the electrode reaction is soluble in the solution, it is possible to obtain independent values of y0 and y20 from current time curves taken near the formal E o for the system. Since DO can be determined separately from data obtained at very cathodic potentials, both 0 and DR can be obtained from cathodic experiments only. When substance R is soluble in the mercury, i t is possible to make the same calculations using Equation 1. These calculations were carried out for the Cd(I1)-Cd(Hg) system used in this work, and a value of 2.303(RT/nF) log 0 = 0.606 volt US. Ag-AgC1 was obtaiwd. For DB, a value of 1.6 X sq. cm./sec. was obsq. tained, compared with 1.61 X cm./sec. obtained from direct anodic
dissolution of cadmium amalgams (11). This procedure should make it possible to calculate 2.303(RT/nF) log 0 to =t2 mv., and D R to about 5% from reduction experiments only. The method should be useful in those cases where it is not feasible to prepare stable amalgam electrodes for direct study. LITERATURE CITED
(1) Alberts, G. S., Shain, I., ANAL. CHEM.35,1859 (1963). (2) Delahav, P., “New Instrumental Methods” in Electrochemistry,” p. 61, Interscience, New York, 1954. (3) Delmastro, J. R., Smith, D. E., ANAL.CHEM.38, 169 (1966). (4) Lingane, P. J., Zbid., 36, 1732 (1964). (5) Macero, D. J., Rulfs, C. L., J . Am. Chem. SOC.81, 2942 (1959).
(6) Reinmuth, W. H., ANAL. CHEM.33, 185 (1961). (7) Shain, I Martin, K. J., J . Phys. Chem. 65.254 11961’). (8) Shain, ’I., Polcyn, D. S.,Zbid., 6 5 , -1849 ---
(1981). \ - - - -
(9) Skobets, ’E. M.,Kavetskii, Tu’. S., Zh. Fiz. Khim. 24, 1486 (1950). (10) Stevens, W. G., Ph.D. Thesis, Uni‘ versity of Wisconsin, 1966. (11) Stevens, W. G., Shain, I., J . Phys. Chem. (in press). (12) Turnham, D. S., J . Electroanul. Chem. 10, 19 (1965). (13) Underkofler, W. L., Shain, I., ANAL. CHEM.35,1778 (1963).
This work was performed in part while W. G. Stevens held an NSF predoctoral fellowship, and also was supported in part by funds received from the U. S. Atomic Energy Commission under Contract No. AT(11-1)-1083.
The Depletion Effect in Pulse Polarography with Stationary Electrodes K. B. OLDHAM’ and E. P. PARRY North American Aviation Science Center, Thousand Oaks, Calif. Normal pulse polarography involves the application of voltage pulses of gradually increasing amplitude to an electrode which is held at a potential (resting potential) more positive than that a t which reduction occurs. The pulse duration i s a t least 50 times smaller than the time between pulses. With reversible systems reoxidation occurs during the time between pulses so that the reductant is fully regenerated. With irreversible systems, no reoxidation occurs, and a depletion effect is observed in unstirred solutions. This paper describes the depletion effect from both a theoretical and experimental point of view and points out its rather wide occurrence. Both stirring and proper choice of resting potential can eliminate this effect in applications of normal pulse polarography.
I
a succession of voltage pulses is applied to an electrode of constant area. The interval between pulses T exceeds the duration 7 of each pulse by a factor of 50 or greater. When employed analytically to study the electrode reaction Ox ne .-,Red (1) N NORMAL PULSE POL.4ROGRAPHY
+
the applied potential during the pulses steadily increases as shown in Figure Present address, Chemistry Department, The University, Xewcastle-uponTyne, England. 1
9 J 360
la, so that initially the potential is inadequate to cause the reduction of Ox. However, toward the end of the experiment the electrode is completely concentration polarized with respect to Ox during all pulses. Between pulses the electrode potential is returned to a resting value at which reaction (1) does not proceed from left to right. All of the theory and many of the experiments described in this communication are, however, concerned with one or other of the situations shown as b or c in Figure 1, in which the potential of the first pulse is already -ufficient to cause complete polarization with respect to Ox. The current measurement in pulse polarography is the difference between the instantaneous cell current measured toward the end of each potential pulse and immediately prior to the pulse. Measured currents are true diffusionlimited currents if the potential pulses are as in Figures 16 or IC. Normal pulse polarography can be used to advantage with solid electrodes. It has been shown, for instance, that in the reduction of silver ion a t a platinum electrode, the same curves can be obtained if the electrode is in stirred solution or is in a quiet solution (4). This is as expected since the silver is reduced for only the very brief period of the pulse but the electrode is kept for a much longer period at a potential where the silver metal is reoxidized so that no net deposit of metal accumulates on the electrode surface. Since the diffusion layer during pulses is thinner
than the shear layer in a stirred solution, stirring has essentially no effect on the polarogram obtained (4). The foregoing is completely valid if the resting potential is chosen so that the metal on the electrode surface is completely oxidized between pulses. However, in an unstirred solution and for a sufficiently irreversible system where a resting potential cannot be chosen a t which oxidation takes place B net depletion will occur during the very short pulse which will not be eliminated during the much longer waiting period between pulses. I n spite of the fact that the interval between pulses exceeds the pulse duration 50 to 100 ti,nei, this depletion effect is quite significant. The object of the present work is t’o describe this effect both experimentally and theoretically and show its wide applicability. The effect ha3 been qualitatively observed in the reduction of nickelous, chromate and iodate ions, and in the oxidation of ascorbic acid at the hanging mercury drop, in the reduction of gold(II1) at the pyrolyt’ic graphite electrode, and lead ion at t’he platinum electrode. Since adequate stirring eliminates the depletion effect, pulse polarographic analyses of irreversible systems should be carried out in the presence of stirring. THEORY
Prior to the first pulse, a uniform concentration C, of Ox exists throughout the cell. Subsequently, the Ox concentration a t a distance z from the VOL. 38, NO. 7, JUNE 1966
867
br POtl. Wove
A
0.93623-
-
0.94 0.92
0.9 0.8 0.7 0.6
Resting Potentiol-
Figure 2. Concentration profiles: AB, at time t = 0; BC, at time t = 7; DE at time t = T according to Equation 6; EF at time t = T according to Equation 10; CD at time t = T 7 according to Equation 1 1
+
In calculoting these profiles, the ratio T/r = 100 has been used. tenfold change of scale a t C(x,t)/C, = 0.90
when the first pulse was initiated. Diffusion theory gives the profile resulting from the decay of this Dirac delta function to be of the following Gaussian form :
electrode and a t a time t after the imposition of the first pulse is given by (6)
I n this equation, D is the diffusion coefficient of Ox and erf denotes the error function. The current a t any instant during the first pulse is readily calculated as
(3) which will be recognized as the familiar Cottrell expression (2). Equation 3 ignores any contribution to the current due to migration and to nonfaradaic processes: experimental conditions during pulse polarography should decrease the importance of these two effects. Integration of Equation 2 between x = 0 and x = 03 gives the total number of moles of Ox reduced during the first pulse as
Figure 2 includes a plot of Equation 2 and shows that virtually all of tkese Ai\', moles are removed from a zone of thickness no greater than 5 6 . At t = T the first pulse ends and the curient immediately falls to zero. The steep concentration profile shown as BC in Figure 2 now commences to decay as Ox diffuses toward the electrode from the bulk of the solution. We are now interested in determining the concentration profile during the interval between the first and second pulses, and in particular a t t = 1. We can write this as a XIacLaurin expansion, in the distance variable, as follow: 868
ANALYTICAL CHEMISTRY
Note the
C(x,t) = The evaluation of the first four coefficients in the series is discussed in Appendix A. I n view of T ' s being a large time interval in comparison with T , no acute curvature can be expected anywhere in the concentration profile when t = T , so that little approximation is introduced for moderate x values in neglecting from Equation 5 the term in 2 4 and those in higher powers of the distance variable. Therefore, from the relations derived in Appendix A, we have
c, -
ANI X A 1 / n D ( t - t*) -52
exp{4D(t
- t*)}
(7)
To determine t* we identify C ( 0 , T ) according to Equation 7 with the exact expression provided by Equation 6. This yields
or, using a power series expansion for arcsin : t*=-
[I + -5T- T
3
691--+ T' 1680T2
. . .] (9)
Equation 6 is an approximation valid a t small x and is exact for x = 0. We now proceed to derive a second approximation for C ( x , T ) which is the more valid the greater the ratio T / T . I n the interval between the first and second pulses the concentration profile changes from a steep deep curve close to the electrode to one which becomes increasingly flat and shallow. I n terms of a concentration us. distance graph, the area enclosed by the lines C = C,, 2: = 0, and C = C(z,t) remains constant and equal to aN1/A. Since T is so much greater than the pulse width 7 it might be expected that the concentration profile at t = T will be virtually identical with that which would have resulted had A N , moles of Ox been removed instantaneously from x = 0 at a time t = t* close to the time t = 0
Hence, under all conditions of pulse polarographic interest] t* is almost precisely equal to one third of the pulse width. Adopting this value, Equation 7 becomes, for t = T :
-3x2 {4D(3T -
T)}
Figure 2 compares the expressions 6 and 10: evidently the agreement is excellent except at the highest values of x , where Equation 6 is known to be erroneous. At time t = T the second pulse commences. This pulse will generate a concentration profile very similar to that which was generated during the first pulse and which is described by Equation 2. However, there are two differences: first, the concentration of Ox adjacent to the electrode is not as
great a t time t = T as it was at time t = 0; and second, the concentration profile a t t = T is slightly curved. These differences are apparent in comparing curves A B and DE in Figure 2 and are reflected in the first and second terms on the right hand side of the following equation which describes the concentration profile during the second pulse and which is derived in Appendix analgous to Equation 7 . Equation 14 will be a better approximation the closer t is to 2T; for t = 2T it may be written
Figure 2 includes a graph of this expression. The current flowing during the second pulse niay be derived analogously to Equation 3 for the first pulse; hence we have (see Appendix B)
i(T
