550
Anal. Chem. 1990, 62,550-552
IDIF = ICOL - IGEN = -2nFADC* [I + 2k=l e~p(-4122~2Dt/l2)1(A3) 1 m
ISUM = ICOL + IGEN = nFADC* [l + 2 (-ilk exp(-k212/(4Dt))] (A4) (sDt)1/2 k=l
2
LITERATURE CITED Anderson, J. L.; Du, T. Y.; Moldoveanu, S. J. Nectroanal. Chem. 1985, 796, 213-226 Bond, A. M.; Henderson, T. L. E.; Thormann, W. J. Phys. Chem. 1986, 9 0 , 2911-2917. Chldsey, C. E.; Feldman, E. J.; Lundgren, C.; Murray, R . W. Anal. Chem. 1986, 58, 801-607. Feldman. E. J.; Murray, R. W. Anal. Chem. 1988. 5 8 , 2844-2847. Fosdlck. L. E.; Anderson, J. L. Anal. Chem. 1986, 58, 2481-2485. Sanderson, D. G.; Anderson, L. E. Anal. Chem. 1985. 5 7 , 2388-2393. Sanderson, D. 0.; Anderson, L. B.; Gross, E. Biochim. Biophys. Acta 1986, 852, 269-278. Seddon, E. J.; Girault, H. H.; Eddowes, M. J. J. Elechoanal. Chem. 1989, 266, 227-238. Howell, J. 0 . BAS Current Separations 1987, 8 , 2-16. Huang, H. J.; He, P.; Faulkner, L. R. Anal. Chem. 1986, 5 8 , 2889-289 1. Bard, A. J.; Crayston, J. A.; Kittleson, G. P.; Shea, T. V.; Wrighton, M. S. Anal. Chem. 1986, 5 8 , 2321-2331. Aoki, K.; Tanaka, M. J. Electroanal. Chem. 1989 266, 11-20. Clough, A. E. Ph.D. Dissertation, The Ohio State University, 1989. Helger, D. N. M.S. Thesis, The Ohio State University, 1987.
(15) Harringtm. M. S.;Anderson, L. E.; Roberts, J. A.; Karweik, D. H. Rev. Sci. Instrum. 1989, 60, 3323-3328. (16) Bartlet, J. E.; Deakln. M. R.; Amatore, C.; Wightman, R. M. Anal. Chem. 1988, 6 0 , 2167-2171. (17) Deakin, M. R.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1986, 215, 49-61. (18) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wlghtman, R . M. J. Electroanal. Chem. 1987, 217. 417-423. (19) Aoki, K.; Morita, M.; Niwa, 0.; Takel, H. J. Electroanal. Chem. 1988, 256, 269-282. (20) Anderson, L. E.; Reillev, C. N. J. Electroanal. Chem. 1986, 72. 477-494. (21) Hubbard, A. T.; Anson, F. C. I n Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1970; Vol. 4. pp 129-214. (22) Anderson, L. E.. unpublished results, The Ohio State University, 1989. (23) Zoski, C. G.; Bond, A. M.; Colyer, C. L.; Myland, J. C.; Oldham, K. E. J . Electroanal. Chem. 1989, 263, 1-21. (24) Heinze, J. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 1096-1103. (25) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706-723. (26) Harrington, M. S.; Anderson, L. E. Bodc of Abstracts; 196th ACS Nat i i a l Meeting of the American Chemlcal Society. Los Angeles. September 1988; American Chemical Society: Washington, DC, 1988; ANYL 192. (27) Clough, A. E.; Anderson, L. E. Bodc of Abstracts; 196th ACS National Meeting of the American Chemical Society, Los Angeles. September 1988; American Chemlcal Society: Washington, DC, 1988; ANYL 193. (28) Harrington, M. S.; Anderson, L. E. Bodc of Abstracts; 198th ACS National Meeting of the American Chemical Society, Miami Beach, September 1989; American Chemlcal Society: Washington, DC, 1989; ANYL 1.
RECEIVED for review October 24,1989. Accepted December 28, 1989.
Determination of Trace-Element Deposition Parameters from Electrolytic Preconcentration Transients Roman E. Sioda Institute of Industrial Chemistry, Warsaw 01-793, Poland Thomas Z. Fahidy* Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
A rapld approxlmate and a rigorous computation scheme for the estlmatkn of the mass transfer coeflldent and the spedfic rate of depodt dissolution are presented. The approach employs experhentally determined preconcentratlon transient data. The approxhnate calculation can be readily carried out via an electronic slide rule, while the rigorous method requires a programmable calculator or a microcomputer. The approach Is Illustrated numericaHy by experimental copper and lead deposnlon transients observed by previous researchers In the lnltlal concentration range of 25 nmol/dm3 to 2.5 pmol/dm3 for copper and 14 nmol/dm3 to 1.4 pmol/dm3 for lead. The computed k , and k , values decrease wlth decreasing kdtld concentrations, suggesting the possiblUty of a fractalklnetlc process mechanism. Equlllbrlum concentrations estlmated from these results are In reasonable-togood agreement wlth values obtained by earlier experimenters.
INTRODUCTION Preconcentration is a standard analytical method ( I ) for the analysis of trace components (analytes) present initially in a sample of a concentration level below a detection or
quantification limit of a chosen instrumental method of analysis. The typical electrodes used in electrolytic preconcentration are the hangmg mercury drop electrode, reticulated vitreous carbon, gold, platinum, and tungsten. When the electrodeposition stage (5-60 min) is terminated, analysis is carried out by standard instrumental methods, e.g. spectroscopy, radiometry, electroanalysis, etc. In the latter case, the same cell is often used. Electrolytic preconcentration is not recommended for long times due to complications, such as nonstationary diffusion, restructuring of the electrode surface and of the deposit, and chemical reactions involving the deposit. Among the many possible applications of the technique, the determination of trace amounts of Cd, Cu, Pb, Hg, and Zn in biological substances and fluids (plants, tissues, fish, blood, and food) merits particular mention. Previous attempts to obtain the governing parameters (2-4) met with limited success, since earlier determinations of rate coefficients and equilibrium concentrations were not sufficiently precise and could not be obtained from a single determination. Only the recent experimental data of Ciszewski and co-workers (5) permit the evaluation of pertinent kinetic and equilibrium parameters at a reasonable accuracy, since the limits of the new determination methods are 2 to 4 orders of magnitude lower than those obtained in previous attempts
0003-2700/90/0362-0550$02.50/00 1990 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 62, NO. 6, MARCH 15, 1990
cited above. Recent experimental data obtained ( 6 ) on the deposition of lead on gold using the novel quartz microbalance technique (7, 8) in the initial concentration range of 3-30 nmol/L confirm a mathematical model of equilibrium concentrations (2,3,9, IO), which serves as a starting point for the current work. The equilibrium conditions and concentration limits for a typical preconcentrationprocess consist of an electrodeposition step
Mn++ ne-
= Mo
(1)
Table I. Summary of the Deposition Process Parameters Estimated from Transient Data by Ciszewski et al. (5)
initial concn, species
(2)
Since the dissolution restores partially the initial substrate
M"+ in solution, the process resembles closely the Nicholson-Shain (11,12) catalytic reaction mechanism. The purpose of the current paper is to present a method whereby the kinetic and mass transport parameters of the process may be determined from the same transient data (such as obtained in the experimental work of Ciszewski et al. ( 5 ) )employed in the evaluation of equilibrium and preconcentration limit conditions, thus extending the scope of the mathematical model.
RESULTS AND DISCUSSION 1. Analysis of the Transient. The transient equation (eq 3, ref 1) is rewritten as d R / d t = a [ ( l - R ) - P(1 - exp{-yc&?))l
(3)
where a = sk,, 6 = k z / k l c o ,and R = (co - c)/co. As shown previously, s is the specific area (cm-') of the electrolytic cell, k l , is the mass transfer coefficient (cm/s) of the deposition process, k , is the specific rate (mol/(cm2-s))of deposit dissolution, co is the initial trace concentration (mol/cm3)in the electrolyte, and y is the ratio of the molar monolayer area constant p (cm2/mol)and the specific area s. In the case of copper and lead ( 5 ) the magnitude of the yco product varies between unity and slightly above one hundred indicating that if co is relatively high (order of 10-9-10-10 mol/cm3), the exponential term in eq 3 might be neglected at some risk. Under such circumstances the recovery (or conversion) is quickly obtained by straightforward integration as
R = (1- P)[1 - exp(-at)]
(5)
lim R = R ,
t-m is first obtained. The solution of eq 3 may conveniently be expressed as a quadrature for the explicit denomination of cy
rR
kl, cm/s
approx method
X lo4 0.1296 0.0797 2.5 X 2.5 x lo-@ 0.0628 1.4 X lo4 0.2072 1.5 X 0.0942 0.0768 1.4 x
lead
rigorous
10"kz, mol/(cm%)
method
approx method
rigorous method
0.1107 0.0794 0.0163 0.2103 0.1173 0.0166
1.296 0.658 0.0974 0.839 0.632 0.0819
1.107 0.655 0.0763 0.852 0.788 0.0875
2. Determination of the Deposition Parameters. In a primitive approach experimental R ( t ) data could be fitted by a regression carrying arbitrary parameters without a mechanistic (i.e. physicochemical)basis; in principle, several competing regressions could be devised whose statistical properties might be similar enough to render them statistically indistinguishable. Consequently, the extraction of rate parameters from the fit would be indeterminate. Further, the numerical work associated with this approach would not be less or simpler than in the approach, described in the sequel, which carries a fundamental mechanistic structure with it. The simplified version of the proposed approach yields a quick estimate of the rate parameters, whose refined values may be obtained by the rigorous version. The higher co the better the quick estimate. (A) Simplified Method. It follows from eq 4 that as t a,R R , = 1 - 0. Moreover, if eq 4 is an acceptable approximation to experimental (R,t) data, the value of R at t , = 1/a is R, = (1 - P)(1 - l / e ) ; e = 2.71828 (7)
-
-
Consequently, if R , is known experimentally, a can be computed from eq 7. The straightforward computation procedure consists of the following steps: (A.1) obtain experimentally R,, then compute = 1- R , and R , = 0.632R,; (A.2) obtain from the transient curve t , corresponding to R, and compute a = l / t a ; (A.3) since a = s k , and P kz/klco,the deposition parameters are calculated as kl = a / s and kz = k,cOP. (B) Rigorous Method. (B.l) From eq 5 calculate P
(4)
It is to be noted that eq 3 cannot be integrated a priori to obtain an analytical form and, consequently, the parameters a and cannot be determined by a nonlinear regression applied to the (nonexistent) analytical form. In a rigorous approach, eq 3 is considered fully and it is noted that by setting the recovery derivative to zero, the important algebraic relationship, allowing the determination of p
1 - R , = P [ 1 - exp(-yc&,)I
mol/dm3
copper 2.5
for cationic species M, followed by the chemical dissolution
Mo + X = Mn++ Y
551
dX
where tR is the time required to reach a certain recovery R. For a specified value of /3, the integral may be evaluated by standard methods of numerical integration (13).
(B.2) using the value of @ hereby computed, obtain several numerical estimates for selected R values of the integral
in eq 6, using experimental (R,tR) data, and compute in each case a value of CY;(B.3) obtain a mean value of a by averaging the individual a values obtained in step B.2; (B.4) obtain mean estimates of kl and k2 according to step A.3. 3. Discussion of the Numerical Illustration. Table I contains the results of the determination of deposition process parameters using the concentration transient data of Ciszewski et al. (5) in the case of copper and lead. The procedure is illustrated numerically in the Appendix. At relatively high concentrations the approximate and rigorous methods yield reasonably close estimates of k , and k2, but as co decreases, the discrepancy becomes large. The decrease of k, and kz with decreasing co might be explainable in terms of fractal reaction kinetics theory (14) which postulates concentration-dependent rate constants: it is quite plausible, in effect, that deposition from trace concentrationsis a fractallike process. Comparison of the values of the equilibrium concentration obtained in the current work with values computed by Ciszewski et al. (5)
552
ANALYTICAL CHEMISTRY, VOL. 62, NO. 6, MARCH 15, 1990
Table 11. Comparison of Computed Equilibrium Concentrations species and initial concn,
eq no. in
approx
mol/cm3 rigorous
mol/cm3
ref 5
method
method
comp.
exptl
2
1.640
4.95
2.02
1.8 f 0.2
5 2
0.796 1.07
1.55 5.28
2.00 1.13
1.1 f 0.5
5
0.530
1.105
1.11
copper (2.5
lO”C,,
Ciszewski et al. I h\
\a I
x 10-11) lead (1.4 X
The principal value of knowing ceq,k l , and k z is in optimizing an electrolytic preconcentration process, e.g. to decrease as much as possible the limit of detection and to increase its sensitivity. The proposed method has this goal for its target.
10-19
Table 111
R
d R , P ) (eq 9)
t ~9 ,
a” (eq 6), 9-l
0.14 0.24 0.42 0.58 0.59 0.65
0.185 10 0.376 34 0.902 68 1.914 31 2.031 21 3.402 14
300 600 1200 2400 3600 7200
6.170 X 6.272 X 7.522 X 7.976 X 5.642 X 4.725 X
“ T h e mean value of a is 6.384 X 10” s-l. Hence, the parameter estimates are k l = (6.384 X 10-‘)/(8.04 X = 0.0794 cm/s and mol/(cm2.s). k, = (0.0794)(2.5 X 1O1O)(O.33)= 6.55 X
shows a certain discrepancy when the simple relationship (eq 2, ref 5 )
is employed. However, the discrepancy becomes minimal when the relationship (eq 5 , ref 5 )
- e x p ( - m + ~ceq)l
APPENDIX: NUMERICAL ILLUSTRATION The numerical calculations are illustrated in the case of copper deposition from an electrolyte with an initial cupric ion concentration of 2.5 X mol/dm3 (Figure 1, curve b, ref 5). Approximate Procedure. From the steady-state portion of the curve R , = 0.67 and ,B = 1 - 0.67 = 0.33. Hence, R, = (0.632)(0.67)= 0.4234. The corresponding time instant is 26 min; hence a = 1/(26)(60) = 6.410 X s-l. Since s = 8.04 X cm-l and p = 3.4 X lo8 cm2/mol in the case of = copper, it follows that k l = (6.410 X 10-4)/(8.04 X 0.0797 cm/s. Consequently, kz = (0.0797)(2.5 X 10-10)(0.33) = 6.58 X 10-l2 mol/(cm2.s). Rigorous Procedure. Since yc, = (3.4 X 108)(2.5 X 10-10)/(8.04 X = 10.572 and R, = 0.67, it follows from eq 8 that 0 = 0.33/0.99916 N 0.33. This is essentiallythe same value as obtained by the approximate procedure since yco is relatively large. Continuing with eq 9 and considering the experimental points denoted by circles in ref 5; Figure Ib, the values given in Table I11 are obtained via numerical integration. LITERATURE CITED (1) Neiman, N. M.: Zolotov Yu. A. Preconcentrafion of Trace Nemenfs;
(2) (3) (4) (5) (6)
(11)
(7)
is used for estimating cq combined with the rigorous method of estimating k , and k z (hence (ce),), as shown in Table 11. The closeness of these (c,) values indicates the validity of parameter determination based on experimental deposition transients and the soundness of the analysis. In view of the common availability of microcomputers equipped with fast and rugged integration software, the handling of eq 9 is essentially a routine operation (in this work a standard Romberg quadrature method was used for rapidity and accuracy), but the approximate method can be retained for a quick preliminary order-of-magnitude estimation as a guide for the more involved computation scheme.
(8) (9) (10) (11) (12)
ceq =
(ce)l[l
(13) (14)
Nauka: Moscow, 1988 (in Russian). Sioda, R. E. Talanfa 1984, 37, 135-137. Sioda, R. E. Taslanta 1985, 32, 1083-1087. Sioda, R. E. Monabcb. Cbem. 1985, 776,49-52. Ciszewski, A.; Fish, J. R.; Malinski, T.; Sioda, R. E. Anal. Chem. 1989, 67,856-860. Schmidt, H.J. Diploma Thesis, Technical University of Darmstadt, FRG, 1989. Melroy, 0.; Kanazawa, K.; Gordon, J. G., 11; Buttry, D. Langmuir 1986, 2, 697-700. Deakin, M. R.; Melroy. 0. J. Electroanal. Cbem. 1988, 239, 321-331. Sioda, R. E. Anal. Chem. 1988, 6 0 , 1177-1179. Sioda. R. E. Anal. Lett. 1983, 76 (AlO), 739-748. Nicholson, R. S.; Shah, I. Anal. Cbem. 1984, 36, 706-723. Bard, A. J.; Faulkner, L. R. Elecfrocbemical Methods; Wiley: New York, 1960; Chapter 11. Hornbeck. R. W. NumericalMetbods; QPI Series: New York, 1975; Chapter 8. Kopelman, R. Science 1988, 241, 1620-1626.
RECEIVEDfor review June 26, 1989. Accepted December 7, 1989. The numerical work was carried out on equipment obtained under a University of Waterloo/Digital Equipment Corporation research agreement.
