (3, 7, 9) and electronic separation of the faradaic and capacitive currents (3, 4) have been the most promising. Techniques of the former type suffer the disadvantages that the extra (preelectrolysis) step is time-consuming and increases error, and that they are limited to substances that can be so concentrated. Techniques of the latter type have been
The factor bi exp(Bt) is the rate constant a t the potential considered. Generally the argument of the error function in regions of interest i s sufficiently large that that function reduces to unity. Equation 2 then becomes k/dpT
To put this shown from (4) that a t graphic Eliz
e
(3)
in perspective, it can be the theory of Kouteckjy the conventional polarofor an irreversible system
k / G
1
5 Q.01
5
ANALYTICAL CHEMISTRY
Thus, it IS to be expected that at SUEciently high scan rates any system can be made to conform to the behavior herein discussed. When it has been ascertained that current a t a particular potentid is independent of scan rate, the rate constant for the charge-transfer reaction can be computed directly from the concentration of the species in solution and the measured current. The transfer coefficient is then obtainable from the slope of a log k us. E plot. In experiments a t very high scan rates, correction for capacitive charging becomes imperative. This can be accomplished readily by measuring current a t several scan rates and performing a straightline extrapolation to the value a t zero scan rate. Linearity of the plot verifies that Equation 3 is satisfied. In studies of this type there are advantages t o employing B bnormally high concentrations of the electroactive species. The analog of this procedure in conventional polarography has been employed by Laitinen and Bubcasky (5). Applications of this technique to the study of the mechanism of electroreduc. tion of cyclo-octatetraene will be reported in the near future (7). LITERATURE CITED
W. H. REIMMUTH Department of chemistry Columbia University New York 27, K.Y .
RECEIVED for review October 3, Accepted October 20, 1960.
1960.
applied almost exclusively a t the dropping mercury electrode, where capillary noise (8) limits sensitivity. In addition, in most variations they require very sophisticated electronic circuitry (8) or electromechanical switching arrangemeat of iimited frequency response and .high inherent noise (4, 8). Recently Juliard and coworkers (1, 6, 6) have utilized small-amplitude a.c. po-
larography a t stationary electrodes. An extension of this technique, in which an electronic phase-sensitive detector allows the rejection of the charging current while retaining a major portion of the faradaic current, has been under study in our laboratories and preliminary results show it to be of extremely high sensitivity. Figure 1 shows a polarogram of 10-6LU Cd++ in 0.1M MC1. RepeG itive runs on the same solution show reproducibility of about 1% and it appears that the sensitivihy of the instrument will allow determinations in solutions at least an order of magnitude more dilute. The technique should be very useful in the study of kinetics of fast electrode processes, since high frequencies, a t which the capacitive current is usually the major a.c. component, are often necessary in such studies. A solution of 1013M Cd++ in 0.1M KC1 a t 620 c.p.s. has a well defined wave for Cd++ when the detector is used; without the
the instrument will be submitted in the near future. LITERATURE ClTED
(1) Adams, R. N., Walker, D. E., Juliard, A. L., ANAL.CHEM.32, 1526 (1960). (2) Barker, G. C., Anal. Chim. Acto 18, 118 (1958). (3) DeMars, R. D., Shain, I., ANAL. CHEM.29,1825 (1957).
(4) Jessop, G., Brit. Patent 640,768
Figure 1. Alternating current polarographic waves at hanging mercury drop with lO-*M Cd f 2 in 0.1OM KCI Alternating potential. 38 c.p.s.1 18 mv. [peakto-peak). Scan rate. 0.6 volt per mlnute
detector the wave is barely discernible atop the large charging current. Detailed studies and a description of
( 1950). ( 5 ) Juliard, A. L., J . Electroanal. Chem. 1,101 (1959). (6) Juliard, A. L., Nature 183, 1040 (1959). (7) Kemula, W., Kublik, Z., Anal. Chhim. Acta 18, 104 (1958). (8) Milner, G. W. C., “Principles and Applications of Polarograph ” pp. 132-3, Longmans, Green & New York, 1957. (9) Nikelly, J. G., Cooke, W. D., ANAL. CHEM.29,933 (1957).
&.,
DOKALD E. SMITH W. H. REINMUTH
Havemeyer Hall Columbia University New York 27, N. Y. RECEIVED for review October 7, 1960. Accepted October 20, 1960.
athematical Expression of the Sup Action o Some Cap ry-Inactive M et a l k he Polar aphic Maximum of SIR: Under special conditions it is possible to express by a simple equation the maximum current of nickel ion, produced by the electroreduction of that ion a t the dropping mercury electrode, as a function of the bulk concentration of the ions of barium, calcium, and magnesium used as maximum suppressors (1) : IP = -kc”*
+I,
(1)
where I, = maximum current-that is, current a t the peak of maximum in the currentvoltage curve-at C concentration of suppressor ion, barium, calcium, or magnesium I , = maximum current in the absence of suppressor ion L = a constant C = concentration of suppressor ion in the bulk of the solution There are two special conditions under which the equation is valid.
where A Cr = change in the total ionic concentration, and AC, = change in
the concentration of the particular suppressor ion. B. Constant capillary characteristics
To satisfy condition A, potassium chloride is added to the solution in relatively large excess of the suppressor ion, so that small changes in the suppressor concentration will have a negligible effect on the total concentration of the indifferent ions which serve as supporting electrolytes. Values of C larger than the minimal required to suppress a certain maximum completely render the equation meaningless. EXPERIMENTAL
Solutions of ca. 2 to 3 X 10-9M in nickel ion and 0.lM in potassium chloride were prepared in distilIed water. Exactly 20 ml. of solution were introduced into the electrolysis cell, consisting of the saturated calomel as the reference electrode, and the dropping mercury (m = 0.0110 gram, t = 3 seconds per drop) as the cathode. Polarograms of the solution, after expulsion of oxygen by nitrogen flow through the solution, were automatically recorded with a Sargent Polarograph Model XV.
First, polarograms in the absence of a suppressor were obtained. Polarorams were run a t least in triplicate. mall increments (0.1 ml. of 0.05M solution) of suppressor ion were added to the solution in the cell. After removal of oxygen by nitrogen and thorough mixing of the solution, polarograms were recorded for each increment of suppressor added. In measuring the current, the top of the pen oscillBtions was read (this represented maximal current during a drop’s life). Correction was made for the dilution effect.
8
Figures 1 and 2 are typical polarograms and graphical representations of the experimental results according to the equation described. All cations were obtained from their respective chlorides. THEORETICAL DlSCUSSlON
A reasonable hypothesis for a qualitative explanation of the empirical equation may be developed on theoretical grounds. The thickness of the electrical double layer a t the interface of the mercury electrode and the solution can be expressed by the equation (4) VOL. 32, NO. 13, DECEM
