Derivative Chro nopotent iometry Reversible and Irreversible Processes at Mercury Pool Electrodes DENNIS G. PETERS and STANLEY L. BURDEN Department o f Chemistry, lndiana University, Bloomington, lnd. Derivative chronopotentiometry has been investigated as a new approach to the measurement of transition times and the study of electrode processes. Instead of the potential E of the working electrode, the rate of change of potential with time, dE/dt, is recorded as o function of time. Derivative chronopotentiometry possesses several unique advantages over conventional chronopotentiometry. First, the rninimum value of dE/df for o derivative chronopotentiogram is quantitatively related to the magnitude of the transition time. Therefore, a chief virtue of derivative chronopotentiometry is that it provides the first direct and absolute method for the evaluation of transition times. Second, the shape of a derivative chronopotentiogram is diagnostic of the kind of electrode process being studied. Third, for irreversible electrode reactions, the kinetic parameter ana can be calculated from the minimum value of dE/dt for a derivative chronopotentiogram.
A
to the application of chronopotentiometry for analytical purposes has been the lack of a precise, well defined, and accurate method for the measurement of transition times. In chronopotentiometry, as in other related electrometric methods of analysis, charging of the electrical double-layer causes the transition time to be longer than the value predicted on the basis of the Sand equation ( I d ) , and prevents the potential of the working electrode from varying in accord with the theoretical potential-time relation (8). Similar effects are produced by other extraneous processes, such as the formation or reduction of an oxide film on the electrode surface or the reaction of adsorbed electroactive substance. Various graphical techniques have been proposed previously for the empirical measurement of transition times in an effort to overcome or compensate for the double-layer charging effect. Delahay and Bereins ( 2 ) , as well as Reinmuth ( I @ , suggested methods for the evaluation of transition times which parallel the practices used in conDETERRENT
530
ANALYTICAL CHEMISTRY
ventional polarography to correct an observed diffusion current for the residual current. A paper by Russell and Peterson ( I S ) presents a comparison of several graphical techniques of measurement. However, these methods require graphical extrapolations to be performed along the portions of a chronopotentiometric wave, where the initial and final relatively rapid changes in potential occur, which are most seriously distorted by charging of the doublelayer. Hence, accurate transition-time measurements are often extremely difficult. Our approach to the problem of transition-time measurements is that of derivative chronopotentiometry. Instead of recording the potential E of the working electrode us. time in the usual chronopotentionietric way, we observe the rate of change of potential with time, dE/dt, as a function of time. Derivative chronopotentiometry possesses several unique advantages over conventional chronopotentiometry. First, the minimum value of dE/dt for a derivative chronopotentiogram is quantitatively related to the magnitude of the transition time. Therefore, a chief virtue of derivative chronopotentiometry is that it provides the first direct and theoretically justifiable method for the evaluation of transition times. The suitability of derivative chronopotentiometry for transition-time measurements is further enhanced because the minimum value of dE/dt is least affected by charging of the doublelayer and because it is unnecessary to record either a complete normal or derivative chronopotentiogram. In addition, the shape of a derivative chronopotentiogram is indicative of the kind of electrode process being studied and, for irreversible electrode reactions, the parameter an, can be determined from the minimum value of dE/dt for a derivative chronopotentiogram. An introduction and discussion of the basic concepts of derivative chronopotentiometry are presented below, and theoretical equations for reversible and irreversible electron-transfer processes are derived and verified. Other applications of derivative chronopotentiometry will be considered in subsequent communications.
EXPERIMENTAL
Reagents. Test solutions of lead nitrate in 0.1F potassium nitrate medium of pH 2, thallous nitrate in 0.1F potassium nitrate, and potassium iodate in 1F sodium hydroxide solution were prepared by weight from analytical reagent grade chemicals. Chronopotentiometric Cell and Electrodes. The chronopotentiometric cell, incorporating a mercury pool cathode recessed into a machined plug made of Teflon, was of a design similar to that described by Morris and Lingane (IO) or by Bruckenstein and Rouse ( I ) , and was surrounded by a water-jacket through which water thermostatted a t 25.00' + 0.05' C. was circulated. Prepurified nitrogen was employed to remove dissolved air from the test solution and to stir the solution between measurements. The test solution was usually deaerated prior t o its introduction into the cell to minimize oxidation of the mercury electrode. Before each trial the flow of nitrogen was diverted over the surface of the solution, the constant - temperature water circulator was stopped, and the test solution itself wa5 allowed 1 minute or longer to become quiescent. High purity mercury was used for the preparation of the mercury pool cathodes. To ensure a high degree of reproducibility of the volume of mercury in the cup made of Teflon, and consequently a reproducible electrode area, the mercury was measured into the cup from a microburet. The platinum wire auxiliary electrode and the saturated calomel reference electrode (S.C.E.) were each separated from the test solution by a cracked-glass saltbridge tube (9) filled with the supporting electrolyte solution. We determined the effective area of the mercury pool cathode to be 1.38 i 0.01 by using conventional chronopotentiometric data obtained for the reduction of thallium(1) in 0.1F potassium nitrate solution and by using a value for the chronopotentiometric constant (ir1I2/ AC) of 388 amp. cm. mole-' for this reduction a t 25' C. This latter value was established experimentally during the present study from chronopotentiometric measurements with a planar platinum cathode and is in perfect agreement with the value reported by Bruckenstein and Rouse (I). Instrumentation. A block diagram of the electrical circuitry used in this investigation to obtain and record
CI I 1 I I
c,1 II
CONSTANT CURRENT SOURCE
Figure 1.
Block diagram for derivative chronopotentiometer
R i = 1 0 kilohms; Rz = 1 megohm; R3 = 1 0 0 kilohms; Rq and R5 = 10 kilohmr Helipot (0.25% linearity, 3% tolerance); 1.010 megohm; CI = 0.2 pf; C? = 2 pf.;Ci = 1.011 pf; Cy = 0 . 0 1 9 9 8 pf.
the derivative chronopotentiograms is shown in Figure L. The basic requirement to be met by this circuit is that it produce an output signal which can be measured with a precision and accuracy of 1% on a conventional X-I' recorder. For chronopotentiometric waves encountered in routine work the value of (dEjdt),,,,, which is of theoretical significance in derivative chronopotentiometry, ranges from about 2 mv. second-' (corresponding to a transition time of 40 seconds) to approximately 10 mv. second-' (corresponding to a transition time of 8 seconds) for a diffusion-controlled, reversible, one-electron process. If an unamplified signal from a chronopotentiometric cell were fed into a derivative amplifier designed to give 5 1 % accuracy for a (dE/dt),,, of 10 niv. second-l, the output voltage would be of insufficient magnitude to be measurable to ilyo accuracy with a conventional X - Y recorder. Consequently, it is necessary and desirable to amplify the signal from the chronopotentiometric cell. In the present work, it was more satisfactory to amplify the signal before rather than after the derivative was taken to maintain a favorable signal-to-noise ratio. Such amplification can be accomplished, however, only a t the sacrifice of requiring an increased frequency response for the derivative amplifier. In turn, this increased frequency response increases the noise generated by the derivative amplifier. In practice, therefore, a compromise must be made to achieve the optimum signal-to-noise ratio. Stabilized or solid-state operational amplifiers (George A. Philbrick Researches, Inc., Dedham, Mass.) were used in all portions of the circuit through which the signal passed before entering the derivative amplifier, and the derivative amplifier itself was a UPA-2. Since a relatively large amount of a.c. pick-up is introduced into the circuit from the cell and the various cell leads, two twin-T filters, tuned to reject 60c.p.s. noise, were incorporated into the network. The first was placed immediately ahead of the input voltage follower (Pa) and the other immediately after the derivative amplifier (UPA-2). This placement allowed both
*
of these filters to be working into high impedance components, thus providing maximum rejection. The input follower (P2) amplifier served both to minimize the current drawn from the chronopotentiometric cell and to improve the operation of the input twin-T filter. The output follower was an unstabilized K2-X operational amplifier. The use of the solid-state P2 amplifier as the input follower (along with its required extra power supply) was unnecessary, but it was employed becauqe a chopperstabilized K2-X amplifier was not available a t the time the instrument was originally constructed; however, a chopper-stabilized K2-X amplifier would be an excellent substitute. The USA-3 amplifier, used to amplify the signal before the derivative-taking operation, was frequency-limited by the 0.2-pf. capacitor C1 across the feedback resistor R2. This greatly reduced the high-frequency noise passed on to the derivative amplifier. The gain of the USA-3 amplifier for the frequency range of interest was approximately 100. A practical derivative amplifier circuit (UPA-2) consisting of an additional input series resistor R, and an additional feedback shunting capacitor C, (with R,C, = RICf to within 1%) was utilized. The derivative chronopotentiograms were recorded on an X - Y recorder (Nodel 500, Electro Instruments, Inc., San Diego, Calif.). The derivative chronopotentiometer was calibrated with known ramp signals. For all input signals (dE/dt) less than approximately 7 mv. second-', the gain of the system had a constant value (+lye) of 0.1095 volt per mv. second-'. Two auxiliary circuits were incorporated into the derivative chronopotentiometric instrument. First, because the maximum output of the USA-3 operational amplifier is =t100 volts and the gain of the amplifier was approximately 100, it was necessary to keep the potential of the working electrode with respect to ground less than +1 volt to prevent the amplifier from limiting. Accordingly, the potential of the reference electrode (S.C.E.) was appropriately biased with respect to ground as indicated in Figure 1. Second, since the portion of a derivative chronopotentiogram around the mini-
Ri
= 19.96 kilohmr; R, =
mum is of primary interest, it was possible to obtain this portion of a derivative wave a t maximum recorder sensitivity after the zero of the Y-axis of the recorder was offset by means of an external bucking-potential. THEORY
Reversible, Diffusion - Controlled Process Involving Soluble Reactants and Products. The potential-time relation describing the chronopotentiometric wave for a reversible electrode reaction for which both the reactants and products are soluble and which occurs under the conditions of semiinfinite linear diffusion was originally derived by Karaoglanoff ( 8 ) . For a reduction reaction and provided virtually none of the reduced species is initially present in either the solution or electrode phases, this equation may be written as
where E is the potential of the working electrode, E1/4 is the quarter-wave potential, and I and 7 are the electrolysis time and the transition time, respectively. In connection with the experimental results described below involving the reductions of thallium(1) and lead(I1) into a mercury pool cathode, the volume of the mercury pool was sufficiently large to minimize the accumulation of either thallium or lead in the amalgam phase and to prevent the violation of the requirement that none of the reduced (product) species be present initially. If the first derivative of the potential with respect to time is taken, the following result is obtained : dE --
at
The latter expression represents the theoretical equation of the derivative chronopotentiogram for the system under discussion. For an oxidation reaction, the relation for the first derivative VOL. 38, NO. 4, APRIL 1966
531
is identical to Equation 2, except that the sign of the derivative is positive. That the derivative chronopotentiogram exhibits a maximum or minimum was determined from the second derivative with respect to time
by equating it t o zero and solving it for t. The above expression for the second
derivative can be zero only if the numerator of the right-hand member of the equation is zero. Thus, +12
- 3-tu2 2
=
0
(4)
and t
4 9
(5)
=--7
This result indicates that the first derivative of the potential-time relation does pass through a maximum or minimum and that this point occurs when the time elapsed from the beginning of the electrolysis is exactly 4/9 of the value for the theoretical transition time given by the Sand equation. Mathematically speaking, this point corresponds to the maximum (most positive value) of the derivative chronopotentiogram for a reduction process and to the minimum (least positive value) in the case of an oxidation reaction. However, in practice it is unnecessary to distinguish between a maximum or a minimum because each is the point on a normal chronopotentiometric wave a t which the rate of change of potential with time is minimal. Therefore, in the present study as well as subsequent reports, we shall adopt the convention of calling such a point the minimum of a derivative chronopotentiogram. If the value of t from Equation 5 is substituted into Equation 2, the magnitude of d E / d t a t the minimum can be evaluated. Performing the substitution and the necessary algebraic rearrangements, we obtained
27 R T
=
-8 nF-7
(6)
When this relation is solved for the transition time r , the result is
The latter relation, Equation 7 , establishes a convenient and explicit method for the absolute determination of transition times because (dE/dt),i. can be accurately measured from a derivative chronopotentiogram. Although Equations 6 and 7 pertain specifically to a 532
ANALYTICAL CHEMISTRY
reduction reaction, the same expressions with positive signs are obtained for an oxidation process. Indeed it suffices for the evaluation of transition times to employ only the absolute magnitude of the quantity (dE/dt),,,. The experimental verification of this technique for the measurement of transition times is described below for the reversible reductions of thallium(1) and lead(I1) a t mercury pool electrodes. Irreversible, Diffusion - Controlled Process Involving Soluble Reactants and Products. Another important application of derivative chronopotentiometry is in the study of irreversible electrode processes. The equation of the chronopotentiometric wave for a totally irreversible reduction reaction involving soluble reactant and product species has been derived by Delahay and Berzins (2) for linear-diffusion processes.
E
.
RT
nFACk",,h
an,/'
2
= --.en
?!f!cm,F h
+
[ - ($)"'] (8) 1
where CY is the transfer coefficient, n, the number of electrons involved in the rate-determining step, k o f . h the heterogeneous rate constant for the forward (reduction) reaction, n the total number of electrons involved in the overall reduction process, and the other terms have their usual chronopotentiometric significance. The first derivative of the potential with respect to time is
The minimum for the derivative chronopotentiogram occurs a t t = '/4-7, and the magnitude of d E / d t a t the minimum is given by the relation
Russell and Peterson (IS) have presented the same equation except for their omission of the negative sign. This equation may be employed to measure either the transition time -7 or the kinetic parameter ana. However, the determination of one of these quantities requires knowledge of the value for the other. For the evaluation of ana by means of derivative chronopotentiometry, the transition time must be obtained independently. For example, if a conventional chronopotentiogram is recorded for a relatively high concentration of the electroactive species, the transition time -7 can be reliably measured according to the usual graphical methods ( 2 , 12, IS), and an. can be calculated from Equation 10 and the observed value of (dE/dt),i, for the corresponding derivative chrono-
potentiogram. Such a procedure for the determination of ana is much more simple and direct than the common technique of plotting the decimal logarithm of [l - ( t / ~ ) ~us.' ~the ] potential E (3). Furthermore, since an, is a constant for a given set of experimental conditions, its evaluation by the above procedure, or by some other independent means, allows Equation 10 to be utilized for the calculation of transition times from derivative chronopotentiometric data. Finally, after the parameter ana is established, one may calculate the heterogeneous rate constant k o j , h ,using Equation 8 and the value of the potential E at t = 0 obtained by extrapolation of a normal chronopotentiometric wave to zero time. As discussed below, the theory of derivative chronopotentiometry for irreversible processes has been tested for the reduction of iodate t s iodide a t a mercury pool cathode in 1F sodium hydroxide medium. RESULTS AND DISCUSSION
The important characteristics of derivative chronopotentiograms are illustrated schematically in Figure 2. A typical conventional chronopotentiometric wave for the reduction of 6.28 x l 0 - T thallous nitrate in 0.1F potassium nitrate supporting electrolyte solution a t a mercury pool cathode is shown as Curve 1. Curve 2 represents the complete derivative chronopotentiogram corresponding t o Curve 1, while Curve 3 is the magnified center portion of Curve 2. We should emphasize that the derivative chronopotentiograms exemplified by Curves 2 and 3 in Figure 2 are inverted in relation to Curve 1. However, because of the qignal-inversion characteristic of operational amplifiers, the direction of plotting or recording derivative chronopotentiograms is entirely arbitrary. In addition, the ordinate ( Y ) axes of Curves 2 and 3 are calibrated in terms of the quantity ( d E / d l ),n. G, which represents the input value of d E / d t (in mv. second-') from the chronopotentiometric cell multiplied by the overall gain G (0.1095 volt per mv. second-') of the electronic circuit. As revealed by Curve 2, a complete derivative chronopotentiogram for the reduction (or oxidation) of a single electroactive substance exhibits two maxima, one peak corresponding to the rapid change in potential due to initial charging of the double-layer and a second, usually larger, peak caused by the rapid potential change which signals the depletion of the electroactive species a t the surface of the electrode. In fact, the idea of derivative chronopotentiometry was originally introduced by Iwamoto (7) as an instrumental technique for locating the points on a chronopotentiometric wave a t which the rate of change of potential with time is
maximal. Iwamoto proposed that it should be possible, a t least in principle, to evaluate the transition time by measurement of the time interval between the two peaks of a derivative chronopotentiogram. Unfortunately, this procedure becomes inaccurate and imprecise as the concentration of the electroactive substance decreases because charging of the double-layer distorts the normal chronopotentiogram and causes the maxima (peaks) of the derivative chronopotentiogram to shift along the time axis as well as to broaden and decrease in absolute magnitude. In the present work, however, the method of derivative chronopotentiometry differs considerably from the technique suggested by Iwamoto because it takes advantage of the unique theoretical and mathematical properties of derivative chronopotentiograms to provide a variety of information. We established in the previous section of this paper that the magnitude of dE/dt a t the minimum of a derivative chronopotentiogram is quantitatively related to the transition time for a reversible process and to both the transition time and the kinetic parameter ana for an irreversible reduction reaction. Although the existence of a well defined minimum for the derivative chronopotentiogram shown as Curve 2 is not evident, its presence is revealed upon amplification (by a factor of 100) of the part of the derivative chronopotentiogram encompassing the location of the minimum (Curve 3). Generally the graphical presentation of a complete, amplified derivative chronopotentiogram is impractical because of the extremely wide range of values over which dE/dt varies. Furthermore, i t is unnecessary to record a complete derivative chronopotentiogram because only the value of (dE/dt),i, is required to evaluate either the transition time 7 or the kinetic paramter ana. Reduction of Thallium(1). T o establish the usefulness of derivative chronopotentiometry for the determination of transition times, the reversible, one-electron reduction of thallium(1) a t a mercury pool cathode was investigated. First, a comparison was made of the shape of an actual recorded derivative chronopotentiogram near its minimum with that of the theoretical curve predicted on the basis of Equation 2. Figure 3 shows a pair of experimental and theoretical derivative chronopotentiograms which coincide quite closely. Within the limitations of the instrumentation, repetitive experimental curves were in excellent agreement with each other as well as with the theoretical equation. However, most important for the success of derivative chronopotentiometry as a technique for transition-time measurements is the fact
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Figure 2. Schematic conventional and derivative chronopotentiograms for thallium(l) reduction Curve 1. Conventional chronopotentiogram for reduction of 6.28 X 10% thollium(l) in 0.1 F potassium nitrate a t mercury pool cathode; i = 0.503 ma. Curve 2. Derivative chronopotentiogram corresponding to Curve 1 Curve 3. Derivative chronopotentiograrn showing magnified portion of Curve 2 near the region of the minimum
that the magnitude of dE/dt a t the minimum of the experimental curve is in almost perfect agreement with the expected theoretical value. Note that the ordinate ( Y ) axis of Figure 3 is calibrated in terms of the parameter (dE/dt) G, which was introduced above in the discussion of Curves 2 and 3 in Figure 2. As a second step in the evaluation of derivative chronopotentiometry, transition-time data for both conventional and derivative chronopotentiometry were obtained for the reduction of 7.83 X l0-V thallous nitrate in 0.1F potassium nitrate supporting electrolyte solution. The experimental results, which represent the averages of from six to fifteen individual trials, are summarized in Table I. The transition ins
times for the conventional chronopotentiograms were measured from the instant the electrolysis was begun to the moment the potential of the mercury pool cathode reached -0.80 volt us. S.C.E. by means of a technique previously described (11). Any correction of these transition times for charging of the double-layer was insignificant and unnecessary for the relatively high concentration of thallium(1) employed. Derivative chronopotentiograms were recorded for the same current densities used in the conventional chronopotentiometric experiments. Transition times for each current density were calculated from Equation 7 and the observed values of (dE/dt),,,in (actually (dE/dt),i.. G) which were readily and directly measurable by visual inspection VOL 38, NO. 4, APRIL 1966
533
of the recorded curves. The values of i ~ l ' ~ / i l determined C by means of the derivative and conventional chronopotentiometric techniques agree well with each other and with the value of 388 amp. cm. mole-' for thallium(1) reduction in 0.1F potassium nitrate at 25' C. reported by Bruckenstein and Rouse ( I ) and confirmed independently in the present investigation. Therefore, the data presented in Table I verify the theory of chronopotentiometry as a precise and accurate method for the determination of transition times. Reduction of Lead(I1). As mentioned previously, when transition times are determined by means of conventional chronopotentiometry, positive deviations from the Sand equation are observed. These errors always increase in magnitude as the concentration of the electroactive substance decreases because charging of the double-layer consumes a larger and larger fraction of the total quantity of electricity required in a chronopotentiometric experiment. On the other hand, the use of derivative chronopotentiometry for transition-time measurements a t low concentration levels not only
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-
-0.8
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2
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12
14
16
18
TIME, SECONDS
Figure 3. Experimental and theoretical derivative chronopotentiograrns for thallium(1) reduction Solid curve represents center portion o f experimental derivative chronopotentiogram for reduction of 6.28 X 1 O-3M thallium(l) in 0.1 F potassium nitrate a t current of 0.480 ma. Dashed curve is corresponding theoretical derivative chronopotentiogram calculated on the basis of Equation 2 for r = 18.32 seconds
offers the advantage of absoluteness but minimizes the effect of double-layer charging because (dE/dt),,, is the point on a chronopotentiometric wave at which the rate of charging of the doublelayer is minimal. Measurements of transition times by conventional and derivative chrono-
potentiometric techniques were compared for the reduction of five different concentrations of lead(I1) at a mercury pool electrode in 0.1F potassium nitrate supporting electrolyte solution of pH 2. A summary of the experimental results, along with the calculated values of i ~ l ' ~ / . - t Cis, given in Table 11. The transition times for the conventional chronopotentiograms were evaluated graphically according to the procedure attributed to Kuwana Table I. Comparison of Derivative and Conventional Chronopotentiometry in ( I S ) , while the transition times listed Measurement of Transition Times for Thalliurn(1) Reduction for derivative chronopotentiometry in Concn. of thallous nitrate, 7.83 X 10-3F in O.1F potassium nitrate supporting electrolyte Table I1 were determined on the basis of solution; electrode area, 1.38 cm.2; temp., 25" C. Equation 7 and the pertinent values Convent,ional of (dE/dt),,,. With the single excepchronopotent,iometry Derivative chronopotentiometry ir1i2/AC,a ~ T ' / ~ / A C ,tion ~ noted, the transition times reported in Table I1 are the averages of at amp. amp. least three trials, second112 second1'2 -( d E / d t ) m i n , cm. cm. For a lead(I1) concentration of 1.02 i, ma. r , seconds mole-' mv. second-' r , seconds mole -1 X l O - 3 M , charging of the double-layer 1.155 13.55 i 0.04 394 6.65 i 0.03 13.04 i 0.06 386 is relatively unimportant and the values 1.087 15.23 i 0.09 393 5.73 f 0.05 15.13 z t 0.14 392 W 2 / A C for conventional and deof 1.027 16.85 i 0.10 390 5 . 2 1 f 0.01 16.64 f 0.04 388 rivative chronopotentiometry are in ... ... 4.13 f 0.03 21.00 =!= 0.15 392 0,925 good agreement with each other as well cm.2 second-' at 25" C. is 388 amp. a Theoretical value based on D = 2.06 X &s with the theoretical chronopotentiosecond"2 cm. mole-'. metric constant of 535 amp. b Calcd. from Equation 7 and the observed average value of (dE/dt),i,. cm. mole-' based on the diffusion coefem.* ficient for lead(I1) of 0.98 X second-' a t infinite dilution. The fact Table II. Comparison of Derivative and Conventional Chronopotentiometry for that the experimental values of ir1'2/ Reduction of Various Concentrations of Lead(l1) AC for the highest concentrations of lead(I1) are actually less than this Supporting electrolyte solution, 0.1F potassium nitrate (pH 2 ) ; electrode area, 1.38 cm.2; temp., 25' C. theoretical value is probably due to a smaller diffusion coficient for lead(I1) Conventional Derivative chronopotentiometry chronopotentiometry in 0.1F potassium nitrate medium. ir112/ A C," ~ T ' ' ~ / A C , However, ~ the experimental values of amp. amp. ir'I2/AC derived from both convensecond1'2 tional and derivative chronopotenticm. cm. ometry increase as the concentration of i, Ma. r , seconds mole-' T , seconds mole-' Lead concn., M lead(I1) is decreased. The positive 15.69 =t 0 . 0 3 530 15.17 f 0.11 520 188.1 1.02 x 10-3 deviations of i W / A C for conventional 17.96 =t 0.21 542 16.81 It 0.27 525 90.1 5.10 x 10-4 18.3 f 0 . 1 581 16.81 i 0 . 2 3 556 38.18 chronopotentiometry can be attributed 2.04 x 10-4 16.8 k 0 . 3 591 14.65 I t O . 1 2 552 1.02 x 10-4 20.27 to the empirical graphical method for 22.Sb 717 15.88 f 0.09 598 8.45 4.08 x 10-5 the determination of transition times second-' at 25" C. is 535 amp. a Theoretical value based on D = 0.98 X which does not adequately correct for second*'*cm. mole-'. the double-layer charging effect. * Single measurement only. Similarly, although the results obtained 534
ANALYTICAL CHEMISTRY
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4/97.
Figure 4. Experimental and theoretical derivative chronopotentiograms for iodate reduction Solid curve represents center portion of experimental derivative chronopotentiogram for reduction of 4.04 X 1 O-3M iodate in 1 F sodium hydroxide a t current of 1.956 ma. Dashed curve is corresponding theoretical derivative chronopotentiogram calculated on the basis of Equation 9 for T = 23.70 seconds and an, = 1.00
from derivative chronopotentiometry are more accurate and precise a t the lower lead(I1) concentrations than those for conventional chronopotentiometry, it is evident that the transition times evaluated by the derivative chronopotentiometric technique are too long. The explanation for such results is that charging of the double-layer causes the rate of change of potential a t the minimum of the derivative chrono( d E / d t )min-to be potentiogram-Le., smaller than expected. Charging of the double-layer prior to reaching the minimum also contributes to the error. We conclude that derivative chronopotentiometry is slightly, but not greatly, superior to conventional chronopotentiometry in providing better constancy of ir1I2/ACas the concentration of the electroactive substance is decreased. However, in comparison to conventional chronopotentiometry, derivative chronopotentiometry allows transition times to be measured in a well defined and theoretically justifiable way and eliminates the empiricism which attends the upe of graphical procedures for the determination of transition times. Reduction of Iodate. As an example of an irreversible process which satisfies the requirements of the theory of derivative chronopotentiometry presented above, we investigated the six-electron reduction of iodate t o iodide a t a mercury pool cathode in 1F sodium hydroxide medium. In Figure 4, an experimental derivative chronopotentiogram for the reduction of 4.04 X iodate in I F sodium hydroxide solution is shown along with the theoretical derivative chronopotentiogram calculated from Equation 9. The magnitude of dE/dt a t the minimum of the experimental curve is in excellent agreement with the theo-
In the present investigation, additional filtering elements, including the twin-T filters, were incorporated into the circuitry. While the arrangement shown in Figure 1 effectively suppresses highfrequency noise, the various filtering components introduce a time lag in the recorded derivative chronopotentiograms. The time lag happens to be more serious for an irreversible process because the minimum of the derivative chronopotentiogram occurs a t t = 1 / 4 ~ , which is closer to the initial large rate of change of potential due to charging of the double-layer than the minimum for a reversible reaction a t t =
retical value predicted by Equation 10 for ana = 1.00. Moreover, the shapes of the experimental and theoretical derivative chronopotentiograms shown in Figure 4 are almost identical except for the displacement of the experimental curve along the time axis. The shape of a derivative chronopotentiogram for an irreversible reaction (Figure 4) is distinctly different from that for a reversible process (Figure 3), and, therefore, this difference offers a criterion to distinguish between reversible and irreversible reactions involving soluble reactants and products. The displacement of the experimental derivative chronopotentiogram along the time axis is much more pronounced for the irreversible reduction of iodate than for the reversible reductions of either thallium(1) or lead(I1). The origin of this displacement is instrumental in nature. The output of an ideal differentiator circuit, designed to respond perfectly to relatively large rates of change of potential, invariably transmits considerable high-frequency noise. To reduce this noise, a practical differentiator circuit is usually employed.
I t is appropriate to mention two other problems relating to the operation and performance of the derivative chronopotentiometer. First, because a limit must be placed on the frequency-response of a practical differentiator circuit, the range of transition times accessible to accurate measurement (* 1%) by means of derivative chronopotentiometry is restricted. In the present work, the characteristics of the derivative chronopotentiometric instrument were tested, and values of (dE/dt),,. up to approximately 8 mv. second-' could be measured with an accuracy of fl%. Depending on the number of electrons involved in the particular electrode process, this allows the accurate evaluation of transition times encompassing the interval from about 8 to 25 seconds with the circuitry shown in Figure 1. For transition times shorter than 8 seconds, the displacement of the experimental derivative chronopotentiogram along the time axis becomes especially severe and the magnitude of (dE/dt),,, begins to deviate significantly from theory due to the imposed frequency-response limit of the instrument. However, the measurement of transition times longer than 25 seconds is possible with the present circuitry, although the onset of natural convective stirring as well as external laboratory vibration may be expected to disturb the diffusion layer when long
Table 111.
Derivative Chronopotentiometric Evaluation of Kinetic Parameter ana for Iodate Reduction Supporting electrolyte solution, 1F sodium hydroxide; electrode area, 1.38 cm.*; temp., 25O i, ma. T , seconds - (dE/dt),i,, mv. second-' an.
c.
0.626 0.596 0.595 0.512
( a ) 1.010 X 10-8F potassium iodate 14.40 f 0 . 0 1 3 . 7 1 f 0.01 16.02 zt 0.02 3.35 f 0.01 15.60 f 0 . 1 0 3.03 f 0.03 21.03 f 0 . 0 8 2.48 i 0 . 0 1
3.260 2.776 2.470 1.955
( b ) 4.040 X 1 O - V potassium iodate 8.40 f 0 . 0 1 6 . 6 1 f 0.05 11.42 =I=0 . 0 1 4.57 f 0 . 0 9 14.36 f 0.09 3 . 4 0 f 0.03 22.71 & 0 . 0 1 2.22 rt 0.01
0.96 0.96 1.09 0.99 0.93 0.98 1.05 1.02 = 1.00 f 0.04
VOL. 38, NO. 4, APRIL 1966
535
transition-time experiments are attempted. The second point concerns the constant-current source employed for the derivative chronopotentiometric measurements. We observed a significant difference between results obtained with a simple constant-voltage current source, consisting of a bank of radio B-batteries, and an electronically regulated constant-current supply incorporating a K2-X operational amplifier. For a battery constant-current source of 180 volts, the back e.m.f. developed across the chronopotentiometric cell varies sufficiently during the course of an electrolysis to cause detectable inconstancy of the current. Such a variation of the current produces a somewhat distorted derivative chronopotentiogram and increases the displacement of the curve along the time axis. I n contrast, the use of an electronically regulated current source minimizes the displacement of the curve and gives an experimental derivative chronopotentiogram in much closer agreement with theory. h test of derivative chronopotentiometry for the evaluation of the kinetic parameter ana for the irreversible reduction of iodate was performed. The experimental data determined for a range of current densities and for two concentrations of potassium iodate in I F sodium hydroxide solution are
listed in Table I11 with the values for an, calculated from Equation 10. Vsing conventional chronopotentiometry (II), we measured the transition time in triplicate for each current density as the interval from the start of the electrolysis to the moment the potential of the mercury pool cathode reached -1.50 volts us. S.C.E. Such a technique for these measurements is acceptable for the relatively high concentrations of iodate studied. The values for (dE/dt),,, were obtained directly from the recorded derivative chronopotentiograms. The average value of ana for the reduction of iodate in 1F sodium hydroxide medium was 1.00 =t 0.04, which is in reasonably good agreement with the results of other workers (4-6, I S ) . The value of an, calculated from plots of E us. log [I - ( t / . ) l / 2 ] for conventional chronopotentiometric waves obtained in this study was 1.10. The reason for the difference in the results obtained by the two techniques is not known a t present. Because of its promise as an analytical and research technique, derivative chronopotentiometry is being subjected to more extensive experimental verification with other simple reversible and irreversible reactions, and further refinements of the instrumentation are being made. In addition, investigations are in progress to determine the possible usefulness of this technique for studies
of adsorption phenomena and of electrode reactions complicated by kinetic and catalytic processes. LITERATURE CITED
(1) Bruckenstein, S., Rouse, T. O., ANAL. CHEILI. 36, 2039 (1964). ( 2 ) Delahav, P., Berziris, T., J . Am. Chem. SOC.75, 2486 (1953). (3) Delahay, P., Mattax, C. C., I b i d . , 76. 874 (19541. 14) f b i d . . I., J . Electroanal. Chem. 5, 467 (1963). (14) Sand, H. J. S.,Phzl. N a g . 1, 45 (1901).
RECEIVED for review December 1, 1965. Accepted January 17, 1966. Appreciation is expressed t o the Kational Aeronautics and Space Administration for partial financial support of this work through a Traineeship held by one of us (S. L. B.).
Polarographic Studies on Reduced Silicomolybdate 6.
P.
SEN and S. N. CHATTERJEE
Research and Control laboratory, Durgapur Steel Plant, Durgapur-3, Burdwan, West Bengal, India Chemically reduced silicomolybdate can be further reduced polarographically in four steps; the first three are reversible and diffusion-controlled and the last is catalytic in nature. The diffusion currents of the first three waves are proportional to the concentration of the reduced silicomolybdate complex alone, while the fourth wave height is dependent on the complex as well as the concentration of molybdate in the solution, The diffusion current data indicate that the complex is composed of silicon and molybdenum in the ratio of 1 to 12. The El/zof all the waves varies linearly with the pH of the solution and is shifted toward more negative potential with increasing pH. The number of electron transfers and hydrogen ion additions in the respective waves is discussed. By measurement of diffusion current of the appropriate wave, silicon and molybdenum can be determined in the range 0.1 to 100 and 5 to 5000 p.p.m., respectively.
536
ANALYTICAL CHEMISTRY
P
reduction of yellow silicomolybdic acid has been investigated a t low p H by several workers. In acetate buffer mixtures, the polarogram of yellow silicomolybdic acid gives ill defined waves ( d ) , the number of which depends on the p H of the solution and the concentration of the reducible species. At p H 1.1 four ill defined waves are produced, while a t p H 4.9 only one wave is recorded. Singlesweep cathode ray polarography (4, 5 ) applied in the determination of silicon as silicomolybdic acid in a citrate buffer at p H 2 containing 4% methyl ethyl ketone gives three waves, and it is suggested that the reduction process is reversible and involves a 4-electron change with the addition of four hydrogen atoms. Reproducible kinetic waves are obtained (7, 8) in the polarographic reduction of molybdosilicic acid in a buffer solution containing hydrochloric acid, sodium formate, and butyl alcohol at a p H of 1.9 to 2.6. By using rotating platinum electrodes ( I d ) three-step reduction of (Y and 0 isomers of silicoOLAROGRAPHIC
molybdic acid is noticed, which corresponds to 2-, 4-, and 6electron reduction, the first two being reversible. As many as six characteristic steps are also reported (15) in the polarographic reduction of 7-molybdosilicic acid, only the first two waves of which are proportional to the silicon concentration. The waves, however, are not reproducible at lower concentration of silicon. The polarographic reduction of yellow silicomolybdic acid does not stop at the silicomolybdenum blue stage and is probably reduced further (3). A comparative study of the different methods used by the authors indicated that in the polarographic estimation of yellow silicomolybdic acid at low p H the waves are not well defined and frequently not reproducible. I n the course of the investigation it was observed that chemically reduced silicomolybdic acid could be further reduced polarographically and the waves obtained were well defined and reproducible at alkaline pH. Polarographic studies of silicomolybdic acid in
