Anal. Chem. 1984, 56, 2147-2153
Registry No. Tinuvin-770, 52829-07-9. LITERATURE CITED (1) Barber, M.; Bordeli, R. S.;Sedgwlck, R. D.; Tyler, A. N. J. Chem. Soc., Chem. Commun. W81, 7 , 325-327. (2) Barber, M.; Bordeli, R. S.;Elliott, G. J.; Sedgwlck, R. D.; Tyler, A. N. Anal. Chem. 1082, 5 4 , 645A-657A. (3) Bennlnghoven, A. Int. J. Mass Spectrom. Ion Phys. 1983, 4 6 , 459-462. (4) Magee, C. W. Int. J. Mass Spectrom. Ion phys. 1983, 4 9 , 21 1-221. (5) Gaskell, S.J.; Brownsey, B. G.; Brooks, P. W.; Green, B. N. Int. J. Mass Spectrom. Ion Phys. 1083, 4 6 , 435-438. (6) Milllngton, D. S.,presented at 31st Annual Conference on Mass Spectrometry and Allied Topics, Boston, MA, 518-13183. (7) Murphy, R. C.; Clay, K. L.; Stene, D. O., presented at 31st Annual Conference on Mass Spectrometry and Allied Topics, Boston, MA, 5/ 8-13/83.
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(8) DeStefano, A. J.; Keough, T., presented at 31st Annual Conference on Mass Spectrometry and Allied Topics, Boston, MA, 518-13183. (9) Ho,9. C.; Fensehu, C.; Hansen, G.; Larson, J.; Daniel, A. Clln. Chem. (Wlnston-Salem, N.C.) 1083, 2 9 , 1349-1353. (IO) Teeter, R. M., presented at 31st Annual Conference on Mass Spectrometry and Allied Topics, Boston, MA, 518-13183. (1 1) Shlyapintokh, V. Y.; Ivanov, V. B. I n "Developments in Polymer Stabilisation-5"; Scott, G., Ed.; Applied Science Publishers: London, 1982; Chapter 3. (12) Bauer, D. R.; Budde, G. F. Ind. Eng. Chem. Prod. Res. Dev. 1981. 20, 674-679. (13) Chattha, M. S.;Cassatta, J. C. J. Coat. Techno/. 1083, 55, 39-46. (14) McLafferty, F. W. "Interpretation of Mass Spectra", 2nd ed.; BenjaminlCummIngs: Reading, MA, 1973; p 42.
RECEIVEDfor review December 8,1983. Accepted May 7,1984.
Analysis of Chronocoulometric Data and Determination of Surface Concentrations James F. Rusling* and Margaret Y. Brooks University of Connecticut, Storrs, Connecticut 06268 Department of Chemistry (U-60),
A general method is described for analysis of data from double-potential-step chronocoulometry by simultaneous nonlinear regression onto exact equations for forward and reverse branches of O-f curves. Parameters determined include diffusion coefficients, surface concentrations, and double-layer charge. Applications to analysis of diff usloncontrolled reductions of Ti(1) and U(V1) and to determination of the surface concentration (I?,) of Cd(I1) adsorbed on mercury from thiocyanate soiutlons are described. With step wldths (7) 10.1 s and data equally spaced on the f axis, improvements in accuracy and precision in determining were realized over those obtained from a conventional iinearplot analysis. Data were most significant in the time range of 25% of T following the initial potential step, and, for determination of for 25% of 7 on either side of f = T . Clustering of data for nonlinear regresslons in these regions when T was 20.1 s. A provided accurate computation of diagnostic test based on deviation-pattern recognltlon was successful in detecting reactant adsorption.
ro
r,,
ro
Since its inception in the 1960s (I-3), double-potential-step chronocoulometryhas made possible significant contributions to the understanding of adsorption of electroactive molecules and ions ( 3 , 4 ) . A particular advantage of the method is that the charge for electrolysis of adsorbed species can be separated from the electrode double-layer charge, enabling the use of chronocoulometry for determining surface concentrations. The method has also found considerable use in elucidating mechanisms of electrode reactions and in determining rate constants of chemical reactions coupled to electron-transfer steps ( 3 , 5 ) . In a typical double-potential-step chronocoulometric experiment, the potential a t the working electrode is held at an initial value (EJwhere no electrolysis occurs, and, at t = 0, is rapidly stepped to a value (Ef)at which the desired electrode reaction takes place at a diffusion-limited rate. At t = T , with T ranging from milliseconds to several seconds, the potential is returned to Ei. The quantity of electricity, or 0003-2700/84/0356-2147$01.50/0
charge (Q), which has passed through the electrochemical cell, is measured vs. t during the period 0 to 27 (Figure 1). The usual mode of analyzing Q vs. t data for electrochemical reactions uncomplicated by chemical steps assumes that plots of Q(t < T ) vs. t1izand Q, = Q(7) - Q(t > T ) vs. 6, where 6 = d 2 (t - 7)112 - t1Iz,are linear and that charging of the electrode double layer and electrolysis of any adsorbed reactant are instantaneous. The linear-plot model is based on semiinfinite linear diffusion to a planar electrode and, when adsorption is involved, approximates a nonlinear function in the theoretical expression for Q with a linear one. As pointed the assumptions of the linear-plot out by its originators (1,2), method are approximately followed within optimum ranges of experimental conditions (e.g., small T for spherical electrodes) and may lead to considerable errors under other conditions. We felt that regression analysis of the data onto the exact theoretical expressions for the Q-t curves would provide a more general approach to analyzing chronocoulometric data and, moreover, would avoid the approximations and limitations of the linear-plot method. For electrode processes uncomplicated by chemical reactions, Q is a linear function of all the parameters in the equations for the forward and reverse potential steps. In principle, then, the data could be analyzed by multiple linear regression. However, in the interest of developing a general method with an approach extending to systems with nonlinear relationships between Q and parameters, and because common factors appear in the forward and reverse equations, we have chosen to evaluate simultaneous nonlinear regression of the data onto equations for forward and reverse branches of the Q-t curves. In this paper, we describe applications of the new method to electrode reactions involving only diffusion and to those featuring adsorption of the electrochemical reactant.
+
THEORY Consider the electrode reaction
0+ne = R
Eo' (1) where only 0 is present initially in the solution at a concen0 1984 Amerlcan Chemical Society
2148
ANALYTICAL CHEMISTRY, VOL. 56, NO. 12, OCTOBER 1984 I
between Q and t for this reaction (2) are
/-
Q(t
7) = 2nFAD,'~2C,*a-1~2[t1~2 - ( t - ,)lj2 + ( T - t)/2rO T ' / ~ D , ' / ~ T-/ ~nFAI', ~] + nFA(I', - T , ) [ ( ~ / Tsin-' ) ( ~ / t ) ' / '(6) ]
J
+
Neglecting electrode geometry terms yields
+
Q, = 2nFAD,1/2C,*a-1/20 n F A ( r , - r J [ l- @ / a )sin-l ( ~ / t ) l /+~nFAI', ]
L
25
50
t . rns
Figure 1. Chronocoulometry of 1 mM Cd(I1) in 0.8 M NaNOJ0.2 M KSCN at T = 0.025 s. Points are experimental data; solid line computed from nonlinear regression onto eq 5 and 6.
was
Q(t < 7) = 2nFAD,1/2C,*a-1/2[t1/2 ~ ~ ~ ~ D , ~ ~ ~Qdl t / (22 r) , ]
+
and for the reverse step
+
Q(t > 7) = 2nFAD,1/2C,*a-1/2[t1/2- ( t - 7)ll2 a ' / 2 D 0 1 / 2 ~ / 2+ r 0T ~ / ~ D , ' / -~ (t )T/ 2 r 0 ] (3) where n is the number of electrons transferred per electroactive molecule or ion, F is Faraday's constant, A is electrode area, Do and D, are the diffusion coefficients of 0 and R, ro is the radius of the electrode, and Qdl is the quantity of electricity needed to charge the electrode double layer. The terms within the brackets of the form account for the geometry of the electrode (6). In eq 2 and 3 they represent so-called spherical corrections, but they are of a similar form and significance for electrodes of other geometries (7). In many applications, these electrode geometry terms are neglected, a procedure which has been considered (1-3) justified at short times of measurement (e.g., T ) is computed, and a plot of Q, vs. 0 is also constructed. For the reaction in eq 1
Q, = 2nFAD,1/2C,*a-1120 + Qdl
(7) In the linear-plot method, the 1 - ( 2 / ~ sin-' ) ( r / t ) l i zterm is approximated by the linear function al ( O / T ~ / ~+) ao. Values of a0 and a1 are found by linear regression of 1 - @/a) sin-' (7/t)'i2VS. 8 / ~ ~ and / ' depend somewhat on the values of 0 / ~ ~ / ~ used. For the special case where rr = 0, the plot of Q, vs. 0 will have an approximate slope of S, = (2nFAD,'/2C,*~-112)(1 + alnFAr,/Q,), where Q, = 2nFAD,1~2C,*a-'~2~1~2, and an intercept Q," = Qd a,nFAr,. A plot of Q(t < T ) vs. t1i2gives an approximate straight line with slope Sf = S and intercept Q" = Qdl + nFAr,. The surface concentration of adsorbed 0 is estimated from
+
tration Co* and R is soluble either in the solution or in the electrode (e.g., as an amalgam). The conditions of the chronocoulometric eiperiment are that Ei >> ED' and Ef I?,, the left-hand side of eq 8 becomes nFA(I', - TI), and an independent way of measuring Qdl is required to determine both I', and rr (2). The approximations and restrictions on measurement time inherent in linear-plot analysis of chronocoulometric data are not necessary for nonlinear regression analysis. In the method described herein, data are simultaneously regressed onto eq 2 and 3 for diffusion-controlled reactions and onto eq 5 and 6 for reactions involving adsorption. Furthermore, these two cases can be distinguished by first fitting each data set to eq 2 and 3 and observing the deviation plot of residuals (48) vs. t , where 4Q = [Q(exptl) - Q(calcd)]/SD and SD is the standard deviation of the regression. A deviation plot with points randomly distributed around the 4Q = 0 axis indicates that the reaction is diffusion controlled; a plot with a nonrandom distribution of residuals shows that at least one factor other than diffusion influences the electrode process (8, 9). To test the hypothesis that a reversible reaction featuring adsorption is involved, inspection of deviation plots following a nonlinear regression of the data onto eq 5 and 6 is necessary. A random deviation plot in this latter case, or deviations which are smaller than the standard error in measuring Q, confirm the hypothesis embodied in the adsorption model and the regression analysis yields a value for surface concentration. EXPERIMENTAL SECTION Chemicals, Apparatus, and Procedures. Water with a specific resistance greater than 10 MQ cm, obtained by passing ordinary distilled water through a Sybron/Barnstead NANOpure water purification system, was used in all studies. Sources of metal ions were thallium(1) chloride, uranyl acetate, and cadmium nitrate. These and all other chemicals were ACS reagent grade. A Bioanalytical Systems BAS-100 electrochemical system wm used for chronocoulometry and cyclic voltammetry (CV). Rise time of the BAS-100 potentiostat for a 1-V step across a series RC ( R = 100 R, C = 1 pF) is 50 ps. Three-electrode cells and procedures described previously were employed (10). The working electrode was a PARC Model 9323 hanging-drop mercury electrode (HDME) with an area (spherical area less internal cross sectional area of capillary stem) of 0.021 cm2. Reproducibility of electrode area was estimated at from CV cathodic peak currents and at &2.6% from Q ( T ) for a solution of 0.54 mM Tl(1) in 0.1 M HC1. A Ag/AgCl reference and a Pt wire counterelectrode completed the cell. All solutions were purged with purified nitrogen for 20 min before and blanketed with nitrogen during each
ANALYTICAL CHEMISTRY, VOL. 56, NO. 12, OCTOBER 1984
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Table I. Parameters Optimized in Nonlinear Regression Analyses eq
V(1)
eq 2 eq 3
2nFAD,'/2C,*/n1/2 as above
V(2)
V(3)
V(4)
V(5)
Pure Diffusion Model d/2D$/2/2ro as above
-
Qdi
as above
~WI,1/~/2r,
-
-
-
~'/~D,'f~/2r,
nFAr,
Reactant Adsorption Model eq 5 eq 6
as above as above
as above as above
electrochemical experiment. For the electrolyte solutions used, cell resistance was between 25 and 120 fl and was almost completely compensated. Initial and final potentials, vs. Ag/AgCl, for systems used for chronocoulometry were as follows: 0.54 mM Tl(1) in 0.1 M HCl, 0 and -550 mV; 0.11 mM U(V1) in 0.1 M HC1, -40 and -250 mV; and Cd(I1) in 0.8 M NaN03/0.2 M KSCN, -155 and -855 mV. (To convert to V VI. SCE, subtract 45 mV.) The electrode was equilibrated with the test solution for 3 min at Ei prior to each chronocoulometricexperiment. All work was done at the ambient temperature of the laboratory (24 & 1 O C ) . Computations, An IBM 3081D computer and the FORTRAN (WATFIV) language in double precision were used for most computations. A general program (CFT4A) for nonlinear regression analysis (8,11) was used, which provides best values of the parameters in a mathemetical model by finding the minimum of the s u m of squares of the residuals. An initial estimate is required for each parameter. In the present work, the program subroutine which calculates values of Q (the dependent variable) was constructed with eq 2 and 3 for the diffusion-controlled model and with eq 5 and 6 for the model describing reactions involving adsorption of 0 and/or R (Po > I'J. Loop indexes in the subroutines directed calculation of Q for data at t < T from eq 2 or 5 and for data with t > T from eq 3 or 6, respectively. Thus, AQ was minimized by iterative, simultaneous optimization of the parameters (Table I), two of which appear in both pairs of equations. (Copies of the subroutinescan be obtained from J.F.R). For most regression analyses, 50 data points equally spaced along the time axis were used. However, for some determinations of surface concentrations, data points were clustered in time ranges of highest relative weight, as described in the text. Unsmoothed data were used for all regressions. It was assumed that the absolute, rather than the relative, error in Q was randomly distributed and that errors in measurement times were negligible. Pointwise variance analysis was effected with a program (VARPAR) included in the nonlinear regression package ( 1 2 ) and modified to employ the subroutines described above. All pointwise variance analyses used error-free simulated Q vs. t data. The program begins by introducing an arbritrary amount of error in the first value of Q and stores the values of parameters obtained from a subsequent nonlinear regression analysis. The first Q is then returned to its original value, error is introduced into the second Q value, and a second nonlinear regression analysis ensues. This process is repeated N times for N data points and the results are used to calculate absolute and standard errors in the computed parameters and to estimate the relative weights of each data point in determining each parameter (8). Relative overall weights for each data point are also obtained. Linear plot analysis of chronocoulometric data employed unweighted linear regression, either of 80% of the data (one point/ms) at the end of each half-cycle or of 50 points equally spaced on the t axis. All plots were linear to the eye, even at measurement times between 1and 10 ms. Values of D, Qd,and nFr, were calculated from slopes and intercepts by using established procedures and corrections (2). Chronocoulometricdata were simulated by using eq 2 and 3 or 5 and 6, respectively. Normally distributed noise reported here as a percentage of Q(T), was computed and added to the simulated charge as described previously (12).
RESULTS AND DISCUSSION Diffusion-ControlledReactions. One-electron reductions of Tl(1) and U(VI), both in 0.1 M HCl, were shown to be strictly diffusion controlled under the experimental conditions
Qdi 4-nFAro as above
Table 11. Results of Regression Analysis" on Chronocoulometric Data Equally Spaced on t Axis for Tl(1) in 0.1 M HC1 at T = 0.1 s data set
102SD, V(1)
V(2)
V(3)
V(4)
PC
0.0174
8
4.50 4.37 4.80 4.64 4.74 4.54 4.53 4.44
0.143
0.422 0.399 0.484 0.415 0.434 0.436 0.427 0.471
0.1060 0.0287 0.0001 0.0977 0.0887 0.0484 0.0670 0.1012
3.2 3.1 3.2 3.2 3.3 3.0 3.7 3.0
mean SD re1 SD, %
4.57 0.15 3.3
0.023 0.050 217
0.436 0.028 6.5
0.067 0.039 57
1
2 3 4 5 6 7
0.0001
0.0216 0.0001 0.0001 0.0001 0.0001
Fifty points were used. All deviation plots were random. of the present study (i.e,, measurement times
