Calculation of reaction order for the homogeneous pyrolysis of organic vapors

Transcription

1 Calculation of reaction order for the homogeneous pyrolysis of organic vapors JOHN TOROK' AND SAMUEL SANDLER Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario Received November 1, 1968 A method is proposed for the derivation of a mathematical equation that relates the kinetic parameters of the propagation reactions of homogeneous free-radical chain reactions to the overall order of the reaction. The equation may be used in evaluating the predominant chain-termination reaction, provided that the mechanism of propagation of the chain is well understood. Application of the technique is demonstrated for the pyrolysis of n-butane. It predicts the predominance of the methyl-methyl radical recombination reaction. Canadian Journal of Chemistry, 47, 2707 (1969) Introduction Goldfinger, Letort, and Niclause (1) have shown that, for pyrolytic reactions, the overall order of the reaction depends on: (a) the order of initiation (first or second order), (b) the order of termination (second or third order), and (c) the type of radical participating in the chaintermination process. They referred to radicals involved in bimolecular propagation reactions as p radicals, and radicals undergoing unimolecular reactions in the propagating steps as p radicals. The various possibilities for the overall orders of such reactions are summarized in Table I. TABLE I Overall orders of reaction for various types of initiation and termination reactions Simple termination PP PP Pi.' First-order initiation Third-body termination PPM PPM uum Second-order initiation Simple Third-body termina- termina- Overall tion tion order PP 2 PP PPM 3{2 PP PPM PPM 162 Although Goldfinger's rule is helpful in establishing which chain-termination reaction is predominant, it neglects the treatment of the homogeneous reaction phenomena frequently encountered in chain reactions and affecting the overall order of such reactions. These include observations such as : (a) the rates of unimolecular 'Present Address: Norton Research Corporation, Chippawa, Ontario. initiation, propagation, or bimolecular termination reactions are proportional to a fractional power of the reaction pressure when in their pressure-dependent region, (b) some radicals, such as ethyl and isopropyl, may act both as P and as p radicals, and (c) with certain reaction conditions 2 or more chain-termination reactions may have comparable rates. The present paper is concerned with the mathematical treatment of the above phenomena. One study of the pyrolysis of n-butane is used to illustrate the effect of the first 2 points, while the third will be treated in a subsequent publication. Derivation of Order Equation Equations permitting the prediction of the overall reaction order for a simple pyrolysis mechanism will be derived first. It will be assumed initially that the initiation and termination reactions are in their pressure-dependent regions in the following generalized scheme. Initiation [I 1 aah + 2 free radicals + (a - l)ah Propagation [2 1 A- + B. + C Reactions [2] and [3] participate in a cyclic process. Termination [4] A. +A. + bah + AZ + bah (P- p termination) [5] A. + B. + cah + AB + cah (P - i.' termination) [6] B. + B. + dah -> BZ + dah (P - P termination) Here A. and B. are free radicals while AH,

2 2708 CANADIAN JOURNAL OF CHEMISTRY. VOL. 47, 1969 BH, A,, B,, AB, and C are organic compounds. In the following derivation the following assumptions are made. Assumption 1, the fundamental hypothesis of Bodenstein that the intermediate products of the reaction are of almost stationary concentration is valid. Assumption 2, the chain lengths are large. Since the rate of chain initiation is equal to the rate of chain termination [7] k, [AH:]" = k4 [A. l2 [AH:Ib tion of radical A. is [9] [A. ] = k1'lzk3 [AH]a'Z/(k32k4[A~]b + Ic21c3k5 [AH]"-' + IcZ2k6 [AH]d-2)112 The overall order of the reaction is equivalent to the slope of the In rate vs. In pressure relationship, or [lo] Order = d In rld In [AH] Substitution of the first part of eq. [8] into eq. [lo] results in + ks[a.][b.][ah]c + k6[b.]2[ah]d [AH] d(k2[as 1) [I11 Order = --. Based on assumption 2 the rates of the two Ic,[A.] d[ah] propagating reactidns, [2] and [3], are approxi- - [AH1 dea.1 mately equal and equivalent to the overall rate of -- [A*] 'qeq reaction, r, that is Equation [9] and its derivative with respect to AH may be substituted into eq. - [I After From eqs. [7] and [8] the steady-state concentra- rearrangement, one gets a - 1 [12] Order = - 2 By substituting into eq. [12] the rate expressions for reactions [4], [5], and [6], as modified by substitution of the second equality in eq. [8], the following expression is obtained, [I31 a- 1 (0.5- b/2)r4+(1.0-c/2)r5 +(IS-d/2)r6 Order = where ri = rate of reaction i. Goldfinger's table (Table I) may be generated from eq. [I31 when conditions such that only one of the three possible termination reactions, that is [4], [5], or [6], may be concerned and, when a equals 1 or 2 and either b, c, ordisequaltooor 1. For most pyrolytic reactions the initiation reaction is first order and the predominant termination reactions are second order. Under these circumstances a third assumption may be made. Assumption 3, the rate constant for the cross combination of free radicals can be calculated from their individual recombination rates. According to the simple collision theory of chemical kinetics r4 + r, + r6 Equation [13] then simplifies to the following expression [I51 Order (0.5k411Z/k2)[A~] + 1.5(k,112/k3) = (Ie411Z/Ic2)[AH:] + (k5'i2/k3) It should be noted that the order in eq. [I51 is a function of ratios of rate constants of the form k3/lc5112 and 1c2/k41/2. Photolytic investigations of free-radical reactions produce a ratio of rate constants of the form l~,,,/k,'~~, where Ic,,, is the rate constant of a propagation step involving a free radical as a reactant and k,is the rate constant of a chain-termination step involving the recombination of two of the sane radicals. In the above mechanism m = 2 or 3 and u = 4 or 5. Thus, a prediction of the overall order may be made if the kinetic parameters for the chainpropagation reactions are known.

3 TOROK AND SANDLER: REACTION ORDER CALCULATION 2709 Reaction Order when One of the Radicals Acts both as a P and as a p Radical, e.g., Pyrolysis of n-butane The ethyl radical may act both as a P and as a p radical. For example, in n-butane pyrolysis, it acts as a p radical in the following reaction and as a p radical in According to Purnell and Quinn (2) reaction [I71 is in its pressure-dependent region at the conditions of their investigation ( "C). It can more properly be represented by the following reaction [I81 C2Hs. + en-c4hlo -> CzH4 + H. + en-c4hlo where 1 > e > 0 Derivation of the order equation for n-butane pyrolysis will demonstrate the effect of an ambivalent radical such as the ethyl radical on the overall order of reaction. The following mechanism differs from the ones proposed by previous investigators (2, 3) in that primary (p-c4hg.) and secondary (s-c4hg.) butyl radicals are treated separately. Investigations of n-butane-2,2,3,3-d4 pyrolysis (4) and comparison of the kinetic parameters for relative p- and s-hydrogen abstraction from n-butane (5, 6) with the kinetic parameters for product distribution in n-butane pyrolysis (2) indicate that the rate of isomerization is negligible compared to the rate of decomposition of butyl radicals in the "C range. The results of an investigation by Kerr and Trotman-Dickenson (7) in this area indicate an appreciable rate of butyl radical isomerization. However, these results do not agree with the findings of other investigators of butyl radical decomposition kinetics (8-10). Mechanism of n-butane Pyrolysis Initiation [I91 C4Hlo -> 2C2H5. Propagation [20 1 p-cdh9. -> CZH4 + C2H5. Termination Assuming that all of the 5 free radicals can participate in recombination and cross-combination reactions and neglecting disproportionation reactions, the following recombination reaction equations may be written CH3. -t C2H6 For the free-radical, cross-combination reactions assumption 3 may be applied and the appropriate rates may be expressed in terms of the rates of reactions [24] to [28]. The relative rates of s- and p-butyl radical formation may be obtained from the product distributions where Q = f(t) only. Using the three assumptions which were applied to the simple pyrolysis mechanism, the rate of initiation = rate of termination where Rl = [CH3.I, R2 = [CzH5. 1, SR4 = [s-c4h9.], PR4 = [p-c4hg.], H = [H.], C = [C4Hlo], and J = a parameter defined as the square root of the rate of initiation (or termination). Since the rate of production of a radical equals the rate of its removal (in propagation reactions only) then and [31] k2,sr4 = k2,r1c r20 = r16 + r18 [21 I s-c~h~. -> C3H6 + CH3. [32] k2,pr4 = k16r2c + k18r2ce

4 2710 CANADIAN JOURNAL OF CHEMISTRY. VOL. 47, 1969 From eq. [29] [371 SR4 = (kzolkzl)(llq)pr4 [341 k2,pr4 = QkZlSR4 [381 H = (kzo/kz,c) From eqs. [311-[34] x (k18ce/(k16c + k18ce))pr4 Taking derivatives with respect to C of eqs. [35]-[38], the following relationships are ob- [361 R2 = (k2,/(k16c + klsce))pr4 tained. Taking the derivative of eq. [30] with respect to C results in the following expression Substitution of eqs. [39],[40],[41],[42], and finally eq. [30] into eq. [43] results in The rate of the overall reaction is equal to the rate of decomposition of butyl radicals. dln r C dr C dpr4 Order = -- = -- = --- dln C r dc PR4 dc Substituting eqs. [35],[36],[37],[38], and [45] into eq. [44], then [46 I J2 = 2J2 order - 2J(k241/2R1 + k2,1/2r2u + kz8lj2h(u - e + I )) where [471 k16c + ek18ce rate of reaction [16] + e rate of reaction [I81 U = - k16c + k18ce rate of reaction [I61 + rate of reaction [18]

5 TOROK AND SANDLER: REACTION ORDER CALCULATION Substitution of eq. [30] into eq. [46] and subsequent rearrangement results in an expression for the order c (Fi + F ~ ) ( Y ~ ~ ~ ) ~ ~ ~ i=28 j=28 c i=24 j=24 [48] Order = i=28 j= 28 (rirj)li2 i=24 j=24 where the values for i, j, 2Fi, and 2Fj are: Value Radical iorj 2(Fi Or J) where 0 < U < 1 and 0 < e < 1 Recent investigators (2, 3) of this reaction in the "C region have found the overall reaction order to be 1.5. Of the 5 radicals participating in the reactions, the recombination of only the methyl and ethyl radical and hydrogen could result in an order of approximately 1.5. Of these hydrogen is the most reactive since Its steady state concentration is thus expected to be much lower than that of methyl or ethyl radicals. Therefore The termination reaction would be one of the predominant chain-termination reactions if the two participating radicals were at comparable concentrations. However, Purnell and Quinn (2) did not find propane among the primary reaction products. This indicates that either reaction [24] or reaction [25] is overwhelmingly operative. It may be seen from eq. [48], that if reaction [24] predominates, the calculated order is 1.5. If, however, reaction [25] predominates then the value of U is needed to calculate the predicted order. Calculation of the Value of U From the expressions for the rate of production of ethane and ethylene by reactions [16], [18], and [20] and from eq. [32], the following expression may be derived. Purnell and Quinn reported values of the left side of eq. [50] at "C between 15 and 120 mm of Hg reaction pressure. They plotted these values as a function of pressure and fitted them with a curve calculated from Kassel's model. This treatment of the data illustrated the pressure dependence of reaction [18]. If natural logarithms of eq. [50] are taken the following relationship is obtained FIG. 1. The variation of the function In {[C2H4]/ [C2H6]- 1 }po with In initial pressure,^,, in the pyrolysis of n-butane at 518 "C (from Purnell and Quinn (2)). Figure I shows a plot of In {[C2H4]/[C2H6] - l}p, vs. lnp, calculated from the curve of Purnell and Quinn. Equation [51] indicates that the slope of the curve is equivalent to the value of e. From the values of e, 2(k18Ce/k16), and C, values of U were calculated and are listed in Table I1 along with the predicted overall reaction orders (assuming the predominance of the ethylethyl termination reaction). With an increase in reaction pressure the rate of reaction [16] increases relative to the rate of reaction [18]. This has the effect of increasing the overall order of the reaction. The order of reaction [18] on the other hand decreases with an increase in pressure and thus compensates for the above effect. The predicted overall order is approximately 1.35 and is relatively independent of

6 2712 CANADIAN JOURNAL OF CHEMISTRY. VOL. 47, 1969 TABLE 11 mechanism. Throughout the above analysis all Overall reaction orders at 518 "C calculated by assuming simple reactions were assumed to be homogereaction [25] as the predominant chain-termination neous. ~f reaction* the experimental order can not be predicted by the calculation procedure outlined, Reaction u it is likely that some of the simple reactions are pressure e calculated Overall partially or completely heterogeneous, or alter- (PO) using r:ip natively that incorrect chain-propagation remm of Hg Fig. 1 [C2H6] eq. [47] actions have been assumed The method outlined may be quite useful when : 3: chain-termination products are identical to the reactant or to products of the propagation reactions, making it necessary to determine the *Calculations are based on the experimental results of Purnell and chain-termination process indirectly, as in n- Quinn (2). butane pyrolysis. However, it is significantly lower than The order equation derived in this publication the experimental value of 1.5 reported by Purnell may be for the quantitative inter~retaand Quinn (2) and Sagert and Laidler (3). tion of the order change resulting from a change Based on the pressure dependence of the in the termination mechanism. The results of a activation energies for the formation of the major study of the kinetics of the thermal decomposition products, Purnell and Quinn (2) predicted the of n-butane at low temperatures indicate that, at predominance of the ethyl<thyl radical termina- 400 OC, the tion reaction (reaction [251). On the basis of the order decreases with a decrease in reaction overall reaction order and using the results of the This in reaction order may be same investigators, the above treatment predicts by the increasing importance of the predominance of the methyl-methyl radical 'eaction [261 reaction [241 at lower termination reaction. In view of this contradictory temperatures. The results of this investigation evidence, further work is necessary for the will be presented in a subsequent publication. establishment of the exact termination mecha- 1. P. GOLDFINGER, M. LETORT, and M. NICLAUSE. nism for n-butane pyrolysis. Contribution a I'ttude de la structure moleculaire. Victor Henry Commemorative Volume. Desoer, Conclusion Liege p J. H. PURNELL and C. P. QUINN. Proc. Roy. SOC. The London, Ser. A, 270, 267 (1962). order of a pyrolytic reaction can 3. N. H. SAGERT and K. J. LAIDLER. Can. J. Chem. 41, quite easily be determined in any kinetic study. 838 (1963). Its variation with temperature and pressure can 4. J. R. MCNESBY, C. M. DREW, and A. S. GORDON. J. Chem. Phys. 24, 1260 (1956). be with the 5. J. G. LARSON and A. KUPPERMAN LONG. Abstracts outlined above for any proposed mechanism, presented to Am. Chem. Soc. meeting. Washington, which then can provide an additional criterion for D.C., March 21-24, J. R. MCNESBY and A. S. GORDON. J. Am. Chem. the proper choice of the chain-termination re- 78, 3570 (1956). action. Such an analysis of the pyrolysis of 7. J. A. KERR and A. F. TROTMAN-DICKENSON. J. n-butane has indicated that the meth~l-meth~l Chem. Soc. 323, 1602 (1960). 8. F. E. FREY and M. J. HEPP. J. Am. Chem. Sot. 55, radical recombination is the major chain (1933). termination process. 9. W. E. MORGANROTH and J. G. CALVERT. J. Am. The of a derived hinges Chem. Soc. 88, 5387 (1966). 10. J. T. GRUVER and J. G. CALVERT. J. Am. Chem. on the choice of a correct chain-propagation SOC. 78,5208 (1956).