reaction), double-layer effects, etc. However, in view of the small magnitude of the deviations and the likelihood that they might simply reflect an occasional error in the calibration procedure, we feel that such speculation is without sound basis. The k , and a values of 0.25 cm sec-’ and 0.55,respectively, for the Cr(CN)e*-/Cr(CN)a4- system are in excellent agreement with the values of 0.24 cm sec-’ and 0.59 recently obtained in these laboratories (12) by fundamental harmonic ac polarographic measurements, not to mention earlier reports of the k , value ranging from 0.22 cm sec-’ to 0.24 cm sec-1(10, 20, 22). Similarly, a recent fundamental harmonic ac polarographic investigation of the Cd*+/Cd(Hg) system (12) yielded k , = 0.15 cm sec-l and a = 0.30 which are identical to the values obtained in the present work. Comparable k , and a values also were obtained in earlier studies (23-25).
We view these results as extremely satisfying because not only is good agreement between second harmonic ac polarographic theory and experiment indicated for appropriate sets of rate parameters, but the apparent rate constants obtained are completely consistent with those obtained by other experimental approaches. Particularly significant in this regard is the establishment of self-consistency between rate parameters obtained from observables responsive to the linear aspects of the faradaic impedance (fundamental harmonic measurements) and those from observables controlled by
-
(22) J. E. B. Randles and K. W. Somerton, Trans. Faraday Soc.. ‘ 48,937 (1952). (23) J . K. Frischmann and A. Tirnnick, ANAL.CHEM.,39, 507 (1967). (24) H.~H.Bauer, D. L. Smith, and P. J. Elving, J. Amer. Chem. Soc., 82,2094 (1960). (25) G. Lauer and R. A. Osteryoung, ANAL.CHEM.,38, 1106 (1966).
faradaic nonlinearity. That the agreement is not fortuitous is suggested by the fact that, including the work of Randles and Whitehouse (6), three quasi-reversible systems have been found to date whose second harmonic response is satisfactorily interpreted by the appropriate second harmonic theory. NOMENCLATURE
A
= electrode area
D1
=
fi C,*
= activity coefficient of species i
Eo
= standard redox potential in European convention = dc component of applied potential
Edo AE
E1,g
=
=
diffusion coefficient of species i initial concentration of oxidized form amplitude of applied alternating potential
= reversible polarographic half-wave potential (planar
diffusion theory) Faraday’s constant = ideal gas constant = absolute temperature n = number of electrons transferred in heterogeneous charge transfer step I(2wt) = second harmonic faradaic alternating current c $ ~ = phase angle of second harmonic current relative to applied alternating potential w = angular frequency t = time = heterogeneous charge transfer rate constant at Eo k, a = charge transfer coefficient r = Euler Gamma Function m = mercury flow rate in mg sec-1 r, = electrode radius
F R T
=
RECEIVED for review August 14, 1968. Accepted October 2, 1968. Work supported by National Science Foundation Grant GP-7985.
Resistance Compensation in Nonaqueous Polarography. Evaluation of Uncompensated Resistance and Effects of Damping in Two- and Three-Electrode Cells William E. Thomas, Jr.,1 and Ward B. Schaap Department of Chemistry, Indiana University, Bloomington, Ind.
A simple, convenient method for calculating uncom-
pensated polarographic resistance is tested in aqueous and in glacial acetic acid solutions and in both twoelectrode and three-electrode cells. Equations are derived and verified which predict a difference between the half-wave potentials of heavily damped and undamped polarograms when uncompensated resistance is present. The sign and magnitude of the difference are dependent upon the relative magnitudes of the external and internal components of the overall cell resistance and are different for two- and threeelectrode cells. In the presence of ohmic distortion, linear extrapolation of the residual current is not satisfactory for the estimation of diffusion currents. A graphical method is presented which allows a better residual current line to be constructed by taking into account the distortion of the wave. RECENTADVANCES in controlled-potential instrumentation have made possible more precise measurement of polarographic half-wave potentials in media of low dielectric constant and high specific resistance. Nevertheless, studies of
136
ANALYTICAL CHEMISTRY
high resistance polarography (1-4) have shown that, with media of high specific resistance or at high current densities, resistance compensation is not complete even with the use of three-electrode, potentiostat-type polarographs. Because the current flow in the immediate vicinity of the dropping mercury electrode appears to be spherically symmetrical, a significant potential gradient may exist in the direction of the reference electrode, even though the reference electrode is out of the path of the main electrolysis current. In the usual cell arrangement, the DME is placed between the reference and 1 Present address, E. I. du Pont de Nemours and Co., Orange, Texas.
(1) W. B. Schaap and P. S. McKinney, ANAL.CHEM., 36,29(1964). (2) Ibid.,p 1251. (3) W. B. Schaap and P. S. McKinney, “Polarography-1964,” Vol. 1, G . Hills, Ed., Macmillan and Co., London, 1966, p 197. (4) L. NemiZ, J. Elecrroanal. Chem., 8, 166 (1964).
counter electrodes at an appreciable distance (20.5 cm.) from both. Schaap and McKinney (1-3) derived equations which relate uncompensated resistance, R,, to the peak height of the first derivative of an average-current polarogram. The derivative method of evaluating uncompensated resistance is dependent upon the relationship between the height of the first derivative polarogram and the voltage scan rate. The effect of R , is to decrease the true scan rate (sensed at the surface of the DME) and thereby to reduce the peak height of the derivative polarogram. This method suffers the disadvantages of being rather tedious for routine use in nonaqueous media and of requiring special electronic circuity for filtering out the oscillations due to the DME and for taking undistorted first derivatives of the polarographic waves. Therefore, an attempt was made to find and to evaluate a more convenient method for measuring uncompensated resistance and for obtaining half-wave potentials free of ohmic potential loss. CALCULATION OF UNCOMPENSATED RESISTANCE
The slope of a polarographic wave decreases with increasing uncompensated resistance and the deviation of the slope of a reversible wave from its theoretical value, with certain limitations, can be used to calculate the effective uncompensated resistance in the cell. The theoretical potential difference between any two points on a reversible polarographic wave can be calculated from the Heyrovsky-Ilkovi; equation. Convenient points for the measurement of the slope are those at which i = ' 1 4 id and i = "4 id. The relation between the true potential (the potential at the DME-solution interface) and the potential applied to the id, in the presence of undropping mercury electrode at compensated resistance, is given by
+
where both potentials are measured us. the reference electrode and ill4, the total current at this point on the wave, is the sum of 'I4 id and the residual current, ires. A similar equation can be written for (E3/4)true. Because the true potential difference between i314 and ill4 is -0.0564/nvolt for a reversible wave at 25 "C, the equations and (Ea/4)true can be subtracted and solved for for (&/&rue the uncompensated resistance, R,, giving
+
{ [(E3!4)spp- (Eli4)appl 0.0564Inj i 3 1 4 - i114
EFFECT OF FILTERING ON HALF-WAVE POTENTIALS IN THE PRESENCE OF UNCOMPENSATED RESISTANCE
Apparent half-wave potentials for average-current (filtered) and regular (unfiltered) polarograms have been observed in this study to be different when the uncompensated resistance is large. The magnitude of the difference in half-wave potentials is proportional to the uncompensated resistance and the sign of the difference may change depending on experimental conditions in the cell. These apparent anomalies can be understood in terms of the relationships between average and minimum resistance and average and maximum current. Because the DME is a very small spherical electrode, the cell current will flow radially from the DME surface. If the specific resistance of the solution or the current density is high, there may be an appreciable ohmic drop in the vicinity of the DME, even on the side toward the reference electrode (I). Using as a model concentric spherical electrodes of radii rl and r2 separated by a medium of specific resistance p , IlkoviE ( 5 ) derived a theoretical expression for the resistance of the solution occupying the volume between the two spheres, i.e.,
R
(Elidtrue= ( E I ~ ~i ~) d~R ,~) ~
R, =
the potential are minimal. In the case of heavily damped waves, the range of accuracy of iR corrections calculated by Equation 1 is much more limited because the use of the average current in the correction depends on the validity of the Ilkovi; equation over the entire lifetime of the drop. The method can also be used to estimate iR corrections to be applied to irreversible waves of unknown theoretical slope if a non-interfering, reversibly reduced pilot ion is added and the uncompensated resistance calculated from the distortion of its wave.
(1)
As written, Equation 1 is merely an application of Ohm's law and is in itself subject only to the requirement of polarographic reversibility. Certain polarographic measurements, on the other hand, may require the validity of the IlkoviE equation, which assumes a constant potential at the surface of each mercury drop during its lifetime. In the presence of uncompensated cell resistance, the potential sensed at the surface of the mercury drop will vary during its lifetime because of variation in the iR loss in the cell. If this variation in potential is significant, the Ilkovi; equation will not be strictly valid and the usual current-time relations will not apply; e.g., the average current cannot be expected to be of the maximum current. Equation 1 can be applied in the cases of both undamped and heavily damped polarographic waves. In the former case, accurate corrections for iR losses can be made over a wide range of values because the measurement of the instantaneous current and the potential are made toward the end of the drop life, when the changes in both the current and
=
-&
(a a> -
which assumes spherically symmetrical current flow. In the case under consideration, rl is the radius of the DME and rl is the distance from the center of the DME to the tip of the reference electrode. When r2 2 10 rl, Equation 2 can be simplified to
The radius of the DME is time-dependent and can be expressed
(4) where rn is the rate of flow of mercury from the capillary, t is the elapsed time of formation of the drop in seconds, and d is density of mercury. The time-dependent resistance has been recognized by a number of workers (6-9) and has been called the internal resistance. Ilkovir (8) has shown that the internal resistance, averaged over the life of the drop, is equal to
( 5 ) D. IlkoviE, Collect. Czech. Chem. Commun., 8, 13 (1936). (6) R. BrdiEka, ibid., p 419. (7) J. Devay, Acta Chim. Hung. Tomus, 35,255 (1963). (8) D. IlkoviE, Collect. Czech. Chem. Commun., 4, 480 (1932). (9) I. M. Kolthoff, J. C . Marshall, and S. L. Gupta, J. Elecrroanal. Chern., 3, 209 (1962). VOL 41, NO. 1, JANUARY 1969
137
where R,, and Rminare the average and minimum resistances, respectively. Kolthoff, Marshall, and Gupta (9) pointed out that the overall cell resistance between the DME and its counter electrode may be considered to be made up of both time-dependent and time-independent components, the latter of which they called external resistance because of its resemblance to resistors added externally in series with the cell. Thus, Reell = Rext
+ Rint
(6)
Combining Equations 3, 4,and 6 gives the expression for cell resistance at any t after the beginning of drop formation :
+ kt-'I3
Rceil = Rex$
+
(8)
ktmax-"a
The internal resistance is dependent only upon the specific resistance of the solution and the capillary characteristics, whereas any changes in cell resistance due to cell geometry or to changes in the anode-to-cathode distance will produce changes in R,,t. Rext>> Rlnt (Two-Electrode Cell). For the limiting case in which Rext >> R,nt,which is often the case in a two-electrode cell, the effect of heavy damping or filtering on the measured EliLis readily derived. In this case the effective cell resistance is constant (independent of time) and the effect of filtering on the half-wave potential is related to the time dependence of the current. For the average, heavily damped current
+i
= (Eldapp
(9)
d
and for the undamped polarogram, the limiting values just before the drop falls are (E,/z)true
= (Eii2)app
+ imaxR
(1 0 )
where i,, and i,,, are the average and maximum currents, respectively, at the half-wave potential. Since ,.i = (@I7) ,,i and the values of should be the same in both cases, the difference in the half-wave potentials observed for undamped and heavily damped polarograms will be
Introducing the relations R., into Equation 12 gives (AEl/P)unfil-fil =
i,,,
- imax)R
=
-I/;
im,,R
ANALYTICAL CHEMISTRY
(12)
Rmio and i,, =
imax
imsxRmin =
'17
imaxRmin (13)
In a three-electrode cell, therefore, the half-wave potential of an unfiltered maximum current polarogram will be more positioe than the half-wave potential of a heavily damped averagecurrent polarogram because the factor ' / a relating the average to minimum resistance overbalances the factor relating the average to maximum current. In a three-electrode cell, the resistance varies as t - " 3 (Equations 3 and 4) and the current varies as t1i6 during the lifetime of the drop. The iR drop therefore varies as t-'i6, which is just the reciprocal of its time dependence in a two-electrode cell. Because the time dependencies of the iR losses in twoand three-electrode cells are both one-sixth order with respect to time, the average iR loss (A&J over the lifetime of the drop will differ from the terminal value by in both cases, but with opposite signs as predicted by Equations 11 and 13. Because of the variation of the potential of the drop in the presence of iR loss, Equation 13 will also be valid over a limited range of corrections. Rint G Rex$(General Case). In the more general case in which both Rext and Rint are significant components of the uncompensated cell resistance, Brdizka (6) has derived the following equation Kav
=
(2:n)2
3Rrnin
I--
+3--
(
Rext 3 J... : (
In
- 3 -
"ex$)
1+-
2Rext
Rmin
(14) which can be expressed Rav
(11)
i.e., the observed half-wave potential for a maximum current polarogram should be more negatioe than the half-wave potential observed for a heavily damped, average-current polarogram simply because i,,, is larger than i,,, while the cell resistance is essentially constant. Equation 11 assumes a reasonably constant potential for the mercury drop during its lifetime and is therefore valid only over a limited range of iR losses. Actually, the potential of the mercury drop in a two-electrode cell will become less cathodic as the drop grows and the current increases, and the time dependence of the iR loss will be the same as that of the cell current. R,,$ >> Rext (Three-Electrode Cell). In the case of the three-electrode polarographic cell in which the DME is placed between the anode and the reference electrode, the effect of ReXtis eliminated and only Rlnt remains uncompensated. If the reference electrode is 0.5 cm or more away from the DME, the uncompensated resistance may be calculated from Equation 3 with an additional factor of */* inserted 138
= '/a
im=X4/3 Rmin)
('17
(AEi/2)app = (AEl/f)unfii-fii = (6/7
- imAmin
(AEli2)unfil-fil = i,,R,,
(7)
Roeilattains its minimum value when t = tmsxjust before the drop falls, Le., (Rce1i)min = Rext
to correct for shielding if the usual blunt-tipped DME is used (3, 9). The value of RIntdecreases as the drop grows and attains its minimum value just before the drop falls. The true half-wave potential can again be expressed in terms of the observed half-wave potentials for both the filtered and unfiltered polarograms. In this case, equating the two expressions gives
Rext f aRmin
(15)
where a is dependent on the magnitude of the ratio RminlReXt. When Rint is much greater than ReXt,a approaches ' 1 3 in agreement with Equation 5. When Rex$is much greater than Rint, a approaches "2. Under the latter conditions, however, the RiDtterm becomes unimportant so that R,, = ReXt (a constant). Substituting Equations 6 and 15 into 12 and rearranging gives for the case in which both Rintand ReXtare present: (E1iz)unf
- (Elidiii
i,, [Rext
=
+ a(Rint)minl imax[Rext
Introducing i,, case (AEl&f-fil
=
=
'17
f (Rint)min] (16)
i,,, and rearranging gives for the general
im,,
[(@/7~
-
1) (Rintlrnin
-
'17
Rextl
(17)
If (Rint)minin Equation 17 is much greater than Re,$, then a = 4/3 and Equation 20 simplifies to Equation 13. If ReXtis much greater than (Riot),in, Equation 17 becomes identical to Equation 1 1 .
Equation 17 is a general expression for the difference which should be observed between the half-wave potentials of filtered and unfiltered polarographic waves. In aqueous solutions and in solutions of low specific resistance, both (R,nt)min and ReXtare usually quite small and AIEL/*in Equation 17 would approach zero. If a two-electrode polarograph is used for polarography involving solutions of moderate or high specific resistance, the uncompensated resistance, R,, may include significant contributions from both Rex$and Rint, but usually Rext >> Rint. If a three-electrode polarograph is used, Rextis eliminated so that R , = Rint. In case the residual current is an appreciable fraction of the total cell current, Equation 17 can be modified to include the time-dependency of the charging current (4). In these studies the residual current was always considerably smaller than the faradaic current and therefore its effect was not observed.
E‘
/
I
E
--+
Residual
EXPERIMENTAL
The instrumentation, procedures, and other experimental details employed in this study are reported fully elsewhere (10). The instrument used was constructed in this laboratory (2) and followed the design of the controlled-potential and derivative polarograph described by Kelley, Jones, and Fisher (1I ) , with the exception that a chopper-stabilized voltage-follower amplifier was inserted between the reference electrode and the potential control amplifier. A recorder with a response of one second (full scale) was used to record all polarograms. Inter-electrode cell resistances were measured with a Serfass conductivity bridge, Model RCM, at 1000 cps. When three-electrode cells were used, the DME was placed between the reference and counter electrodes at distances 2 1 cm from each. Selection and Preparation of Electrolyte Solutions. In order to evaluate the accuracy of the values of uncompensated resistance calculated by Equation 1, the method was tested on systems known to be reversible and for which the halfwave potentials are accurately known. Uncompensated resistances were calculated from experiments with cadmium ion in aqueous potassium chloride solution in which known amounts of common-path resistance were inserted in the cell leads. The calculated and known values of the resistance and the half-wave potential were then compared for both undamped and heavily damped polarograms. These measurements were made with the instrument functioning as a two-electrode polarograph-i.e., with the counter electrode connected to the reference electrode-so that the known, total cell current would pass through the measured cell resistance and produce a known iR drop. Further studies were made using glacial acetic acid as a solvent. In this solvent half-wave potentials are not known accurately, making comparisons somewhat more difficult. Measurements in acetic acid solutions having relatively low resistances were carried out in a three-electrode cell and assumed to be valid, then known amounts of common-path resistance were added, as in the aqueous studies. Repetitive polarograms were also recorded so that the precision with which the resistance-corrected half-wave potentials are measured could be evaluated. Residual Current. A common method of measuring the diffusion current for a conventional polarographic wave is to extrapolate linearly both the residual current and the diffusion current to the half-wave potential and to consider that the diffusion current is given by the difference between these extrapolated currents at this potential. In the presence of a large amount of uncompensated resistance, the polaro(10) W. E. Thomas, Ph.D. Thesis, Department of Chemistry, Indiana University, Bloomington, Ind., June 1966. (11) M. T. Kelley, H. C. Jones, and D. J. Fisher, ANAL.CHEM., 31, 1475 (1959).
Figure 1. Graphical construction of an approximate residual current curve for a polarogram greatly distorted by uncompensated resistance
graphic wave will be seriously distorted and errors due to extrapolation may become intolerable, even when the ratio of faradaic to charging current is large. It has been pointed out (3) that, whenever a polarographic wave is distorted by an iR loss, the true voltage scanning rate sensed at the surface of the DME lags the applied scanning
(3
rate by an amount proportional to R - -Le.,
the true
scanning rate lags the scanning rate applied by the instrument by an amount equal to the rate of change of the iR drop, Before and after the wave, where di/dt S 0, the true scanning rate equals the applied scanning rate and the slope of the polarogram is equal to the slope of the residual current curve. During the rising portion of the wave, however, where dijdt > 0, lag is introduced and the wave is distorted. Thus, extrapolations from linear portions of the polarograms are not valid when extended into the rising portion of the wave, because the true scanning rate is decreased in this region, and the true residual current is less than that predicted by linear extrapolation. In order to obtain accurate diffusion currents, a method was devised which takes into account the distortion of the wave and which allows a more valid residual current line to be constructed. According to the Heyrovsky-Ilkovir equation, the true DME potential for a reversible wave changes by 0.120/n volt between the half-wave potential and the points where i is 1 of id and where i is 99 of id. Figure 1 illustrates the technique of constructing the residual current line for a filtered polarogram recorded for thallium(1) in glacial acetic acid containing 0.036 molar sodium acetate, The diffusion current can be measured by constructing the usual extrapolated lines ( A F and A ’F’); locating the voltages A and A ’ where i = 0.01 id and i = 0.99 id, respectively; moving along the extrapolated lines, in the direction of the Eliz,from each of these points by an amount equal to 0.120/n volt (to points B and B’, which would coincide and represent the halfwave potential if the wave were not distorted by iR loss); and drawing horizontal lines from each of these points (lines BC and B’C‘). The difference between these horizontal lines gives the magnitude of the diffusion current at the half-wave potential as approached from opposite directions and is a good estimate of the true diffusion current. It can be seen from Figure 1 that the measured diffusion current would be in error by the difference between CC‘ and line FF’ if the residual current line were simply extrapolated in the usual way rather than constructed in the manner described here. In a typical experiment carried out in glacial
z
z
VOL. 41, NO. 1, JANUARY 1969
139
Table I. Comparison of Experimentally Evaluated and Known Added Increments of Uncompensated Resistance Experimental Total cell uncompensated Added external resistznce, cell resistance Applied Eli2 Resistance-corrected resistance, ohms X lo-* ohms X lo-‘ (Eq. l), ohms X volts us. SCE E112volts CS. SCE 0.0 0.0
A. AQUEOUS SOLUTION, 0.0
UNFILTERED POLAROGRAMS, TWO-ELECTRODE OPERATION^
0.33 ( R d ) 2.08 7.13
1.75 6.80
0.0
0.34 2.16 7.29
-0,598 -0.613 -0,683 -0,880
-0.598 -0.600 -0.598 -0.595
B. GLACIAL ACETIC ACID
SOLUTION, UNFILTERED POLAROGRAMS, TWO-ELECTRODE OPERATION* -0.316 22.8 -0.264 22.8 ( R i n d -0.378 51.2 -0.264 51.2 ( R o e l l ) 11.0 62.2 -0.395 63.4 -0.259 -0.456 87.2 36.0 91.6 -0.260 -0.500 108.6 57.4 109 -0.260 a Solution contains 0.1M KC1 and lOWM Cdz+. Cell resistance measured between DME and reference electrode is 3300 ohms. The first experiment listed utilized three-electrode operation with essentially no uncompensated resistance. The last three experiments utilized twoelectrode operation in which the total cell resistance is uncompensated and equals Roell(3300 ohms) plus the value of the external series resistor, b Solution contains 0.097Msodiumacetate and 1.1 X 10-4MTl+; specific resistance = 1.34 X 105 ohm cm. The first experiment represents three-electrode operation in which the uncompensated resistance = Rint = 22.8 X lo4 ohms. The remaining experiments involve twoelectrode operation in which the uncompensated resistance equals Reell( = 51.2 x lo4ohms) plus the value of the added external series resistor.
0.0 0.0
acetic acid, this difference can amount to an error of 10 to 20 %. The recorded polarogram prior to point A is a good representation of the residual current up to that point, and point C accurately represents the residual current flowing at the half-wave potential. Therefore, a straight line drawn connecting points A and C serves as a reasonable approximation of the residual current for the first half of the wave. Extending this line beyond C as a straight line parallel to a line connecting points C’ and A’ provides an approximation of the residual current in the second half of the wave. After point D,the residual current is again parallel to the experimental curve. Drawing the line DE parallel to A’E’ thus completes the residual current line. Calculations. The calculation of uncompensated resistance from both filtered and unfiltered polarographic waves was made using Equation 1. In the case of unfiltered waves, the maxima of the current oscillations were used in the calculations and, since the radius of the mercury drop is at its maximum when the current is maximum, the calculated quantity is the minimum resistance. The half-wave potential was measured from the same polarogram and corrected for uncompensated resistance by adding the term AE = imaxRmin in which the maximum current, ,i is the sum of (id)&2 and the properly extrapolated residual current flowing at the half-wave potential. In the case of filtered waves, the measured currents and calculated resistances are the values averaged over the life of the drop. RESULTS
Evaluation of Uncompensated Cell Resistance. A comparison of uncompensated resistance calculated by means of Equation 1 with known values is made in Table I. The first set of data given (A) was obtained with an aqueous solution containing 0.1M potassium chloride as supporting electrolyte and 10-3M cadmium ion as the electroactive species for which the half wave potential is known to be -0.598 volt US. the SCE. The first polarogram was run with a three-electrode cell, so that there would be no uncompensated resistance, and the other three polarograms were obtained with a two-electrode cell in series with which were added the indicated series resistors. The uncompensated resistance evaluated from the un140
ANALYTICAL CHEMISTRY
damped polarograms using Equation 1 agreed well with the known total cell resistance (columns 2 and 3). Correction of the applied half-wave potentials of the resistance-distorted waves by addition of the quantity imsxR(Equation lo), where ,i is the maximum current flowing at the half-wave potential and R is the experimental uncompensated resistance, gave half-wave potentials which agreed well with the accepted value for distortions as large as 0.3 volt. A second similar series of experiments was conducted with glacial acetic acid as solvent (part B of Table I). Again, progressively larger resistors were placed in series with a twoelectrode cell. The changes in the experimental uncompensated resistance calculated from undamped poiarograms closely paralleled the differences in the known values of the added resistors (compare runs 2-5). To evaluate the validity of the correction of the half-wave potential, it is necessary to assume that the corrected half-wave potential for thallium obtained at the lowest resistance value using a three-electrode cell is accurate (first experiment of part B), and then to compare with this value, the half-wave potentials obtained as higher resistances are added in series with the two-electrode cell. The corrected half-wave potentials are in good agreement with the three-electrode, low-resistance value, the largest difference being 5 mV in corrections exceeding 200 mV. The evaluation of uncompensated resistance and the correction of half-wave potentials for iR losses appear to be accurate for corrections as large as 300 mV in the case of undamped polarograms. With heavily damped, average current polarograms, agreement with theory was observed only for corrections up to about 100 mV. This result confirms the observations of McKenzie ( I 2 ) , who found that simple iR corrections calculated from the slopes of damped, averagecurrent polarograms were accurate for corrections of 30 to 40 mV, but were appreciably in error when the corrections became as large as 150 mV. Effects of Resistance on Filtered and Unfiltered Polarograms. The predictions of Equations 11, 13, and 17 concerning the differing effects of the internal and external components of (12) H. A. McKenzie, Rec?.Pure Appl. Chem., 8, 53 (1958).
Table 11. Comparison of Observed and Calculated Differences in the Half Wave Potentials of Unfiltered and Filtered Polarograms in the Presence of External or Internal Uncompensated Resistance Calcd (AElidunf-fii A: Eq. 11 Evaluated uncompensated B: Eq. 13 resistance, ohms X im,x at Em, pa mV A. UNCOMPENSATED RESISTANCE IS “EXTERNAL” ; TWO-ELECTRODE OPERATIONn 3.92 -3.3 -1.9 0.345 (R,xt) 2.16 (Rext) 3.93 -15.4 -12.1 3.92 -46.7 -40.8 7.29 (Rext) B. UNCOMPENSATED RESISTANCE IS “INTERNAL”; THREE-ELECTRODE OPERATION^
0.249 24.5 30.4 85.6 ( R i n t ) m i n 0.247 10.3 14.4 40.8 ( R i n t ) m i n 0.247 7.5 9.1 25.9 ( R i n t ) m i n 0.236 6.7 5.1 15.0 ( R i n t l m i n a Cell solution consists of aqueous 0.1M KC1 and 10-aM Cd2+in all three experiments. (The external resistance was varied by addition of various resistors in series with the cell.) * Cell solution is glacial acetic acid containing 10-4M T1+ and varying concentrations of sodium acetate added to change (Rindmin.The sodium acetate molarities are 0.010, 0.037, 0.063, and 0.097, respectively, in the four experiments. Table 111. Comparison of Observed and Calculated Differences in Half Wave Potentials of Unfiltered and Filtered Polarograms as a Function of Ratio of Internal to External Uncompensated Resistance. Total observed Added external Total external (Reeldmin Calcd resistance, resistance: [ = (R,)min Rextl Ratio Observed (A&/dunf-fil ohms x ohms X ohms X (RJlminiRext a (I) (AEl/z)unf-filmV (Eq. 17) mV 22.8 ( R i n t ) m i n m 1.334 8.1 7.4 0 0 0.445 51 . 2 ( R c e l d m i n 1.363 1.3 0.2 0 28.4 63.4 0.360 1.367 7.3 0.0 11.0 40.6 0.249 1.371 91.6 36.0 68.8 -7.4 -11.3 0.209 1.372 108.9 -19.6 -17.2 57.4 86.1 Cell solution: glacial acetic acid containing 0.097M sodium acetate and 1.1 X 10-4M T1+; specific resistance = 1.34 x 105 ohm cm. First experiment utilized three electrode operation and was run to evaluate ( R i n t ) m i n . All subsequent experiments represent two-electrode operation in which the observed uncompensated resistance should equal Reeii (51.2 X lo4ohms) plus the value of the added external series resistor. The value of i,, at the El/*was about 0.33 pA in all polarograms. b The values in column 2 are the differences between 22.8 X lo4ohms (line 1, column 3) and the values for (Rce&.+, given on the corresponding lines in column 3.
+
4
Uncompensated resistance on the half-wave potentials of filtered and unfiltered waves were also tested experimentally. In media of low specific resistance--e.g., aqueous solutions-(Ri)min should be negligibly small. The condition Rex$>> can therefore be realized by conducting twoelectrode polarography in aqueous solutions with the addition of large external resistors in series with the cell. For the verification of Equation 11, an aqueous solution containing molar cadmium ion and 0.1 molar potassium chloride was used. External resistors were inserted into the counter electrode lead and the instrument was operated as a two-electrode polarograph. The results are given in part A of Table 11. The signs of the differences in half-wave potentials are correct and the agreement between the experimental AEiiz and the values calculated by means of Equation 11 is reasonably good over the limited range of iR losses investigated. In a solvent of low dielectric constant, such as acetic acid, the specific resistance is large and gives rise to a large internal, time-dependent resistance. Using a three-electrode, controlled-potential polarograph, compensation for Rext should be complete and all the uncompensated resistance should be of the internal type. In this case addition of external resistors in series with the cell is ineffective and does not increase the uncompensated resistance. Rather, the composition and specific resistance of the cell solution must be varied in order to vary the internal resistance and the uncompensated resistance. To
verify Equation 13, polarograms were recorded for thallium (I) ion in glacial acetic acid solutions containing several different concentrations of sodium acetate as supporting electrolyte. The iR-compensating, three-electrode polarograph and cell were used so that the condition (Rt),i, >> Rextwould obtain. A comparison of the experimental deviations in the half-wave potentials with those calculated from Equation 13 is given in part B of Table 11. The differences again have the correct sign (opposite from those in Part A), and the quantitative agreement is fairly good over the range of uncompensated resistances investigated. To evaluate Equation 17 under conditions other than limiting, a solution containing 1.1 X molar thallium@) ion and 0.1 molar sodium acetate in glacial acetic acid was used so that (RJminwould be large and so that the ratio (ROmin/Rext could be varied by adding external resistors of various sizes in the counter electrode lead of a two-electrode cell. The specific resistance of this solution was high (1.34 X l o 5 ohm-cm) and consequently the internal resistance was also high. Polarograms were recorded using the instrument first as a threeelectrode polarograph, in which case the only uncompensated resistance would be internal resistance. Without adding any external resistors, polarograms were next recorded using the instrument as a two-electrode polarograph and the uncompensated resistance again evaluated. It was found that Rext of the solution in the cell was of the same order of magnitude VOL. 41, NO. 1, JANUARY 1969
141
as (Rint)minmeasured in the previous experiment. Continuing with two-electrode operation, various external resistors were placed in the counter electrode lead in order to vary the ratio (Ri,&in/Rext. In these experiments Rint had to be maintained constant because it was not possible to change RiDt without changing the composition of the solution and therewith the half-wave potential. Differences in the halfwave potentials for various ratios of (R&,in/Rext are given in Table 111. Values of a were interpolated from tabulated data given by BrdiEka ( 6 ) for various ratios of (RinJmin/Rext.Although the values given in Table 111 were more difficult to measure than in the limiting cases, the agreement between the experimental and theoretical differences in half-wave potentials is satisfactory. The differences are seen to become more negative with increasing proportion of ReXtand also to pass through zero and to change sign as predicted. These results support the conclusions of our previous studies
that resistance compensation is not complete in three-electrode cells containing solutions of high specific resistance. In addition, they show that the magnitude of the uncompensated resistance in a three-electrode cell, as well as the apparent half-wave potential, varies significantly with the degree of damping employed in the recording of the wave. When corrections for iR losses in the cell approach or exceed 100 mV, it is advantageous to record undamped, maximum-current polarograms using a fast response recorder, regardless of the type of cell used. RECEIVED for review August 19, 1968. Accepted October 7, 1968. Work supported by the U.S. Atomic Energy Commission under Contract No. AT(l1-1)-256. The authors are also grateful to E. I. du Pont de Nemours and Co., and Lubrizol Corp., who sponsored fellowships held by one of us (W.E.T.). (A.E.C. document No. COO-256-89.)
Nonunity Electrode Reaction Orders and Stationary Electrode Polarography Mark S . Shuman Department of Chemistry, Texas Christian University, Fort Worth, Texas 76129
The theory of diffusion-controlled electrode processes, mO ne c? qR, where one or both of the stoichiometric numbers m and q are integers greater than unity, has been considered for the triangular-wave chronoamperometric experiment. Two cases were considered in detail, 2 0 ne e R and 3 0 ne R. Theoretical current-potential curves for these reactions are lower in height and broader than the curve for unity m and q. Satisfactory correlations have been made between the theory and the dissolution of mercury into cyanide solutions.
+
+
+
*
THEpolarographic oxidation of mercury into alkaline cyanide solutions ( I , 2) appears to be a diffusion-controlled process in which the overall electrode reaction order is higher than first order. The current is limited by diffusion of cyanide, and the electrode reaction can be written Hg mCN- + Hg(CN),+*”
+ 2e
with stationary electrode polarography and nonunity reaction orders corresponding to the reactions 21-
e Iz + 2e
(4)
+ 2e
(5)
and 31-
F?
Is-
have been proposed (5). However, theoretical current-potential curves for these reactions and this electroanalytical method have not been available for comparison with experiment. The work presented here outlines the general method for obtaining theoretical current-potential curves when the electrode mechanism is Reaction 3 and m and q have any integral values. Results are given for the important cases of a second-order reaction (m = 2, q = 1) and a third-order reaction (m = 3, q = 1). THEORY
This oxidation appears as an mth-order electrode reaction with polarographic current-potential characteristics given by the following equation [see for example, Ref. (3)].
Boundary Value Problem. For a reversible reduction and Reaction 3, the boundary value problem for diffusion to a planar electrode is (6, 7 ) :
- -at
The potential-time relationship for any nonunity reaction order represented by the general scheme
+
mO neeqR (3) is known for the constant-current experiment (4). Recently, the oxidation of iodide ion in acetonitrile has been studied (1) J. Revenda, Collect. Czech. Chem. Commun., 6, 453 (1934). (2) I. M. Kolthoff and C. S . Miller, J . Amer. Chem. Soc., 63, 1405 (1941). ( 3 ) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, 1966, p 177. (4) W. H. Reinmuth, ANAL.CHEM., 32, 1514 (1960).
142
ANALYTICAL CHEMISTRY
DO a2- v o ax
(7) For t
= 0, x
2 0,
(5) G . Dryhurst and P. J. Elving, ibid., 39,606 (1967). (6) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, Chap. 3. (7) R. S. Nicholson and I. Shain, ANAL.CHEM., 36,706 (1964).
