1719
Anal. Chem. 1983, 55, 1719-1723
Applications of a Computerized Method for Mechanistic Classif icat i10 n of 0na- EIect ron Potentiost at ic Current-Po t ent iaI Curves James F. Rusling Department of Chemktry, U-60, University of Connecticut, Storrs, Connecticut 06268
An automated, computierlred method for mechanlstlc classlflcatlon based on devlation-pattern recognition has been appled to one-electron ireductions and oxldatlons In dc polarography, normal-pulse polarography, normal-pulse vollammetry, and rotatlng-dlslr voltammetry. Thirty-six sets of data representlng a verlety of worklng electrodes and experimental conditions were classified In accord wlth prevlously proposed mechanisms. Elght pollarograms representlng two reductlon systems were found tal have mechanlstlc nuances not previously recognlzed, although publlshed mechanlsms were approxlmately conflrmed. The automated method rapldly classlfled data for simplle mechanisms wllthout lnterventlon of the chemist, allowlng addltonal studies to be focused on the more complex systems. The combinailon of nonlinear regresslon analysls and tlevlatlon-pattern irecognition provldes a degree of Insight Into the lnterpretatlon of potentlostatlc / - E curves not prevlously aivallable wlth conventlonal data analysts.
We have recently developed an automated, computerized method for mechanistic analysis of one-ebctron potentiostatic current-potential (i-E) curves based on nonlinear regression and deviation-pattern recognition ( I ) . Guided by a hierarchical tree, the procedure reaches the final classification by a series of binary decisions. Confirmatioii of class is provided by nonlinear regression analysis. This method gave 100% correct classifications of computer-simulated data for mechanisms leading to a single wave when random noise was less than 0.85% of the limiting current. In this paper, applications of the classification method to dc polarography, normal-pulse polarography (NPP), normal-pulse voltammetry (NPV), and rotating-disk-electrode voltammetry (R,DV) are described. Data were obtained with a variety of working electrodes, in aqueous and acetonitrile solutions, and include both reductions and oxidations.
EXPERIMENTAL SECTION Chemicals and Solutions. Acetophenone was Fisher certified grade; 3-formylpyridine was obtained from Aldrich Chemical Co. Both compounds were distilled under reduced pressure in an atmosphere of nitrogen before use. Ferrocene was a gift from S. L. Suib, University of Connecticut, and was purified by vacuum sublimation. Hemin (bovine, crystalline),homarine hydrochloride (97%), and 1,2,3,4-tetrahydrocarbazole (99%) were obtained from Aldrich Chemical Co. and used as received. Trigonelline hydrochloride was from Sigma Chemical Co. and also used as rewas a gift from J. ceived. 9-Methyl-1,2,3,4-tetrahydrocarbazole M. Bobbitt, University of Connecticut. Stock solutions (5-20 mM) were prepared in water or acetonitrile. To avoid decomposition, hemin solutions were prepared immediately before the experiment. All other stock solutionE5 were stable for more than 2 weeks at 5 "C. Apparatus and Procedures. Dc polarography, NPP, NPV, and RDV were carried out with a Princeton Applied Research Corp. (PARC) Model 174A polarographic analyzer attached to
a Houston Omnigraphic 100 X-Y recorder. A few RDV scans were performed with a PARC Model 170 electrochemistry system. A Sargent synchronous rotator (Type KYC-22,1800 rpm) was used for RDV. Cell resistances were estimated with a Yellow Springs Instruments Model 31 conductivity bridge. The dropping mercury electrode (DME) for dc polarography and NPP had a flow rate of 1.88 mg/s and a natural drop time ( t )in 1 M KC1 of 4.00 s at a mercury-column height of 70 cm. Unless otherwise noted, polarography was carried out with a controlled drop time of 0.5 s. A PARC Model 9323 hangingdrop-mercury electrode (HDME), a Pt disk electrode (A = 0.004 cm2),and a carbon paste electrode (A = 0.07 cm2)were used in normal-pulse experiments. The construction of the Pt and carbon-paste electrodes has been previously described ( 2 ) . The working electrode for RDV was constructed from a rod of glassy carbon (Normar Industries GC, in diameter, A = 0.08 cm2) tightly fitted into a cylindrical Teflon collar of thickness 0.3 cm. Preparation of the surfaces of Pt and carbon-paste electrodes prior to each scan has been described previously ( 2 ) . After construction of the glassy carbon electrode, its surface was prepared by polishing on a metallographic polishing wheel, successively, with 400 and 600 grit silicon carbide paper, Metadi 0.25-pm diamond polishing compound on Buehler microcloth, and ultrasonication in spectrograde methanol for 2 min. The final pretreatment, carried out prior to each scan, consisted of polishing with a slurry of 0.3 pm alumina on felt cloth on the wheel for 2 min, followed by ultrasonication for 2 min in water (3). Glassy carbon electrodes polished in this way were allowed to stand in air for 0.5 to 2.0 h before use and were held at a potential of 0.0 V vs. SCE for 2 min before the start of the scan. All electrochemical experiments were carried out in threeelectrode cells of the Metrohm type employing Pt-wire counterelectrodes. With aqueous solutions, a PARC Model K0065 SCE was the reference electrode. In acetonitrile, a Ag/Ag+ (0.001 M) electrode served as the reference. Salt bridges filled with supporting electrolyte and terminating in a Vycor frit (aqueous solutions) or a medium porosity glass frit (aqueous and acetonitrile solutions) were used to connect reference electrodes to analyte solutions. Assembled cells had resistances between 400 and 900 Q. Reproducibility of potential and current measurements with the above types of cells was reported previously (2,4). Prior to the initiation of the potential scan, all solutions were purged with purified nitrogen ( 2 , 4 )for 5-15 min. All experiments were carried out at the ambient temperature of the laboratory (25 f 2 O C ) . Computations. Calculations were carried out with a Radio Shack TRS-80 Model I microcomputer (48K) in the Level I1 BASIC language. The program for mechanistic classification,DPRF, and its operation with simulated reduction waves has been described previously ( 1 ) . In the present paper, DPRS is also used for classification of oxidation waves. The major modification required involves changing 0 = exp[(E - E,/,)/S] (cf. Table I, ref 1)to 0 = [exp(E,,, - E)/S]. Currents of the oxidation waves are treated as positive quantities. Potentials are calculated from an initial value, Eo,and the potential increment E, from
E = Eo + Ei(N - 1) where N is the data point number. With Ei retaining the correct sign, eq 1is used for both oxidations and reductions. Mechanisms considered and parameters used in DPRS are given in Table I. Because of the time required, five-parameterregression analyses were run with a separate program (5) at times of low computer
0003-2700/83/0355-1719$01.50/00 1983 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 11, SEPTEMBER 1983
Table I. Mechanistic Classes and Regression Parameters Used in Classifications parameters, units class no. 1
2 3 4 5 6 7 8
mechanismsa EC2 DIFF -R (CE,EC) DIFF-I EC2A D-M QR(3 param.) (2 param.) w2
V(11 E,,,, V b vs. SCE E,,,, V b vs. SCE E,,,, V b vs. SCE E,,,, V b vs. SCE EL,,, V b vs. SCE E , V vs. SCE E " , V b vs. SCE
V(2) RT/F, V RT/F, V RT/F, V RT/F, V RT/2F, V RT/aF, V rn/k"sH (RT/F),, V V(5)=E,,,,,, V
f"
V(4) = (RT/F),, V
NO MECH
See ref 1 for details of classification procedure. limiting current carried by the first wave. a
V(3) i, ,PA i, ,PA i,, r A i, , PA i, 3 PA m/k 'SH Ell, ,I
V vs. Ag/Ag+ (0.001 M) in acetonitrile solutions.
9
v
Fraction of total
Table 11. Classification of Dc Polarographic Reductions compound
c,
mM
ace tophenone
1.0 2.0 3-formylpyridine 0.4 1.4 2.4 trigonelline 0.4
electrolyte 0.1 M NaOH
0.2 M NaOH, + G b pH 12.3, phos. buffer
0.8
1.2 homarine
0.4
pH 9.3 BR, " 5X % Triton X-100 pH 12.3, phos. buffer
0.8
uo
,2+
hemin e
1.2 0.20 0.24 0.56 0.74 0.20 0.48 0.91 1.3 1.3 0.16 0.20 0.40 0.60 0.60 0.66 0.66 0.66 1.00 1.00
0.02 M HClO,, 3 M NaClO, 0.1 M HCI
0.1 M HC1, G 0.1 M HC1
+Gb no G b 0.1 M NaOH
final parameters SEa S D x V(1) V(2) V(3) used? l o 3
class found
reported mech (ref)
EC 2 EC 2 EC2 EC 2 EC2 EC2 EC2 EC2
-1.600 -1.602 -1.276 -1.270 -1.268 -1.388 -1.384 -1.352
0.0242 0.0248 0.0251 0.0260 0.0261 0.0270 0.0262 0.0267
1.23 N 2.66 N 0.73 N 2.44 N 3.92 N 0.45 N 0.92 N 3.96d N
4.5 5.6 1.8 7.6 9.0 1.7 3.0 15.7
EC2 (6)
EC 2 EC 2 EC2 DIFF-R DIFF-R NO MECH NO MECH DIFF-R w2f w2f w2f w2f DIFF-R DIFF-R DIFF-R DIFF-R NO MECH D-M D-M D-M NO MECH NO MECH
-1.326 -1.323 -1.322 -0.162 -0.160
0.0242 0.0232 0.0232 0.0260 0.0246
0.47 N 0.90 N 1.32 N 0.265 N 0.370 N
1.4 3.6 6.2 3.3 2.8
EC2 (4)
-0.152
0.0236
0.536
N
4.9
DIFF-R (9,lO)
-0.483 -0.456 -0.458 -0.457
0.0248 0.0244 0.0244 0.0259
0.518 0.661 1.351 1.882
N N N N
1.2 2.1 4.8 4.3
DIFF-R (11)
-0.730 -0.730 -0.730
0.0139 0.0141 0.0138
0.679 0.699 0.686
N Y Y
4.3 3.7 3.7
EC2 (7) EC2 (4)
DIFF-R (8)
filmg (12, 1 3 ) D-M (14,15) aggreg.g
Standard error used to obtain final deviation plot when SD < SE (0.95). Gelatin added to the solution (0.004%). t = 2 s. e Current-sampled or Tast mode used. f See Table 111. g See text for detailed explanation. a
" Britton-Robinson buffer.
usage. Deviation plots for these curves were analyzed by the program D P R ~which , contained the routines for deviation-pattern recognition from DPRS. In all nonlinear regression analyses, the absolute errors in the currents were assumed to be randomly distributed. Currents from experimental curves were measured from a linear extension of the residual current line and digitized at potential increments (Ei) of 5-10 mV. Eo and Ei were adjusted so that between 24 and 33 data points were obtained between about 5 and 95% of the limiting current. When the i-E curve was sufficiently well defined to yield accurate values of the base line and plateau slopes, their difference, b (cf. Table I, ref l),in wA/V was used as a correction, where necessary, for the slight potential dependence of the limiting current. Estimates of limiting current (il)at the half-wave potential and the standard error (SE) of the
current measurements were also made for use in
DPRS.
RESULTS AND DISCUSSION Dc Polarography. For 28 dc polarograms representing EC2, DIFF-R, and D-M reactions, classifications were in accord with the published mechanisms (Table 11). Reductions of acetophenone, 3-formylpyridine, trigonelline, and homarine under the conditions listed in Table I1 are all correctly classified as EC2 reactions. Surfactants were used in some solutions to suppress weak polarographic maxima which, although not visibly apparent, were reflected by a type A deviation plot (cf. ref 1) obtained from the initial fit to the EC2 equation. This resulted in an eventual NO MECH or W2 classification of the polarograms obtained without sur-
ANALYTICAL CHEMISTRY, VOL. 55, NO. 11, SEPTEMBER 1983
1721
Table 111. Five-Parameter Regression Analyses of Data for Polarographic Reduction of Uranyl Ion C, mM
electrolyte
v(i)
-v(2)
v(3)
-v(4)
v(5)
0.56 0.74 0.74 0.48 0.91 1.3 1.3
0.02NI NaClO,, 3 M NaC10,
0.601 0.153 0.153 0.534 0.401 0.281 0.325
0.146 0.129 0.129 0.169 0.163 0.157 0.160
0.0212 0.0125 0.0124 0.0212 0.0202 0.0204 0.0212
0.287 0.154 0.154 0.198 0.196 0.192 0.192
0.0632 0.0272 0.0272 0.0231 0.0228 0.0222 0.0228
0.1M HCl 0.1 M HCl t G
a Characteristics of deviation plot found by DPRI. R = random;NR = nonrandom. parameters.
factant, as predicted from studies with simulated data ( I ) . Thus, in any case where a type A deviation plot resulted after the initial fit, the following message was printed ***** DATA MAY FALL INTO THIS CLASS IF DISTORTED E Y A WEAK POLAROGRAPHIC MAXIMUM. TO GUARD AGAINST THIS, POLAROGRAMS SHOULD BE RUN IN THE PRESENCE AND ABSENCE 01F MAXIMUM SUPPRESSORS. Electroreduction of uranyl ion represents a case in which the mechanism changer3 with concentration of the electroactive species. In media of low acidity (pH 11)and at low concentrations (C) of uranyl ion, a reversible, diffusion-controlled reduction of U(V1) to U(V) has been reported (8-10). Accordingly, at C < 0.25 mM and in supporting electrolytes containing 0.02 M HCIOl or 0.1 M HC1I)IFF-R classifications were obtained (Table 11). However, at C > 0.4 mM NO MECH (perchlorate media) or W2 (HCI) classifications were obtained. On the otheir hand, plots of limiting current against concentration (C < 1.3 mM) were strictly linear, giving correlation coefficients of 0.9999 in both media. By conventional standards, such linearity would be taken as evidence that the electrode reaction is controlled by diffusion. Disproportionation of U(V) at the electrode to U(V1) and U(IV), reported at higher acidities and uranyl concentrations @-IO), would yield a nonlinear dependence of limiting current on concentration. The results could possibly be explained in terms of adsorption phenomena, since the W2 model can be used as an empirical model fair a polarographic reduction involving weak adsorption (16). Although the W2 equation gave good fits for the uranyl reductions in 0.1 M HC1, all polarograms obtained in perchlorate media gave poor agreement with this model (Table 111). Results in the HC1 solutions could be rationalized by postulating an adsorption prewave about 30 mV positive of a main diffusion-controlledwave. Interpreting the W2 model in this way, limiting currents for the first wave would be 0.372,0.547,and 0.562 pA at 0.48,0.91, and 1.3 mM uranyl, respectively, consistent with a limiting concentration of about 0.9 mM for foirmation of an adsorbed layer of product on the electrode surface (17).However, no evidence of adsorption of U(V1) or lJ(V) was found i n perchlorate or HC1 media by either cyclic voltammetry or double-potential-step chronocoulometry (13). Addition of gellatin to the HC1 solutions caused deviatioin plots derived from the W2 fit to be nonrandom (Table 111),arguing against distortion of the uranyl waves by a weak maximum. The most reasonable explanation for the observed results at higher uranyl concentrations is the increasing influence of disproportionation of U(V) as the uranyl concentration is increased, resulting in mixed control of the reaction by diffusion and disproportionation. Under the conditions used, however, the rate of disproportionation is small (8-10) as confirmed by its negligible effect on the limiting currents. Tlhis example illustrates the danger in placing too much confidence in results of regression analyses obtained with empiricd models. Thus, the W2 equation gave good fits to the uranyl/HCl polarograms, but the parameters
D P ~SD x 103 NR NR NR R R R NR
2.41 3.33 3.33 1.75 3.39 7.86 5.16
Duplicate fit with different initial
have no real physical meaning. Additional experimental results were needed to show that the uranyl polarograms did not correspond to overlapped adsorption-diffusion waves. Thallium(1) in 0.1 M HC1 was classified DIFF-R at C < 0.5 mM and also at C > 0.5 mM when the supporting electrolyte contained gelatin. A t C > 0.5 mM in 0.1 M HC1 without gelatin, NO MECH classifications were obtained. At C > 2 mM in 0.1 mM HC1 and at C > 0.2 mM in 1.0 M HC1, polarographic maxima were observed. Such maxima in chloride media had previously been reported and were attributed to adsorption of Tl(1) (18). The adsorbed species has recently been identified as a film of TlCl which forms on the mercury electrode when the Kspof TlCl at the electrode surface is exceeded by the ion product (12). The film causes deviations from diffusion-controlled behavior at higher concentrations of Tl(1). Gelatin apparently inhibits film formation so that the reaction again becomes diffusion controlled (Table 11). Current-sampled polarograms of 0.66 mM hemin in 0.1 M NaOH were classified D-M; two of these required the standard error modification of the deviation plot (1)for classification. At 1.0 mM hemin, NO MECH classifications were obtained. The equation used for the regression analyses (cf. Table I, ref l),derived by Jordan and Bednarski (14,15),assumed equal adsorption of oxidized and reduced forms. When a more rigorously derived equation (19),which does not assume equal adsorption, was used to fit the data, nearly identical deviation plots and standard deviations were obtained. Cyclic voltammetric investigations of the reduction of hemin in 0.1 M NaOH at a HDME (13) revealed at least one cathodic and two anodic adsorption peaks of unequal heights. The complexity of the CV data can be related to unequal adsorption of hemin and its reduced form as well as aggregation of these species (14, 15,20,21). A detailed discussion is beyond the scope of this paper and will be presented separately (13). In polarography, however, because of the renewable nature of the DME, the D-M model (1)is followed more closely than in CV, especially at lower concentrations where aggregation is not as severe. Nevertheless, a small prewave is observed in polarograms of 1.0 mM hemin which is not as evident at 0.66 mM. We conclude that adsorption and aggregation of reactants and products invalidate the D-M model at the higher concentration. As an additional check on the validity of the classifications, calculated values of V(2) were compared to theoretical values. For the 18 polarograms classified EC2 or DIFF-R (Table 11), the average value of V(2) (i.e., RT/F) was 0.0250 f 0.0012 V. A t test showed no significant difference at the 98% confidence level from the theoretical value at 25 "C of 0.02569 V. For 0.66 mM hemin, the average V(2) was 0.0139 V, slightly larger than the theoretical value of 0.012 85 V at 25 "C. NPP, NPV, and RDV. Eleven sets of normal-pulse data obtained with platinum, mercury, and carbon-paste electrodes were classified in agreement with published mechanisms (Table IV). Variation of the time between pulses for pulse voltammetric oxidation of ferrocene had little effect on either
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 11, SEPTEMBER 1983
Table IV. Classification of NPP, NPV, and RDV Reductions and Oxidations compound
C, mM
rxn
electrolyte
class found
final parameters SE V(1) V(2) V(3) used?
SDX
lo3
reported mech (ref)
Normal Pulse acetophenonea TI(I)& THC 9-MeTHC ferrocene
2.0 0.20 0.40 1.0 0.95 3.0 0.20f 0.20h 0.49 0.49 O.4gg
RED RED
0.1MNaOH 0.1 M HCl, + G
Ox Ox Ox
0.2M LiCIO,e 0.2M LiCIO,e 0.2 M LiCIO,e
EC2 DIFF-R DIFF-R DIFF-R EC2 EC2 DIFF-R DIFF-R DIFF-R DIFF-R DIFF-R
-1.616 -0.492 -0.490 -0.498 0.586 0.511 0.131 0.136 0.132 0.128 0.136
0.0234 0.0258 0.0258 0.0275 0.0249 0.0250 0.0256 0.0264 0.0257 0.0254 0.0264
14.94 4.71 9.24 25.0 70.5 5.68 22.8 23.7 55.8 56.7 59.7
N N N N N N N N N N N
128 11 27 120 323 21.5 43.1 37.3 64.7 164 93.5
EC2(6) DIFF-R EC2 ( 2 ) EC2 ( 2 ) DIFF-R
Rotating Disk Voltammetry Fe(CN )," -
0.49: Ox 0.5M KNO, QR 0.208 0.0712 1.73 Y 43.3 QR (23, 24) 0.208 0.0711 1.83 N 61.8 0.49' QR 0.227 0.0690 1.52 Y 45.3 0.49' QR Working electrode was HDME. a Working electrode was DME. 1,2,3,4-Tetrahydrocarbazole, carbon-paste working electrode, t = 2 s. 9-Methyl-1,2,3,4-tetrahydrocarbazole, Pt disk working electrode. e Solvent was acetonitri1e:water (cJ:I). f t d = 0.5 s, carbon-paste electrode. g t d = 2.0 s, carbon-paste electrode. h t d = 5.0 s, carbon-paste electrode. Glassy carbon electrode. the classification or the final values of parameters V(1) and V(2). The average value of V(2) for these data was 0.02563 f 0.00103 V, which a t test showed did not differ significantly a t the 95% confidence level from the theoretical value. In general, i-E curves obtained with Pt and carbon electrodes had residual current lines and plateaus with larger slopes than those for mercury electrodes. If residual current lines and plateaus could not be obtained over a large enough potential range (about 150 mV), uncertainty in the difference, b (cf. Table I, ref l),of their slopes tended to be large. In such cases the value of b was set equal to zero, which it was for all the oxidations in Table IV. For oxidations of the tetrahydrocarbazoles, attempts were made to differentiate between different types of EC2 mechanisms. Previous studies showed (2)that THC undergoes a radical-parent dimerization following electron transfer, whereas 9-MeTHC dimerizes by either a radical-radical or a radical-parent pathway. When NPV data for these two compounds were regressed onto the appropriate equations (25) nearly identical deviation plots and standard deviations were obtained as when the data were regressed onto the usual EC2 equation ( I ) , corresponding to radical-radical dimerization. In this case, the S / N of the i-E curves is too large to allow differentiation between such closely related mechanisms. Since the automatic classification program is not equipped a t this time to handle the more complex expressions which describe irreversible and quasi-reversible normal-pulse voltammograms, with possible depletion effects (26,27),RDV was used to investigate a quasi-reversible oxidation. Rotating-disk electrode voltammograms of ferrocyanide ion were classified QR, as expected (Table IV). The final parameters for three voltammograms show reasonable agreement. The average value of the standard formal potential in 0.5 M K N 0 3 was 0.458 f 0.011 V vs. NHE, in the right range for the ferrocyanide redox couple (cf. Eo = 0.36 V (28)and E"' (1.0 M KC1) = 0.474 V (23) vs. NHE). Average values for V(2) (RT/aF) and V(3) ( m R / k o g hwere ) 0.0704 f 0.0012 and 1.69 f 0.16, respectively. The larger standard deviation in V(3) was in part predicted by studies with simulated polarograms ( I ) . In conclusion, this study demonstrates the capabilities of the automated deviation-pattern recognition procedure for mechanistic classification of potentiostatic i-E curves obtained
under a wide variety of experimental conditions. The results confirm those previously obtained with computer-simulated polarograms ( I ) . The method rapidly classified data for simple mechanisms without intervention of the chemist, allowing additional investigations to be focused on the systems with more complex behavior. Thirty-six sets of data were classified into one of six (Table I) simple classes corresponding to the published mechanisms. Eight polarograms from two reduction reactions were classified NO MECH or W2 and required further investigation. These few differences from earlier published results are not attributed to erroneous prior interpretations but rather to the additional insight into the data provided by the powerful combination of nonlinear regression and deviation-pattern recognition in conjunction with modern instrumentation. ACKNOWLEDGMENT The author thanks Margaret Brooks for polarograms of hemin, Anne Ho for pulse voltammograms of Tl(I), Barry Scheer for pulse voltammograms of tetrahydrocarbazoles, John Gammerino for construction of carbon-paste and glassy carbon electrodes, S. L. Suib for the gift of purified ferrocene, and J. M. Bobbitt for the gift of 9-methyl-1,2,3,4-tetrahydrocarbazole. Registry No. U022+,16637-16-4;T1,7440-28-0; THC, 942-01-8; g-MeTHC, 6303-88-4; Fe(CN):-, 13408-63-4;acetophenone, 9886-2; 3-formylpyridine,500-22-1;trigonelline, 535-83-1;homarine, 445-30-7; hemin, 16009-13-5;ferrocene, 102-54-5. LITERATURE CITED (1) Rusling, J. F. Anal. Chem., preceding paper in this issue. (2) Kulkarni, C. L.; Scheer, B. J.; Rusling, J. F. J . flectroanal. Chem. 1982, 140, 57-74. (3) Evans, J. F.; Kuwana, T. Anal. Chem. 1979, 57, 358-365. (4) Rusling, J. F. J . Nectroanal. Chem. 1981, 725, 447-458. (5) Meites, L. "The General Non-linear Regression Program CFT4A"; privately published, 1983. (6) Evans, D. H. In "Encyclopedia of the Electrochemlstry of the Elements": Bard, A. J., Lund, H., Eds.; Marcel Dekker: New York, 1978; Vol. XII, pp 1-259. (7) Rusling, J. F. Ph.D. Thesis, Ciarkson College of Technology, Postdam, NY. . . , 1979. . .. .
(8) Mastragostino, M.; Saveant, J. M. f/ectrochim. Acta 1968, 73, 751-762. ( 9 ) Nelson, F.; Kraus, K. A. J . Am. Chem. SOC. 1951, 73,2157-2161. (IO) Orlemann, E. F.; Kern, D. M. H. J . Am. Chem. SOC. 1953, 75, 3058-3063.
Anal. Chem. 1983, 55,1723-1728 (11) Meites, L. “Polarographlc Techniques”, 2nd ed.; Interscience: New York, 1965. (12) Elliot, M.; Murray, R. W. Anal. Chem. 1978, 48, 259-267. (13) Brooks, M. Y.; Rusling, J. F., Universlty off Connecticut, unpublished results, 1983. (14) Jordan, J.; Bednarskl, T. M. J . Am. Chem. SOC. 1084, 86, 5690-5691. .... ... (15) Bednarskl, T. M.; Jordan, J. J . Am. Chem. SOC. 1967, 89, 1552-1558. (16) Meites, L.; Lampugneni, L. Anal. Chem. 11373, 45, 1317-1323. (17) Heyrovsky, J.: Kuta, ,I. “Princigies of Polaroaraghy”; Academic Press: New York, 1966. (18) Barker, G. C.; Faircioth, R. L. I n “Advances in Polarography”; Longmuir, I., Ed.; Pergammon Press: New York, 1960; Vol. I,pp 313-329. . . .-. (19) Gelb, R. I.J . Nectroanal. Chem. 1988, f9, 215-218. (20) White, W, I. I n “The1 Porphyrlns”; Dolphln, D., Ed., Academic Press: New York, 1979; Vol. V, pp 303-339. (21) Davis, D. G. I n “The Porphyrlns”; Dolphln, D., Ed.; Academic Press:
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New York, 1979; Vol. V, pp 127-152. (22) Kuwana, T.; Bublitz, D. E.; Hoh, G. J . Am. Chem. SOC. 1980, 82, 581 1-5817. (23) Jordan, J. Anal. Chem. 1855, 27, 1708-1711. (24) Engstrom, R. C. Anal. Chem. 1982, 54, 2310-2314. (25) Nadjo, L.; Saveant, J. M. J. Necfroanal. Chem. 1973, 44, 327-366. (26) Oldham, K. B.; Parry, E. P. Anal. Chem. 1988, 40, 85-69. (27) Bond, A. M. “Modern Polarographic Methods in Analytical Chemistry”; Marcel Dekker: New York, 1980; pp 236-287. (28) Lingane, J. J. “Electroanalytical Chemistry”, 2nd. ed.; Interscience: New York, 1958; p 644.
RECEIVED for review March 29,1983. Accepted May 26,1983. This work was supported by the University of Connecticut Research Foundation and partially by Public Health Service Grant No. 1-RO1 CA-33195-01.
Bayesian Sltatistical Methods for Use in Mass Spectral Assignrnent Lothar M. Karrer, Heather L. Gordon, Stuart M. Rothstein,* Jack M. Miller, and Timothy R. B. Jones Department of Chemistry, Brock University, St. Catharines, Ontario, Canada L2S 3A1
We apply Bayeslan statistlcal methods to estimate the mole fractlons of species present In the linear model which describes the observed mass spectral data. We contrast our approach to the usual1 practlce In mass spectrometry whlch employs least squares, whlch we argue Is not statlstlcally sound. We describe our computer program, with emphasls on explotlng subroutiries from commonly avallable program ilbrarles. Appiicatlonrr involve the deconvolutlon of overiapping spectra resultlng from both successlve loss of hydrogen and gain of hydrogen via ion/molecuk reactlons.
expectation values imposes an additional condition on the coefficients in the model, eq 1 P
cis,= 100
p=l
S
W P=) s=l cxsisp
(1)
where ,i is the peak intensity due to substance s, xs is the unknown mole fraction of substance s, and there are S species present. It has been suggeeited that least-squares procedures be employed for a quantitative analysis of mass spectral data (1-4). However, because the sum of the P observations is normalized to, say, 100% P
El, p=l
= 100
(2)
the random errors tp =
I p - E(I,)
(3)
are not statistically independent and probably do not have equal variances and covariances. To ignore these consideraill lead to invalid estimates of the mole fractions (5-7). tions w Futhermore, the requirement that the random errors have zero
(4)
which may also be overlooked. For the purposes of discriminating among various plausible models, where some mole fractions are hypothesized to be zero, usually mass spectroscopists (e.g., ref 8) have confined themselves to the model which gives the lowest R factor, where
R The objective of this paper is to report appropriate statistical methodology for estimating the mole fractions which appear in the linear model which is used to describe mass spectral data. This model represents the expected value of the p t h peak intensity as
s = 1, ..., s
-
P p=l
IIpCalCd - IPI
(5)
where IpcalCd are predicted intensities, using least-squares estimates of the mole fractions. A sum of relative errors or of absolute errors may be used. This approach is undesirable as one can always lower the R factor by introducing additional variable parameters in the model. Furthermore, the statistical method to determine whether or not an additional parameter gives an R factor which differs significantly from that of a model with fewer parameters, Hamilton’s test (9, IO), is invalid because of the lack of independence in the random errors. It was pointed out to us by the referees of an earlier draft of this manuscript that mass spectral data consist of several responses (peak intensities), measured in each run of the spectrum, and normally several runs are taken. This suggests the approach of Box and Draper (7) to estimate the mole fractions, taking care to account for the various linear dependencies in the data (e.g., eq 2). Central to this approach is Bayes’ theorem, where under the assumptions detailed in the Theory section below, the joint (posterior) density function of the mole fractions is obtained. The mole fractions which maximize the density, are estimates of the mole fractions of species present in the spectra. Furthermore, a joint confidence region for the parameters in the model is obtained. If this region includes areas where one or more x are zero, then alternative models which ignore those species are also consistent with the data.
0003-5!700/83/0355-1723$0’1.50/0 0 1983 American Chemical Soclety
