tiometry the following substitution can be made io = io/s
(11)
where io is the constant applied current density. Rearranging Equations 4 and 8 ryith the substitution of Equation 11 leads to explicit expressions for the concentration of R a t the electrode surface. Inverse transforming these and setting the results equal to zero gives relations for the transition time in the two cases. The general form of these expressions is rather cumbersome; however, as in the case of potential scan, the deviations from behavior to be expected under semi-infinite linear diffusion conditions are divisible into contributions from curvature and finite elcctrode volume. Qualitativrly, the same relative imp0rtanc.e attaches to the two corrections as in the potential scan case. A simplified equation taking account only of spherivity can be obtained by replacing the hyperbolic cotangent of Equation S by unity. TIP result is
( d 7 D I r "I ]
case in which the reactant is soluble in solution. The qualitative prediction of Equation 12 is a shortening of id; from t h a t predicted for linear diffusion, with the deviation increasing as T becomes longer. OTHER CASES
Corrections for sphericity of the electrode can be obtained from the simplified (coth = 1) form of Equation 8 for any type of voltammetric technique by substituting explicitly the form of io or f obtained from appropriate boundary conditions. The conditions under which the approximation is valid are a t least qualitatively the same as in the potcntial scan case. For situations in which potential is controlled and the electrode reaction is non-K'crnstian, invcrsr transformation yields an integral equation rather than explicit solution. Hon-ever, mathematical approaches are available w e n for these complex cases ( 1 , 1 2 ) . The nature of the relations herein derived also lends qualitative substantiation to the application of negative spherical corrections to the Ilkovi5 equation in the polarography of amalgams (f 6 ) .
LITERATURE CITED
(1) Delahav, P.. J . Am. Chem. SOC. 75.
R. D., Shain, I., ~ ~ N A L . CHEX.29,1825 (1957). (4) Frankenthal, R. P., Shain, I., J . d m . Chem. SOC.78, 2969 (1956). ( 5 ) Kemula, W..Kublik, 2.. Anal. C h i n . Acta 18, 104 (1958). ' ' (6) Lee, T. S., University of Chicago, Chicago, Ill., private communication, 19.56.
( 7 , I&mantov, G., Delahay, P., J . d m . Chem. Soc. 76, 5319 (1954). (8) Mamantov, G., Papoff, P., Delahay, P., Ibid., 79, 6358 (1957). (9) Xicholson, M. M., Ihid., 76, 2539
(1954). (10) Randles, J. E. B., Trans. Faraday Soc. 44, 327 (1948). (11) Reinmuth, W. H., J . Am. Chem. Soc. 79, 6358 (1957). (12) Reinmuth, W. H., J . Phya. Chem., in mess (March 1961). (13) 'Sevcik, A,, Collectton Czechoslav. Chem. Communs. 13,349 (1948). (14) Shain, I., Lewison, J., AKAL.CHEM 33, 187 (1961). (15) Shain, I., Martin, K. J., Ross. J. W., Ahstracts, p. 4B, Division of Analytical Chemistry, 138th Meeting, .4CS, New 1-ork. 3'. Y.. 1960. (16) Strehlowv,H., von Stackelberg, 11, 2. Elektrochem. 34, 51 (1950). RECEIVED for review September 27, 1960. Accepted Xovember 14, 1960. I
(12)
wliich may be compared Fvith the result of llnmantov and Delahay ( 7 ) for the
gesting this work and informing us of some of his data prior to its publication.
ACKNOWLEDGMENT
Thanks are due Irving Shain for sug-
-
Stripping Analysis with Spherical Mercury Electrodes IRVING SHAlN and JOHN LEWINSON Chemisfry Department, Universify o f Wisconsin, Madison, Wis.
b The concentration distributions of metal within a hanging mercury drop electrode, which result from the processes taking place during the preelectrolysis step of a stripping analysis, have been considered. This concentration was shown to b e relatively uniform, and an equation by which it can b e calculated was derived. This, combined with the theory of the stripping process in a recent report by Reinmuth, permits the calculation of theoretical current voltage curves which are useful in the development of new methods of stripping analysis. The theoretical and experimental curves were compared for the stripping analysis of thallous ion.
R
on the use of hanging mercury drop electrodes for stripping analysis have emphasized the remarkable sensitivity ( l O - 9 V ) of this electroanalytical method. This SCENT R E P o n T b
high sensitivity is a result of a preelectrolysis step in which the sample is concentrated in the mercury electrode by electrodeposition. The amalgam thus formed is then stripped and the electrodissolution can be followed by any of several methods, including galvanostatic and potentiostatic methods (fO),voltammetry with linearly varying potential (6, 7 , Q), square wave polarography ( I ) , and oscillographic polarography (8). Those methods based on voltamnietry with linearly varying potential have proved to be particularly applicable for general analytical work because the cells, apparatus, and techniques are direct (,\tensions of polnrographic practice. For analytical purposcs, thr (.oncentration of the sample in the bulk of the solution can be correlated to the stripping process by calibration with known solutions. I n the development of new analytical procedures, however, i t is frequently desirable to predict
the general behavior of amalgams during the stripping process. This would be particularly useful in the detection and investigation of anomalies caused by formation of precipitates and intermetallic compounds during the preelectrolysis step, and irreversible behavior during the stripping process. Mamantov, Papoff, and Delahay (10)derived equations describing the stripping process for the case of reversible electrodissolution from a plane mercury electrode when carried out with either constant current or constant potential methods. Because these authors used the hanging mercury drop electrode for their mpcriments, the rcsults at long times (60 to 180 seconds) showed considerable deviation from theory because of the curvature of the surface and the noninfinite nature of the diffusion process inside the electrode. Since the pre-electrolysis times used in analysis ordinarily are even longer, both sources of deviation must VOL 33, NO. 2, FEBRUARY 1961
187
be considered in calculating the concentration distributions within the electrode. I n general, the limited volume of the spherical electrode results in much higher amalgam concentrations than are obtained a t equivalent electrolysis times with a plane electrode. This paper presents a n investigation of these effects. Similarly, both the finite volume of the electrode and the curvature of the surface must be considered for the electrodissolution process. The theory for this part of a stripping analysis has been developed by Reinmuth ( 1 2 ) and an esperiniental evaluation of this theory is also included in this paper.
IE
li . 1
* E
LLO \
s
0
4
C
EXPERIMENTAL
All materials mere reagent grade and were used without To remove further purification. oxygen from t h e cell, high purity nitrogen ITas used without further purification. Apparatus. A modified Sargent Model XV Polarograph (E. H. Sargent Co., Chicago) was used t o obtain some of t h e curves. T h e chart speed was increased t o 10 inches per minute, t h e recorder response was increased t o 1 second full-scale, and t h e voltage scan was modified SQ that 1, 2, or 3 volts could be traversed in 60 seconds. For this work the rate of voltage scan was 33.3 mv. per second. An external source of potential was placed in series with thc voltage scan potentiometer so that the initial potential could be selected independently. Other measurements were made on an instrument based on the analog computer amplifiers manufactured by G. A. Philbrick Researches, Inc (Boston), incorporating some of the ideas suggested by DeFord (4). The hanging mercury drop assembly and the cell have been described previously (6). The electrode radius was 0.062 em.
J
I
I
I
I
0
0.2
0.4
0.6
0.8
I
1.0
r/ro
Materials.
RESULTS A N D DISCUSSION
Pre-electrolysis Step. During the pre-electrolysis step of stripping analysis, t h e potential is set a t some cathodic value such t h a t t h e rate of mass transfer determines the rate at which t h e sample is electrodeposited into t h e hanging mercury drop electrode. T h e solution is stirred in order to increase the rate of mass transfer. After a carefully timed interval, the stirring is stopped to allow the solution to come to rest, and then the potential is scanned linearly toward anodic values. If the stirring is carefully controlled during the electrodeposition step, the flux, F,, of material entering the drop is constant. Under these circumstances, the concentration distribution of the metal in the electrode can be obtained by solving Fick's second law written
188
ANALYTICAL CHEMISTRY
Figure 1 . Concentration gradients within a hanging mercury drop electrode a t various times after the start of pre-electrolysis Times, in seconds A, 240; 8, 120; C, 60; D, 30; E, 10
for spherical diffusion, with the following boundary conditions : t = 0, C R = 0, 0
r T
=
0, D R
= ro, D R
(%)
< r < ro = 0, t
(I)
>0
(2)
i (F) F , nk, > =
=
t
0
(3)
where t is the time, CR is the concentration of the metal in the amalgam, r is the distance from the center of the electrode, r0 is the electrode radius, D R is the diffusion coefficient, F , is the flux of C R a t the electrode surface, i, is the cathodic pre-electrolysis current, n is the number of rlectrons, d is the area, and F is the faraday. The finite volume of the electrode is reflected in the second boundary condition, and the third boundary condition relates the constant flux a t the electrode surface to the pre-electrolysis current. The solution of an analogous problem in heat conduction is given by Carslaw and Jaeger ( 2 ); in terms of the electrochemical parameters, the rewlt iq
in the electrode which IS dependent on the flux and the time, modified by a correction term which vanes according t o time and distance. From data given by Carslaw and Jaeger, it can be calculated that the correction term is essentially independent of time after about 25 seconds, and that further electrolysis merely raises the concentration-distance curve at a rate dependent on the flux (Figure 1). The pre-electrolysis times ordinarily used in stripping analysis always esceed 3 minutes and frequently 15 and 20-minute electrolysis times are required for very dilute solutions (IO-* to 10-~,1f). After about a 4-minute electrolysis (curve A , Figure l), the concentration a t the center of the electrode is about 65y0 of the value a t the electrode surfacr, and as the total amount of material in the drop increases with longcr electrolysis times, this percentage increascs. A comparison of this concentration distribution with that calculated for semi-infinite linear diffusion ( I O ) confirms that for the same pre-electrolysis times. significantly higher amalgam concentrations can be produced in a hanging mercury drop electrode than in a mercury pool rlec trode of the same area. Since higher amalgam concentrations lead to more sensitive stripping analyses, the advantages of hanging mercury drop electrodes over mercury pool electrodes are obvious. After the pre-electrolysis, a period of 20 to 30 seconds normally is allowed for the solution to come to rest in order to improve the reproducibility of the stripping analysis. During this period, the flux of the material entering the electrode drops sharply. For a mercury pool electrode, this penod results in a decrease in C R a t the electrode surface as the metal diffuses further into the mercury. On the other hand, with a spherical electrode of limited volume, the concentration gradient merely becomes more uniform. From equations given by Crank (3), this process can be described as a function of time if it is assumed that the flux FO drops to zero when the stirring is stopped :
:
n = l
(4)
where cyn, for n = 1, 2. . . . . .. are the positive roots of tan CY = cy. The form of this equation is that of a uniform concentration distribution with-
where f(r) is the concentration distribution a t the moment the stirring stops, and t is the time now counted from the same moment. The an's are the positive roots of the transcendental equation roan
cot
roan
=
1
the experimental conditions normally used in stripping analysis, the concentration of the metal within a spherical electrode is relatively uniform, and can be calculated from the number of coulombs involved in the electrodeposition process:
148-
CE =
0
02
04
06
08
IO
r/ro
Figure 2. Concentration gradients within a hanging mercury drop electrode at various times after stirring is stopped Times, in seconds: A, 30; 8, 20; C, 10; D , 0
the first six roots of which are given by Crank (3). The assumption that Fodrops t o zero when the stirring ceases introduces a relatively small error Measurements on a 10-4;V solution of thallous ion showed t h a t the current dropped rapidly when the stirring mas stopped, but that the additional amount of material deposited during this 30second interval could amount to several per cent of the total for a 5-minute pre-elertrolj-sis. This source of error could be eliminated by opening the cell circuit for the 30-second interval, and then starting the voltage scan from the equilibrium potential. For longer pre-electrolysis times, this Source of error was negligible. Because of the complexity of S(T) (Equation 4)) and since the flux does not entircly drop t o zero when the stirring is stopped, it is not useful to pursue the rigorous treatment any further. Nevertheless, some conclusions regarding the concentration distribution within the electrode can be obtained b y approximating Equation 4 by a straight line I
f ( r ) = kr
(6)
where k, an average slope measured from Figure 1, is essentially independent of the flux for normal pre-electrolysis times. On substitution of Equation 6 into Equation 5, the indicated integration can be performed and the result indicates t h a t C R approaches a constant value very rapidly. Concentration distribution curves based on the integrated form of Equation 5 are shown in Figure 2 for various times after the stirring is stopped. Although these curves are only approximate, they indicate that under
,
,
VOLTS, vs S C E -014 P
z 2
0
c a a
3 i,t
m z
-
4mzFr,8
where CE is in moles per cubic centimeter, i, is in amperes, and T , is in centimeters. It should be noted t h a t the right side of Equation 7 is equal t o the first term in Equation 4. Stripping Process. On scanning t h e potential toward anodic values, a typical peak shaped current voltage curve is obtained as t h e metal is re-oxidized from t h e amalgam. An equation for this current voltage curve has been derived by Reinmuth ( 1 2 ) , who considered both the limited volume of the hanging mercury drop electrode, and the curvature of the electrode surface. For the times and scanning rates normally used in analytical work, only the effect of the surface curvature must be considered. The equation is indentical in form to t h a t previously derived for voltammetry with linearly varying potential at a hanging mercury drop electrode ( I 1 ), except that the term which corrects for the curvature of the electrode surface decreases the calculated current. This correction term can amount to as much as 20% of the peak current in some cases. An alternate method of calculating a theoretical current voltage curve is to use the graphical data presented previously (6), again noting that the spherical correction should be subtracted for the case of stripping analysis. An experimental evaluation of these conclusions regarding both the preelectrolysis step and the stripping process was performed. Several concentrations of thallous ion in 0,lM KC1 were analyzed and the results were compared to theory. The amalgam concentrations were calculated from Equation 7, using the cathodic pre-electrolysis currents (corrected for the blank) which were measured for the more concentrated solutions. For more dilute solutions (lo-* and 10-7Al) the preelectrolysis currents can be taken as proportional to the concentration, if care is taken to maintain the mass transfer process (stirring and cell geometry) as constant as possible. The validity of this procedure was demonstrated previously (6). An example of the results is shown in Figure 3 which
g 0
=-I
Figure 3. Current-voltage curve for anodic stripping of thallium, using voltammetry with linearly varying potential Solution: 1.00 X 1 0-jM TI+, 0.1 M KCl; r a t e of voltage scan, 33.3 mv./sec.; pre-electrolysis time, 5 min. a t -0.7 volt VI. S.C.E. - Experimental 0 Theoretical
indicates t h a t the theory adequately predicts both the shape and magnitude of current-voltage curves for stripping analysis with the hanging mercury drop electrode. LITERATURE CITED
(1) Barker, G. C., Anal. Chim. Acta 18, 118 (1958).
(2) Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” p. 242, 2nd ed., Oxford University Press, London, 1959. (3) Crank, J., “The Mathematics of Diffusion.” DD. 92. 331. Oxford University Pies$,’London, 1956. (4) DeFord, D. D., Division of Analytical Chemistry, 133rd hleeting, ACS, San Francisco, Calif., April 1958. (5) DeMars, R. D., Shain, I., ANAL. CHEM.29, 1825 (1957). (6) Frankenthal, R. P., Shain, I., J . A m . Chem. SOC.78,2969 (1956). ( 7 ) Kemula, W., Kublik, Z., Anal. Chim. Acta 18, 104 (1958). (8) Kemula, W., Kublik, Z., Rocznzki Chem. 30, 1005 (1956). (9) Kemula, W., Kublik, Z., Clodowski, S., J.Electroanal. Chem. 1,91 (1959). (10) Mamantov, G., Papoff, P., Delahay, P., J . A m . Chem. SOC 79,4034 (1957). (11) Reinmuth, W. H., Ibid., 79, 6358 (1957). (12) Reinmuth, W. H I ANAL. CHEM. 33,185 (1961) RECEIVEDfor review October 17 1960. Accepted November 18, 1960. This work was supported in part by funds received from the Research Committee of the Graduate School of the University of Wisconsin. Other support was obtained from the United States Atomic Energy Commission under Contract No. AT(11-1)-64, Project No. 17.
VOL. 33, NO. 2, FEBRUARY 1961
189
