Verification of a diffusion current equation accounting for convection

1980

ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978

constituents in each ash fraction. A shortcut procedure for the determination of the “alkali fraction” may be employed in which the water-ash slurry is neutralized to pH 7, followed by determination of soluble sulfate, Na, K, Ca, and Mg. This should give more realistic results for most applications than does a published procedure involving extraction with deionized water (12).

(6) 0. E. Manz in U . S . Bur. Mines I n f . Circ., 8650, G. H. Gronhovd and W. R. Kube, Compilers, pp 204-19 (1974). (7) C. E. Brackett in U . S . Bur. Mines I n f . Circ., 8640, J. H. Faber, W. E. Eckard, and J. D. Spencer, Compilers, pp 12-18, (1974). (8) J. D. Eye and T. K. Basu, J . Water Pollut. Control Fed., 42 (2), R125 (1970). (9) F. N. Rhoad, S. Afr. Pub Paper Mfr., 32, 62 (1969). (10) P. H. Tufte, E. A. Sondreal, K. W. Korpi, and G. H. Gronhovd in U.S. Bur. Mines Inf. Circ., 8650, G. H. Gronhovd and W. R. Kube, Compilers, pp 103-33 (1974). (11) Anon., J . Air Pollut. Control Assoc., 27, 948 (1977). (12) D. C. Martens, Compost Sci., 12 (6), 15 (1971). (13) J. P. Capp and D. W. Gillmore in U.S.Bur. Mines Inf. Circ., 8640, J. H. Faber, W. E. Eckard, and J. D. Spencer, Compilers, pp 258-68 (1974). (14) N. Singh and S. B. Mather, NML Tech. J . , 15 (2), 42 (1973). (15) H. S. Simons and J. W. Jeffery, J . Appl. Chem., I O , 328 (1960). (16) D. M. Bibby, Fuel, 56, 427 (1977). (17) P. Schreiter, Siiikatfechnik, 19, 358 (1968); Chem. Abstr., 70, 108100s ( 1969). (18) D. G. Shannon and L. 0. Fine, Environ. Sci. Technoi., 8, 1026 (1974). (19) C. Block and R. Dams, Environ. Sci. Techno/., I O , 1011 (1976). (201 , , R. L. Davison. D. F. S. Natusch, J. R. Wallace. and C. A. Evans, Environ. Sci. Techno/., 8, 1107 (1974). (21) D. F. S. Natusch, J. R. Wallace, and C. A. Evans, Science, 183, 202 119741. (22) k.-W.’Linton, A. Loh, D. F. S. Natusch, C. A. Evans, and P. Williams, Science, 191, 852 (1976). (23) T. L. Theis and J. L. Wirth, Environ. Sci. Techno/., 11, 1096 (1977). (24) G. L. Fisher, B. A. Prentice, D. Silberman, J. M. Ondov, R. C. Ragaini, A. H. Bierman. A. R. McFariand, and J. 6.Pawley, A m . Chem. SOC.. Div. Fuel Prepr., 22(4), 149 (1977). (25) “Coal Humic Substances and Their Application to PollutionControl in the Synthetic Fuels Industry”, S. E. Manahan, R. E. Pouison, J. B. Green, and D. S. Farrier, Report of Investigations, U.S. Department of Energy, Laramie Energy Technology Center, Laramie, Wyo., 1978.

SUMMARY Dissolution of the lignite fly ash sample studied was shown to be a function of solution acidity and the intermixing of fly ash components. The amount of base obtained from fly ash through direct reaction with acids is much greater than that obtained from dissolution in pure water, a fact of significance in the chemical utilization of fly ash. Fly ash dissolves in discrete steps with increasing acidity, thereby indicating the presence of specific fractions within the material. Determination of total available base in fly ash can be accomplished via dissolution in dilute mineral acid, followed by either back titration with NaOH, or determination of dissolved sulfate, Na, K, Ca, and Mg. LITERATURE CITED J. B. Green, Potential of Humic Acids as Stack Gas Scrubbing Media for Removal of Sulfur Dioxide and Fly Ash, Ph.D. Dissertation, University of Missouri Department of Chemistry, Columbia, Mo., 1977. J. A. Campbell, J. C. Laul, K. K . Nielson, and R. D. Smith, Anal. Chem., 50, 1032 (1978). J. M. Ondov, W H. Zol!er I. O l m z , N. K. Aras, G. E. Gordon, L. A. Rancitelii, K. H. Abei. R. H. Filbv, K . R. Shah, and R. C. Raaaini, Anal. Chem.. 47, 1102 (1975). D. R. Dressen, E. S. Gladney, J. W. Owens, B. L. Perkins, C. L. Wienke, and L. E . Wangen, Environ. Sci. Techno/., 11. 1017 (1977). W. D. James, M. Janghorbani, and T. Baxter, Anal. Chem., 49, 1994 (1977).

RECEIVED for review July 14. 1978. Accepted September 8, 1978. This research was supported by the U.S. Department of the Interior, Office of Water Research and Technology, Allotment Grant A-106-MO.

Verification of a Diffusion Current Equation Accounting for Convection and Capillary Shielding at the Dropping Mercury Electrode Timothy E. Cummings University of Miami, Coral Gables, Florida 33 124

Philip J. Elving” University of Michigan, Ann Arbor, Michigan 48 109

A previous paper ( I ) reported on the effects of DME orifice contact area and shielding by a blunt-ended capillary (combined effect is referred to as the effective contact area) on observed ac and dc polarographic response as compared to theoretical prediction. The results indicated a difference between the effective contact area, measured from ac polarographic capacitive current data, and dc polarographic diffusion current data for cadmium(I1) in KC1 and iron(II1) in K2C204aqueous media. Although possible reasons for the discrepancy were advanced, the data did not permit quantitative elucidation of the discrepancy source. In order to determine more precisely the causes of the effective contact area-which is ca. two orders of magnitude larger than the orifice contact area-the effective electrode area of a tapered-tip capillary DME has been evaluated as a function of mercury column height and drop-time using ac and dc polarography. Additionally, the possible influence of stirring effects, due to mechanical drop dislodgment or natural drop-fall, on the observed faradaic response has been con-

Examination of ac polarographic capacitive currents for 0.1 M KCI and dc polarographic diffusion currents (id) for cadmium(I1) at a tapered dropping mercury electrode (DME) indicates that the extent of shielding by the capillary is the same for faradaic and nonfaradaic processes. The shielded area is equivalent to the area defined by projection of the capillary tip onto the mercury drop surface. When using a mechanical drop-knocker, stirring effects persist for ca. 0.15 s after drop-birth, compared to ca. 0.05 s under conditions of natural drop-fall. A modified form of the diffusion current equation for an expanding sphere, which accounts for both capillary shielding and convective stirring at drop-birth, accurately describes the id-t behavior on single drops after ca. 0.2 s. Concentration depletion at drop-birth does not exist at tapered capillaries. The modified equation also accurately describes the id-f behavior at nontapered capillaries when mechanical drop dislodgement is used; under conditions of natural drop-fall, concentration depletion at drop-birth causes negative deviation from prediction. 0003-2700/78/0350-1980$01 O O / O

C

1978 American Chemical Society


sidered in the formulation of a modified equation for the diffusion current a t an expanding sphere electrode.

EXPERIMENTAL Chemicals. Reagent grade CdC12.2.5H20(J. T. Baker) was dried at 110 "C for several days. Reagent grade KC1 was used. Mercury for electrodes was chemically purified and distilled. Instrumentation. The electrochemical cell, potentiostat, recorder, and digital voltmeter have been described ( 1 ) . AIternating current polarographic capacitive current measurements were made with a Princeton Applied Research Model 122 lock-in amplifier. The DME capillary was drawn from 4-mm o.d., 1-mm i.d. Pyrex tubing; its length from tip to the point at which the glass began to taper from a 4-mm o.d. was 7.4 cm. The capillary orifice diameter and capillary tip o.d. were measured with a Leitz microscope, using an ocular micrometer, which was calibrated by means of a Bausch &- Lomb Model 31-16-90 stage micrometer ruled to 0.01 mm. Procedures. Solutions were deaerated with prepurified nitrogen for 30 min; a nitrogen atmosphere was maintained in the cell during experiments. The cell was kept at 25.0 f 0.1 "C. Reported potentials are vs. an aqueous saturated calomel electrode. Direct Current Polarography. For 1.022 It 0.004 mM Cd(I1) in 0.1 M KCl solution, the Cd(I1) diffusion current (id) at -0.750 V was measured as a function of mercury column height ( h )and drop-time ( t ) . At each combination of h and t , the currents for 8 to 27 drops were measured (the number of drops depended upon the current reproducibility at the end of the drop-life);an identical procedure used for background alone permitted background subtraction. Alternating Current Polarography. The capacitive current (iac) for 0.1 M KC1 solution for a 10-mV p-p, 100-Hz applied ac modulation voltage was measured at -0.350 V for each of 14 drop-times at 5 column heights; the number of drops measured a t each combination of h and t depended upon the current reproducibility. Measurements on the apparent in-phase component of the capacitive current, due to iR loss with concomitant phase shift of the working electrode ac voltage relative to the applied ac voltage, showed that there was no measurable iR loss, Le., the in-phase signal-to-noise ratio, which was less than two, indicated an apparent in-phase component which was less than 1% of the out-of-phase component; hence, the measured out-of-phase signal was equal to the ac capacitive current. Mercury Flott-Rate. Mercury flow-rates (m) were determined at open circuit for three combinations of h and t by collecting the drops for periods of 30 to 40 min. At h of 49.6 and 77.6 cm, the mean m for 2.26-s drop-time were 0.269 and 0.434 mg/s, respectively; at 49.6 cm and 3.19 s, m was 0.272 mg/s. The resulting corrected column-heights (h,) of 45.94, 74.48, and 46.35 cm, respectively, yielded an average m / h , of 5.84 x mg/cm s. Measurements of m at 77.6 cm and 2.66 s 3 and 18 days later yielded 0.435 and 0.433 mg/s, respectively. THEORY Mercury Flow-Rate. The mercury flow-rate, m,at time, t , was evaluated from the determined m / h , (or K ) and Equation 1: where h is the uncorrected column height and h, is the height corrected for back-pressure, hb.

m = Kh

-

3.1K/(mt)'i3

(1)

Contact Area. The evaluation of the effective contact area (A,) depends upon the assumed shape of the drop. Although several studies (2-4) found the spherical drop assumption to be sufficiently accurate, at least for drop-area accuracy of 0.5% or better (depending on drop size), two cinematographic studies ( 3 , 5 )found that the photographically measured drop area (A,) is not exactly proportional to t2j3. Smith (3),using capillaries with drop-times of 40 to 70 s, found that the power o f t which best described the data lay between 0.70 and 0.72; Newcombe and Woods ( 5 ) , using a capillary with 1.7-s drop-time, found the best power of t to be 0.72. Nonspherical Drop-shape. Although the cinematographic studies did not necessitate evaluation of A , - a log A, vs. log t plot of Equation 2, where T,,and are determined, could

1981

be used for slope evaluation-the electrochemically effective area will not equal the geometric area, so that Equation 3 rather than Equation 2, must be used for electrochemical data.

A, = sOtJ1

(2)

Spherical Drop-shape. If the drop is assumed to be spherical at the time of current measurement, Equation 4, due to Mohilner et al. ( 6 ) ,is used for evaluation, where j,, is the current density and d is the density of mercury. Equation 4 is similar in form to Equation 3. where is assumed to be 213.

where the term in parentheses is the gross calculated spherical area (AJ, Le., A, - A , = A, the effective electrode area. Faradaic Current. For a DME operating with natural drop-fall, the Ilkovic equation (Equation 5 ) , where n is the number of electrochemical equivalents per mole of species electrolyzed, D is its diffusion coefficient in cm'js and C is its bulk concentration in mM, fairly accurately describes the observed diffusion current ( 7 ) ;however, Taylor, Smith, and Cooter (8) showed that the instantaneous current does not obey Equation 5.

From ca. 0.1 s to drop-fall, id is roughly proportional to t1!3 (8): Le., the current is better described by assuming spherical diffusion. Markowitz and Elving (9) found that the spherical diffusion derivations due to Matsuda (10) were most rigorous; of these, the one accounting for shielding (Equation 6 ) or that assuming no shielding (Equation 7),depending upon time after drop-birth, has been shown to best describe the id-t relation a t a blunt-tip capillary ( I ) .

id = 709 nD'12C7722i3t'/6(1 + 23."'l't''6/m1/3)

(6)

id = 709 nD1/2Crn2:3t'I6(1 + 36,3D1!2t1'6/m1/3) ( 7 ) Extension of Equation 7 to include the third term in the series expansion (10)yields Equation 8. The last term in Equation 8 contributes significantly, Le., 170or greater, only for t1/6/rn1'3 greater than 2.

id = 709nDl/2Crn2/3~t'!6(1 + 36.3D1/2t116/m1/3 + 3430 t

/ rn

j3)

(8)

Effectice Contact Area Correction. None of the theoretical equations presented accounts for experimentally measured id-t behavior over the entire drop-life; currents deviate significantly from either Equation 6 or 7 at times shortly after drop-birth, and slopes of id/t'I6 vs. t 1 / 6plots are much greater than predicted even a t 5 s ( I ) . However, Cummings and Elving ( I ) derived Equation 9, which accurately describes the id-t relation at a controlled drop-time, blunt-tip capillary to times as short as 0.3 s. id = 709nD1'2Crn2/3t1/6(1+ 36.3D'/2t1/6/rn1/3) [ l - 117.8A,/(mt)2/3] ( 9 )

Diffusion Coefficient Ecaluation. Since Equations 6, 7, and 9 are quadratic equations in D1l2,D1j2m ay be evaluated by Equation 10, where B is 23.5 or 36.3 and Zdf is given by Equation 11. If A,, is assumed to be zero, Equation 11 reduces to the definition of Idfor instantaneous current

D1/2 = [ ( I + 4Bt1!61d'/709nrn1/3)1/' - l]mli3/2Bt1/6 (10)

1982


Id’

= id/C(m2i3t”6- 117.8A,/t”2)

(11)

Zero should be used for A, in Equation 11 if B is 23.5 in Equation 10, since this B corresponds to Equation 6, which already accounts for shielding. Conuection Due to Stirring. Stirring effects due to natural drop-fall ( 1 1 ) or mechanical dislodgement ( 1 ) have been postulated as possible causes for current enhancement early in drop-life. If such convective effects occur shortly after drop-fall, concentration a t the electrode surface will not be depleted until cessation of stirring and consumption of the electroactive material in excess of bulk concentration, which was convectively delivered to the surface. Thus, any terms in Equations 6 to 9 which involve time of formation of a diffusion layer should measure that time from the start of diffusion layer growth rather than from drop-birth; however, terms which involve electrode area or radius dependence on time, must be measured from drop-birth. In the case of Equations 6 to 9, the first t1/6on the right-hand side is due to the time-dependence of the electrode area ( t 2 l 3 )multiplied by that of the current density ( t - ’ l 2 ) ; hence, the latter dependence must be corrected for the time after drop-birth a t which a diffusion layer begins to form ( t o ) , Le., the first t116 term is replaced by t 2 / 3 / ( tExamination of Matsuda’s derivation (10)regarding the power series indicates, although never so stated expansion in D1”2t1/6/m1/3 by Matsuda, that the t1l6 in this series is due to the timedependence of the diffusion layer thickness ( t 1 j 2 )divided by that of the drop radius (t’/3). Thus, the t1’2term must be corrected for to,and all t’l6 in the power series expansion are replaced by ( t - t , ) 1 / 2 / t 1 /raised 3 to the appropriate power, e.g., t’j3 in Equation 8 becomes ( t - t o ) / t z ’ 3 .Correction of Equation 9 for t o yields Equation 12. id

+

= 709nD1’2C(mt)2’3/(t - t,)‘i’2[1 36.3D1”(t

-

t,)’~z/(mt)1/3][l - 117.8A,/(mt)2/3] (12) Solution for D’/* in Equation 1 2 yields Equation 13, where 36.3 and I d * is given by Equation 14.

B is

D’” = {[I4- 4BId”(t - t , ) ” 2 / 7 0 9 ~ ( ~ t ) ’ ~- 311] 1 ~ 2 ( m t ) 1 : 3 / 2 ~ ( t- t p (13) 1,” = i d ( t - t,)”2/C[(mt)2’3- 117.8A0]

(14)

Statistical Analysis of the Data. For least-squares fits to Equation 3 or 4, the precision and accuracy to which the equation and associated determined parameters describe the data, are measured by the standard deviation of the fit (sy) given by Equation 15, where 6, is the difference between the observed i, at ti and that predicted by the fitted line. n

sy =

[ ( L y ) / ( n- 2 ) ] ’ / 2

(15)

1

For data analysis using Equation 3, reported uncertainties in the T , were calculated from sy. For data analysis using Equation 4, reported uncertainties in the slope, intercept, and derived parameters were calculated from sy and standard deviations for m/h,, t , and h of 2.1 x 10-5mg/cm s, 0.01 s, and 0.02 cm, respectively, based on the relation between the variance of related parameters and the equations for linear least-squares slope and intercept calculation (12). R E S U L T S A N D DISCUSSION: E F F E C T I V E CONTACT AREA Alternating current polarographic capacitive current data (Table I) for 0.1 M KC1 a t -0.350 V were used for effective contact area (A,) evaluation. Grahame (13) reported the differential capacitance of 0.1 M KC1 a t -0.44 V vs. 0.1 N calomel electrode (which is within f 2 mV of -0.350 V) to be 39.44 fiF/cm*; he does not indicate that A, or drop non-

Table I. Alternating Current Polarographic Capacitive Currents as a Function of Drop-Time and Mercury Column Heighta h, cm

t, s 2.26 2.32 2.39 2.47 2.53 2.62 2.70

2.79 2.90 3.00 3.10 3.26 3.40 3.56

49.6 0.669 0.684 0.701 0.716 0.732 0.752 0.770 0.786 0.813 0.833 0.855 0.885 0.913 0.93Bb

56.6 0.752b 0.772 0.789 0.806 0.822 0.846 0.866 0.887 0.91 1 0.932 0.957 0.992 1.024 1.O4Bb

63.6 0.823 0.840 0.858 0.877 0.892 0.920 0.942 0.964 0.992 1.013 1.044b 1.080

70.6 0.887 0.903 0.929 0.948 0.964 0.994 1.014

1.036 1.069 1.094 1.124 1.16gb 1.109 1.197 1.141b 1.238

77.6 0.956b O.97lb 0.994 1.011b 1.030 1.057 1.084 1.107 1.143 1.172 1.204 1.247 1.281 1.323

a E,, = - 0.350 V; E,, = 5-mV peak; f = 100 Hz; current magnitudes are given in M Apeak current, Points not used in least-squares fits to Equations 3 and 4.

sphericity was considered in calculation of the electrode area; however, since he used a natural drop-fall, A , should not be significant at the time of measurement, e.g., the capillary 0.d. was only 100 fim and the final drop area for the 40-fim i.d. was ca. 2.5 X lo-’ cm2 ( 4 ) . Nonspherical Drop-Shape. Attempts to fit the Table I data at each h to Equation 3 by a nonlinear least-squares method ( 1 4 ) gave poor results, apparently due to very large correlation among the T , being determined and the small number of data used for a three-parameter fit. For three of the h, K~ was ca. 0.5 and T ? was less than -0.2 PA, while a t 56.6 cm and 77.6 cm, r1were 0.63 and 0.73, respectively, and K~ were -0.13 and +0.021 PA, respectively. although the standard deviations of the fits, sy,(Equation 15) were less than 3 nA, the 95% confidence intervals on the T , were so large that the results are meaningless. A single fit to the data a t all h gave T,,, K’, and r2of 1.16 f 0.16,0.60 f 0.09, and 4.18 f 0.16, respectively (uncertainties cited are 95% confidence intervals based on sy and the T , interactions). Although sy was only 5 nA, the uncertainties are large; however, the negative deviation of all data at 49.6 cm from prediction by ca. 0.5% (probably due to the different lock-in sensitivity a t this h ) biased the results. Re-evaluation, using only the four largest heights gave H,,,rl,and r2 of 0.96 f 0.20, 0.71 f 0.14, and 0.015 f 0.006, respectively. T h e r1 of 0.71 agrees with the values of Smith (3) and Newcombe and Woods ( 5 ) . The positive r2 indicates that a mercury surface is already present at drop-birth; since 7r2 is proportional to its area through the ac current density (hence, CdJ, an area of 1.2 x cm2 is calculated using Grahame’s Cdl of 39.44 pF/cm2. Such a large initial drop mass seems unlikely; more probably, the positive x 2 is due to the nature of the data fit. Examination of the deviations between observed and predicted currents showed that, in general, the five smallest t at each h deviated negatively from prediction while the large t deviated positively. Since Equation 3 is simply lactimes an equation relating electrode area to drop mass, it would appear that no single equation can adequately relate area and drop mass over a wide range of drop mass. Although evaluation of Cdl from the slope, i.e., x o of Equation 3, might seem a reasonable approach for testing the determined fit, no simple mathematical relation is available which relates drop area and the 0.71 power of drop mass; since “0 represents the product of j,,, which is related to Cdl, and the unknown constant for the area-(mt)’” relation, Cd cannot be evaluated from H,,. This means that, unless Cdl were accurately known

ANALYTICAL CHEMISTRY, VOL.

50, NO. 14, DECEMBER 1978

1983

Table 11. Parameters Derived from Linear-Least Squares Fit of ac Capacitive Current to Equation 4' h, cm

slope, fiA/mg2'3

intercept, fiA

C,, fiF/cm2

104A,, cm2

s y rnAb

49.6 56.6 63.6 70.6 77.6 mean std. dev. rel. std. dev., 7%

1.052 (0.013) 1.054 (0.014) 1.049 (0.014) 1.051 (0.012) 1.050 (0.014) 1.051 0.002 0.18

-0.0829 (0.0107) -0.0755 (0.0125) -0,0777 (0.0138) - 0.0844 (0.0131) -0.0882 (0.0164)

39.44 (0.49) 39.52 (0.52) 39.33 (0.53) 39.40 (0.45) 39.37 (0.53) 39.41

6.70 (0.87) 6.08 (1.01) 6.29 (1.12) 6.81 (1.06) 7.13 (1.33) 6.60 0.42 6.3

1.5 1.2 2.1 2.3 2.8

- 0.0817

0.0051 6.3

0.07 0.18

Experimental data are shown in Table I, which indicates those points excluded from the fit. Numbers in parentheses are standard deviations for the corresponding value. The nature of sv is described in the text, for the conditions used to obtain i,, A , could not be obtained from a fit to Equation 3 without resort to some method permitting evaluation of the constant relating drop area t o (mt)rl. Spherical Drop-Shape. Analysis of the Table I data using Equation 4 (Table 11) yields an average A, of 6.60 0.19 X cm2and an average Cdl of 39.41 0.03 pF/'cm2,based on slopes of i, vs. (mt)2'3and in excellent agreement with Grahame's 39.44 FF/cm2 (13). The relative standard deviation of the mean, 0.08%, which is much smaller than the uncertainty of the experimental data, indicates the precision to which Cdl can be determined and, hence, the reproducibility * ' ~ As would be expected, of the observed i, vs. ( r ~ t ) slope. because of the lengthy extrapolation to the intercept, the relative standard deviation of A,, 2.8%. is considerably larger. Validity of the Spherical Drop ilssumption. Several considerations indicate the spherical assumption to be satisfactory: (a) The largest drop employed in the present study is only 40% of the maximum stable drop volume for the capillary employed (cf. section on Significance of the Contact Area); for a capillary of 80-pm i.d.: the spherical assumption results in a 0.0676 error in area calculation when the drop is 4076 of maximum volume and the surface tension is 425 dyne/cm ( 4 ) . (b) Deviations between observed and predicted currents for the least-squares fits of Table I1 show no systematic trend with drop area; if the drop shape deviated significantly from spherical, a systematic trend in the deviations would appear on imposing the spherical assumption, Le., abscissa of (mt)'Iia,on the least-squares fit. (c) The excellent agreement seen between the Table I1 Cdl and Grahame's value, which was also based on the spherical assumption and was obtained from a drop of much larger volume (up to 3 times that of the largest drop in the present study). Accuracj, of the Method. The sensitivity of intercept derived quantities to the number of points fitted and, more importantly, to the distance of the points from the origin is shown in Table I11 for the best-case value, h = 49.6 cm, for which the data lie closest to the origin. These results emphasize the importance of using data a t as many drop-times and over as wide a time range as possible for Cdl or A, evaluation. The small A, of 2.67 X cm2 obtained from the four-point fit over the time range of 2.995 to 3.405 s is due to the nature of the deviations involved, Le., the last four points in the fit to the thirteen points from 2.260 to 3.405 s show deviations of 3, 1, -1, and -2 nA. Significance of the Contact Area. T o compare the determined A , magnitude with the magnitudes of possible contributing sources, e.g., orifice contact area and shielding by the glass tip, the inner and outer diameters at the tip were measured microscopically. The inner and outer diameters of 25.8 and 254 pm yield orifice and cross-sectional areas of 5.2 X and 5.1 X cm2. Use of Tate's law (Equation 21 of Ref. 4) to evaluate the orifice diameter from the maximum

*

*

Table 111. Influence of Time Range of Data and Number of Points Fitted on Parameters Derived from Least-Squares Analysis of ac Capacitive Current Using Equation 4'

3

time range, s

C, fiF/cm2 lo'&, cm' 2.260-2.470 39.10 (3.64) 6.25 (5.87) 1.6 2.260-2.530 7.14 (4.15) 1.4 39.70 (2.59) 2.260-2.620 40.15 (1.89) 7.77 (3.03) 1.3 2.260-2.705 40.07 (1.43) 7.62 (2.32) 1.1 2.260-2.790 39.40 (1.17) 6.70 (1.95) 1.4 2.260-2.905 39.59 (0.94) 6.95 (1.58) 1.3 2.260-2.995 39.89 (0.80) 7.37 (1.34) 1.4 2.260-3.105 39.89 (0.67) 7.37 (1.14) 1.3 2.260-3.265 39.62 (0.57) 6.97 (0.99) 1.5 2.260-3.405 39.44 (0.49) 6.70 (0.87) 1.5 2.320-3.405 39.32 (0.26) 6.53 (0.44) 1.5 2.390-3.405 39.30 (0.29) 6.49 (0.52) 1.6 2.470-3.405 39.36 (0.34) 6.58 (0.62) 1.7 2.53 0- 3.405 39.10 (0.35) 6.14 (0.66) 1.6 39.02 (0.44) 2.620-3.405 5.98 (0.83) 1.7 2.705-3.405 39.07 (0.56) 6.09 (1.07) 1.8 2.790-3.405 39.06 (0.76) 6.06 (1.47) 2.0 2.905-3.405 38.03 ( 0 . 6 1 ) 4.08 (1.23) 1.3 2.995-3.405 37.31 (0.38) 2.67 (0.78) 0.6 average 39.29 6.39 std. dev. 0.68 1.21 rel. std. dev., % 1.7 19.0 Experimental data are those shown in Table I for I1 = 49.6 cm, which indicates those points excluded from the fit. The nature of sy is described in the text,

I

Figure 1. Spherical mercury drop suspended from capillary of 254-pm outer diameter: drawn to scale. Conditions: 49.6 cm uncorrected cm3drop mercury column height: 2.25 s after drop birth: 4.46 X volume: 6.09 X cm2 spherical drop area, A , . Dimensions are shown in Mm: that for 135 represents an arc. The region of the drop extending from the capillary orifice to 40 pm below the orifice and from the axis of the bore to either edge of the capillary represents the area defined by projection of the capillary onto the drop surface

1984


measured drop volume obtainable (2.86 X cm3) at h of 49.6 and 77.6 cm yields 27.2 pm, which is within 5% of that determined microscopically. Figure 1 is a scale drawing of the capillary tip and a mercury drop. The surface area directly below the tip projection of cm2 is within 1.3 standard deviations of the de5.6 X termined A , (Table 11). Since this area does not change significantly over the range of h and t used, e.g., the area is 5.3 x cm2 at 77.6 cm and 3.56 s, the projected tip areas are within 1.4 standard deviations of all determined A,; hence, there is no significant difference between these A , magnitudes and that of the drop area defined by projection of the tip onto the drop surface. Because the drop is 40 pm below the outer edge of the tip, such a large A, cannot be explained by assuming the layer of solution between mercury and glass to be too thin for formation of the electrical double layer, Le., 95% of the double layer potential for a 0.1 M solution is dissipated within 32 A of the outer Helmholtz plane (15);however, the insulating glass tip unquestionably interferes with the bulk solution electric force lines between counter and working electrodes, and may, therefore, prevent establishment of a double layer on that portion of the mercury surface which is within some critical distance from the tip. From the experimental data and determined A , for a blunt capillary ( I ) , the shielded area includes regions of the mercury surface which are 49 pm away from the capillary tip, if a spherical drop shape is assumed. An additional possible source for part of the apparent A , is retraction of the mercury into the capillary when the drop is detached. Lingane (26) noted that the cell current was zero for both faradaically active and inactive solutions for times ranging to greater than 0.1 s between drop detachment and birth of a suceeding drop: under such conditions, the mercury might retract as much as 6 mm into the capillary. Newcombe and Woods ( 5 ) also observed delay times from a few milliseconds to a few minutes, which were due to either mercury retraction at drop detachment or a stationary state in which mercury remained at the orifice for a short period after prior drop detachment. The extent of retraction should depend on h, immediately after drop dislodgement. For the capillary used, hb is 49.2 cm when the orifice radius is used in the relation, 27 l r g d , where y is the surface tension and g is standard gravity; for h of 49.6 and 77.6 cm, h, are 0.4 and 28.4 cm, respectively, at drop-birth. Assuming a 3-mm retraction at h of 49.6 cm, the retraction volume is 3.6% of the drop volume at 2.26 s; this should result in significant variation of determined m / h cwith h, which was not observed hence, retraction, if present, is likely to be slight. Although the data are somewhat ambiguous and irreproducible, periods of no drop-growth cannot exceed 0.06 s at h of 49.6 cm; accordingly, retraction cannot exceed 0.002 cm, so that essentially a static mercury ribbon obtains at the orifice during this period. A 0.06-s delay in drop growth would by 2% for t of (2.26 - 0.06)s: while at a true decrease (mt)2/" t of 3.345 s would be decreased by 1.3%. Such a change in the i, vs. plot would decrease the determined A , a t 49.6 cm from 6.7 X to 5.2 X cm2 and would lower the slope-determined Cdl from 39.4 to 39.0 pF/cm2, in disagreement with Grahame's result (13). Thus, under the experimental conditions employed, the determined A, is apparently a true shielded area, as defined by projection of the capillary tip onto the drop surface.

RESULTS AND DISCUSSION: CADMIUM(I1) DIFFUSION CURRENT Single Drop Current-Time Relation. The i-t relation for the first drop with natural drop-time (Figures 2 and 3) indicates that an A, of 6.6 X cm2 is too large to properly correct the observed current for shielding and to yield the

i.5

0.2

8.C

cc

5.-

L

I

, ' b , - :

5

'3

r . J -2

Figure 2. Variation of cadmium(I1) diffusion current ( i d )with time ( t ) during a single drop-life. Conditions: h = 49.6 cm; m l h , = 5.84 X m g / c m s; 1.022 mM Cd(I1) in 0.1 M KCI. (A) First natural drop after drop knocker shut-off; k = 1. (B) Same as A; k = 1 - 0.00023/AS. (C) Same as A; k = 1 - 0.00066/A,. (D) Theoretical behavior based c m / s ' / z ( 77). The Darameter on Equation 8 with 0"' = 2.65 X k is a means of area correction

t, 0.2

3

t 1.0

I

5

0.5

1.0

2.0

5 .O

I

I

I

I

I

I

1

1.3

tlprn':

I

I

1.6

I

9

1 I.

gl/yrngi/3

Figure 3. Variation of cadmium(I1) diffusion current ( i d )with time ( t ) during a single drop-life. Conditions: h = 77.6 cm; m l h , = 5.84 X m g / c m s. 1.022 mM Cd(I1) in 0.1 M KCI. (A) First natural drop after drop-knocker shut-off; k = 1. (B) Same as A; k = 1 - 0.00023/AS. (C) Same as A; k = 1 - 0.00066/A, (D) Theoretical behavior based on Equation 8 with 0"' = 2.65 X cm/s'" ( 17). The parameter k is a means of area correction

behavior predicted by Equation 7. The best fits indicate that A, decreases from 2.64 f 0.34 X cm2 a t h of 49.6 cm to 1.96 f 0.34 X lo-* cm2 at h of 77.6 cm. Such a trend is physically reasonable, since the area defined by projection of the capillary tip onto the drop surface decreases as the drop radius increases, Le., with increasing t a t constant h or with increasing h at constant t . Depletion ut Drop-Birth. It is generally presumed that, at a tapered-tip capillary, depletion effects caused by preceding drops are negligible, since the tip does not significantly shield the solution. Comparison of Figures 4 and 5 for the second natural drop with Figures 2 and 3, respectively, indicates that, during the first 10 s of drop-life, the current for the second natural drop is lower than that for the first natural drop.


+ L

1.2

08

/

1

I

1 ~

.6

2C

2.4

Figure 4. Variation of cadmium(I1)diffusion current (id)with time ( t ) during a single drop-life. Conditions: h = 49.6 crn; rnlh, = 5.84 X mg/cm s; 1.022 mM Cd(I1) in 0.1 M KCI. (A) Second natural drop; k = 1. (B) Same as A; k = 1 - 0 . 0 0 0 5 6 / A S . (C) Same as A; k = 1 - 0 . 0 0 0 6 6 / A S . (D) Theoretical behavior based on Equation 8 with 01/2 = 2.65 X cm/s"' ( 77). The parameter kis a means of area

correction +,

51

0.5

0.3 I

I

1.0

I

s

5.0

2.0

-_--

C

9 .0

1

- e-5_ I

_-

I -

/ .

I

/ 3

I

i

I

I

I

I

I

Figure 5. Variation of cadmium(I1)diffusion current ( i d ) with time ( t ) during a single drop-life. Conditions: h = 77.6 cm; m l h , = 5.84 X mg/cm s; 1.022 mM KCI. (A) Second natural drop; k = 1. (€3) Same as A; k = 1 - 0.0052/A,. (C) Same as A; k = 1 - 0 . 0 0 0 6 6 / A , (D) Theoretical behavior based on Equation 8 with D"' = 2.65 X cm/s"' ( 77). The parameter k is a means of area correction

Possible causes for this difference are (a) concentration depletion present a t birth of the second drop, which is not present at that of the first drop due to drop-knocker induced stirring which homogenizes the solution, and (b) convective enhancement of the current on the first drop due to dropknocker induced stirring. If concentration depletion existed a t birth of the'second drop, the contact area corrected current would be smaller than that predicted by Equation 7; then, the excellent agreement between the contact area corrected data a t h of 49.6 cm and theoretical prediction (cf. curves B and D of Figure 4) could only he due to over-correction for contact area, i.e., a too large A,. The effect of a too large A, on contact area correction at any time after drop-birth decreases with increasing h, since A,/A, a t constant t decreases with increasing h; however, comparison of Figures 4 and 5 shows that, for a given A,, the observed current at h of 77.6 cm is always higher relative to theoretical than that at h of 49.6 cm. Thus, the good agreement between observed and

1.2

'E (T

1985

2.4

- To;,'ym,;;3 ,'Yrg'j

Figure 6. Effect of contact area and stirring correction on cadmium(I1) diffusion current ( i d ) variation with time ( t ) at a tapered capillary. Conditions: m l h , = 5.84 X mg/cm s; 1.022 mM Cd(I1) in 0.1 M KCI. (A) First natural drop; k = 1; t o = 0 . 2 s. (6) Same as A; k = 1 - 0.00066/A, (C) Second natural drop; k = 1; t o = 0.05 s. (D) Same as C k = 1 - 0.00066/As; to = 0.05 s. (E) Theoretical prediction based on Equation 8 with D"' = 2.65 X cm/s"* ( 1 7 ) . The unprimed curves are for h of 49.6 cm; the primed curves are for h of 77.6 cm. The arrows and associated numbers on the curves indicate t , which are at identical abscissa values for A and B or C and D. The parameter k is a means of area correction

predicted behavior on second natural drops is not due to overcorrection for A,, and no significant concentration polarization exists at drop-birth. Apparently, the too large currents obslrved on correction of first natural drops for A, of 6.6 X cm2 are principally due to mechanical dropknocker induced convection. Conuection. MacNevin and Balis (2) and Lingane (16) have reported current enhancement by stirring induced by a rapidly flowing DME. Using a capillary with m and t of 2.5 mg/s and 2.2 s, Lingane observed a peak at ca. 0.5 s in the i-t curve for Cd(I1) in ammoniacal buffer. Although m in the present study are very much smaller than that used by Lingane, similar hut lesser stirring effects may be reasonably expected at times shortly after natural drop-fall; however, without correction for A,, convective enhancement at short times after drop-birth would be masked by the relatively large influence of A, on A early in drop growth. In fact, examination of Lingane's data (Figure 5 of Ref. 16) suggests a plateau at times to ea. 0.05 s after drop-birth for conditions of small m and large h (such that h, and, hence, m are relatively independent of h b and, consequently, do not vary significantly with t ) ; detection of such a plateau under conditions used for the other capillaries, Le., a time-dependent m, is precluded by the nature of the plot, i/t' >C us. t'lfi. Stirring effects at drop-birth would mean that concentration polarization at the Hg surface could not occur until cessation of the stirring and the i d - t dependence would be better described by Equation 12. Since mechanical drop-knocker induced stirring is unquestionably more vigorous and longer lived than stirring due to natural drop-fall, the value of t o in Equation 12, which best describes experimental data, will be expected to he larger when using a drop-knocker (and for the first natural drop) than for natural drop-fall (and for the second natural drop). Initial tests of Equation 12 on the data of Figures 2 to 5 for A, of 6.6 X cm2 indicated that t , is ca. 0.2 s for the first natural drop following mechanical drop dislodgement and ca. 0.05 s with natural drop fall, e.g., second natural drop; the results using these t , are shown in Figure 6. .4lthough no attempt was made to optimize to,Figure 6 clearly shows that the experimental id-t behavior closely follows Equation 12,

1986


Table IV. Determined Diffusion Coefficient for 1.022 mM Cd(I1) in 0.1 M KC1 as a Function of Mercury Column-Height ( h ) ,Drop-Time ( t )and Assumed Shielded Area ( A , )

49.6 2.26 2.04, 52.6 2.26 2.13 56.6 2.26 2.24 59.6 2.31, 2.26 62.6 2.26 2.40 65.6 2.47 2.26 68.6 2.26 2.56 71.6 2.26 2.63 74.6 2.26 2.70, 77.6 2.26 2.77, 77.6 2.45 2.84 77.6 2.71 2.91 77.6 2.98 3.00 77.6 3.40 3.09 average ( n = 1 4 ) std. dev. ( n = 1 4 ) rel. std. dev. ( n = 14), 70 average ( n = IO)^ std. dev. ( n = rel. std. dev. ( n = l o ) , %h

4.20 4.19 4.18 4.16 4.16 4.14 4.15 4.14 4.14 4.1 3 4.17 4.19 4.25 4.28

2.96b 2.95 2.95 2.93 2.93 2.92 2.93 2.92 2.92 2.91 2.94 2.95 3.00 3.02

2.66c 2.66 2.66 2.66 2.66 2.66 2.67 2.66 2.66 2.66 2.68 2.69 2.72 2.73

2.54d 2.54 2.55 2.54 2.55 2.54 2.56 2.55 2.55 2.55 2.57 2.58 2.61 2.62

2.81f 2.72’! 2.80 2.71 2.79 2.71 2.77 2.68 2.77 2.68 2.76 2.67 2.77 2.67 2.76 2.66 2.75 2.65 2.75 2.65 2.75 2.67 2.75 2.67 2.77 2.67 2.76 2.70 2.94, 2.67 2.56 2.77 2.68 0.03 0.03 0.02 0.02 0.02 1.04 0.92 0.93 0.72 0.76 2.93 2.66 2.55 2.77 2.68 0.02 0.00, 0.00, 0.00, 0.02 0.02 0.56 0.14 0.20 0.19 0.78 0.82 Literature value, D1’2 = 2.65 t 0.025 X cm/sl’z( I 7). Based on Equation 5. Based on Equation 6. Based on Equation 7. e Based on Equation 9 with A , = 0.00023 cm2. Based on Equation 9 with A , = 0.00066 cm2. e Based on Equation 1 2 with to = 0.2 s and A , = 0.00066 cm2. Statistics for points at t = 2.26 s only. when the A, determined from ac capacitive current data and Equation 4 is used, provided that the proper value of t o is chosen. Sample calculations on the data of curve B’ in Figure 6. using to of 0.15 s rather than 0.20 s, showed that the starting point of B’ would be shifted along the abscissa from 0.810 to 0.906, a t which point B’ would be within 0.2% of theoretical curve E; additionally, for abscissa values greater than 1.1.Le., t greater than 0.7 s for first natural drop (curve B’) and t greater than 0.45 s for second natural drop (curve D’), curves B’ and D’ would be virtually coincident, e.g., deviations of 0.4%, 0.2%, and 0.2% at abscissa values of 1.11,1.43, and 1.69. Since this change in t o from 0.20 to 0.15 s would make curve B’ parallel theoretical curve E after ca. 0.4 s, it seems reasonable to assume that, for the first natural drop a t h of 77.6 cm, stirring effects cease after ca. 0.15 s (cf. Conclusions). Cadmium(I1) Diffusion Coefficient. Diffusion current data for Cd(I1) in 0.1 M KCl as a function of h and t are given in Table IV. The precision of id is considerably poorer than for similar data a t a blunt capillary (I), because of a low frequency vibration (period of ca. 1 min) which caused the range of id measured at any h and t to be about 1%. This lower precision is probably due to a tapered capillary’s sensitivity to vibration as compared to a blunt capillary. Diffusion coefficients (Table IV) calculated from id using Equations 5 to 7 and 9 show that either Equation 9 with A, = 2.3 X cm2 or Equation 6 gives good agreement with the literature value ( I 7). Use of the experimentally determined A , of 6.6 X cm2 results in a D”2 which is statistically significantly different and which markedly decreases with increasing h. Use of Equation 6, which accounts for shielding, yields results within one standard deviation of the literature value, despite the fact that, for h of 49.6 cm, the shielded area is predicted as including the region defined by an angle of 53” relative to the bore axis (cf. the 35” angle of the capillary o.d. in Figure 1); however, Equation 6 assumes some faradaic activity a t the drop surface in the shielded region, so that the equivalent A,, i.e., an effective area of no faradaic activity equivalent to the shielding effect, is 3.7 X 10 cm2. The positive trend of 0’’’ with increasing h and/or t indicates the

0.2) 0.6

,

2.63“ 2.63 2.63 2.62 2.62 2.62 2.63 2.62 2.62 2.62 2.63 2.64 2.66 2.66 2.63 0.02 0.57 2.62

/

, ~

.C

I

I 1.4

1 1.8

Figure 7. Effect of contact area and stirring correction on cadmium(I1) diffusion current (id) variation with time ( t ) at a blunt capillary. Conditions: m l h , = 1.38 X lo-* mg/cm s; h = 49.7 cm; 0.212 mM Cd(I1) in 0.1 M KCI. (A) First natural drop; k = 1; t o = 0.1 s. (B) Same as A; k = 1 - 0.00083/AS. (C) Second natural drop; k = 1; t o = 0.05 s. (D) Same as C; k = 1 - 0.00083/A,. (E) Theoretical prediction based on Equation 8 with D1”= 2.65 X cm/s”* ( 17). The arrows and associated numbers on the curves indicate t , which are at identical abscissa values for A and B or C and D. The parameter k is a means of area correction increasing amount of surface which Equation 6 assumes is shielded but which, in the case of a tapered capillary, is not shielded. Evaluation of D’ using Equation 12, which accounts for stirring, yields results within one standard deviation of the literature value; however, this D’i2 systematically increases with decreasing h or increasing t . Nonetheless, the ability of Equation 1 2 to describe the id-t behavior over a wide range of t , independent of h, coupled with the ability to evaluate D’” with a precision and accuracy of 1% is dramatic. The with decreasing h or increasing t systematic increase of D‘’? is probably due to the constant t , of 0.2 s used for all h and t (cf. previous section); however, since best-fit values of t o were not available a t all h, it was necessary to assume a constant to.


Faradaic Response at a Blunt Capillary. T o test the ability of Equation 12 to predict behavior at a blunt capillary, the data of Cummings and Elving (1) were re-evaluated. Profiles of id-t for the first natural drop (Figure 7) indicate that t, in Equation 12 is 0.1 s; the experimental data corrected for A, and t , (curve B of Figure 7 ) parallel prediction for times greater than 0.4 s. Because of the extreme shielding by a blunt capillary, particularly toward concentration homogenization after drop-fall (even using mechanical drop dislodgement), the negative deviation from theory a t t less than 0.4 s is probably due to concentration depletion near the electrode surface caused by previous drop electrolysis. Concentration depletion a t the second natural drop on a blunt capillary, which is not present a t a tapered capillary, is also evident from Figure 7 . Significance of Combined Stirring and Shielding. The presence of both capillary shielding and convective stirring makes quantitative measurements a t short drop-time treacherous. Although A, can be determined with a precision of ca. 6 % (better precision may be possible if drop-times shorter than those in the present study are included), which yields a n effective electrode area precision of 1% or better a t drop-times of 2 s or greater, there is no presently available method for t, evaluation. Since A, involves a finite uncertainty and t , must presently be determined by trial-and-error, the best-fit value to t , in Equation 12 will depend on A,; consequently, the best-fit value of t o at each of several A, must be determined, and the pair of A , and t , yielding the optimum fit over a wide time-range chosen. To minimize the influence of t o and A, on the observed id, short drop-times and/or small h should be avoided. For example, a t 4 s after drop-birth with m of 0.27 mg/s, the error cm2 is 5 7 ~in in neglecting t o of 0.2 s and A, of 6.6 X id/m2/3t1and 2.570 in t' 6 / m 13 ; neglecting only t , decreases the error in i d / m 2 / 3 t 1 t 6 to 2.570; however, at t of 1 s, even neglecting only to results in a 1070 error in both terms. Larger m will not reduce the errors associated with neglect of to,but will reduce the uncertainty in A for a fixed t associated with the uncertainty in A,. Convective current enhancement caused by stirring due to rapid drop expansion. Le., large m, generally does not become significant until m is between 1 and 2 mg/s (2, 16; citations in Ref. 16). The latter means t h a t h should not be made too large for wide-bore capillaries; however, small h results in a significantly timedependent m. Variation of m during drop-life, which is not considered in the theoretical derivations, may result in a positive deviation from theory (9,11, 16) due to the drop's being smaller at short times than t h a t implicit in the theories which assume a constant m ; thus, the extent of concentration polarization is smaller a t short times since less electrolysis has occurred, and the observed currents over the whole drop-life are greater than predicted by an57 of the theories. T o minimize this effect, h should be large so that hb does not result in a significant time variation of h, (hence. m ) . T h e best compromise between a large h and a not-too-large m is t o use either a narrow-bore or a long capillary. Under these conditions, an optimum m can be achieved, so that (a) drop-time is sufficiently long to minimize the effect of to,if unknown or neglected, (b) A, is sufficiently large to make the error in A due to uncertainty in A, small, and (c) m is kept below the magnitude which causes convective stirring, while using an h sufficiently large to make m virtually time-dependent.

CONCLUSIONS T h e experimental results demonstrate that the effective contact area (A,) is the same for both faradaic and nonfaradaic processes. T h e magnitude of A, a t tapered capillaries is

1987

defined by projection of the tip onto the drop surface, despite the fact that this includes regions of the surface which are 40 to 50 p m from the capillary. Consequently, A,, must be presumed to be due to capillary interference with the electrical force lines, which prevents formation of a n electrical double layer on the upper region of the drop with concomitant prevention of faradaic activity in that region. Based on data analysis using Equation 12, there is very little depletion transfer from drop to drop a t a tapered capillary, even with natural drop-fall; in the case of a blunt capillary using mechanical drop dislodgement, there is also no depletion, a t least after 0.4 s. Although Equation 12 can accurately describe the observed id-t behavior on single drops, the present unavailability of an experimental method for evaluation of t o makes application of Equation 12 a tedious trial and error approach. Since t, probably varies with the striking force of the drop-knocker, extreme care should be exercised in the use of short drop-times for quantitative analysis, e.g., in pulse polarography. Dayto-day variations in the striking force or changes in physical arrangement which alter capillary movement during dislodgement may change to, resulting in erroneous concentrations as determined from calibration curves run under different conditions. Although the cautions given are appropriate and important for area-dependent observables with short drop-times, reference should be made to the existence of area-independent observables, which many prefer to use for certain nonanalytical purposes even when employing relatively long drop-times or stationary electrodes, e.g., current or charge ratios as obtained in double potential-step experiments, in-phase/quadrature current ratios in ac polarography or voltammetry (cot 6 value), and peak widths or potentials in differential pulse and ac polarography. In quantitative analysis, measurements generally involve the use of standards, usually run immediately prior to and/or following the unknown. Internal standards are at times included in the unknown solution. In either case, peak height ratios can be used, which are drop area-independent. With such area-independent observables, the legitimate warnings regarding short drop-time experiments developed in the present paper may not be relevant. In fact, the results presented suggest that t h e aforementioned area-independent parameters should be employed, when possible, for both fundamental research and analytical determinations; when the available area-dependent parameter must be used or can give complementary or corroborative information, consideration of A, and t , should be made. In applying Equation 12 to experimental data by trialand-error to obtain to,it is reasonable to assume that the best ] [(tfit is obtained when the [id(t - to)' 2 / h ( m t ) 2 ' 3vs. t,)1'2/(mt)1/3]plot parallels prediction rather than being simply based on the value of to, which gives minimum deviation from calculated currents based on a published D. In this regard, the selection of t , should not be overly biased by results a t short t , for which the data are extremely sensitive to to, e.g., a change of t , a t 0.3 s from 0.20 t o 0.18 s causes a shift on both axes.

LITERATURE CITED T. E. Cummings and P. J. Elving, Anal. Chem., 50, 480 (1978). W. M. MacNevin and E. W. Balis, J . Am. Chem. Soc., 65, 660 (1943). G. S. Smith, Trans. Faraday Soc., 47, 63 (1951). J. W. Perram, J. B. Hayter, and R. J. Hunter, J . Elecfroaflal. Chem., 42, 291 (1973). R. J. Newcombe and R . Woods, Trans. Faraday Soc., 5 7 , 130 (1961). D. M. Mohiiner, J. C. Kreuser, H. Nakadomari, and P. 0.Mohiiner, J . Nectrochem. Soc., 123, 359 (1976). J. Kuta and I . Smoler, in "Progress in Polarography", Vol. 1, P. Zuman and I. M. Koithoff, Ed., Interscience, New York, 1962, p 43. J. K. Taylor, R. E. Smith, and I. L. Cooter, J . Res. Natl. Bur. Stand., 42, 387 (1949). J. M. Markowitz and P. J. Elving, Cbem. Rev., 58, 1047 (1958). H. Matsuda, Bull. Chem. SOC. Jpn., 36, 342 (1953).

1988


(11) J. M. Markowitz and P. J. Elving, d . Am. Chem. Soc., 81, 3518 (1959). (12) D. P. Shoemaker and C.W. Garland, "Experiments in physical Chemistry". 2nd ed., McGraw-Hill, New York, 1967, pp 26-29, 34. (13) . . D. C. Grahame. Technical Reoott Number 1 to Office of Naval Research. March 9, 1950. (14) D. W. Marquardt, J . Soc. Ind. Appl. Math., 11, 431 (1963). (15) P. Delahay, "Double Layer and Electrode Kinetics". Wiley-Interscience, New York, 1965, pp 43-44.

(16) J. J. Lingane, J . Am. Chem. SOC.,7 5 , 788 (1953). (17) D. J. Macero and C. L. Rulfs. J . Electroanal. Chem., 7, 328 (1964).

RECEIVED for review July 12, 1978. Accepted September 14, lg7&The authors thank the Science Foundation) which helped support the work described.

Theoretical and Experimental Evaluation of Cyclic Staircase Voltammetry Lee-Hua L. Miaw, P. A. Boudreau, M. A. Pichler,' and S. P. Perone" Department of Chemistry, Purdue University, West Lafayette, Indiana 47907

The theory for Cyclic Staircase Voltammetry (CSCV) for the reversible case has been described and verified experimentally in this study. Voltammetric instrumentation was assembled and interfaced to a laboratory minicomputer which provided experimental control, data collection, ensemble averaging, and Fourier filtering functions. Experiments were conducted to test the effects of potential step size, A€, switching potential, E,,, and sampling parameter, a , on peak separation, A€p, ( E , , Epc),and peak current ratio, y, (ipa/ipc). Various data handling and data processing procedures were evaluated to provide the minimum of distortion in measuring peak separations and peak currents. Realistic error levels are reported. The results of the multiple experiments reported here confirm the predictions of CSCV theory.

-

In 1960, Barker and Gardner (I) reported improvements in polarographic performance obtainable by application of potential-step techniques in electrochemical analysis. Since then progress has been made in refining and extending pulse methods. Blutstein and Bond (2) described advantages of fast sweep differential pulse voltammetry over linear sweep and alternating current techniques applied to irreversible systems. Osteryoung and co-workers (3)compared the sensitivity and performance of several pulse voltammetric stripping methods a t the thin-film mercury electrode. Also, several new approaches for the complete elimination of charging current in voltammetric measurements have been reported (4-6). T h e appearance of the laboratory computer as a control device in electrochemical experiments has greatly facilitated the application of potential-step experiments in voltammetric investigations. Cyclic staircase voltammetry (CSCV) is a potential-step technique which is particularly important because of the widespread use of the corresponding analogue technique, cyclic voltammetry (CV). for mechanistic studies of electrode processes. Because of the distinct advantage of CSCV for discriminating against charging current, this technique should soon replace CV as the method of choice in mechanistic studies. Thus, a systematic examination of theoretical and experimental features of CSCV would seem t o be timely, and such a study is reported here. I Present address, Standard Oil Company, 4440 Warrensville Road, Warrensville Heights, Ohio 44128.

0003-2700/78/0350-1988$01 O O / O

The theory of staircase voltammetry (SCV) was first reported by Christie and Lingane (7) for reversible electrode reactions. Zipper and Perone (8) and Schroeder and coworkers (9) extended the theory by including the effect of varied current sampling time on SCV current-voltage waveforms. Schroeder's group (IO) also applied staircase voltammetry t o irreversible and quasi-reversible electrode processes. Ryan ( 1 1 ) generated theoretical CSC\' voltammograms using digital simulation and reported the use of CSCV in the study of kinetic mechanisms. Recently Reilley, et al. (12)described a deconvolution procedure employing the fast Fourier transform approach, where the influence of the data acquisition parameters can be removed from the SCV waveform allowing a direct comparison between data generated from SCV and classical linear sweep voltammetry. In the present work, we have presented a complete, systematic, theoretical, and experimental study of CSCV for reversible electrode processes (13). Included in the study are the effects of step height, sweep rate, switching potential, and sampling time. In addition, we have evaluated data processing approaches in the analysis of CSCV current-voltage curves.

EXPERIMENTAL Reagents. Chemicals used in this study were analytical reagent grade (Mallinckrodt Chemical Works, St. Louis, Mo.) and were used as received. Ferrioxalate solutions were prepared by dissolving ferric ammonium sulfate in 0.4 M potassium oxalate. Mercury was quadruply distilled and then scrubbed with 10% nitric acid by drawing air through the mercury layer with a vacuum aspirator. Water used was purified by passage through a mixed bed ion-exchange resin followed by distillation. Solutions were deaerated with high purity nitrogen. Traces of oxygen in the nitrogen were removed by passage through two gas washing bottles containing chromous chloride and zinc amalgam, one containing deionized water, and a fourth containing 0.4 M potassium oxalate. Cell and Electrodes. The cell was a 50-mL glass bottle with threaded Teflon top with holes drilled that snugly fit electrodes and nitrogen gas tubes. The working electrode was a Metrohm E-410 micrometer type hanging mercury drop electrode. Drop area was 1.80 f 0.05 mm2 throughout the study, and a new drop was used for each experiment. A Coleman 3-710 saturated calomel electrode was used as reference and the counter electrode was a platinum helix. The cell was mounted on a metal ring stand and placed inside a metal box which also housed the electronic circuitry. The temperature of the cell was laboratory ambient (23 f 2 "C). Blanks were run for each set of experimental conditions and were subtracted from voltammograms t o account for changes in background. C 1978 American Chemical Society

Verification of a diffusion current equation accounting for convection

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