Anal. Chem. 1904, 56. 1967-1970 Lubman, D. M.; Rettner, C. T.; Zare, R. N. J . Phys. Chem. 1982, 86, 1129. Altshuller, A. P.: Cohen, J. R. Anal. Chem. 1980, 32, 802. Dunn, T. M., prlvate communication. Kantrowltz, A.; Grey, J. Rev. Sci. Insfrum. 1951, 22, 328. Anderson, J. B.; Andres, R. P.;Fenn, J. B. A&. Chem. Phys. 1988, 10, 275. McClelland, G. M.; Saenger, K. L.; Valentlnl, J. J.; Herschbach, D. R. J . Phys. Chem. 1979, 83. 947.
1967
(38) Beck, S. M.; Monts, D. L.; Liverman, M. G.; Smalley, R. E. J . Chem. Phys. 1979, 7 0 , 1062.
RECEIVED for review March 19,1984. Accepted May 10,1984. We gratefully acknowledge financial support from a Cottrell Research Grant, a Petroleum Research Fund Type G Grant, and a University of Michigan Rackham Award.
Chronoamperometric Determination of Diffusion-Layer Thicknesses at Hydrodynamic Electrodes Kenneth W.Pratt Center for Analytical Chemistry, National Bureau of Standards, Washington, D.C. 20234
A new technlque Is described by whlch dlffuslon-layer thlcknesses at hydrodynamic electrodes are measured wlthout knowlng the electrode area, sdutlon concentration, OT number of electrons In the electrode reactlon. Comparison of the chronoamperometrlc current, obtalned In qulescent solution, wlth the llmitlng current obtalned at the same electrode In hydrodynamic voltammetry ylelds a characterlstk “equlvalent tlme”. Thls parameter Is directly related to the dlffuslon-layer thlckness at the electrode. Experlmental diffusion-layer thlcknesses are measured at rotating dlsk and vlbratlng wlre electrodes uslng thls technlque. The values agree wlh those obtalned from llmllngturrenl measurements to wlhln 5 % at the rotating dlsk and 16% at the vlbratlng wlre electrode. Factors contrlbutlng to these errors are evaluated.
at time t = 0 from a value Eo,where no reaction occurs at the electrode, to El, the value used in the steady-state measurements. These currents are described by eq 1 for the steadystate case and by eq 2, the Cottrell equation, for the transient case
Using the same electrode and solutions in both experiments ensures the equality of n, A, D , and C. Graphs of eq 1 and 2 are presented in Figure 1 by solid lines. The values of ISs and I(t)were calculated by using representative values for n, A, D, C , and 6. For t = tr,the point at which I ( t ) = I,,, the right members of eq 1and 2 are equal. Simplification yields the following expression:
6 = (7rDt?1/2 The analytical sensitivity of hydrodynamic electrodes in voltammetry is governed by the thickness of the diffusion layer, 6, established at the electrode. The value of 6 is used to intercompare the performance of electrodes of different geometries and to optimize a given electrode (I). Experimental values for 6 generally are calculated by using eq 1 nFADC 18,= (1) 6 where I,, n, F, A, D, and C represent the steady-state limiting current, number of electrons in the electrode reaction, Faraday constant, electrode surface area, diffusion coefficient, and bulk concentration of the electroactive species, respectively. Calculation of 6 depends on each of these parameters. Frequently, the exact electrode area is not available, due to surface roughness or partial fouling of the electrode. In contrast to the above method, the technique described here allows the value of 6 to be determined for a hydrodynamic electrode without measuring its surface area. Prior knowledge of the hydrodynamic conditions prevailing a t the electrode is not required. The determination is based on a comparison of the steady-state and transient currents observed at the same electrode in a single solution. The steady-state limiting current, I-, is measured under the hydrodynamic conditions of interest at a potential E,, corresponding to the masstransport limited reaction of the electroactive species a t the electrode. The transient faradaic response of the electrode in quiescent solution, I(t),is then recorded for a potential step
(3)
Equation 3 is used to calculate the value of 6, using the experimental value for t ! The value for D may be calculated (2) from the equivalent conductance of the electroactive species at infinite dilution. Neither A, n, nor C needs to be experimentally determined. The value of 6 calculated by using this technique represents the average value for the entire electrode surface. In most cases, the local value of 6 varies a t different points on the electrode, as a result of varying hydrodynamic conditions. Only those electrodes satisfying the condition of uniform accessibility (3), such as the rotating disk electrode (RDE), have equal values of 6 at all points on the electrode surface. Flanagan and Marcoux (4) first introduced the concept of t’, the ”equivalent time”, in a theoretical study of the chronoamperometric response of tubular electrodes. Measurementa of t’values for actual electrodes were not presented. The present contribution represents the first experimental comparison of chronoamperometric data, obtained in quiescent solution, with data obtained by hydrodynamic voltammetry at the same electrode. Other combinations of independent electroanalytical measurements, obtained on a single electrode system, have previously been used in measurements of the value of n (5) and D (6) for electrochemical systems. The present method for the measurement of 6 represents a third member of this family of “paired” techniques. EXPERIMENTAL SECTION Apparatus and Reagents. Experimental studies were conducted using a Pt disk electrode (Model DD20,Pine Instrument
Thls artlcle not subject to U S . Copyright. Published 1984 by the American Chemlcal Society
1988
ANALYTICAL CHEMISTRY, VOL. 56, NO. 11, SEPTEMBER 1984
1.8
Table I. Experimental Values of 6 at the RDEa w,
rpm
1800 3600 4800 7200 9600
6 from I,,pm
530.6 752.9 863.0 1050 1145
45.5 22.8 16.9 11.4 8.45
16.16 11.39 9.94 8.16 7.49
6 from
t’, pm 16.6 11.8 10.1 8.33 7.17
Table 11. Experimental Values of 6 at the VWE”
-
10
t’, ms
“Electrolyte: 1.00 mmol/L KI in 0.10 mol/L H2S04.
1.00.8
I,, PA
,
I
15
20
! 25
30
I
35
,
40
45
t (ms)
Figure 1. Theoretical currents obtained in chronoamperometryand
hydrodynamic vottammetry at a single electrode: values calculated for n = 1 equlv mol-’, F = 96487 C equlv-l, A = 1 cm2,D = 1 X cmz s-‘, C = 1 pmol ~ m - 6~ =, IX 1 0 - ~cm, k , = 0.1 cm s-l. Solid lines indicate I ( t ) and I , for a reversible reaction; dashed llnes Indicate I ( t ) and I , for an irreversible reaction at k , = 0.1 cm s-l. Co., Grove City, PA) and a Pt wire electrode. The geometric surface areas of these electrodeswere 0.4588 cm2and 0.0136 cm2, respectively. In the measurements of I,,, these electrodes were used as a RDE and a vibrating wire electrode (VWE) (7). The VWE was vibrated at a frequency of 400.0 Hz. The disk electrode was polished with 0.3-pm alumina prior to use. A Pt wire and saturated calomel electrode (SCE) served as the counterelectrode and reference electrode. All potentials are reported in volts (V) VS. SCE. The SCE was isolated from the cell solution by a salt bridge filled with 0.1 mol/L H2SOI. The 3-mm porous Vycor tip of the salt bridge was placed within 3 mm of the working electrode to minimize uncompensated solution resistance. Measurements of I(t)and I , were performed with a commercial potentiostat (Model 174A, Princeton Applied Research Co., Princeton, NJ) in the pulsed and DC modes, respectively. For the experiments using the disk electrode, a high-current potentiostat was constructed using the control amplifier and voltage follower of the 174A and an external current-to-voltage converter of conventional design. The two channels of a DC-coupled stereo amplifier (Model 215, Southwest Technical Products Co., San Antonio, TX), used as booster amplifiers for the control amplifier and current-to-voltage converter, were connected according to standard practice (8). Cross-talk between the two booster amplifiers was not observed. Measurements of the potential step obtained in pulsed operation showed that potentiostatic control of the working electrode at the find potential (El) was established within 200 ps of the voltage step produced by the pulse generator of the potentiostat. Using the booster amplifiers increased the current capability of the potentiostat to 2 A. Experimental chronoamperograms and I , values were displayed on an oscilloscope with 10-MHz frequency response. The oscilloscope was triggered by the leading edge of the voltage pulse applied to the potentiostat. The vertical input was connected to the output of the current-to-voltageconverter. The time base was calibrated by using the 10-kHz NBS standard frequency. A digital voltmeter with f0.1% accuracy was used for all voltage measurements. Experimental measurements and electrode preconditioning were performed in 1.00 mmol/L KI with the RDE and in 0.500 mmol/L KI with the VWE. These solutions were prepared in 0.10 mol/L H,S04, the supporting electrolyte. All solutions were prepared from reagent-grade chemicals and doubly distilled, deionized water. The initial (Eo)and final (El) potentials for the chronoamperometric step were +0.200 V and +0.800V, respectively. At +0.800 V, iodide was oxidized to iodine at a mass transport limited rate, while no reaction occurred a t +0.200 V. Procedure. The electrode was f i t preconditoned by repeated pulsing between +0.200 V and +0.800 V in KI solution. I,, was then measured at +0.800 V under the hydrodynamic conditions shown in Tables I and 11. Next, two chronoamperometric
6 from
6 from
app,cm
I,,, PA
t’, ms
I,,, pm
t’, pm
0.0202 0.0274 0.0487
18.35 28.90 47.37
9.6 4.1 1.6
6.93 4.40 2.68
5.0 3.1
7.6
“Electrodevibrated at 400.0 Hz in all experiments. Electrolyte: mmol/L KI in 0.10 mol/L H2S04.
0.500
measurements were performed: first, with the electrode in 0.10 mol/L HzSO,; and second, with the electrode in the KI solution used in the measurements of I,,. The first measurement, with the electrode in pure supporting electrolyte, served as a background correction for the double-layer charging current resulting from the potential step. I @ ) ,the faradaic current, was given by the differencebetween the transient recorded in KI solution and that recorded in the supporting electrolyte. The intersection of I(t) with I , fixed the experimental value of t! The background correction constituted up to 10% of the totalexperimentalcurrent at t = t’ for the VWE. The correction was negligible in the experimentsusing the RDE, for which characteristic6 and t’values were greater. For the chronoamperometricmeasurements,the electrode was pulsed from +0.200 V to +0.800 V using a periodic train of 50-ms pulses. Except as noted, the period of the pulse wave form was 5.0 s, corresponding to a duty cycle of 1.0%. Values for S were calculated from I,, by using eq 1 and from t’ by using eq 3. The diffusion coefficient of iodide was taken cm2s-l, calculated from literature values for the as 1.937 X equivalent conductance of iodide at infinite dilution (9) and corrected to 23 OC by linear interpolation. In the calculation of 6 from I,, by eq 1, the measured (geometric) electrode area was used for A.
RESULTS AND DISCUSSION Experimental results are shown in Table I for the RDE at rotation speeds ( w ) from 1800 to 9600 rev m i d . Analogous results for the VWE are shown in Table I1 for three values of app,the peak-to-peak vibration amplitude. For the RDE, the values of 6 calculated by using the two methods agree within 5% a t all rotation speeds studied. For the VWE, the values of 6 calculated from t’are 10% to 16% greater than the corresponding values calculated from I,,. At the RDE, values of t ’were reproducible to within 1% at each rotation speed. This uncertainty in t’ corresponds to an uncertainty of 0.5% in the value of 6 obtained from eq 3. At the VWE, the background correction for I(t) limited the accuracy of the determination of t’ to an uncertainty of 2-5%, with larger uncertainties noted at smaller values o f t ! Values presented in Tables I and I1 represent averages obtained for 10 determinations of t’at each value of w or aqp. Several factors contribute to uncertainties in the value of 6 , as calculated from experimental values of t’ using eq 3. These include the response time of the potentiostat, convection in the chronoamperometric experiment, nonlinear diffusion, depletion effects, and limitations imposed by the heterogeneous kinetics of the electrode reaction. Each factor influences the experimental value of t’, thereby affecting the calculated value of 6.
ANALYTICAL CHEMISTRY, VOL. 56, NO. 11, SEPTEMBER 1984
The response time of the potentiostat increases the observed value of t ’by a constant time delay. This positive error in t ’increases in relative magnitude as the value of t ’decreases. In the experimental determinations, the maximum effect resulting from this effect was 14% at t’ = 1.6 ms at the VWE and 2.4% at t’ = 8.45 ms for the RDE. Convection during the chronoamperometricexperiment also causes a positive error in t Convection increases the value of I ( t ) above that resulting from diffusion alone. Thus, the intersection of I ( t ) with I,, occurs later than in quiescent solution, and the experimental value of t’ is greater. The relative effect of convection is greater at larger values of t ’, because the diffusional flux is inversely proportional to the square root of time, while the convective flux is constant. In the experimental work, this effect was of minor importance. The only source of convection in the experimental chronoamperometric determinations is natural convection. This typically causes deviations of I(t) from the predicted Cottrell value only a t t > 1 s (10). The effect of nonlinear diffusion on t ’depends on the geometry of the electrode. For a tubular electrode, the diffusion layer extends inward from a concave surface, and I( t ) is less than the predicted current from the Cottrell equation (10). For cylindrical and spherical electrodes, the diffusion layer extends outward from a convex surface, and I(t ) is greater than the predicted current from the Cottrell equation (11). These deviations in I(t), which result from nonlinear diffusion, affect the values of t’ in a manner analogous to that noted above for convection. The effect is more significant at large values of t’and at electrodes of small radius. Limitations on the use of the Cottrell equation at nonplanar electrodes are discussed by MacDonald (12). At the VWE, nonlinear diffusion contributed to the discrepancies in the values of 6 calculated from experimental t’ values (see Table 11). This influence of nonlinear diffusion on 6 and t ’was estimated by calculating I (t ) from the series approximation to cylindrical diffusion at small t (II), using ro = 63.5 pm, the radius of the VWE. These values were compared with corresponding values of I ( t ) predicted from the Cottrell equation. The value predicted from the equation for cylindrical diffusion was 2.4% greater at t = 1.6 ms and 6.0% greater a t t = 9.6 ms. The corresponding changes in t’were 4.8% at t’= 1.6 ms and 12.0% at t’= 9.6 ms, based on a constant value of I,,. Depletion effects result from the use of a repetitive wave form in the chronoamperometric measurement, If the diffusion layer is not completely replenished during the period between pulses, the value of I ( t ) in the succeeding pulse is decreased. Hence, the apparent value of t ’is reduced. This effect was noted experimentally at both the disk and wire electrodes. The experimental value of t ’decreased ca. 10% when the period of the pulse wave form was decreased from 5.0 s to 0.10 s. Smaller decreases in t’were noted a t periods of 1.0 s and 0.5 s. The effect was absent a t periods of 2.0 s or longer. The heterogeneous kinetics of the electrode reaction significantly influence the value oft! For an irreversible reaction, I, and I ( t ) are given by eq 4 (13) and 5 (14),respectively. In these equations, kf represents the forward, potential-dependent rate constant, and erfc represents the complement of the error function (14).
Table 111. Effect of Electrode Kinetics on the Equivalent Time“
Kb 1 10d 100 1000 10000
’
(4)
I ( t ) = nFAkfC exp[k?t/D] erfc [kft1/z/D1/2] (5) Graphs of eq 4 and 5 are shown in Figure 1by broken lines. The currents were calculated by using the same values of n,
I969
PRlW
RC
0.59148 37.534 3246.1 318950 31837000
1.8582 1.1792 1.0198 1.0020 1.0002
“Calculated from eq 7. b K = kfJ(D/6). c R = tfi,/tf. dSituation depicted by dotted lines in Figure 1. A, D, C, and 6 used previously in the graphs of eq 1 and 2. The value of kf was chosen as 10 times the value of D/6, the mass transfer coefficient (1). Presentation of eq 1,2,4, and 5 on a single graph permits direct comparison of the effects of irreversible reaction kinetics on I,,, I@), and t’. From Figure 1,it is evident that the effect of irreversible reaction kinetics on I,, is greater than that on I( t ) a t t = t ’. Hence, t’irr,the equivalent time obtained for an irreversible reaction, is greater than the value of t’obtained for a reversible reaction. The value of t may be calculated from eq 6, which results from equating the right members of eq 4 and 5
The ratio of t{nto t’can be expressed as an implicit function of the ratio of kf to ( D / 6 ) 1 -K + l - e x p [ P R / ~ ]erfc [KR1/2/.lr1/2] (7) where
and
R = t’i,,/t’
(9)
Equation 7 results from eq 6, using the definitions of K and R and the definition of t’from eq 3. Table I11 gives values of R calculated for various values of K . In the experimental work, the error in t ’due to electrode kinetics was minimized by choosing an electroactive species, iodide, characterized by fast kinetic parameters. Also, the value of kf was maximized by measuring I,, and I ( t ) at a potential well out in the limiting current region for the oxidation of iodide to iodine. The concept of the equivalent time and the associated relationship between t ’and 6 have interesting consequences in comparisons of the sensitivities of a given electrode in pulse and hydrodynamic voltammetry. The faradaic currents measured by these two techniques are equal when the current transient, obtained in quiescent conditions in the pulsed technique, is sampled at t = t’. Hence, the two sensitivities, defined by the faradaic current obtained per unit concentration of electroactive species, are equal at t = t ’. For a pulse sampling delay greater than t ’, the sensitivity of hydrodynamic voltammetry is greater. For a delay less than t’, pulse voltammetry shows greater sensitivity. The utility of this increased sensitivity, however, is often limited, since the nonfaradaic charging current also increases at small sampling delays and ultimately determines the detection limit. In conclusion,experimentalmeasurements of the equivalent time of a hydrodynamic electrode are useful for the calculation of the diffusion-layer thickness. It is not necessary to know the electrode area. Influences of experimental variables, including the heterogeneous kinetics of the electrode reaction, on the accuracy of the technique have been determined.
1970
Anal. Chem. 1984, 56, 1970-1973
Registry No. KI, 7681-11-0; H2S04,7664-93-9; Pt, 7440-06-4.
LITERATURE CITED (1) Ibl,
N.; Dossenbach, 0. I n “Comprehensive Treatise of Electrochemistry”: Yeager, E., Bockris, J. O’M., Conway, B. E., Sarangapanl, S., Eds.; Plenum Press: New York, 1983;Vol. 6, pp 139-140,
223-227. (2) Kolthoff, I. M.; Lingane, J. J. ”Polarography”; Intersclence: New York. 1975:Vol. 1: DD 50-53. 61. (3) Albery, W. J.; Bruckbnsteln, S . J . Nectroanal. Chem. 1983, 144, 105.
(4) Flanagan, J. B.; Marcoux, L. J . Phys. Chem. 1974, 78, 718. (5) Malachesky, P. A. Anal. Chem. 1060, 4 7 , 1493. (6) Hitchman, M. L.; Albery, W. J. Nectrochim. Acta 1972, 17, 787. (7) Pratt, K. W.; Johnson, D. C. Nectrochim. Acta 1982, 2 7 , 1013.
(8) MacDonald, D. D. “Translent Techniques In Electrochemlstry”; Plenum Press: New York, 1977;pp 32-33. (9) Falkenhagen, H., Kelbg, G., Schmutzer, E., Eds. “Landolt-Bornsteins Zahlenwerke und Funktlonen”, 6th ed.;Springer-Verlag: Berlin, 1960; Vol 2,part 7, pp 257-259.
(10) Oesterling, T. D.; Olson, C. L. Anal. Chem. 1967, 39, 1546. (1 1) Delahay, P. “New Instrumental Methods in Electrochemistry”: Interscience: New York, 1954;pp 61-62,67-70. (I2) MacDonald, D. D.“Transient Technlques In Electrochemistry”; Plenum Press: New York, 1977;p 20. (13)Delahay, P. ”New Instrumental Methods in Electrochemistry”: Interscience: New York, 1954;p 224. (14) MacDonald, D. . “Translent Technlques in Electrochemistry”; Plenum Press: New York, 1977;pp 77-78.
RECEIVED for review March 15,1984. Accepted May 7,1984. Certain commercial equipment, instruments, or materials are identified in this report to specify adequately the experimental procedure. Such identification does not imply recommendation or endorsement by the National Bureau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
High-Temperature and High-pressure Decomposition and Comprehensive Analysis of Steel by Direct Current Plasma Atomic Emission Spectrometry Lancelot A. Fernando
Allegheny Ludlum Steel Corporation, Research Center, Brackenridge, Pennsylvania 15014
Complete decomposltlon of steel samples Is achieved In a Teflon (Du Pont) container-metal bomb apparatus (Parr) using a HNOS-HCI-HF acids system at 180 OC. This procedure was found to be free of voiatillratlon losses in addition to being adaptable for routlne, rapid analyses. The resulting salt-free matrlx provides a favorable environment for concentration measurements by direct current plasma atomic emission spectrometry, ellmlnating difficulties found wlth the matrlx resulting from the more conventional fusion method. Major and minor element concentratlons have been determined by using the same sample solution, and results are presented for Ai, 6, Ti, Mn, Si, Mo, Cr, Ni, P, and Sn. Also, results from fusion and simple acld dissolution are compared with the bomb results.
The ability to determine trace levels of elements in steel is becoming increasingly important with the realization that some trace elements have a dramatic effect on the properties of steel. Analytical techniques commonly used for trace analysis of steel, such as wet chemical methods, atomic absorption, and the more recent plasma emission spectrometry, require the sample to be in solution. The conventional sample decomposition methods are simple acid dissolution and the use of fused-salt media. Both of these approaches do, sometimes, present difficulties. Simple acid dissolution would be inadequate for samples containing compounds which are attacked slowly, if at all, by the common liquid reagents. The analysis of trace aluminum in steel is a good example of such a problem. ASTM (I)recommends a sodium hydrogen sulfate fusion for this analysis. In general, however, several disadvantages attend the use of fluxes (2). These include the possibility of incomplete
digestion as well as the undesirably high salt content of the resulting solution, which may cause difficulties in subsequent analysis steps. An alternate decomposition procedure was sought that would give complete aluminum digestion and, at the same time, be simple and rapid. Preferably, the procedure should be adaptable to a comprehensive analytical scheme. The high-pressure acid digestion bomb (Parr) method was investigated in this regard. The apparatus consists of a cup and cover made of Teflon (registered Trademark of Du Pont) contained in a stainless steel bomb. These sturdy digestion bombs enable digestion to be carried out at elevated pressures and temperatures without contamination and volatilization losses. Coal, soils, and foodstuffs (3-5) are some of the materials which have been digested by using the bomb method. Materials which evolve a large volume of gas when reacted with acids are not usually decomposed in this fashion. In the current procedure this difficulty was overcome by simply carrying out the initial part of the digestion in an open vessel. Only after the reaction subsided was the reaction mixture subjected to high pressure and temperature. The solutions resulting from the Parr bomb digestion of steel samples were analyzed for several elements by direct current plasma atomic emission spectrometry (DCP-AES). In addition to the Parr bomb procedure, an investigation of the fusion method (1) is also presented here. Analytical results from the bomb method, the fusion method, and simple acid dissolution are compared. EXPERIMENTAL SECTION Apparatus and Reagents. The acid digestion bomb used is the 23-mL No. 4749 bomb manufactured by Parr Instrument Co., Moline, IL. All concentration measurements were made with the Spectrallletrics, Inc., Spectraspan I11 instrument. This consists of
0003-2700/84/0356-1970$01.50/00 1984 American Chemical Society
