The relative importance of several elements present in white bond papers has been assessed; this may be of interest to anyone wishing to establish a forensic data base on such papers. For example, the recent study by Brunelle et al. (9) on a wide variety of papers including white bonds demonstrated that distinguishing paper manufacturers is often possible with a mostly qualitative "fingerprint" of 23 elements: Na, C1, Sc, Ti, Cr, Mn, Fe, Co, Cu, Zn, As, Br, Mo, Ru, Sb, Ba, La, Ce, Sm, Ta, W, Au, and Hg. If our analysis of the Lukens data is indicative, the most important element for both manufacturer and paper grade categorization was not measured: Al. And the concentration of aluminum, being a major component of many fillers and opacifiers, may be much less sensitive to variation over time than many of the trace elements. We wish to emphasize that pattern recognition is in no manner limited to the information taken from one specific instrument nor to one kind of information. The physical characteristics of the papers involved in the Lukens study, and the reflectance spectra, would undoubtedly improve the predictive ability of the data. Nor are "refined" data necessary, for the techniques used are insensitive to the units of measurement and may indeed be used to extract the interfeature correlations, eliminating the need for elaborate and costly decorrelation procedures.
ACKNOWLEDGMENT The authors express their gratitude to James L. Booker, Director, State Crime Laboratory, State of Wyoming, for
his assistance and encouragement and to Maynarhs deKoven for his valuable observations.
LITERATURE CITED (I)E. R. Kowalski, "Pattern Recognition In Chemical Research," in "Computers in Chemical and Biochemical Research," Vol. 2, C. E. Klopfenstein and C. L. Wilkins, Ed., Academic Press, New York, N.Y., 1974. (2)E. R. Kowalski and C. F. Bender, J. Amer. Chem. SOC., 94, 5632 (1972). (3)8. R. Kowalski and C. F. Bender, J. Amer. Chem. SOC.,96, 916 (1974). (4)A. H. Jones, Anal. Chem., 37, 1761 (1965). (5) J. J. Manura and R. Saferstein, J. Ass. Offic. Anal. Chem., 56, 1227 (1973). (6) D. E. Bryan, V. P. Gulnn, R . P. Hackleman, and H. R . Lukens. "Development of Nuclear Analytical Techniques for Oil-Slick identification (Phase I),'' U. S. At. Energy Comm. Rept, GA-9889, Gulf General Atomic Incorporated, January 21,1970. (7)R. L. Brunelle and M. J. Pro, J. Ass. Offic. Anal. Chem., 55, 823 (1972). (8)H. R. Lukens. H. L. Schlesinger, D. M. Settle, and V. P. Guinn, "Forensic Njn Activation Analysis of Paper," U.S. At. Energy Comm. Rept. GA10113, Gulf General Atomic Incorporated. May 22. 1970. (9)R. L. Brunelle, D. Washington, C.. Hoffman and M. J. Pro, J. Ass. Offic. Anal Chem., 54, 920 (1971). (10)R. L. Williams, Anal. Chem., 45, 1076A (1973). (11)E. R. Kowalski and D. L. Duewer. 'Documentation for ARTHUR (Batch)," Chemometrics Society Report 1, Chemornetrics Society, August 1, 1974. (12)Any numerical statistics text. (13) N. J. Nilsson, "Learning Machines." McGraw-Hill. New York, N.Y., 1965. (14)C. F. Bender and B. R. Kowalski, Anal. Chem., 46, 294,(1974). (15)P. C. Jurs, E. R. Kowalski, and T. L. Isenhour, Anal Chem., 41, 21
(1969).
RECEIVEDfor review August 23,1974. Accepted November 25, 1974. We gratefully acknowledge the financial support of the National Science Foundation (GP-36578X).
NOTES
Reexamination of Experimental Data on the Ti ne-Dependence of Alternating Current Polarographic Waves w th Metal lonMetal Amalgam Redox Couples lvica Ru&'
and Donald E. Smith2
Department of Chemistry, Northwestern University, Evanston, Ill. 6020 1
In the middle of the past decade, theoretical predictions based on the stationary sphere diffusion model (2, 2) suggested existence of the so-called "spherical diffusion induced time-dependence" of the ac polarographic current magnitude. In measurements with the dropping mercury electrode (DME), this effect will be manifested as a mercury column height dependence. With pure diffusion-controlled waves, the effect was predicted to be rather small, if not negligible, with metal ion-metal ion couples, while its magnitude was suggested to be too large to ignore with metal ion-metal amalgam systems ( 2 ) . Experiments supported qualitatively the theoretical predictions (3-5), but quantitative theory-experiment comparisons with metal ion-metal amalgam systems suggested that the stationary sphere model is in this case insufficiently accurate for treatment of data obtained with the DMR ( 5 ) ,unless some On leave f r o m the Center for M a r i n e Research, R u d e r Boskovi c I n s t i t u t e , Zagreb, Yugoslavia, 197'2-75. To w h o m correspondence should be addressed. 530
form of semiempirical correction is introduced (6, 7 ) . The failure to achieve good quantitative theoretical interpretation of the data in question left some uncertainty regarding the importance of electrode sphericity in controlling the observed time-dependent ac response, relative to other mass transfer perturbations, such as electrode streaming (2-4, 8 ) ,shielding ( 9 ) ,and depletion (20). Recently, dc and ac polarographic boundary value problems based on the more exact (for the DME) expanding sphere electrode model have been treated successfully (22-24) using the digital simulation method ( 2 5 ) .The most detailed of the aforementioned data on metal ion-metal amalgam system time-dependence have been reexamined using rate laws based on the expanding sphere model. Data obtained from two-electron processes were compared to predictions based on both single-step and two-step heterogeneous charge transfer. Presentation and discussion of this reappraisal of published data are the purposes of this Note.
ANALYTICAL CHEMISTRY, VOL. 47, NO. 3, M A R C H 1975
I20
I 20
I16
0
c 0 [L
I12
108
IC4
IO0
-0560
-0520
-0480
E d C (voits vs
1.25
-0440
sc E
IW
-0400
E d c (volts vs S C E )
1
I
c l
I 20
2
c
0 ._ c
I15
0
0
a
a
I10
I05
E,,,(volts v s S C E )
€,,(volts vs S C E )
Figure 1. Ratio of fundamental harmonic current magnitudes at two different drop lives vs. dc potential for several metal ion-metal amalgam
systems
+
3.0 X 10-3MCu2+in l.0MNH3 1.0MNH4CI,(C) 1.0 X 10-3MCd2+in System: (A) 3.0 X 10-3MTl+ in 1 . O M K N 0 3 -t 0.1MNaC2H302f 0.1MHC2H3O2,(6) 1.OM KCI 0.1MNaC2H302 0.1MHC2H302,(0)1.0 X 10-3MCd2+in 0.5MHCi. Applied. 0 = 20 Hz., A = 40 Hz., 0 = 80 Hz., 10 mV peak-to-peak sine wave; dc potential varied incrementally. Measured. Ratio of fundamental harmonic faradaic current components in phase with applied alternating potential at end of natural drop lives, t2 and tl, of: (Figure lA), t2 = 9.92 sec, tl = 5.31 sec; ( 1 6 and 1C) t2 = 10.0 sec, tl = 5.26 sec; (lD), t2 = 10.11 sec, tl = 5.63 sec. Other parameter values. Mercury flowrates, m ( t 2 ) and r n ( t 1 ) : (1A.6.C) m ( t 2 ) = 0.764 mg sec-', m(t1) = 1.455 mg sec-'; (1D)m(t2)= 0.703 mg sec-'. f i t l ) cm2 sec-' and oxidized cmz sec-', DCd(Hg) = 3.06 X = 1.266 mg sec-'. Diffusioncoefficients: &(H~) = 1.30 X loW5cm2 sec-', D c ~ (=~ 2.26 ~ ) X = results of stationary sphere calculation given in Reference (5).(-) = results of expanding sphere calform diffusion coefficients same as Ref. 5. * culation based on Mechanism l or Mechanism II with Ez0 - €1' 2 0.50 volt. (- - -) = results of expanding sphere calculation based on Mechanism II EZo- El0 = -0.26 volt (6),0.00volt (C,D). -.- -) = results of expanding sphere calculation based on Mechanism 11, EZo- El0 = 0.10 volt
+
+
(a
a)
(-a
THEORY Mechanisms considered in this presentation are the simple single-step heterogeneous process with amalgam formation, 0
+
ne e R(Hg)
(1)
and the two-step heterogeneous process 0
+ e aY
t
e e R(Hg)
(11)
Derivations of the theoretical ac polarographic rate laws for these mechanisms were effected by combining digital simulation procedures and analytical methods to solve the dc and ac parts, respectively, of the boundary value problems. The approaches have been outlined in detail in previous publications (11-14). FORTRAN programs developed to provide predictions of these rate laws are available from the authors on request.
perimental details. Briefly, the data in question were obtained with several metal ion-metal amalgam systems whose apparent heterogeneous charge transfer rate constants are sufficiently large that, with the low frequencies employed (20,40,80 Hz.), the ac polarographic responses are characterizable as diffusion-controlled ( 5 ) .Depletion effects ( I O ) were eliminated through the use of first drop experiments. The specific data format of interest here is the dc potential dependence of the ratio of instantaneous ac polarographic peak currents a t two different mercury drop lives (column heights). This observable is predicted to be unity with diffusioncontrolled systems in absence of spherical diffusion effects (2). Consequently, deviations from unity in such observables serve to highlight the spherical diffusion induced time dependence ( 5 ) . Diffusion coefficient values were taken from the same literature sources ( I , 17-20) employed by Delmastro and Smith. However, the amalgam diffusion coefficient values presented by Cooper and Furman (19, 20) had been calculated from the approximate Ilkovic equation (expanding plane model). For present purposes, these values were recalculated using the Koutecky-Matsuda expanding sphere equations for the dc limiting current (9, 2 1 ) and the Cooper-Furman limiting current data.
EXPERIMENTAL
RESULTS AND DISCUSSION
Ac polarographic data presented here are taken directly from the work of Delmastro and Smith ( 5 , 16), which provides full ex-
The ac polarographic current ratio data of Delmastro and Smith are given in Figure 1,together with various theANALYTICAL CHEMISTRY, VOL.
47, NO. 3, M A R C H 1975
m
531
I30
r
B ~
I
* I
I 2 5 1
0 0 5 volt
Jo -,:25vOtl
I20 120
0
0
+
115
0
c
LL
0
LL I10
105
I10 -
t
\
-0680
-0660
E d c (volts vs S C
-0640
E
loo
100
200
3’00
400
)
Figure 2. Illustrations of effect of diffusion coefficients on ac ratio profiles ( A ) Same as Figure 1C: DCd(Hg) = 3.06 X IO@ cm2 sec-’ (Curve l), = 1.52 X cm2 sec-’ (Curve 2) as calculated for Mechanism I. (E)Effect of variation in amalgam species diffusion coefficient on ac ratio magnitude at two specific dc potentials as calculated for Mechanism 11, where standard potentials fulfill the El0 = 0.000 volt, and t1 = 5.31 sec., t2 = 9.92 sec., m(t1) = 1.45 mg sec-’, m(t2) = 0.764 mg sec-’, DO = Dy= 0.735 X relationship cm2 sec-l
-
oretical profiles for this observable. The theoretical profiles include the stationary sphere predictions for Mechanism I (dotted curves), which were given in the original publication (51, together with expanding sphere predictions based on Mechanisms I and 11. The two-step case (Mechanism 11) is considered only where this possibility is a t least remotely conceivable. In evaluating the implications of Figure 1, it should be kept in mind that the current ratio profiles represent a relatively high resolution data presentation format. For example, the scatter in experimental points obtained with different frequencies a t a given dc potential (which we take as a manifestation of random experimental uncertainty) corresponds to relative average deviations from the mean of less than 1%of the total alternating cell current in the majority of cases. With the exception of the thallium system, a common feature of the comparisons provided in Figure 1 is that the expanding sphere theoretical basis provides significantly better agreement with experimental results than the stationary sphere model. The latter observation applies regardless of whether one focuses on the conventional singlestep interpretation for the redox processes in question, or considers the two-step possibility (where relevant). The thallium system is an exception in the sense that the differences between the stationary and expanding sphere model’s predictions are relatively small in this case, differing by the same order of magnitude as the experimental dispersion, and yielding roughly comparable theory-experiment agreement. Although not shown in Figure 1, the expanding sphere model also yields improvement in the case of Pb(I1) in 1.OM KN03, reducing the theory-experiment disparities observed with the stationary sphere model ( 5 ) by approximately a factor of two. Despite the expanding sphere model’s success in affording moderate to substantial improvement in the interpretation of the observed ac polarographic time-dependence, some theory-experiment differences which remain appear to exceed random measurement uncertainty. Perhaps the most consistent example of this is the larger experimental time-dependence a t positive extremities of the profiles, relative to theoretical predictions. Previous publications which deal with higher-order mass transfer perturbations 532
*
ANALYTICAL C H E M I S T R Y , VOL. 47, N O . 3, M A R C H 1975
attending the DME (1-10) tempt one to assign these small residual disparities to factors such as streaming (2-4, 8), and shielding (9). However, while plausible, this viewpoint is not readily defensible because other sources of theoryexperiment disparity appear equally viable. For example, in the course of selecting diffusion coefficients from the literature for use in theoretical rate laws, a wide variety of sources was perused (17-20, 22-33), and some non-trivial disagreement in reported values was noted. Consequently, effects of diffusion coefficient fluctuations on predicted time-dependences were calculated. I t was concluded that current ratio curves, such as shown in Figure 1, are quite sensitive to diffusion coefficient values, so that the quality of theory-experiment agreement can be significantly influenced by one’s choice of diffusion coefficient. This is illustrated in Figure 2. Figure 2A shows current ratio profiles predicted for Mechanism I and conditions corresponding t o Figure 1C. The lower profile uses the original FurmanCooper Cd(Hg) diffusion coefficient value (expanding plane model) while the higher profile uses our recalculated value (expanding sphere model). The latter gives the best agreement with the data. Figure 2B gives an example of the effect on the current ratio of varying the amalgam diffusion coefficient for two specific dc potentials, as predicted by Mechanism 11. Clearly, a factor of two variation in diffusion coefficient can have a profound influence on the theoretical predictions. Indeed, by freely varying Do and D R within reasonable proximity (&50%)of the experimental values used in Figure 1, considerably better theory-experiment agreement than shown in Figure 1 could have been realized, although the significance of such an exercise is questionable. The above-mentioned literature survey indicated that consistency in reported diffusion coefficient values for solution-soluble species (DO’S),is reasonably satisfactory in the present context, and should not be a source of concern. However, reported amalgam diffusion coefficients (DR’s), including the recalculated values used in Figure 1, show disparities that are sufficiently large to make highly speculative any attempts to definitively interpret residual theory-experiment disagreement shown in Figure 1, since the latter fall within the uncertainty associated with Dn. In addition to possible systematic error in assignment of parameter values, mechanistic uncertainty further clouds in-
terpretation of some of the theory-experiment disparities in Figure 1. In particular, the question of whether the usually-accepted single-step model (Mechanism I) is appropriate for the Cd system can be raised. For example, the possibility that there is evidence in faradaic impedance data for a stepwise electron transfer with Cd(II)/Cd(Hg) has been discussed recently ( 3 4 ) . Although the cadmium system ac waves under consideration fulfill the phenomenological requirements for a single-step process in the context of wave shape, half-width, etc. ( 5 ) ,this may not apply to the higher resolution observable embodied in the current ratio data of Figure 1. For this reason, we have compared in Figure 1C and 1D the Cd(II)/Cd(Hg) current ratio data to the theory for reversible stepwise electron transfer using Ezo - El0 = 0.00, 0.10, and 0.50 volt. The zero volt standard potential separ;.tion is not a highly realistic assignment in view of other data, but is included to provide a better picture of the effect of the standard potential separation on the theoretical profiles in question. The 0.10- and 0.50-volt separations arc? consistent with most other data. The 0.50-volt case yields predictions which are identical to the singlestep model. No firm quantitative conclusions can be reached from this exercise, since the relevant single-step Eo-values are unknown. Nevertheless, it should be evident that, relative to the predictions of Mechanism I, some improvement in theory-experiment agreement can be obtained by treating the Cd system as a stepwise process with E20 - El" as small a:; 100 mV, without destroying theoryexperiment consistency regarding other observables. A more definitive judgment concerning the merits of using Mechanism I1 can be made with the Cu(I)/Cu(Hg) case. The ac wave from which the copper system data (Figure 1B) were obtained is the second wave in the polarographic reduction of Cu(l1) (5).Because the Cu(II)/Cu(I)and Cu(I)/ Cu(Hg) waves appear reasonably well-resolved, the singlestep theory should be applicable (35) and, indeed, it does give reasonable int,,erpretation of the data (solid curve, Figure 1 B ) . However, theoretical predictions based on Mechanism 11, using known standard potentials for the two redox steps ( 1 7 ) , actually provide some improvement in theoryexperiment agreement, relative to Mechanism I predictions (dashed curve, Figure 1B). Evidently, the two copper waves are not sufficientiy separated (EzO - El0 = 0.26 volt) to remove all differences between the single-step and two-step mechanism predictions, at least with regard to the observable in question. The general conclusion one reaches from considerations of the type formulated in the previous paragraph is that uncertainties in mechanistic details and parameter values preclude one's assigning with any degree of certainty the residual theory-experiment disparities in Figure 1 to any specific factor or factors, including perturbations in the mass transfer regime (2-4, 8, 9 ) which have been ignored in the theoretical derivation. The practical consequences of the latter observation are not serious. What is most important is that a self-consistent scheme now is available for theoretically interpreting with reasonable accuracy the ob-
served ac polarographic time-dependence with the kinds of facile amalgam-forming processes under consideration. That is, in terms of Mechanism I, or Mechanism I1 (when it is rigorously justified), we find that invoking rate laws based on the expanding sphere boundary value problem to calculate first the diffusion coefficients from dc polarographic data, and then the ac polarographic wave timedependence, leads to theoretical predictions which closely match the experimental observations. Small theory-experiment departures do remain which, from a pragmatic viewpoint, are inconsequential for most purposes. Finally, the satisfactory level of success of the expanding sphere theory can be taken as an indication that electrode sphericity is the major source of the ac polarographic time-dependence with the systems under consideration, a conclusion which was less compelling in the original investigation (5).
LITERATURE CITED T. Biegler and H. A. Laitinen, Anal. Chem., 37, 572 (1965). J. R. Delmastro and D. E. Smith, Anal. Chem., 38, 169 (1966). G. H. Avlward, J. W. Hayes, D. E. Smith, and H. L. Hung, Anal. Chem., 36, 2218 (1964). G. H. Aylward and J. W. Hayes, J. Electroanal. Chem., 8, 442 (1964). J. R. Delmastro and D. E. Smith, Anal. Chem., 39, 1050 (1967). T. G. McCord and D. E. Smith, Anal. Chem., 41, 131 (1969). T. G. McCord and D. E. Smith. Anal. Chem.. 42. 126 11970). H. Strehlow and M. von Stackelberg, 2. flekfrochem:, 54, 51 (1950). H. Matsuda, Bull. Chem. SOC.Jap., 26, 342 (1953). J. Kuta and I. Smoler, "Progress in Polarography," P. Zuman, Ed., with collaboration of I. M. Kolthoff, Vol. 1, Interscience. New York, N.Y., 1962, Chap. 3. I. Ruzic and S. W.feldberg, in preparation. J. W. Hayes, I. Ruzic, D. E. Smith, G. L. Booman, and J. R. Delmastro, J. ElecEoanal. Chem., 51, 245, 269 (1974). I. Ruzic, D. E. Smith, and S. W. Feldberg, J. Electroanal. Chem., 52, 157 (1974). I.Ruzic and D. E. Smith, J. flectroanal. Chem., 58, 145 (1975). S. W. Feldberg, in "Electroanalytical Chemistry," Vol. 3, A. J. Bard, ed., M. Dekker, New York, N.Y., 1969, pp 199-296. J. R. Delmastro, Doctoral Dissertation, Northwestern University, Evanston, 111. 1967. D. S. Polcyn and I. Shain, Anal. Chem., 38, 370 (1966). I. Shain and K. J. Martin, J. Phys. Chem., 65, 254 (1961). N. H. Furman and W. C. Cooper, J. Amer. Chem. SOC., 72, 5667 (19501. ---, W. C. Cooper and N. H. Furman, J. Amer. Chem. SOC., 74, 6183 (1952). J. Koutecky, Czech. J. Phys., 2, 50 (1953). M. von Stackelberg, M. Pilgram, and V. Toome. 2. Eiektrochem., 57, 342 (1953). C. L. Rulfs, J. Amer. Chem. Soc., 76, 2071 (1954). D. J. Macero and C. L. Rulfs, J. Amer. Chem. SOC., 81, 2942, 2944 (1959). H. Strehlow, 0. Madrich, M. von Stackelberg, 2. Elekfrochem., 55, 244 (1951). I. M. Kolthoff and J . J. Lingane, "Polarography, Vol. 1, 2nd ed.. Interscience, New York, N.Y., 1952, pp 409-41 1. J. H. Wang and F. M. Palestra, J. Amer. Chem. Soc., 76, 1584 (1954). C. N. Reilley, G. W. Everett, and R. H. Johns, Anal. Chem., 27, 483 (1955). G. Meyer. Wed. Ann., 61, 225 (1897); 64, 742 (1898). M. Wogau, Ann. Physik., 23, 345 (1907). E. Cohen and H. R. Bruins, 2. Phys. Chem., 109, 397 (1924). F. Warschedel, 2. Physik., 84, 29 (1933). M. von Stackelberg, and V. Toome, 2. Eiektrochem., 58, 226 (1954). A. R. Despic, D. R. Jovanovic, and S. P. Bingulac, Nectrochim. Acta, 15, 459 (1970). H. L. Hung and D. E. Smith, J. Nectroanal. Chem., 11, 237, 425 (1966). \
RECEIVEDfor review August 26, 1974. Accepted December 9, 1974. This work was supported by NSF Grant GP28748X.
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533
