J. Phys. Chem. 1986, 90, 6261-6266
6261
Direct Numerical Approach to Complex Reaction Kinetics: The Addition of Bromine to Cyclohexene in the Presence of Pyridine Roberto Ambrosetti,*lS Giuseppe Bellucci,lb and Roberto Bianchinilb Istituto di Chimica Quantistica ed Energetica Molecolare del CNR and Istituto di Chimica Organica della Facoltd di Farmacia, Universitd di Pisa, Pisa, Italy (Received: July 16, 1985)
A rigorous numerical method for dealing with complex reaction mechanisms, including both equilibria and reaction kinetics, is presented and applied to the kinetics for the addition of bromine to cyclohexene in the presence of pyridine. The rea xion was followed spectrophotometrically by the stopped-flow technique in 1,2-dichloroethane at 25 "C. The simplest model fmnd capable of fitting the kinetic data, while explaining the nature and distribution of the products and being consistent with previously available information, includes three equilibrium reactions and five reaction rates. In particular, the inclusion of a brominating route involving the pyridine-bromine 1:l complex is necessary for a satisfactory fit of the data. This reaction escaped kinetic detection until now because of the concurrent addition of the same pyridine-bromine complex to the double bond to yield N-(trans-2-bromocyclohexyl)pyridinium bromide. The latter in turn is in equilibrium with the corresponding tribromide, which is very stable in the solvent used, tribromide also acts as an effective brominating agent. Nonlinear least-squares fitting of the experimental data was performed through repeated nun,erical integration of the kinetic differential equations, allowing for equilibrium conditions at each integration step.
Introduction Like so many chemical reactions, the addition of bromine to olefins involves a mechanism much more complex than its simple stoichiometry suggests. In fact, considerable experimental evidence has been accumulated2-' in favor of reaction paths involving the formation of intermediate charge-transfer complexes (CTCs) between bromine and olefin, whose collapse to products can be assisted by protic hydrogen bonding solvents or by bromine itself when the reaction is carried out in aprotic nonpolar solvents. In the latter case, the reaction exhibits a second-order dependence on bromine concentration, and the second molecule of halogen appears to help in the heterolytic fission of the bromine-bromine bond of the CTC, giving a chargedispersed Bry ion. This is shown in Scheme I by equilibrium El and parallel reactions R1and R2, which are first and secand order, respectively, leading to an overall second- or third-order rate law (at least when the experimental conditions allow neglect of the actual fraction of bromine present as CTC). In the presence of an organic base, typically pyridine (Py),the course of the bromination is affected considerably. Both the regioand the stereochemistry of the addition can be quite different.68 Incorporation of pyridine is also possible, giving salts (B+Br- in (1) (a) Istituto di Chimica Quantistica ed Eqergctica Molecolare del CNR, via Risorgimento 35,1-56100, Pisa, Italy. (b) Istituto di Chimica Organica della Facolti di Farmacia dell' Universiti, via Bonanno 6,1-56100, Pia, Italy. (2) (a) Dubois, J. E.; Garnier, F. Spectrochim. Acta, Part A 1967, 23A, 2279. (b) Dubois, J. E.; Garnier, F. Tefrahedron Lett. 1966, 3047. (c) Dubois, J. E.; Garnier, F. Chem. Commun. 1968, 241. (d) Garnier, F.; Donnay, R. H.; Dubois, J. E. Chem. Commun. 1971, 829. (3) (a) Yates, K.; McDonald, R. S.;Shapiro, S. A. J . Org. Chem. 1973, 38,2460. (b) Mcdro, A.; Schmid, G. H.; Yates, K. J . Org. Chem. 1977,42, 3673. (4) (a) Sergeev, G. B.; Serguchev, Yu. A.; Smirnov, V. V. Russ. Chem. Rev.1973,42(9), 697. (b) V'Yunov, K.; Gynak, A. I. Rws. Chem. Rev.1981, 50, 151. (5) Bellucci, G.; Biahchini, R.; Ambrosetti, R. J . Am. Chem. SOC.1985, 107, 2464. (6) Perugini, R.; Ruzziconi, R.; Sebastiani, G . V. Guzz. Chim. Itol. 1983, 113, 149. (7) (a) Barili, P. L.; Bellucci, G.; Marioni, F.; Morelli, I.; Scartoni, V. J . Org. Chem. 1972, 37, 4353. (b) Barili, P. L.; Bellucci, G.; Marioni, F.; Morelli, I.; Scartoni, V. J . Org. Chem. 1973, 38, 3472. (c) Bellucci, G.; Ingrosso, G.; Marioni, F.; Mastrorilli, E.; Morelli, I. J. Org. Chem. 1974, 39, 2562. (d) Barili, P. L.; Bellucci, G.; Marioni, F.; Scartoni, V. J . Org. Chem. 1975, 40, 3331. (8) (a) Heasley, V. L.; Griffith, C. N.; Heasley, G. E. J. Org. Chem. 1975, 40, 1358. (b) Heasley, G. E.; McCall Bundy, J.; Heasley, V. L.; Amold, S.; Gipe, A.; McKee, D.; On, R.; Rodgen, S. L.; Shellhamer,D. F. J. Org. Chem. 1978,43, 2793.
SCHEME I
Scheme I) that are stable with respect to conversion into dibr~mide.~J~ A closer look at the chemical interactions occurring in a reaction mixture containing cyclohexene, bromine, and pyridine in 1,2dichloroethane has revealed the involvement in the addition reactions of three different brominating species: molecular bromine, the 1:l pyridine-bromine CTC, and the tribromide ion, the latter arising since Br- acts as a powerful scavenger for available bromine (the formation constant for equilibrium E3is certainly larger than 2 X lo7 M-' in 1,2-di~hloroethane).~ In an aprotic nonnucleophilic solvent each of the three brominating reactants could lead both to dibromide and to N-(bromoalky1)pyridinium bromide products. A complete (as far as we know) picture of the above reaction should therefore involve, as Scheme I shows, not less than three equilibrium reactions (El, E2, and E3),while the equilibrating species can be involved in not less than six reactions. The equilibria can be studied separatelyF1l and the reactions occur, under proper experimental conditions, on a time scale accessible with fast kinetics apparatus, although only R, R2 and R4can be followed individually. In a previous investigation9 we showed that, for specially chose11 reagent concentration ratios (50 5 [Py]/[B,]5 100) and under quite drastic simplifying assumptions, the very complicated re-
+
(9) Bcllucci, G.; Berti, G.; Bianchini, R.; Ingrosso, G.;Ambrosetti, R. J . Am. Chem. SOC.1980,102,7480. (10) Bellucci, G.;Berti, G.; Bianchini, R.; Ingrosso, G.; Yates, K. J. Org. Chem. 1981,46, 2315. (1 1) (a) DHondt, J.; Dorval, C.; Zeegers-Huyskens, Th. J . Chim. Phys. Phys. Chem. Biol. 1972,69, 516. (b) Aloisi, G. G.; Beggiato, G.; Mazzuccato, U. Trans. Faraday SOC.1970, 3075.
0022-3654/86/2090-6261$01.50/00 1986 American Chemical Society
6262 The Journal of Physical Chemistry, Vol. 90, No. 23, 1986
action system shown in Scheme I can be treated simply as twoseries first-order reactions of Py-Brz and Br3-, whose rate equations are easily handled by the usual integration procedures. However, this method did not work for a somewhat more complicated caselo and for less stable base-Br2 CTCs.I2 The purpose of the present paper is to report on a numerical approach to the problem that is able to take into account the complete reaction scheme so far outlined without simplifying assumptions. Such an approach has shown Scheme I to be valid over a much broader set of experimental conditions than it was previously possible to test.13 Furthermore, this approach makes it possible to evaluate the amount of products formed through each reaction path, such as the relative amounts of dibromide (A) arising from reactions Rz, R4,and R6. Besides the application to the present reaction, which in itself is of some interest since olefin bromination is a very common chemical reaction, the method outlined here appears to show promise for a very broad class of mechanistic investigations. Numerical Approach
In the following, reference will be made to the quantity so = tOO(S/n)'/Z
(1)
which is the square root of the variance of the fitting, expressed in percent transmittance units (S is the sum of the squared transmittance residuals; n is the number of degrees of freedom, or the number of data points minus the number of fitting parameters). The so value was taken as a measure of the goodness of the fittings: lower so values indicate a better fit. Of course, minimization of S by least-squares methods14 is equivalent to minimization of so. The estimation of S requires computing first the concentrations as a function of time. To do so, we write a system of differential equations describing the time evolution of concentrations, without considering intervening equilibria (eq 2-8 for the present case), d[011 -dt
- -k3[01] [Py.Brz] - k4[Ol] [Br3-] d[ Ol.Brz] = -kl[Ol.Brz] - kZIOl.Brz][Br2] dt
(3)
d [Py.Br,l = -k3[01] [Py.Brz] - ks[Ol] [Py.Brz] dt
(4)
d [Br-] -- k3[01] [Py.Br2] + k4[01] [Br3-] + 2k5[01][Br3-][Py] dt
and a system of algebraic equations describing the equilibrium (12) Bellucci, G.; Bianchini, R.; Ingrosso, G.; Ambrosetti, R. 2nd Eur. Symp. Org. Chem., Stresa, June 1-5, 1981; Abstracts, p 220. (13) Like reactions R, and R2,5reactions R, through R6 may be not single-step processes but ones involving discrete CTC intermediates. Although hypothesized for reaction R., (see for instance ref 23), these intermediates could not be observed directly in that reaction, probably because of their extremely low concentration. Furthermore, the resulting multistep reactions would be kinetically indistinguishable from the corresponding single-step processas. Therefore we consider the formulation given in Scheme I more convenient, since it allows direct reference to the bromine-containing species that were actually observed. (14) Hamilton, W. C. Statistics in Physical Science; Ronald: New York, 1965.
Ambrosetti et al.
.:".. ,. . ... . t* ..._ .....: .
2c 'kT
+
,
,
*,
*
,
,
I '
,
*
,
,
, ,
Figure 1. Reaction of bromine with cyclohexene in the presence of a 30-fold excess of pyridine (run 2 of Table I). Above: absorbance recorded at 350 nm in a 2-cm cell. Below: residuals of the fitting (run 2a) on an expanded transmittance scale (time scale identical for the two plots).
between the instantaneous (time-dependent) concentrations (eq 9-1 1). This can be done by simple inspection of the steps involved (9)
in the props d mechanism (Scheme I for the present case). Note that no independent differential equation is required to describe consumption of Brz. This arises because no reaction occurs through free bromine., Consequently, Br2 must be determined through the equilibrium conditions. Equations 2 and 5 can be ignored when olefin and pyridine are present in large excess. Of course, one should devise a method for taking into account all such equations simultaneously. The direct solution of the eighth-degree system of the equilibrium equations, followed by substitution of the equilibrium concentrations into the differential equations, is a near-to-impossible procedure. A possible solution to the problem, which was successfully implemented in the course of this work, can be found by recognizing that the set of differential equations 2-8, even ignoring equilibrium conditions, is too complicated to allow its integration by analytical methods. Numerical integration is certainly required and even desirable because of its generality. Once numerical integration is accepted, equilibrium conditions may be added as an additional constraint after each integration step. If the step is small enough, the concentrations are never allowed to depart significantly from those at equilibrium. The meaning of a "small-enough" step needs to be found in practice. Integration can be performed by using some step size and then repeated by using half the step, and so on until a stable limit solution is reached. Furthermore, since during typical kinetic runs the reaction rates change considerably, a variable-step method was found much more efficient and adopted. The number of integration steps required to a v e r a kinetic run ranged from about 200 to a few thousand. The R ~ n g e - K u t t a ' ~method was adopted as a simple and efficient integration procedure. While other methods are known16 to be necessary for difficult cases arising when fast, opposing reactions compensate each other nearly exactly ("stiff equations" (15) Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry; Van Nostrand: London, 1956, p 486. (16) (a) Gardiner, W. C., Jr. J . Phys. Chem. 1979,83, 37. (b) Warner, D.D.J. Phys. Chem. 1977, 81, 2329.
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6263
Numerical Approach to Reaction Kinetics
TABLE I: Initial Concentrations and Best-Fit Parameters for the Addition Reaction of Bromine to Cyclohexene (c-Hex) in the Presence of Pyridine (Py) at 25 'C in 1,2-Dichloroethane Monitored at 350 nm'
run
[BrZ]X lo4, M 5.2
[Py]X [c-Hex] X [Py]/ free lo2, M lo2, M [Br2] Br2, 7% 1.08
15.3
32.5
5.2
1.58
1.08
30.4
19.0
5.0
2.53
6.54
50.6
12.8
5.0
5.07
6.54
101.3
6.8
k4,
M-' s-I
k59
M-2 s-l
k6,
M-'
4d
0.96 (1) 2.5 0.98 ( i j 0.0 0.97 (1) 2.5 1.00 (1) 0.0 1.37 (2) 0.0 0.92 (2) 2.5 0.97 (2) 0.0 0.96 -4.7 (1.0) 0.99 (3) 2.5 1.00 (3) 0.0 0.80 (1) 2.5 0.42 (3) 2.5
0.0
a9
12.9 (9)
0.96 (2)
7.7 (6)
4b 4c
2.5
9.0 (4) 9.2 (4j 1.3 (3) 7.3 (3)
0.0 7.9 7.9 7.9 6.6 6.5 6.3
A0
s-l
15.5 (3) 15.6 (3j 12.5 (2) 12.5 (3) 7.8 (4) 12.1 (4) 12.0 (4) 12.1 (4) 11.6 (5) 11.4 ( 5 ) 13.1 (4) 28.2 (24)
la Ib 2a 2b 2c 3a 3b 3c 4a
0.795
k3, M-I s-l
(2) (2) (2) (2) (2) (2)
0.0150 0.0154 0.0193 0.0204 0.0329 0.0124 0.0138 0.0130 0.0187 0.0211
so
(3) (2j (2) (2) (4) (2) (3) (3) (4j (3)
0.0 -0.116 (5)
% distrn % dibromide yield A fromb of R 2 R4 R6 B+Br-b
0.54 24 0.57 24 0.78 13 0.82 2.30 1.05 5 1.10 1.13 1.42 2 1.43 2 1.45 2 4.85 2
49 50 54
27 26 33
30 31 37
56
39
38
60 60 62 98
38 38 36 0
43 39 46 66
'Different numbers indicate different experiments; different letters indicate different fitting schemes tried for the same data. Enclosed in parentheses are least-squares estimates of standard deviations, referred to the last significant digit of the parameter. Lack of a standard deviation denotes a value kept fixed in that fitting. Parameters not appearing in the table were kept fixed in all fittings to the values reported in the text. Ao is the reference value for zero absorbance; so is the square root of the variance of the fitting in percent transmittance units and is taken as a measure of the goodness of the fittings. Runs 2c and 4d are included to show that k6 is essential to obtain good fittings. Run 4c shows the adverse effect of not including Ao as a fitting parameter: even if so does not increase, a 20% decrease in k4 is observed. All other runs show the mild effect of parameter k5, while run 3c shows that its value cannot be determined from the fitting of data taken at a single pyridine concentration. bComputed values. Weighted average of the runs showing the lowest so. The standard deviation of the average was computed from the difference from the mean of individual values for each run included in the average. case), we found the simpler Runge-Kutta procedure adequate for the present case. A very small integration step was necessary around the maximum of the measured absorbances (Figure l), where the relatively fast disappearance of an absorbing species (PyBrJ was nearly exactly compensated for by the formation of a differently absorbing one (Br3-). An approximated procedure (described in detail in the Supplementary Material), involving the initial neglect of the disappearing amount of bromine with respect to [Ol] and to [Py], was adopted for solving the system of equilibrium equations. The overall degree of the system was thus reduced from the original 8 to 2, allowing straightforward solution. Iteration was needed for correcting the approximations, and convergence was normally attained at the third iteration. However, some difficult cases required more than ten iterations. This was essentially due to the limited numerical precision available (nine digits on the Commodore 3032 microcomputer used). The problem was amplified by the fact that formation constants spanned more than 7 orders of magnitude. Overall, adjustment for equilibrium conditions typically required 3 times more computer time than an integration step. In order to find the minimum of S in eq 1, the so-called "normal equations" approach was This method, like all others that do not depend on a systematic search of the minimum, requires the evaluation of the derivatives of the residuals with respect to the fitting parameters. Since we lacked an analytical expression for the fitting function, we resorted again to a numerical approximation, evaluating the derivatives as an incremental ratio. Each parameter in turn was given an increment, the entire integration procedure was repeated, and the differences in the residuals were divided by the increment of the parameter, so obtaining the required approximation for the derivative. Typical increments were 0.1 times the current estimate of the standard deviation on the parameters. However, the particular choice of the increment was not found to be critical. The full integration procedure had therefore to be carried out p + 1 times (p is the number of adjustable parameters) for each iteration of the "normal equations" procedure. Iteration was necessary since the dependence of the fitting function on the parameters was nonlinear for most of them. Again, at least three iterations had to be done to obtain convergence. The whole (17) Wentworth, W. E. J . Chem. Educ. 1965, 42, 96.
procedure was implemented in a Basic program running on the same microcomputer performing data acquisition from the stopped-flow apparatus used for the kinetics measurements. A typical fitting required no less than lo6 floating-point operations and was completed in about 1 h, although calculations with small integration steps required proportionately more time. Comparison fittings with a very small integration step and extended precision (14 decimal digits) performed on larger computers (either a Gould 32/87 or an IBM 3033) confirmed the results obtained on the microcomputer and allowed tailoring of the required integration step. A constant weight was assumed for all experimental values. This is to be considered acceptable since, to a very good approximation, the photcmetric error on transmittance is constant. For an orderly formulation of the computational algorithm, we found it useful to include as formal parameters the initial concentration of all species shown in Scheme I and their molar extinction coefficients, the kinetic constants ( k , to k,), the formation constants (Kif to K,?, the start time of the reaction, and the reference level for zero absorbance (see later). However, only a maximum of five parameters out of the involved 25 could actually be refined at a time, since no complete information on many of them was contained in the kinetic runs alone. At the end of this section, we would like to comment on the fact that, although numerical integration of fitting models containing differential equations has been suggested for chemical kinetic^,'^,'^ its use does not seem as widespread as it probably merits. In particular, we are not aware of any other application of a mixed differential-algebraic model and its implementation on an inexpensive personal computer. It should be stressed that the numerical method reported on so far reconducts the present complex case within the well-known frame of nonlinear least squares (NLLSQ). Therefore all results known to be valid for NLLSQ can be taken for granted also ill the present case. Thus, for instance, the correlation between the parameters can be kept under control by simply looking at the estimate of the correlation coefficients, which are obtained easily by the normal-equations method.I4 Of course, no information on the correlation between the refined parameters and the ones kept (18) Bard, Y.Nonlinear Parameter Estimation; Academic: New York, 1914. (19) Guthrie, J. P. Con. J . Chem. 1982, 60, 765.
6264 The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 TABLE II: Observed Rate Constants for the Reaction of 5.0 X lo-* M Cyclohexene with 2.38 X lo-' M N-(tram-2-Bromocyctohexy1)pyridinium Tribromide in the Presence of Pyridine (Py)in 1,2-Dichloroethaneat 25 OC
102 212 636
2.43 5.06 15.15
1.11 1.16 1.38
fixed was available from the fittings. Application to the Bromination of Cyclohexene in the Presence of Pyridine
The experimental conditions for the kinetic measurements, and the best-fit parameters obtained under a few different assumptions, are given in Table I. The total range of pyridine-bromine concentration ratios explored was from 15 to 100. Independently measured values were available for many of the 25 formal parameters. This was the case for the initial concentrations and the molar extinction coefficients of all species appearing in Scheme I (at 350 nm, t = 33 for Br,, 165 for B'Br-, 2550 for B+Br3-, 650 for Py.Brz, 400 M-I cm-I for c-Hex-Br,, and 0 for all others). For the formation constants the following values were used: Klf = 0.47,5 Kzf = 273,lI8 and K3' = 2 X lo7 M-Is9 All such parameters were kept fixed in the fittings reported in Table I.2o The rate constant kl was kept equal to zero, since no first-order term was found in the bromination of cyclohexene in the absence of pyridine down to a lo4 M concentration of the halogems This pathway is usually observed in brominations carried out in protic solvents. The rate constant k2 was fixed to a value of 5.2 X lo5 M-' s-I, deduced from the value of the third-order rate constant (2.4 X lo5 M-, s-l) measured for the bromination in the absence of pyridine, taking into account the value of the formation constant of the 01-Br, CTC (K1').' The latter complex has been definitely shown to be an essential intermediate of the bromination of cyclohexene carried out by molecular brominee5 The value of kS used in runs l a , 2a, 3a, 4a, 4c, and 4d came from separate reactions in which pre-formed tribromide was allowed to react with cyclohexene in the presence of various amounts of pyridine (Table 11). Such data were treated by neglecting the reactions of both free bromine and of the Py-Br, complex formed through equilibria E3 and E2, whose concentrations were always very low. The observed disappearance of Br3- then follows a second-order rate law whose kobsddepends linearly on the concentration of pyridineI0 (12) + k5[Pyl The data of Table I1 yielded k5 = 2.5 (4)M-2 s-l and k4 = 1.01 kobsd
=
k4
(3) M-l s-l, the latter being in fair agreement with the results of Table I. The rate constant k3 for the reaction of the PyeBr, CTC to give the bromopyridinium adduct B'Br- was treated as a fitting parameter, starting from the previously determined value? The rate constants k4 and k6 for the formation of dibromide from the tribromide ion and the Py-Br, CTC, respectively, were also treated as fitting parameters, using as initial estimate for the former the one obtained from eq 12. Typical correlation coefficients between the fitting parameters are given for run 4a of Table I: r(k3,k4) = -0.66; r(k3,k6) = 0.83; r(k3,A0) = -0.55; r(k4,k6) = -0.71; r(k4,Ao) = 0.84; r(k6,A0) = -0.35. Only the correlation between k4 and Ao and that between k3 and k6 can be considered significant, although still acceptably low.21 (20) Trial fittings (not shown here) showed that K,'and K t could be varied within the limits of the corresponding uncertainties5-"* with no significant effect on the parameters. No significant effect was likewise observed when K: was increased to 2 X lo8. This is not surprising, since, because of its very high value, any further increase of 4'would lead to no significant change in the extent of association of bromine and bromide over most of the reaction.
Ambrosetti et al. A typical fitting is presented in Figure 1. The residuals of this fitting actually exhibit some systematic trend that could be taken as an indication for some additional factor not included in Scheme I. However, these residuals were about as large as can be expected from the overall error of stopped-flow measurements, and we do not believe that significant additional information can be extracted from them. It can also be observed that the differences between the values of the rate constants obtained in different runs exceeded 3 times the respective standard deviations obtained from each single fit.,, This is not surprising, since stopped-flow measurements have only moderate reproducibility. Furthermore, some of the reactions involved in Scheme I show a marked dependence on the composition of the reaction medium.23 Therefore we consider as more reliable the estimate of the standard deviations given in Table I for the average rate constants obtained by the weighted average of the values corresponding to the best fit of each experiment (runs l a , 2a, 3a, and 4a). The inclusion of the separately measured value of the rate constant k5 did not improve the fit significantly (compare runs l a to lb, 2a to 2b, 3a to 3b, and 3c and 4a to 4b in Table I). Furthermore, when k4 was given a fixed value (k4 = 0.98 M-' s-l) equal to the average value resulting from the best fits of Table I and k5 was allowed to vary, a negative value, with a high standard deviation, was obtained for the latter parameter (run 3c). Attempts at varying both k4 and k5 at once led to divergence of the fitting procedure. All this shows that k5 cannot be obtained from the fitting of data taken at a single pyridine concentration. However, the dependence of the kOMvalues (eq 12 and Table 11) on the concentration of pyridine defined a nonzero value for k5. It is to be noted that the introduction of a reference value for zero absorbance as a fitting parameter (Ao)was essential for consistent fitting of the data. See for instance run 4c,which gives an unacceptably low k4 value as a consequence of forcing Ao to zero. The nonzero Ao value was probably necessary in order to account for long-term drifts and slight mechanical instabilities from shot to shot. Since in the presence of excess pyridine and olefin the concentrations of Ol.Br2 and of Py-Br, are proportional both reactions R Iand R6 are first order in Br, (eq 13 and 14); thus they are kinetically indistinguishable.
- d[Ol.Br,]/d~ = kl[Ol.Brz] = k,K1f[O1][Brz]
(13)
- d[Py.Br,] /dt = k6[Py.Brz] = k6Kzf[Py][Br,]
(14)
Fittings equivalent in goodness to those reported in Table I were actually obtained by introducing k l in place of k6 as a fitting parameter. However, only k6 (not k , ) was reasonably constant in fittings of runs performed with varying pyridine concentration, thus showing that reaction R6, and not R,, is the one actually involved. This is consistent with the previous report5 on the complete lack of any contribution from a first-order kinetic term in the bromination of cyclohexene in 1,2-dichloroethane without added pyridine. On the other hand, putting k6 = 0 increased so significantly (compare run 2c to runs 2a and 2b and run 4d to runs 4a, 4b, and 4c). Furthermore, in all fittings k6 was satisfactorily constant and consistently well-defined, as shown by the large ratio of its values to the relative standard deviations. This confirms that reaction R6, involving the formation of dibromide (A) through the Py.Br2 route, is actually operating. This reaction could not (21) A unity correlation coefficient between any two parameters indicates total interdependence between their values. A positive correlation coefficient indicates that both will tend to change together (both increasing or both decreasing) as the result of measurement errors; the reverse holds for negative correlation. (22) Any difference between parameters was considered significant whenever it exceeded 3 times the average estimated standard deviation on the parameters. Due to the uncertainties introduced by the nonlinearity of the fitting scheme, we conservatively take this as a 95% probability criterion." (23) Bellucci, G.; Bianchini, R.; Ambrosetti, R.; Ingrcsso, G. J . Org. Chem.
1985.50, 3313.
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6265
Numerical Approach to Reaction Kinetics
loot 80
I
40
I3
ti\
2otl\
2
7 ‘
2
0 0
0.5
-7
1
I
5
I
10
I
15
I
20
Time (see)
Figure 2. Computed concentrations against time for the reaction of the
pyridinebromine 1:1 complex with cyclohexene under typical preparative conditions (complex 0.1 1 M, cyclohexene 0.27 M), obtained with the average rate constants of Table I: (1) Brz; (2) Br3-; (3) Py.Br2; (4) Br-; (5-7) trans-1,2-dibromocyclohexaneformed respectively by Br2, Py.Br2, and Br,-.
perimentally both from absorbance measurements and through a titrimetric method (computed, 1.87 X M; experimental, 1.82 x M). The kinetic model was thus shown to be valid under conditions drastically different from those to which it had been adjusted. It can be observed that the yield of the pyridine incorporation product formed under preparative conditions (17%) was much lower than those calculated in runs 1-4 of Table I. Furthermore, in contrast with the kinetic runs, most of the total dibromide (68%) arose by the reaction’ of Br2 (R2) and only 11% and 21% respectively by the reactions of Py.Br2 (R6) and of Brq (R4). This was not unexpected, since at the relatively high (0.11 M) concentration of pre-formed Py.Br2 employed, the computed initial concentrations of Br2, Py.Br2, and pyridine are respectively 1.83 X 9.17 X and 1.83 X M. These correspond to initial reaction rates of 21.7 M s-l for path R2 and 0.191 M s-l for path R,. The initial reaction rate for path R3 (0.319 M s-l) is also lower than that for path Rz. This accounts for the relatively low formation of adduct B’Bi and consequently for the relatively low contribution of the Br3- pathway to the formation of the dibromo adduct. Caution is needed, however, in extrapolating the results obtained in this particular case to other similar bromination reactions. In fact relatively small structural changes in the substrate as well as in the reaction conditions may well lead to a quite different distribution of the products obtained with molecular bromine or the Py.Br2 complex as brominating reagent.7J0 Conclusions
be included in the simplified kinetic treatment reported in the previous invatigation? although its involvement was inferred from a consideration of the product distribution obtained in the bromination of 1,3-butadiene in the presence of pyridine.1° It should also be pointed out that the value obtained for k3 is in good agreement with the one found with the above-mentioned simplified treatment. Since with appropriate substrates the addition reactions of the different brominating species involved in Scheme I exhibit different stereo- and regioselectivity, it is important to know what fraction of the dibromo adduct is formed by the Br2 (R2), the Brc (R4), and the Py.Br2 (R6) routes. This information is not directly available from experiment. However, its computation on the basis of the assumed mechanism was feasible within the numerical framework presented here. Although individual concentrations at each time were thus obtained, we only give in Table I the final distribution of dibromide from routes R2, R4, and R6 for some runs. The calculated yields of the pyridine incorporation product (B’Br-) are also given in Table I. In principle it would be possible to obtain these yields experimentally from the measured final absorbances. However, the small absorbance values found, as well as the possibility of long-term drifts during the slow final stage of the reactions and of contributions from small residual amounts of Br3-, prevented us from obtaining sufficiently accurate values of these yields. A titrimetric method (see the Experimental Section), used successfully for the determination of higher bromide concentrations, also gave unaccurate results for the kinetic runs. On the other hand, a reliable check of the calculated and experimental yields of B’Br- was made in a preparative reaction. In fact, the values of the rate constants have been used to anticipate the entire kinetic course of a reaction carried out under typical preparative conditions: 0.1 1 M pre-formed Py.Br2 complex was used to brominate 0.27 M cyclohexene in 1,2-dichloroethane at 25 OC. The computed concentrations of reagents and products against time are shown in Figure 2, which also shows the formation of dibromide through the three routes R1,R4, and %. Of course, the extremely high absorbance of the reaction mixture, even in a 1-mm cell, and the very high reaction rates made it impossible to compare directly the experimental and computed kinetic course of the reaction. We could only check the computed final concentration of N-(trans-2-bromocyclohexyl)pyridinium bromide, which was in very good agreement with the one estimated ex-
The present results show that the proposed numerical approach is able to cope with a complex kinetic model such as that given in Scheme I for the bromination of an alkene in the presence of pyridine. In particular, this approach has demonstrated the involvement of a reaction of the Py.Br2 complex leading to dibromo adduct. This reaction has escaped observation under the previously applied9 simplified treatment, which, however, still maintains its validity as an easily understood clue to the most relevant aspects of this important reaction. Beyond the present reaction, which is reported here as a typical example, the general fitting method outlined could be applied with success to many problems in chemical kinetics. Chemistry frequently suggests a reaction scheme felt too complex to be used as a fitting model. The most frequently adopted strategy is one of drastic simplification, leading to a subset of the overall scheme, to be checked by experiments performed under some extreme conditions. While this procedure is of great value in testing parts of the mechanistic scheme and in giving initial estimates for the rate and equilibrium constants, we feel that only fitting the complete model to experiments run under the most extended range of conditions will really allow a critical and meaningful test of a mechanistic hypothesis. Of course, while failure to fit the data is certainly enough to disprove a mechanism, success in fitting them does not exclude alternative mechanisms that could fit just as well. However, this is a much more general problem, and the method adopted here at least allows for a fast and easy turnover from one reaction scheme to another. Several promising applications of the present approach are being presently investigated. Experimental Section
Preparation and purification of reagents and solvent, as well as product analysis, followed exactly the procedures described previ~usly.~ Kinetic measurements were carried out on a Durrum stopped-flow kinetic spectrophotometer? collecting the data with a Commodore 3032 microcomputer through an analog-digital int e r f a ~ e .Most ~ computations were carried out on the same microcomputer. Reactions referred to in the text as ‘preparative conditions” were carried out as follows: A 0.1 1 M solution of Py.Br2 prepared either by dissolving the freshly precipitated CTC” or by mixing Br2 and pyridine solutions
6266
J. Phys. Chem. 1986, 90, 6266-6270
in 1,2-dichloroethane was mixed with a 0.27 M solution of cyclohexene in the same solvent using the mixing device of the stopped-flow apparatus in order to ensure reproducible mixing conditions without concentration gradients. After 5 or more minutes the reaction mixture was withdrawn and combined with that obtained from several identical shots. The concentration of N-( trans-2-bromocyclohexy1)pyridiniumbromide in this solution was determined both spectrophotometrically, after appropriate dilution, at several wavelengths between 300 and 360 nm and by titration with 0.01 N aqueous Hg(C104)2in the presence of dip h e n y l c a r b a ~ o n e ,in ~ *comparison ~~ with a blank containing an (24) SchBniger, W. Mikrochim. Acta 1955, 123; 1956, 869.
equal concentration of pyridine. A maximum difference of 3% was found between the two analytical methods. Acknowledgment. This work was financed in part by a NATO grant, in part by a grant from the Consiglio Nazionale delle Ricerche, and in part by a grant from the Minister0 della hbblica Istruzione. We thank Prof. K. Yates for helpful discussion of a preliminary manuscript. Registry No. Cyclohexene, 110-83-8.
Supplementary Material Available: Description of the fitting algorithm by means of commented flow charts and a partial listing of a computer program (21 pages). Ordering information is given on any current masthead page.
Nlechanlsm of the Photochemistry of p-Benzoquinone in Aqueous Solutions. 1. Spin Trapping and Flash Photolysis Electron Paramagnetic Resonance Studied A. Ike Ononye, Alan R. McIntosh, and James R. Bolton* Photochemistry Unit, Department of Chemistry, University of Western Ontario, London, Ontario, Canada N6A 5B7 (Received: January 6, 1986; In Final Form: June 27, 1986)
The primary photochemistry of pbenzoquinone (BQ) in aqueous solution has been investigated at pH 7 by steady-state and transient electron paramagnetic resonance (EPR) spectroscopy, using the spin trap 5J-dimethyl-1-pyrroline 1-oxide (DMPO). The results strongly suggest a mechanism in which the triplet state of BQ abstracts a hydrogen atom directly from water forming p-benzosemiquinone (BQH') releasing a free hydroxyl radical (OH'), which then reacts with another molecule of BQ to form the adduct BQ-OH'. The ultimate products are hydroquinone (BQH,) and 2-hydroxy-p-benzoquinone(HO-BQ). A mechanism as in reactions 1-8 of Scheme I11 is proposed which accounts for the observations.
Introduction Quinones are found widely in biological systems. In particular, substituted quinones, such as ubiquinone and plastoquinone are found in organelles in which electron transport occurs, e.g., photosynthetic and mitochondrial membranes. Many of these biological systems are exposed to sunlight. Thus, in addition to an inherent interest in the photochemistry of quinones in aqueous solutions, there is a need to rationalize the mechanism in view of the possible biological significance of quinone photochemistry. Although the final products, I and I1 (see Scheme I), of the aqueous reactions of the unsubstituted p-benzoquinone (BQ) have long been identified,14 there has been considerable disagreementw over the primary step that leads to the principal intermediate, the semiquinone radical (BQH'). The contention has been over the suggestionlo that the primary step involves hydrogen abstraction from H 2 0 by excited BQ [which should lead to a free hydroxyl radical (OH')]. Another conceivable mechanism is that of photosolvation of 3BQ to form a hydrated intermediate (BQ-H,O) which could then react with BQ to form BQH' and BQ-OH'. Such a photosolvation process has been advanced"J2 for the primary photochemistry of disodium anthraquinone-2,6-diiulfonate in aqueous solutions and has been proposed as the primary process in the photochemistry of BQ.' The major problem has been the lack of convincing evidence to prove or disprove the presence of free OH' in this system. Doubts have also been cast as to whether the triplet of BQ has enough energy to oxidize water.9 We therefore began a study of the mechanism of this reaction with the objective of resolving the controversy that surrounds it. Our method of approach in this study involved performing experiments on a system which is known to yield free OH' radicals and then comparing the results with those obtained by carrying out similar experiments on the BQ system. The system chosen Publication No. 358, Photochemistry Unit, University of Western Ontario.
SCHEME I: Illustration of the Products of the Photochemistry of p -Benzoquinone in Aqueous Solutions
qu i n o n e
radical
(BO)
0 (11) 2-hydroxy-
p-benzoquinone (HO-BO 1 to generate free OH' radicals was the UV photolysis of a dilute solution (1% or less) of H202in water." (1) Lcighton, P. A,; Forbes, G. S.J. Am. Chem. SOC.1929,51,3549-3561. (2) Poupe, F. Collect. Czech. Chem. Commun. 1947, 12, 225-236. (3) Joschek, H.-I.; Miller, S.I. J . Am. Chem. SOC.1966, 88, 3273-3281. (4) Kurien, K. C.; Robins, P. A. J. Chem. SOC.( E ) 1970, 855-859. (5) Hashimoto, S.;Kano, K.; Okamoto, H. Bull. Chem. SOC.Jpn. 1972, 45, 966. (6) Kano, K.; Matsuo, T. Chem. Lett. 1973, 1127-1 132. (7) Shirai, M.; Awatsuji, T.; Tanaka, M. Bull. Chem. SOC.Jpn. 1975,48, 1329-1 330.
0022-3654/86/2090-6266$01.50/0 0 1986 American Chemical Society
