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INTERNATIONAL JOURNAL OF CHEMICAL KINETICS, VOL. IV,

345-362 (1972)

Computational Modeling of the Mechanisms of the Free Radical-Chain Reaction of Alkanes with Oxygen. The Oxidation of Isobutane, n-Butane, and Isopentane. D. L. ALLARA A N D D. EDELSON Bell Telephone Laboratories, Incorporated, Murray Hill,New Jersey 07974

AND

K. C . IRWIN Stanford Research Institute, Menlo Park, California 94025

Abstract A general computational method for obtaining complete solutions of time-dependent kinetic equations has been developed and applied to free radical-initiated reactions of alkanes with oxygen. The method has been applied to the low-temperature, peroxideinitiated oxidations of isobutane and isopentane. Using available independently measured and estimated values for the rate constants and activation parameters for each of the 20 proposed reaction steps for the oxidation of isobutane, the rates and products have been calculated for both the liquid phase and gas phase in the range of 1OO0-155"C. The calculated rates and products of oxidation agree with published experimental values. The oxidation of isopentane was examined by a 32-reaction model. The rate constants were estimated using values for the appropriate rate steps in the oxidation of n-butane and isobutane. The calculation of the oxidation rate and products agree with our experiments.

1. Introduction Efforts to account rigorously for the observed rates of oxidation a n d product formation of simple alkanes under controlled conditions [ 11 usually involve too many reaction steps to permit the analytic derivation a n d testing of an exact rate law. Although the differential equations corresponding to any kinetic scheme can be written in exact form, valid solutions are obtainable only for certain special simplified forms. T h e general approach has been to make assumptions regarding the relative magnitude of the rates of the various steps in the proposed mechanism and then to restrict the initial conditions for such values that the entire set of equations could be approximated by one of the simpler forms for which a n exact analytic solution is available. Thus, in recent low-temperature studies ( 100°C) of the peroxide-initiated oxidations of isobutane [2] and n-butane [3],

345 @ 1972 by John Wiley

& Sons, Inc.

346

ALLARA, EDELSON, AND IRWIN

models for linear and branched alkanes, approximate expressions such as (1)

-d[Oz]/dt

(2)

Ro

=

= RO= Rillz k [ i s o b ~ t a n e ] ~ ' ~

R,li2k'[n-butane]

+ Ri/Pa

have been derived for the steady-state region to explain the experimental results. Here, Ri is the rate of initiation and k, k', and a are composite rate constants whose relatiomhip to the rate constants of the individual reactions in the proposed scheme is known only approximately. Although values for nearly all the individual rate constants can be estimated from previous work, a valid test for their participation in the proposed mechanism is vitiated by the constraints required to obtain a solution. For example, these constraints restrict termination to one particular reaction and require the propagation rate constants of all peroxy radicals to be the same. Recently, computer programs have been developed which are able to generate complete time-dependent solutions of the differential equations encountered in chemical kinetics [4]. These programs have been used successfully in aeronomic and aerodynamic programs [5] and are applied in this report to the low-temperature, peroxide-initiated oxidations of isobutane [2] and isopentane. For isobutane a previously proposed mechanism is modified and tested, and for isopentane the rates and products are predicted using the oxidations of n- and isobutane as models.

2. Isobutane The important proposed reactions for the oxidation of isobutane between 50" and 155°C are based on a previous report [2], and their corresponding rate constants (evaluated at 1OOOC), Arrhenius parameters, and rate expressions are given in Table I. Formation of i-butyl radicals by attack at the methyl groups has been considered previously at temperatures above 260" [6] but not below 155" [2]. Reactions of i-butyl and i-butylperoxy radicals are considered here and listed in Table I. Unless otherwise specified, values of the A-factors were calculated from specified rate constants and activation energies using the Arrhenius rate expression k = Ae-"iRT. Termination reactions involving alkoxy and alkyl radicals are assumed to be negligible because of their low steady-state concentrations. The values for the rate constants and parameters in Table I were substituted into the rate expressions. Numerical values for Ri, [i-BuH] and [OZ]were specified and the equations were solved for the concentrations of radicals, reactants, and products as functions of time. The computations were carried out on a Honeywell 635 computer. A. Liquid Phase Oxidations In order to compare calculated rates of oxygen uptake with experimental results at 100" [2], R , = 4.80X10-9 M/sec was chosen for the calculations, corresponding to a value of 0.00353M for the concentration of tert-butyl peroxide

ALKANES WITH OXYGEN

347

(t-Bu204, the average concentration in the actual experiments. A value of 10-3M was used as the approximate solubility of O2 in the liquid phase, and was maintained a t this value by replenishment from a gas reservoir. However, any errors in this assignment are not significant since lowering this concentration by a factor of 10 does not significantly alter the overall rates during the steady-state period. Since no values were available for the rates of terminations involving methylperoxy radicals, reactions (16-18), a value of 2X 10' mole-sec was assigned to the self-termination reaction, reaction (18), by assuming a value similar to the measured rate constant of the primary n-BuOz. radical [8] (extrapolated value at 100" of 3.8Xl07 mole-sec based on E = 2 kcal/mole) but slightly lowered because of the higher bond strength in the CH302- radical for the C-H bond broken in the termination reaction. Runs were also made using higher and lower values (see below). Absolute values for the propagation rate constants for the peroxy radicals were not available, reactions (7-12). However, reasonable relative values of the different propagation rate constants could be estimated. Keeping the relative values of each of the rate constants for reactions (7-12) fixed, the absolute values of the set were adjusted to fit the experimental rate data in neat liquid solution (where the fewest reactions occur). These values were then used in the subsequent calculations and are reported in Table I. Experiment and theory are compared in Figure 1 where calculated values of Ro/R;'I2as functions of isobutane concentration for the liquid phase are included in the previously reported plot of observed rates in CCl, solution, neat hydrocarbon, and in the gas phase [2]. Typical plots of calculated species concentrations as functions of time are also shown in Figure 2. The complete steady state is reached in about 50-100 sec. A plot of --d[02]/dt against time shows that the maximum rate is reached in 100 sec, corresponding to the attainment of the steady state. The calculated rate for the oxidation of neat hydrocarbon is the same as the observed rate since the propagation reactions were adjusted to fit this point. Between 7.46M and 1.0M hydrocarbon, the predicted rate is within a factor of 2 of the observed rate in CC14. Around 0.6M, the experimental and observed values merge. The slope of the plot in Figure 1 gives the order of the rate with respect to [i-BuH] for the empirical relation Ro/Ri112o( [i-BuH]". The calculated order in hydrocarbon increases from 1.0 to 1.3 as [i-BuH] decreases from 7.46M to 0.3M and then decreases to 1.1 as [i-BuH] decreases to 0.10M. This predicted behavior agrees with the proposal that at low concentrations of isobutane the rate should be higher than first order in isobutane [2] eq. (1). Since the higher than first-order term in hydrocarbon has been attributed to intervention of methylperoxy radicals in termination [2], the contributions of these radicals to chain termination were determined by comparing the values of the rate terrris (defined in Table I), k l A + l[CH302-] ~ [i-Bu02-], k l M M [CH3O2.l2and k t [CH302~ ~ ] [t-BuO2*] with the similar expressions involving only the i-and t-BuO,. radicals. These results are shown in Figure 3; the total contributions are equal to Ri = 4.80X10-9 M/sec. The effects of a tenfold higher rate of initiation on the contributions are also shown. At high concentrations of iso-

t-B~zOz-+ 2t-BuO. t-BuO. i-BuH -+ t-BuOH t-BuO. i-BuH 3 t-BuOH t-Bu. 0 2 -i t-BuOz. i-Bu. 0 2 i i-Bu02. CH3. 0 2 -+ CH302. t-BuOz. i-BuH -i t-BuO2H t-BuOz. i-BuH -+ t-BuOzH

0 2

2

-+

-+0 2

i 0 2

-+

0

-+02

-+

-+

-+

4

0 2

0 2

+ + t-Bu. + + i-Bu. + + + ++ + + t-Bu. i-Bu. ( 9 ) i-Bu0.L. + i-BuH i-BuO2H + t-Bu. (1 0) i-BuOz. + i-BuH i-BuOzH + i-Bu. (1 I ) CHaOZ. + i-BuH + CHJOZH + t-Bu. (12) CH302. + i-BuH CH302H + i-Bu. (13) 2t-Bu02. -+ + t-BuzOz (14) t-BuOz. + i-BuOz. -+t-BuOH + i-PrCHO + (1 5) 2i-BuOz. + i-BuOH + i-PrCHO (16) CH30z. + t-BuOz. + L-BUOH + CHZO (1 7) CH302. + i-Bu02. 4- CH3OH + i-PrCHO (18) 2CH302. + CH30H + CHz0 (1 9) Zt-BuOz. + 2t-BuO. (20) t-BuO. CH3. + (CH3)zCO

(1) (2) (3) (4) (5) (6) (7) (8)

Reaction (kd)

+

fit

b

Source

9 A p ~ ~ 10.2 11.1 Apr~ 11.1 9.4 8.4 8.7 8.2 8.6 8.5 10.3 13.5

9.4

7.9 ~A,T

Source log A

p + 4~. 9 ~ Y E p ~ ~ E p ~ ~ + 4 . 90 E pT ~ E,,TT 4.9 0 9.0 k 2.0 2.0 2.0 2.0 I 2.0 9.0, ( E ~ T T ) 14 *

E

4.1 7.1 -0 -0 -0 16

Value of rate constant at IOO'C, E, 1. mole sec Source kcal/mole

LI 6.8X10-' b k,T 3.2X105 c k,r 5.1 X104, (0.161k,~) e ko -10' e ko -lo9 c ko -10' k p ~1.1 ~ fit kpTr 1.4X lo-', (1.25 X lo-' . k p ~ ~ ) 1 z kpiT 5.5, ( ~ ~ P T T ) kprr 7X lo-', ( 5 k P r r ) E kpMT 5.5, ( 5 k p T T ) kpMz 7x10-', (5kp~1) ktTT 1.3X104 i ktTr 1.9X107, (1/2ktrr) ktrz 3.8X107 m ktMr 1 .OX lo7, ( 1 / 2 k t ~ ~ ) k t M r 2.7X lo7, ( k t M M . ktlr)"' n ktMM 2.0x10' ' keTT 1.2X1o5, ( 9 k L T T ) kd 2.1X105 P

Ri

Rate constant

TABLE I. Reaction steps and their rate constants and Arrhenius parameters for the oxidation of isobutane in the liquid phase.

2i:

U

z

2 9

t: 0

m

E:

9

P

9

t-

$

-

-

R. Hiatt, T. Mill, K. C. Irwin, and J . K. Castleman, J . Org. Chem., 33, 1421 (1968). Estimate A = lo7 g, E = 4.1 kcal/mole; from CH30. data, T. Berces and A. F. Trotman Dickenson, J . Chem. SUG., 348 (1961). Estimated from results on the decomposition of f-Bu202 in alkanes at 100°C, ref. [12]. Calculated on the basis that the difference between k , and ~ k,l is due solely to the difference in activation energies after correction for the different numbers of hydrogens. Assumed to be diffusion controlled. f Estimated from experiments on the oxidation of n-butane and mixtures of n-butane and isobutane, ref. [3]. Calculated on the basis that the difference between k p T r and kprT is due solely to the differences in activation energies. A The value of 9 represents a statistical correction for 9 primary hydrogens for the one tertiary hydrogen in isobutane. ~ ~ generalization that primary ROs propagates about five times as fast as tertiary Ron. ; B. S. Middleton Estimated to be 5 k p from and K. U. Ingold, Can. J . Chem., 45, 191 (1966). 7 Average of literature values extrapolated to 100"; D. G. Hendry, J . Amer. Chem. SOG., 89, 5433 (1967); J. A. Howard and K. U. Ingold, Can. J . Chem., 46, 2655 (1968); J. R Thomas, J . Amer. Chem. SOG., 87, 3935 (1965); K. Adamic, T. A. Howard, and K. U. Ingold, Chem. Commun., 505 (1969). Average of values from references in footnotej. Most literature values for primary and secondary RO2. radicals give E 1-3 kcal. Estimated to be the extrapolated value for n-BuOz. termination at 100" (using E = 2 kcal/mole), ref.[€!]. Estimated to be the extrapolated value for the termination rate constant for the n-BuOz., ref. [8], but arbitrarily lowered by a factor of 2 to correct for the higher C-H bond strength in the CH302. radical. ' Estimated from results of the induced decomposition of t-BuOnH; A. Factor, C. A. Russell, and T. G. Traylor, J . A m r . Chem. Suc., 87, 3692 (1965); R. Hiatt. J. Clipsham, and T. Visser, Can. J . Chem., 42, 2754 (1964); R. Hiatt, T. Mill, K. C. Irwin, and K. C. Castleman, J . Org. Chem., 33, 1421 (1968). From k,/kd = 1.5 for t-BuO. in i-BuH, ref. [ 121. '1 E d - E , = 10 kcal/mole, average of values measured for t-BuO. radical in solutions of cyclohexane, ref. [ 1 I].

6

H

2

3 z

s

G!

z

E>

350

ALLARA, EDELSON, AND IRWIN

100

c

CCP,

SOLUTION

\ ,;.:

EXPERIMENTAL

d,cpl /'

d /'

/ /

GAS PHASE

0.01 I 0.01

I

I

I

1

1

1

1

1

I

I

1

1

1

1

1

l

I

1.o

0.1

,

I I I I I ,

10

[i-BUH] ,tvJ Figure 1. Calculated and observed relations of f t ~ / R ; *to / * [i-BuH] in oxidations of isobutane at 100°C.

+

butane, the major termination reaction is t-BuOz. i-BuOz-. Reactions of C H 3 0 2 -only become important below I M , and the reaction of 2t-BuOz- radicals contributes less than 20% a t all concentrations. Reactions of i-BuOz. radicals were not considered in the low-temperature mechanism previously [2]. Figure 3 also shows the contributions of each of the radicals to the overall abstraction of hydrogens from isobutane for Ri = 4.8X10-9 and 4.8X10V M / sec. The contributions were calculated from the steady-state values of the rate expressions ( k p ~ kpTI) ~ [t-BuOz-] [i-BuH], ( k p r T k , r r ) [i-BuOz.] [i-BuH], (kpMT k p ~ [CH302-] ~ ) [i-BuH] and (/car k,~)[t-BuO.1 [i-BuH]. Except a t very low isobutane concentrations, chain propagation is due almost entirely to the t-BuOz- radical.

+

+

+

+

35 1

ALKANES WITH OXYGEN

-lEL -6

I

I

I

-4

1

I

I

I

-2 0 L O G T I M E , SECONDS

I

2

I

I 4

Figure 2. Species concentration as a function of time for the liquid phase M/sec and [i-BuH] = 1M. oxidation of isobutane at 100°C with R; = 4.8 X

The calculated product distribution for the liquid phase oxidation of neat isobutane at 100°C for Ri = 9.OX 10W M/sec are given in Table 11. The actual experimental results from two runs obtained under the same conditions of concentration, temperature, rate of initiation, and length of experiment are also given. T h e agreement is quite satisfactory considering the precision of the experiments. Since the previously reported value of 15.8 f 1.5 kcal/mole activation energy for propagation by the t-BuOz. radical ( E p T T , reaction (7), Table I) appeared to be too large compared to other similar reactions [2], a value for E p T was ~ calculated. Using the activation parameters and the relations between them in Table I, values of the rate of oxygen uptake were calculated under the conditions of actual experiments of 50" and 80°C. Figure 4 shows the results for E p = ~12, ~14, and 16 kcal/mole at each temperature. The experimentally observed rates intersect the curves at 15.5 and 16.6 kcal/mole at 50" and 80"C, respecitvely. The average value of E p T T 16.0 kcal/mole agrees with the previously reported value of 15.8 kcal/mole [2]. The corresponding value of the A-factor, l./mole-sec, is an order of magnitude higher than that expected for similar atom transfer reactions [9].'

-

In this regard the calculated value for log A p 1 ~reaction , (9), 10.2, appears high and suggests hat E,IT may be less than E,TT, contrary to the assumption in Table I.

352

ALLARA, EDELSOK, AND IRWIN

toor o L

0 0

I

/ 2

,

3

i

%

Figure 3. Relative contributions of termination and propagation reactions in Iiquid phase oxidations of isobutane at 100°C.

TABLE 11. Product yields for the liquid phase oxidation of isobutane at 100°C." Yield,

yo

Product

Calculated

Experimental

(CHa)&O i-BuOzH t-BuOzH i-BuOH t-BuOH (CHs)&HCHO HZCO CH3OzH CHsOH

0.1 1.2 96.5