5796
J . Phys. Chem. 1989, 93, 5796-5802
Removal of Parameter Correlation in Nonlinear Regression. Lifetimes of Aryl Halide Anion Radicals from Potential Step Chronocoulometry Artur Sucheta and James F. Rusling* Department of Chemistry (U-60), University of Connecticut, Storrs, Connecticut 06269-3060 (Receioed: November 28, 1988: In Final Form: March 17, 1989)
Gram-Schmidt orthogonalization of an ECE (electron transfer/chemical step/electron transfer) reaction model in single potential step chronocoulometry removed correlation between parameters and greatly improved convergence properties when using the Marquardt-Levenberg nonlinear regression algorithm. The orthogonalized model was applied to estimate lifetimes (rate constants for decomposition) of intermediate anion radicals in the reduction of aryl halides in dry N,N-dimethylformamide. The ECE model gave a better fit to the charge vs time data than an alternative EC model for 3’-chloroacetophenone, 4-chlorobenzophenone, and 9-chloroanthracene. The orthogonalized ECE model eliminated divergence problems and converged 3-4 times faster than the nonorthogonal model, making real time kinetic analyses possible on a PC/AT microcomputer. The largest determinable rate constant (about IO3 s-’ in the present study) is limited by the assumption of a time-independent charging current and the time window observed by the instrumentation available.
Introduction Chronocoulometry involves a rapid step in potential at a working electrode in an electrochemical cell containing the solution of interest. At the initial potential, no faradaic charge passes through the cell. The step potential is usually chosen such that faradaic charge transfer between the electrode and the reactant occurs a t a fast rate. Following an appropriate time interval, a reverse step to the initial potential may be employed. The charge (Q) flowing through the electrochemical cell is measured vs time during the application of the potential step program. The output of the experiment is a plot of Q vs t . Nonlinear regression analysis has proved useful for obtaining thermodynamic and kinetic parameters from potential step chronocoulometric data. It has been used to estimate diffusion coefficients of reversibly reduced metal ions, surface concentrations of electroactive adsorbates,’ and chemical rate constants for anodic hydroxylation of tetrahydrocarbazole cation radicals2 Nonlinear regression has also been used to distinguish between very closely related electrode reaction mechanisms.*s3 Closed form expressions are available to describe the Q-t response for many common types of electrochemical reaction^.^ Mathematical models describing experimental behavior are assumed here to be derived from theoretical considerations. Nonlinear regression analysis seeks the best values of physically meaningful parameters in such a model with respect to a given set of data. Reliable general programs are available for this task.+* In applications, the user writes a code for the desired mathematical model into the “model” subroutine of the program and provides the program with data and “best guesses” of starting values for the parameters. The program then seeks a minimum in an appropriate error sum, usually by making use of the principle of least squares. This principle holds that the best values of the parameters will be foundS at the minimum in the error sum S. n
S = xw,[y,(measd) - y,(calcd)12 j=
I
In eq 1, the j\(measd) are the measured experimental quantities (I) Rusling, J. F.; Brooks, M. Y . Anal. Chem. 1984, 56, 2147-2153. (2) Rusling, J. F.; Brooks, M. Y.; Scheer, B. J.; Chou, T. T.; Shukla, S. S . Anal. Chem. 1986, 58, 1942-1947. (3) Arena, J. A,; Rusling, J. F. J . Phys. Chem. 1987, 91, 3368-3373. (4) Murray, R. W. In Physical Methods of Chemistry; Weissberger, A,,
Ed.; Wiley: New York, 1971; Part Ila, pp 591-644. (5) Meites, L. CRC Crit. Reu. Anal. Chem. 1979, 8, 1-53, (6) Bard, Y. Nonlinear Parameter Estimation; Academic Press: New York, 1974. (7) Christian. S. D.; Tucker, E. E. A m . Lab. (Fairfield. Conn.) 1982, 31-36. (8) Marquardt, D.W . J . SOC.Ind. Appl. Math. 1963, 11, 431-441.
and the y,(calcd) are those computed from the regression model for each of n data points. The error sum S is minimized by systematic iterative variation of the parameters by an appropriate algorithm. The wj in eq 1 are weighting factors, taken to equal unity in this paper. The purpose of the model subroutine is to supply values of y,(calcd) to the main program. The approach to minimum S starting from a set of initial “best guesses” for m parameters can be conceptualized as the journey of an initial point toward the global minimum (So)on an error surface in an ( m 1 )-dimensional coordinate system. One axis of the coordinate system corresponds to the values of S, and the others correspond to the parameters. The minimum S is called the convergence point. A problem that complicates some applications of nonlinear regression is correlation between the parameters of the model. Two interdependent parameters in a regression model are said to be correlated. Their final values will depend on one another and on the starting point chosen for the analysis. If correlation is very strong, convergence and/or unique estimates of each parameter may be difficult to achieve. Partial correlations have more subtle influences. Models with partial correlations may often be used successfully with slowly converging algorithms such as the steepest descent method5s6 employing conservative convergence criteria. However, extensive computing time and degraded precision in parameters compared to an uncorrelated model are the costs that may have to be paid. Upon use of rapidly converging regression methods such as the Levenberg-Marquardt a l g ~ r i t h m problems ,~~~ caused by correlation may be amplified. Often convergence cannot be obtained a t all or rcquires a tedious search for a set of initial parameters from which convergence is attainable. A method for removing correlation of parameters by transforming the model into an equivalent but uncorrelated form should be of general use. Such transformations would allow retention of the general program format for nonlinear regres~ion,~-~ since only the model subroutine would be affected. One case of parameter correlation arises in determining chemical rate constants in electron transfer/chemical step/electron transfer (ECE) type reactions from chronocoulometric data. Returning to the error surface concept, parameter correlation can be equated with nonorthogonal parameter axes in the coordinate system of the error surface.6 In this paper, we explore the effect of orthogonalization of these axes on convergence, computation time, and precision of parameters. This method of removing correlations is applied to a model for ECE reduction in chronocoulometry to determine lifetimes of anion radicals of three aryl halides. Our interest in this problem arises both from a general aim of good precision, accuracy, and ease of extraction of kinetic parameters from electrochemical data and from a specific goal of understanding the decomposition of aryl halide pollutants.
0022-365418912093-5796$01.50/0 0 1989 American Chemical Society ,
+
Removal of Parameter Correlation in Nonlinear Regression Anaerobic microbiological dehalogenation of polychlorinated biphenyls (PCB's) was recently found in river sediments9 Such natural reductions may occur by pathways similar to in vitro reduction. In particular, we wish to measure rates of decomposition of anion radicals that are intermediates in reductions of toxic aryl halides. As a first step, we rigorously test the orthogonalized model on anion radicals of aryl halides with estimated" decomposition rates in the range 1-103 s-I. Direct electrochemical reduction of aryl halides (ArX)'os" follows the ECE pathway in Scheme I . Various complicating
SCHEME I ArX
+ e = ArX'-
+ + -
ArX'Ar' Ar-
k
Ar'
Eol
+ X-
(2)
Eo2
(4)
SH
+ S-
(5)
nuances to this mechanism have been including the homogeneous electron exchange (eq 6a) and hydrogen atom transfer from the solvent (SH) to radical Ar':
+ ArX'Ar' + S H
Ar'
+ ArX ArH + S'
= Ar-
-
relation12 between pairs of parameters? (iii) Does removal of parameter correlation improve convergence in nonlinear regression analysis? To answer these questions, Gram-Schmidt orthogonalizationi3 was used to derive an equivalent model from the explicit model for a single potential step chronocoulometric response for the ECE reaction. The orthogonalized model consists of a linear combination of orthogonal and normalized functions derived from the explicit ECE model. In our chronocoulometric experiment, the initial potential (Ei) of the working electrode is chosen so that no electrolysis occurs. At time t = 0, the potential is rapidly pulsed to Ef > Eolin ECE reductions of ArX, a single cathodic voltammetric peak results. Single potential step chronocoulometry with an intial potential well positive of this peak and a final potential beyond the peak will result in a Q-t response reflecting the rate of decomposition of anion radical ArX'-.2-4 We previously used nonlinear regression of single potential step chronocoulometric data collected at carbon paste electrodes to obtain rate constants for the ECE anodic hydroxylation of several tetrahydrocarbazoles.2 Eight different systems analyzed gave pseudo-first-order rate constants between 17 and 35 s-, with an average relative standard deviation of 17%. A nonlinear regression program based on a modified steepest descent method required up to 2000 iterations for convergence. Initial studies on reductions of aryl halides showed that because of the partial correlation among parameters the ECE model used with the faster Levenberg-Marquardt algorithm required a large number of computational cycles. Regression analyses diverged for some sets of initial parameters. Orthogonalization of the model eliminates these problems and allows rate constants to be found in a few minutes on a microcomputer. Orthogonalization Many models describing the charge response in chronocoulometry can be expressed in closed form as explicit functions of time. These models can be considered as linear combinations of basis functions which may contain nonlinear parameters. W e posed the following questions: (i) Can nonorthogonality of the basis functions be responsible for poor convergence properties of the ECE model? (ii) Does nonorthogonality affect coupling or cor(9) Brown, J. F.; Bedard, D. L.; Brennan, M . J.; Carnahan, J. C.; Feng, H.; Wagner, R. F. Science 1987, 236, 709-712. (10) Andrieux, C. P.; Saveant, J. M.; Zann, D. NOW.J . Chem. 1984,8, 107-1 16. ( I 1,) (a) Nadjo, L.; Saveant, J. M . J . Electroonal. Chem. 1971, 30,41-57. (b) M Halla, F.; Pinson, J.; Saveant, J. M. J . Electroanal. Chem. 1978, 89, 347-361.
where Qdlis the double-layer charge, F is Faraday's constant, A is electrode area, and ro is its radius. C*ArXand DAIX are respectively the bulk concentration and the diffusion coefficient of ArX. The last term on the right-hand side of eq 7 corrects for electrode sphericity. The rate constant for aryl halide decomposition is k (eq 3). Nonlinear regression with eq 7 employs the four parameters bo = k , b, = 2FAcChx(Dhx/x)'/2, b2 = Qdl,and b3 = Dhx'/2/yo. The model in eq 7 was used previously to analyze data for anodic hydroxylation of tetrahydrocarbazoles. A modified steepest descent regression algorithm led to long convergence times and indications of parameter correlation for eq 7 . As mentioned, correlation of the parameters can be thought of as lack of orthogonality between the parameter axes in the error surface plot. Such correlations can be related to the dot products of the appropriate pairs of unit vectors representing parameter axes in the error surface space. The value of the dot product will be unity for total correlation and close to zero for zero correlation. A correlation matrix containing such evaluations of correlations for all pairs of parameters in its [row, column] elements can be constructed from results of a completed regression analysis.12 This is a symmetric matrix with diagonal elements of unity. A representative correlation matrix for eq 7 regressed onto a set of simulated data (Table I) shows off-diagonal elements with absolute values close to one, indicating significant nonorthogonality and partial correlation between pairs of parameters. The model in eq 7 was orthogonalized by the Gram-Schmidt procedure. The object of the orthogonalization is to transform the model to the form Y, = Blh,(tj) + B2h2(t,) + B,h,(tj;Bo)
(8)
where the hk(t,) are the new orthonormal functions and the Bk are the new set of four adjustable parameters. The starting basis functions were derived from the overall form of eq 7:
h(t)= 1 (constant)
f2(t) = t (linear dependence)
f3(t) = 2 - ( r / 4 k t ) ' I 2 erf ( k t ) ' l 2 (term containing erf)
(9)
In development of the orthonormal basis function set, the next step is to accept one of the starting functions without a change. Hence, the first orthogonal function gl(t) was made identical with f i ( t ) . Normalization of gi to unity was then performed. The normalized function hl took the form h l ( t ) = gl(t)/N,
where N , is a normalization factor such that
Solving for NI gave the first function of the orthonormal basis: h , ( t ) = g l ( t ) / N , = n-l/*
(12)
(12) CET Research Group, LTD. Nonlinear Least Squares Program NLLSQ (and program manual); privately published, Norman, OK, 1981. (1 3) Arfken, G. Mathematical Methods for Physicists; Academic Press: Orlando, FL, 1985; pp 516-519.
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
5798
Sucheta and Rusling
TABLE I: Correlation Matrices and Regression Parameters for Nonorthogonal and Orthogonal ECE Models"
ECE Model (Eq 7)b k, bl -0.8 746
bo 60
1 .oooo
b,
1
.oooo
b,
true
calcd
true
0.8134 -0.9808
0.4275 -0.8099 0.8338
10.00
10.04 f 0.14
5.1 1
1 .oooo
b2
s-l
b2
1 .oooo
b3
106D,cm2/s calcd 5.10 f 0.05
Orthoaonal ECE Model (Ea 8)' k, Bo
B" 1.0000
-9.622
X
1 .oooo
BI B2
lo-"
lo60, cm2/s
s-I
B,
B,
true
calcd
true
calcd
3.484 X lo-" -2.170 X IO-"
3.507 X 8.994 X IO-'' 5.903 X
10.00
10.04 f 0.14
5.11
6 . 5 f 1.5
B1
1 .oooo
1.oooo
B?
"Same data set simulated from eq 7 with normally distributed random error of 0.30% of the maximum charge (Qma,). For each regression analysis,'50 data points uniformly spaced on the time axis were analyzed with 7 = 250 ms. Equivalent starting values of parameters. bTwenty-two iterations, 291 s. 'Four iterations, 82 s. According to Gram-Schmidt rules, the second member of the orthogonal base assumed the form
g2(0 = f 2 ( d - a21h,(t)
(13)
Here, the second term on the right-hand side removes the component of f 2 in the direction of f l . Imposing the orthogonality requirement of vanishing scalar product of basis functions
produced the azl constant n
a21 = C f 2 ( t j ) hl(tj) = n-'/'(t)
(15)
j= 1
where ( t ) is the average value o f t . Substitution for a21in eq 10 gave
g2(0 = - ( 2 )
(16)
Finally, normalization of g2 yielded
The last orthogonal function takes the form
g3(t)
=f3(l)
- a32h2(t) -
a31h1(t)
(18)
The coefficients a32 and a31 were found from the requirement of orthogonality between g, and both h2 and hi. A complication arose since the function & contained the nonlinear parameter k , represented here as Bo. Thus, a32 and a31 were functions of Bo and had to be evaluated each time Bo changed during the process of optimizing the parameters in regression analysis. Also, the normalization factor N 3 depended on Bo and had to be reevaluated whenever the latter changed. To simplify the regression calculations, the following expression was derived:
Finally
h3(t;B0) = g3(t;B0)/N3(B0)
(20)
Thus, the orthogonal model for ECE chronocoulometric data is given by eq 8 with h , , h,, and h, from eq 12, 17, and 20. The parameters to be used for regression analyses are Bo = k (the same as bo in the nonorthogonal model), B l , B2, and B3. Experimental Section
Chemicals. 3'-Chloroacetophenone, 4-chlorobenzophenone, and 9-chloroanthracene were from Aldrich, supplied in 98%, 99%, and
75% (Tech.) purities, respectively. The former two compounds were used as received. 9-Chloroanthracene was recrystallized four times from ethanol before use. The purified material showed a single spot by thin-layer chromatography and had a melting point of 105 OC (lit. 106 oC).14a Tetrabutylammonium iodide (TBAI) and tetrabutylammonium perchlorate (TBAP) were from Eastman Kodak and were used as received. N,N-Dimethylformamide (DMF) was Burdick & Jackson "distilled-in-glass" grade. It was purified with and stored over activated alumina as described previously.'4b Chronocoulometry was done after stirring a small amount of alumina placed in the cell to scavenge impurities. Apparatus and Procedures. A Bioanalytical Systems BAS- 100 electrochemical analyzer was used throughout this work. Data were shipped to the hard disk of an IBM-PC/XT compatible microcomputer by means of BAS-100 hardware and software. The water-jacketed electrochemical cell was thermostated a t 20.0 A 0.2 OC. A three-electrode cell was used with a hanging mercury drop electrode (HMDE) as working electrode ( A = 0.0182 cm2), a Pt wire counter electrode, and a homemade, miniature aqueous saturated calomel electrode (SCE) as the reference. The S C E terminated in a tube containing an internal agar plug saturated with KCI, ending in a glass frit. This S C E contacted the nonaqueous solution in the cell by a salt bridge filled with 0.1 M solution of the electrolyte used in DMF. Resistance of the cells containing D M F and electrolyte was about 800 ohms. Initial potentials for chronocoulometry were chosen positive of the foot of the first peak in cyclic voltammetry. Final values were chosen to be 5C-90 mV more negative than the standard potentiall' of each compound. These initial and final potentials were as follows: 3'-chloroacetophenone, -1.530 and -1.880 V vs SCE; 4-chlorobenzophenone, -1.430 and -1.720 V; and 9-chloroanthracene, -1 SO0 and -1 300 V. After removal of oxygen from the solution with a stream of purified, DMF-saturated nitrogen, a waiting time of 10 or 20 s at the initial potential was used prior to the step to the final potential. All potentials are reported vs SCE. Computations. Nonlinear regression analysis was done with a general nonlinear least-squares program (NLLSQ~,)employing the Marquardt-Levenberg algorithm. The version used was slightly modified for compilation with the Borland Turbo Basic compiler. Double-precision calculations were accomplished on an IBM-PC/XT compatible microcomputer with 4.77-MHz clock speed, configured with an Intel 8087 math coprocessor, or an IBM-PC/AT compatible with 12-MHz clock speed. All regression analyses assumed that the standard error in Q(t) was independent of the size of Q ( t ) . Starting values of model parameters were chosen with the aid of a computer graphics subroutine to compare (14) (a) Handbook of Chemistry and Physics, 53rd ed.; CRC Press: Cleveland, 1972. (b) Rusling, J. F.; Arena, J. V. J . Electroanal. Chem. 1985, 186. 225-235.
Removal of Parameter Correlation in Nonlinear Regression
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5799
TABLE 11: Rate Constants Estimated from Regression Analyses of Simulated Data (0.3%Noise”) onto Orthozonal ECE Model (Eq 8) k , s-l true calcd T , ms Ar.6 ms rsd:% log ( k r ) 10.0 10.1 f 1.2 1000 20 0.22 1.o 9.5 f 0.8 250 5 0.20 0.4
TABLE 111: Comparison of Goodness of Fit of Experimental Data Sets to Diffusion and ECE Models” diffusion model ECE model compound trial rsd,b % x2 rsd,b % x2
10.6 f 2.5 50.0
100
500
1000
49 f 7 49 f 4 53 f 5 136 110 94 103
f 33
f 12 f9 f 14
100
2
0.18
0.0
1000 250 50
20 5 1
0.19 0.18 0.22
1.7 1.1
500 250 100 50
10 5 2 1
0.20 0.20 0.17 0.19
0.4
4-chlorobenzophenone
1.7
1.4 1.o
250 100 50 25
5 2 1 0.5
0.21 0.19 0.17 0.18
2.1 1.7 1.4 1.1
800 f 240 1380 f 370 1000 f 110
100 50 25
2 1 0.5
0.19 0.21 0.23
2.0 1.7 1.4
Data simulated from eq 7 with normally distributed random error of 0.30% Qmax. For each regression analysis, 50 data points uniformly spaced on the time axis were analyzed. Initial k values were up to 70% different from true values but were similar for a given k . Calculated k values given with standard errors. Interval between data points. Relative standard deviation obtained by division of the standard deviation of the regression by
emax.
computed and experimental response curves prior to initiating regression analysis. The form of the ECE model required evaluation of an expression involving the error function erf (x). This was accomplished by using the following expansion: ( ~ ’ / ~ / 2 ) [ e (r xf ) ] / x = exp(-x2) 1F,(1,3/;~2) where
+ ( a / b ) z + a(a + l ) / [ b ( b + I)]z2/2! + ...
is Kummer’s confluent hypergeometric function computed directly in a subroutine in such a way that its error was smaller than round-off error in double-precision calculations.
Results and Discussion Validation of Orthogonalized Models. Precision and accuracy of chemical rate constants obtained from chronocoulometric data depend on the time range of the data analyzed. If the total time of the experiment is T , it was previously shown2 that the best precision for ECE pathways is found for k in the range 0 I log ( k r ) I 2 , with errors slightly larger a t the ends of this range. Under these conditions, data simulated from eq 7 with normally distributed noise added at 0.05% of maximum charge demonstrate the reliability of both nonorthogonal (eq 7 ) and orthogonalized (eq 8) models in estimating rate constants by nonlinear regression analysis. Both models give k within 0.4% of its correct value with a standard errorlSaof about 1.4% (Table I). The orthogonal model converged 3-fold faster and in fewer iterations compared to the nonorthogonal model. Correlation matrix elements indicate that correlation between parameters is completely removed in the orthogonal model. However, diffusion coefficients (D)are estimated with about 20% standard error when the orthogonalized model was used because the parameter D is a function of several ( I 5 ) (a) Standard errors of parameters are computed by the regression program as the difference between the “best-fit” parameter and its extreme value on the standard error of the error surface given by S, = So[1 + I / ( n - m ) ] , where m is the number of parameters. (b) Christian, S . D. J . Chem. Educ. 1965, 42, 604-607. (c) Brooks, M . Y . Ph.D. Thesis, University of Connecticut, Storrs, CT, 1983.
1
0.13
2
0.15 0.12 0.13
3 4
0.4
370 f 170 499 f 75 510 f 41 509 f 43
I F , ( a , b ; z= ) I
3’-chloroacetophenone
9-chloroanthracene
25 25 31 20
0.078 0.094 0.093 0.092
31 25
0.080 0.038
0.044 0.040
0.045
1 2 3 4 5
0.22 0.10 0.16
6
0.14
38 25 25 31
1 2 3
0.054
31
4 5 6 7 8 9 10 11 12 13
0.14 0.17
0.27
0.042 0.040 0.034 0.064 0.044 0.039 0.25 0.17
0.10 0.10 0.1 10
4.6 5.6 22 17 31 27 21 9 3.5 2.5 7.4 5.9
0.047
0.041 0.27
0.036 0.033 0.034
0.048 0.037 0.033 0.20 0.16 0.098 0.10 0.108
1.7 1.7 17 13 1.7 0.18 0.5 1 3.5
5.9 5.9 22 3.5 10 12 17 22
IO 10 3.5 2.5 3.5 3.5
9.0
“Nonlinear regression on 50 equally spaced data ponts on t axis with values: 1000 ms for 3’-chloroacetophenone, 250 ms for 4-chlorobenzophenone, 53 ms for 9-chloroanthracene with first point analyzed at 4 ms. bRelative standard deviation obtained by division of the standard deviation of the regression by the maximum value of charge. T
of the orthogonalized parameters and its error is propagated. Extensive tests with data for 2 I k I 1000 s-I simulated from eq 7 for data acquisition rate of 1 point/ms showed that good precision and accuracy for k were found by regression of 50 data points equally spaced on the t axis onto eq 8 when r was chosen to give 0 I log ( k r ) I 2. As shown by analyses of data with normally distributed noise at a relatively large value of 0.3% of the maximum charge (Table 11), precision and accuracy in k degraded in some cases a t the ends of this range. For example, poor results were found for k = 1000 s-l when log ( k r ) 2 1.7 and for k = 500 s-I when log ( k r ) = 2.1. This suggests that for best results r should be chosen from a preliminary estimate of k so that log ( k r ) is in the middle or low end of the 0-2 range. In all cases, the orthogonal model gave much faster convergence and dramatically decreased absolute values of off-diagonal elements in the correlation matrix as illustrated in Table I. Thus, we could apply eq 8 to nonlinear regression analysis of experimental data with confidence. Chronocoulometric Reduction of Aryl Halides. The orthogonalized model was used to analyze data for direct electrochemical reduction of 3’-chloroacetophenone, 4-chlorobenzophenone, and 9-chloroanthracene in DMF. Experiments were designed such that log ( k r ) would fall between 0.3 and 1.7 from preliminary estimates of k . In some previous estimations of k for the decomposition of the anion radicals of these organohalides, a oneelectron EC mechanism was assumed to hold under experimental conditions of double potential step chronoamperometry and derivative cyclic voltammetry.I0 However, individual response curve shapes were not analyzed and alternative mechanisms were not considered. To test the alternative hypotheses of EC or ECE mechanism under our experimental conditions, all data were analyzed by nonlinear regression onto (i) a model assuming diffusion-controlled electron transfer characteristic of the expected EC response on the forward pulse, (ii) the orthogonalized ECE model, eq 8, and (iii) the explicit nonorthogonal ECE model, eq 7 . Final pulse potentials for single step chronocoulometry experiments were chosen to fall between the cyclic voltammetric peaks for reduction of the parent compound and the product benzophenone or acetophenone.10,” Except for significant im-
5800 The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
Sucheta and Rusling
TABLE I V Results of Nonlinear Regression Analysis of Chronocoulometric Response
compound 3'-chloroaceto~henone
trial
C*.M. mM
1
0.25 0.25 0.37 0.37
2 3 4 4-chlorobenzophenone
1
2
3 4 5
6 9-chloroanthracene
1
2 3 4 5 6 7
8 9 10
11 12
13
10sD, cm2/s
k f (r,'s-'
2.193 f 0.065 2.292 f 0.077 2.1 18 f 0.069 2.147 f 0.068
1.98 1.98 2.96 2.96 2.96 2.96
10.90 f 0.31 9.02 f 0.17 10.49 f 0.20 11.04 f 0.23 11.07 f 0.23 10.50 f 0.22
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 1.49 1.49 1.49
(4.07 f 0.28) (5.8 f 2.3) X (5.74 f 0.43) (5.78 f 0.39) (9.2 f 2.4) X (4.45 f 0.26) (4.42 f 0.29) (4.83 f 0.34) (2.64 f 0.37) (3.22 f 0.51) (7.2 f 1.4) X (6.9 f 1.2) X (7.5 f 1.7) X
electrolyteb TBAI
2.5 2.8 2.2 2.2
0.22 0.16 0.18
TBAI
0.10
0.13 0.1 1 0.22 0.17 0.23 0.23 0.26 0.21 0.25 0.24 0.29 0.28 0.15
IO2 IO2
X
X
lo2
X
IO2
lo2 X X X
lo2
lo2 lo2
IO2 IO2 IO2 IO2
X X
TBAP
0.16
10'
0.16
"Sta dard error from nonlinear regression analysis; experime tal co ditions as in Table 111. bO.lOO M solution in DMF. TABLE V: Summary of Rate Constants and Convergence Properties
av ratio eq 7/eq 8 iterations seconds 4.5 f 0.5 4.4 f 1.1
compound 3'-c hloroacetophenone
av k f s,' s-' 2.19 f 0.07
k(lit.1, s-' (ref) 5 (17)
methodb ( T , "C) CV ( e )
4-chlorobenzophenone
10.5 f 0.8
8.5 (10) 40 (10) 42 (18)
LSV (25) DPS (25) DCV (23)
3.8 f 2.7
3.0 f 2.2
DPS (25)
4.8 f 2.9
3.0 i 1.7
9-chloroanthracene
550 f 180
160 ( I O )
"Standard deviation of the mean. bAbbreviations: CV, peak current ratio in cyclic voltammetry; LSV, peak potential-sweep rate plots in linear sweep voltammetry; DPS, double potential step chronoamperometry; DCV, derivative cyclic voltammetry. CAmbient temperature. provements in convergence time with eq 8 and several cases in which eq 7 did not yield convergence, results of regressions with eq 7 and 8 were identical. Comparisons of goodness of fit for the diffusion and ECE models were made by comparing relative standard deviations (rsd's) of the regression, xz tests on the residual^,^*^ and visual comparison of residual (deviation) plots. Except for 2 of 13 sets of data for 9-chloroanthracene, all rsd and x2 values were significantly smaller for the ECE model (Table 111). Experimental and computed values of charge were in excellent agreement when the ECE model was used (Figure 1). However, deviation plots provide a more discriminating graphical test to aid in choosing the correct model. Deviation plots in this case are plots of the relative deviation, i.e., [Qj(measd) - Q,(calcd)]/SD, vs t. Such plots should have a random distribution of points around zero relative deviation if the model correctly fits the data, which should be free of unaccounted for systematic error. If the model is incorrect, the plot will show a nonrandom pattern. The shape or "deviation at tern"^ for a nonrandom plot can be established by fitting simulated data containing little or no error to the incorrect model. When such low-noise simulated data from eq 7 were fit to the diffusion model, definitive patterns dependent on k emerged, as found p r e v i o u ~ l y . ' These ~~ patterns had essentially the same shapes as the deviation plots obtained from experimental data with similar rate constants for the three compounds of interest (Figure 2). Random deviation plots resulted when the experimental data were fit to the ECE model, eq 8 (Figure 3). Thus, summary statistics (Table III), deviation pattern r e c o g n i t i ~ n and , ~ final random deviation plots indicate that the ECE model fits better than the EC (diffusion) model under our experimental conditions. However, the larger the k
0.5 l.1
r"
14
0.0 0
50
I
1
100
150
J
200
250 time (msl
Figure 1. Chronocoulogram for 1.98 mM 4-chlorobenzophenone in 0.1
M TBAI/DMF. Solid line marks the values obtained from nonlinear regression onto the orthogonal ECE model; + symbols are the experimental data. the more difficult it was to distinguish the model ( E C E or diffusion) giving the best fit. This is a consequence of a time window and data collection rate for which a very fast chemical reaction will go unobserved, resulting in apparent diffusion-controlled behavior. Relative standard deviations (rsd's) of the regressions for 3'chloroacetophenone and 4-chlorobenzophenone were smaller than 0.1% for the ECE model in all cases (Table 111). All but one of the analyses for 9-chloroanthracene gave rsd 5 0.2%. Rate constants and diffusion coefficients for the three aryl halides were obtained with good precision (Table IV). Diffusion coefficients
The Journal ofPhysical Chemistry, Vol. 93, No. 15, 1989 5801
Removal of Parameter Correlation in Nonlinear Regression
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Figure 2. Deviation plots from nonlinear regression onto the diffusion model for (a) 0.25 mM 3'-chloroacetophenone, (b) 1.98 mM 4-chlorobenzophenone, and (c) 0.5 mM 9-chloroanthracene. Diamonds are experimental points. Solid lines are theoretical deviation patterns obtained by regressing ECE data simulated from eq 7 onto the diffusion model. Rate constants for simulated data are (a) 2, (b) 10, and (c) 500 s-', and random noise is 55 X 10-5Q,al.
were similar to those reported previously for similar sized molecules in DMF,I6 although the accuracy and precision of D by this analysis are probably not high (cf. Table I). Rate constants agree reasonably well with previous estimates (Table V), although estimates by different electrochemical techniques are not in especially good agreement. Reproducibility of the previous measurements was estimated'O at about h0.2 on log k. This is considerably larger than the &3% rsd for 3'-chloroacetophenone and &7% for 4chlorobenzophenone. It is possible that previous results contain (16) (a) Wawzonek, S.;Gundersen, A. J . Electrochem. SOC.1960, 107, 537-540. (b) Given, P. H. J . Chem. SOC.1958, 2684-2687. (c) Given, P. H.; Peover, M. E.: Schoen, J. J . Chem. SOC.1958, 2674-2684. (d) Bergman, 1. Trans. Faraday SOC.1954, 829-833. (17) Gores, G . J.; Koeppe, C. E.; Bartak, D. E. J . Org. Chem. 1979, 44, 380-385. (18) Aalstad, B.; Parker, V. D. Acta Chem. Scand. B 1982, 36, 47-52.
Figure 3. Example of random deviation plot obtained from regression of chronocoulogram in Figure 1 for 1.98 mM 4-chlorobenzophenone onto the orthogonal ECE model (eq 8).
systematic error since they involved CV or potential step experiments over a wide range of times in which the effect of chemical reaction has varying amounts of influence. Systematic differences in k's determined from pulsed response ratio have been noted previo~sly.'~It is most likely caused by inclusion of pulse widths for which the chemical reaction does not have sufficient influence on the response for an accurate value of rate constant to be determined.2 All goodness-of-fit criteria are consistent with an ECE pathway for direct reduction of the aryl halides studied here. In previous work,1° 9-chloroanthracene gave a visually acceptable fit to the EC mechanism for double potential step chronoamperometric current ratios over a range of pulse widths. In the same report, 4-chlorobenzophenone gave an exceedingly poor fit to the E C mechanism. In neither case was the alternative ECE model tested. For 3'-chloroacetophenone, the reported rate constant (Table V) was estimated by single potential step chronoamperometry using a very negative final potential where acetophenone was reduced, so that the ECE model could be used. In this case, an apparent number of electrons transferred of about 2.2 was obtained a t t > 1 s. Bulk electrolysis of 3'-chloroacetophenone at a platinum electrode at a potential close to what we used for chronocoulometry gave 1.02 faradays of electricity per mole, but analysis of the solution showed only 74% conversion to acetophenone. For bulk electrolysis of 4-chlorobenzophenone a t a potential just past its first CV reduction peak, the electricity passed varied between 1.2 and 2 faradays/mol depending on time of electrolysis and size of mercury pool electrodes.'1a The sole product was benzophenone. These results are not fully consistent with an E C process, which should provide 1 faraday/mol and quantitative yield of product. A second unresolved issue is that reactions on large electrodes in bulk electrolyses do not always follow the same pathway as reactions on analytical electrodes in shorter time experiments.*O Certainly, the aryl halide reductions examined here may proceed by a one-electron EC pathway under conditions of fast CV where the second charge transfer is not given time to occur. However, the reactions appear unequivocally to follow the E C E pathway under the conditions of our chronocoulometric experiments. Speculations about the chemical nature of the second charge transfer following the rate-determining radical anion decomposition (eq 2) are beyond the scope of this paper, and further work is needed to make definitive conclusions. Several possibilities are presented in eq 4-6. Use of Orthogonalized Models. The orthogonalized expression in eq 8 for the ECE pathway yielded a remarkable improvement in time and number of cycles (Table V) for convergence of nonlinear regression analyses. A 3-4-fold improvement was found in all cases. Regression analyses with the compiled program using eq 8 were typically completed in 40-60 s on an IBM/PC-AT (19) Herman, H. B.; Blount, H. N . J . Phys. Chem. 1969, 73, 1406-1413. (20) Zuman, P.In Organic Electrochemistry, 2nd ed.; Baker, M., Lund, H., Eds.; Marcel Dekker: New York, 1983; pp 151-157.
5802
J . Phys. Chem. 1989, 93, 5802-5808
compatible microcomputer. Thus, this procedure is applicable to real time, on-line kinetic analysis and model testing. Precision in the rate constants of the two more slowly decomposing anion radicals was excellent. However, a visible trend seen in Tables 111 and IV is the deterioration of precision in k at higher rates. While work is being done to fully explain the cause of such behavior, we feel that it is mainly a limitation of electrode size and data acquisition rate of 1 point/ms. Analysis of data for faster reactions is more heavily influenced by instrumental errors in the measured charge. Also, deviation from the model due to the time dependence of charging of the electrode double layer (Qdl) is most pronounced in the first few milliseconds after the potential step. This also contributes to degraded precision for larger rate constants where most of the kinetic information is contained a t short times.* This limitation may possibly be removed by using ultramicroelectrodes, decreasing the time window into the submillisecond range and increasing the rate a t which data are collected. Such an approach may also require incorporation of the time dependence of Q d ] in the model.
Orthogonalization necessitates deriving an orthogonal form of the model. After the values of regression parameters are obtained by use of such a model, deconvolution to recoup the original set of parameters is required. Fortunately, the nature of GramSchmidt orthogonalization should eventually enable fully automated transitions between the original basis set and the orthogonal one. It is not necessary or practical to use an orthogonalized model for every problem to be solved by nonlinear regression analysis. However, our results demonstrate the usefullness of orthogonalization when serious correlation between parameters creates problems in convergence of nonlinear regression analyses. Since the transformation is made on the model, the method is compatible with currently used general programs for nonlinear regression. Acknowledgment. This work was supported by U S . PHS Grant ES03 154 awarded by the National Institute of Environmental Health Sciences. Registry No. 3'-Chloroacetophenone, 99-02-5; 4-chlorobenzophenone, 134-85-0; 9-chloroanthracene, 7 16-53-0.
Isomerization and Decomposition of Pyrrole at Elevated Temperatures. Studies with a Single-Pulse Shock Tube Assa Lifshitz,* Carmen Tamburu, and Aya Suslensky Department of Physical Chemistry, The Hebrew University, Jerusalem 91904, Israel (Received: November 30, 1988)
The thermal decomposition of pyrrole was studied behind reflected shocks in a pressurized driver single-pulse shock tube mol/cm3. Under these conditions the nitroover the temperature range 1050-1450 K and overall densities of -3 X gen-containing products found in the postshock mixtures were cis-CH,CH=CHCN, HCN, CH2=CH-CH2CN, transCH,CH=CHCN, CH,CN, CH2=CHCN, C2H5CN, CHcC-CN, and small quantities of C6HSCN, C6HSCH2CN, CH2=C=CHCN and CH,C=C-CN which began to appear at the high end of the temperature range. Products without nitrogen were CH,C=CH, CH=CH, CH2=C=CH2, CH4, C2H4, and small quantities of C4H6, C4H4, C4H2, C6H6, C6H5C=CH, and C6H5CH3which appeared only at high temperatures. The main reaction of pyrrole under these conditions is a simultaneous unimolecular bond cleavage in the 1-5 (1-2)-position and a hydrogen atom transfer, followed by electronic rearrangement and (1) isomerization to cis-crotonitrile, (2) dissociation to HCN C3H4(mainly propyne) and (3) isomerization to allyl cyanide, with a branching ratio of approximately 3.5:1.5:1. The overall process proceeds with a rate constant of k = 10'4.83exp (-75 X 103/RT) s-l, where R is expressed in units of cal/(K-mol). The second major reaction of pyrrole is a dissociation to acetylene and CH2=C=NH. Decomposition and isomerization of the initiation products as well as additional free-radical reactions lead to the formation of a plethora of reaction products. Arrhenius rate parameters for their formation are reported and a general pyrolysis scheme is suggested.
+
Introduction W e have recently published a series of studies on the thermal reactions of five-membered ring ethers: furan,' tetrahydrofuran? and 2,3- and 2 , 5 - d i h y d r o f ~ r a n , ~discussing ,~ their pyrolysis mechanism and reporting Arrhenius parameters for the production rates of the pyrolysis products. The pyrolysis of pyrrolidine, the nitrogen analogue of tetrahydrofuran, was also recently investigated behind reflected shock^,^ stressing the differences and similarities between the nitrogen- and oxygen-containing heterocyclics. Pyrrole is the nitrogen analogue of furan. Its skeleton appears in a large number of naturally occurring structures as well as in
the major constituents of fuel n i t r ~ g e n . ~As , ~far as we are aware its thermal reactions a t elevated temperatures have never been studied in the past. In addition to the interest in the basic chemical kinetics of these compounds, they are of great relevance to combustion technology. Their thermal behavior both under pyrolytic and oxidative environments simulate the reactions that occur during the combustion of a large variety of nitrogen-containing fuel molecules. This article presents a continuous effort in elucidating the kinetics and mechanism of the thermal reactions of oxygen- and nitrogen-containing heterocyclics. It discusses in detail the thermal reactions of pyrrole with a special emphasis on the differences and similarities with its oxygen analogue, furan.
( 1 ) Lifshitz, A.; Bidani, M.; Bidani, S.J . Phys. Chem. 1986, 90, 5373. ( 2 ) Lifshitz, A.; Bidani, M.; Bidani, S. J . Phys. Chem. 1986, 90, 3422. (3) Lifshitz, A.; Bidani, M. J . Phys. Chem. 1989, 93, 1139. (4) Lifshitz, A,; Bidani, M.; Bidani, S.J . Phys. Chem. 1986, 90, 601 1. ( 5 ) Lifshitz, A,: Bidani. M.; Agranat, A,; Suslensky, A. J . Phys. Chem. 1987, 91. 6043.
(6) Sneider, L. R.; Buell, B. E.; Howard, H. E. Anal. Chem. 1968, 40, 1303. (7) Speers, G. C.; Whitehead, E. V. In Crude Petroleum in Organic Geochemistry; Eglington, G., Murphy, M . T.J., Eds.; Springer Verlag: Heidelberg, FRG, 1969; p 638.
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0 1989 American Chemical Society
