13544
J. Phys. Chem. 1996, 100, 13544-13547
Some Observations on the Thermochemistry and Kinetics of Peroxy Radicals Sidney W. Benson Hydrocarbon Research Institute, Chemistry Department, UniVersity Park, MC 1661, UniVersity of Southern California, Los Angeles, California 90089-1661 ReceiVed: February 13, 1996; In Final Form: May 21, 1996X
1
Recent studies from one laboratory over the past decade of the equilibrium reaction R• + O2 h RO2• for seven different radicals have led to discrepancies in six cases in ∆H1° of from 3 to 7 kcal/mol when compared to estimates from group additivity. These studies have utilized flash photolysis to generate R• radicals and have measured the decay of [R•] by mass spectrometry using near threshold photoionization. This decay was fitted to a double-exponential rate expression. Only one additional reaction was introduced in addition to 1 and -1, namely, the wall decay (w) of R• radicals, measured in the absence of O2. Reexamination of the differential equations has shown that they are not compatible with the reported data and in fact lead to negative values of K1 and k1. It has not been possible using the equation given by the authors of the papers under consideration to derive K1 from kw and the exponents of the double-exponential decay. It is shown that HO•2 elimination is important for EtO2•, iPrO2•, and tBuO2• at the temperatures used. If this is the only additional reaction, it is further shown that the authors’ values of ∆H1 are still not compatible with their equations. Peroxy radicals RO2• are key species in the mechanisms of all oxidative and combustion systems.1 At the same time they have been among the most difficult radicals to study experimentally. This difficulty arises in part from the lability of these radicals toward reversible unimolecular dissociation into R• + O2 and then to reversible isomerization into hydroperoxy alkyl radicals •R-HO2H, both of these reactions occurring at comparable rates at temperatures below 450 K. Early efforts to measure the heats of formation of RO2• radicals have made use of reversible reactions such as the nearly thermoneutral2,3
Br + RO2H h HBr + RO2• Perhaps the most reliable values have come from measurements of the heat of formation of the HO2• radical by a variety of methods.4 ∆fH°298(HO2;g) ) 3.5 ( 0.5 kcal/mol is probably the best established of all the RO2• radicals.5 Combining this value with ∆fH°298(H2O2;g) ) -32.6 kcal/mol4,5 and ∆fH°298(H;g) ) 52.1 kcal/mol6 yields a bond dissociation enthalpy, DH°298 (HO2-H;g) ) 88.2 ( 0.6 kcal/mol. There are good reasons to believe that all alkyl hydroperoxides have bond dissociation enthalpies that are described by DH°298(RO2-H;g) ) 88.6 ( 0.6 kcal/mol,5 and we shall adopt this as a benchmark. Over the past decade a series of studies on the equilibria of alkyl free radicals with O2 7-10 have yielded values of DH°298(R-O2•;g) that show differences from group additivity (Table 1) as large as 7 kcal/mol.7 Since this is far beyond the usual deviations associated with the use of group additivity in such systems,5,6 it is of some interest to explore the methods used to arrive at these results. The system used to study the equibibria 1
R• + O2 h RO2• employs a flash photolysis technique in which the radicals R• are generated by photolysis of a suitable reagent to form low X
Abstract published in AdVance ACS Abstracts, July 15, 1996.
S0022-3654(96)00448-0 CCC: $12.00
TABLE 1: Comparison of Measured and Estimated Values of ∆H°298(R• + O2 h RO2•) R• H CH3 C2H5 iPr tBu allyl
•CH OH 2
-∆H°/(obs), kcal/mol
-∆H°(est GA), kcal/mola
48.6 ( 0.5 32.2 ( 1.5 32.4 ( 0.7 >30.3 34.1 ( 0.5 39.7 36.7 ( 1.9 30.2 ( 1.1 18.2 ( 0.5 16.3 ( 0.3
[same] 30.0 ( 1.1 31.3 ( 1.1 32.2 ( 1.3 31.4 ( 1.1 18.6 ( 1.1 15.7 ( 1.1
ref 4, 5 15 8 3 9, 17 7 9 2 11, 12 5, 16, 18
a The values shown are based on using DH°298(CH3-H) ) 105.1; DH°298(Et-H) ) 100.4; DH°298(iPr-H) ) 97.0; DH°298(tBu-H) ) 93.5; DH°298(allyl-H) ) 88.5; and DH°298(H-CH2OH) ) 95.0 kcal/ mol.
concentrations of R•; thus CH3COCH3 photolysis at either 254 or 193 nm could be used to generate CH3 radicals. In the presence of sufficient added O2, CH3O2• radicals would be formed, and if the temperature were sufficiently high, then the above equilibrium would be realized. The authors employed mass spectrometry to observe R• using near threshold photoionization to form ions, which could be measured by ion-counting techniques. In addition to reaction 1 and its inverse -1, the authors report a slow disappearance of their radical signal, which they ascribe to an irreversible wall reaction of the radicals. Thus their system includes one more reaction: w
R• 98 products (?) In the absence of any O2, reaction w is the only reaction of radicals, and by following the reaction under these conditions, a first-order loss of R• is observed with measured rate constant k w. In a typical system in which only these three reactions occur we can write a set of two, linear, coupled differential equations: © 1996 American Chemical Society
Thermochemistry and Kinetics of Peroxy Radicals
J. Phys. Chem., Vol. 100, No. 32, 1996 13545
d(R•)/dt ) -k1(R•)(O2) + k-1(RO2•) - kw(R•)
(1)
d(RO2•)/dt ) k1(R•)(O2) - k-1(RO2•)
(2)
Such a system can be solved by standard techniques, and the solutions take the form of a double-exponential decay:
(R•) ) Ae-m1t + Be-m2t (RO2•) ) C(e-m1t - e-m2t)
}
(3)
where the initial decay characterized by m1 is always faster than the final decay, described by m2, and where we have employed the boundary condition on [RO2•] that at t ) 0 [RO2•] ) 0. We also note that at t ) 0, [R•] ) [R• ]0, the maximum concentration of R• radicals. Thus A + B ) [R•]0. Note also that C is negative since d(RO2•)/dt > 0 at t ) 0. On substituting the values of [R•] and [RO2•] from eqs 3 into eqs 1 and 2, respectively, we can obtain two equations involving only first-order terms in the exponentials exp(-m1t) and exp(-m2t). On setting the coefficients of these exponentials equal to zero (since the equations cannot otherwise be satisfied at all values of t), we obtain a total of four equations involving the ratios A/B and C/A, m1, and m2 and the three rate constants k1, k-1, and kw as well as the O2 concentration. On eliminating the ratios A/B and C/A from these equations, we are left with two identical quadratic equations: one is quadratic in m1 and the other in m2. Thus m1 and m2 are the two positive roots of these equations. Both these equations have the same form:
m2 - m(k1′ + k-1 + kw) + k-1kw ) 0
(4)
where k1′ ) k1(O2). The authors of refs 7-10 have never reported eq 4. They instead gave an equation for K1 in terms of A/B, m1, O2, and m2, their experimentally obtained variables:
K1 )
(A/B)(m1 - m2)2 [m2(A/B) + m1] (O2) 2
K1′ )
[(m2/m1) + B/A]
2
T, K
M × 10-16, molecules cm-13
O2 pressure × 105, atm
kw, s-1
m1, s-1
m2, s-1
692 672 672 652 652 632 632 612 602 592
5.83 5.84 5.84 5.83 5.83 5.83 5.83 5.81 5.82 5.82
23.0 9.06 7.55 9.20 4.67 6.09 4.09 2.83 3.50 1.32
8.2 13.2 13.2 24.8 24.8 24.2 24.2 23.6 28.1 29.5
1110 1150 848 889 735 559 774 350 460 392
189 173 147 128 106 50.7 84.5 15.4 31.0 60.8
TABLE 3: Conditions and Results of Experiments To Measure Equilibrium Constants9 T, K
O2 pressure × 105, atm
They used eq 5 to calculate all their values of K1, although they have never presented a formal derivation for it. The authors do not report individual rate constants for k1 and k-1 but only the measured values of A/B, m1, m2, kw, and K1 at each temperature. Tables 2 and 3 are examples. Since, however, kw is measured independently, it is a very simple matter to obtain explicit values of k1′ ) k1(O2) and k-1 directly from eq 4. The product of the two roots of eq 4, namely, m1m2, is given by the constant term
m1m2 ) k-1kw
(7)
k-1 ) m1m2/kw
(8)
This gives for k-1
The sum of the two roots of eq 4, m1 + m2, is given by the coefficient of m in eq 4:
m1, s-1
m2, s-1
C2H5 + O2 T C2H5O2 Experiments 1.34 23.1 100 3.04 20.7 145 3.97 22.0 205 4.24 22.8 180 4.39 18.9 182 9.39 18.9 331
17.8 16.0 36.6 26.5 38.4 56.3
550 560 560 560 570 570 580
t-C4H9 + O2 T t-C4H9O2 Experiments 0.368 46.8 185 100 0.397 45.6 166 0.817 45.6 276 0.817 41.4 346 0.826 45.9 345 1.60 45.9 505 1.64 40.9 342
37.3 39.0 48.7 49.7 63.6 65.4 73.7
m1 + m2 ) k1(O2) + k-1 + kw
(9)
k1′ ) m1 + m2 - kw - m1m2/kw
(10a)
(m1 - k2)(kw - m2) kw
(10b)
(5)
(6)
kw, s-1
609 617 624 634 654 654
)
Dividing top and bottom of the right-hand side by (m1A/B) 2 (note that m1 > m2) and setting K1′ ) K1(O2), we obtain
(B/A)(1 - m2/m1)2
TABLE 2: Conditions and Results of Experiments To Measure Equilibrium Constants:7 Experiments on the i-C3H7 + O2 T i-C3H7O2 Equilibrium (K6 × 10-3)
The equilibrium constant K1 is given by
(O2)K1 )
k1′ kw(m1 + m2) - (kw2 + m1m2) ) k-1 m1m2
(11)
which on rearrangement gives
)
(m1 - kw)(kw - m2) m1m2
(12)
For most of the data reported on m1, m2, and kw in all their various studies,7-10 eq 12 yields negative values of K1. Clearly there is a gross discrepancy between equations 12 and 5. Which is correct? I have been unable to derive eq 5, but a simple argument will reveal a gross discrepancy between the chemical mechanism used by the authors to describe their system and their experimentally obtained values of m1, m2, and A/B. For an immediate insight into the problem, consider eq 12. Since m1 is greater than m2, we see that in order for K1′ ) K1(O2) to be positive m2 must be less than kw! In fact the only way a positive result can be obtained for K1′ from eq 12 is if
m1 > kw > m2
13546 J. Phys. Chem., Vol. 100, No. 32, 1996
Benson
As we shall see, this is not in accord with experimental observation. A more quantitative view of this situation is obtained in the following paragraphs. Instead of using the quadratic eq 4 to derive values of k1′, k-1, and then K1 from the experimental values observed for m1, m2, and kw let us derive the values for m1 and m2 from a hypothetical system in which we select values of k1′ and k-1 and use a typical experimental value for kw. The roots m1 and m2 are given by the familiar formula
( [
])
4kwk-1 b mi ) 1 ( 1 2 b2
1/2
(13)
where b ) k1′ + k-1 + kw. We see that the term inside the radical 4kwk-1/b2 is much less than unity since kw is usually very small compared to k1′ in all the systems studied. Hence we can expand the square root term, (1 - x)1/2 ≈ 1 - x/2, and obtain
mi )
(
)
2kwk-1 b 1(12 b2
}
(14)
so that m1, the larger root, is given by (positive sign)
m1 ≈ b ) k1′ + k-1 + kw while (negative sign)
m2 ≈ kw(k-1/b) < kw
(15)
(16)
At time tm when this steady state is reached, d(RO2•)/dt ) 0 and [RO2•] has its maximum value [RO2•]m, after which it falls exponentially to zero; the exponent is given by m2t (eq 3). Note that at this time the term in eq 3, representing the exponential rise of [RO2•], exp(-m1t), has decayed to a very small value. Inserting the steady state value of [RO2•] in eq 1, we obtain •
•
d(R )ss/dt ) -kw(R )
(18)
At [RO2]ss ) 1/4 [R•], representing about 20% conversion of [R•]0 into [RO2•]ss, we require k1′ ) k-1/4 and
m2 1 ) < 0.80 kw 1.25 + kw/(1.25k-1)
(19)
And finally at [RO2]ss ) 4[R•], corresponding to about 80% conversion of [R′]0 into [RO2•]ss, we would need k1′ ) 4k-1 and
m2 1 ) < 0.2 kw 5 + kw/(5k-1)
(20)
Thus in all cases m2 < kw! For only one of the seven different radicals studied by the authors, that of allyl,7 was there a significant number of data points, 9 out of 12, for which it was found that m2 < kw. Interestingly enough, this was the only case which agreed strikingly well with the results of prior work5,11,12 and with group additivity. In Table 2 are shown the data reported on the reaction of iPr + O2,7 while in Table 3 are shown the data for Et + O2 together with tBu + O2.9 Discussion
This last equation for m2 is very important since it tells us that m2 < kw. In contrast to this conclusion, most of the data presented show m2 > kw (Tables II and 3). Is this a reasonable conclusion? Let us consider physically what happens after the flash when the radicals R• are able to react with O2. [R•] will decay from its initial value of [R•]0 at t ) 0 with a rate given by the sum of k1′ + kw (eq 1) since (RO2•)0 is zero. The values of k1′ and k-1′ are usually both much larger than kw so that [RO2•] quickly builds up to a quasi-equilibrium with [R•], where it reaches a steady state given by
[RO2•]ss/[R•] ) k1(O2)/k-1 ) K1(O2) ) K1′
k-1 m2 1 ) ) < 0.5 kw k1′ + k-1 + kw 2 + kw/(k-1)
(17)
so that we might expect the slow final decay of [R•] described by exp(-m2t) to have m2 ) kw. However eq 17 is exact only at tm. After tm it is approximate since it neglects the fact that after tm as R• decays with the usually slow rate constant kw, it is replenished by the continued dissociation of [RO2•], which constitutes a buffer reservoir for [R•]. Hence m2 will be much smaller than kw, as demonstrated by eqs 15. How much smaller than kw can m2 be? Let us consider a hypothetical case corresponding to the most sensitive conditions for measuring m1 and m2 with accuracy. Let us pick a case where k1′ ) k-1 so that K1′ ) 1 and the stationary state of RO2•, [RO2•]ss, ) [R•] so that about half of the initial [R•]0 are present as [RO2•]ss, and we have neglected the small loss of R• via kw. From eqs 15 we see that (k-1)/b is given by (setting k1′ ) k-1):
We can well ask how it is possible for the observed slow decay of radical concentration with rate constant m2 to be faster than kw, the observed decay rate of R• alone. One obvious answer is that there may be an irreversible reaction of RO2• radicals. This could be the known isomerization reaction of RO2• which proceeds by abstraction of an H atom from a β-carbon atom to form a hydroperoxy alkyl radical and subsequent faster cleavage to olefin + HO2. This is not possible for the allyl radical, the methyl,8 or the halogenated methyl radicals,10 but it is possible for ethyl, isopropyl, and tert-butyl.9 The missing reaction could also be a wall reaction of the RO2• radicals. In none of the cases studied were the walls coated to prevent the well-documented decay of RO2• radicals on either Pyrex or quartz. The addition of such a reaction, r
RO2• 98 products leads to a quadratic equation similar in form to eq 4:
m2 - bm + c ) 0
(21)
b ) k1′ + k-1 + kw + kr
(22)
c ) k-1kw + kr(k1′ + kw)
(23)
m1 + m2 ) k1′ + k-1 + kw + kr
(24)
m1m2 ) k-1kw + kr(k1′ + kw)
(25a)
with
so that
where
[ (
) k-1kw 1 +
)]
kw kr K ′+ kw 1 k-1
(25b)
Thermochemistry and Kinetics of Peroxy Radicals
J. Phys. Chem., Vol. 100, No. 32, 1996 13547
The net effect is to increase kw by an additive term, kr (eq 24), or by a multiplicative term which is (eq 25) approximately [1 + krK1′/kw]1, since we can usually neglect kw/k-1 relative to K1′, which will generally be in the range 5 > K1′ > 0.25. An explicit solution of eqs 24 and 25a leads to
k1′ )
(m1 - kw)(m2 - kw)
k-1 )
(kr - kw) (m1 - kr)(m2 - kr) (kw - kr)
kr > kw; m2 > kw and kr > m2
case II:
kr < kw; m2 < kw and kr < m2
(26)
(27)
then
we see that case II amounts to the situation in which kr is a relatively small correction to the original system, while in case I with m2 > kw, kr is a major correction to the original system. Most experimental points fit the condition m2 > kw and hence case I! (Tables 2, 3). The authors7 have considered the effects of the reaction
iPrO2• f C3H6 + HO2• and dismissed it on what seem to be insufficient grounds. They made an effort to analyze for C3H6 in the reaction products of an experiment done at 500 K in large excess of O2. They found very little above an expected background. However their O2 experiments were carried out in the range 592-692 K, much above 500 K, and this reaction with an appreciable activation energy is much faster. It is actually a complex reaction that starts as a rate-determining isomerization r: r
f
(CH3)2CHO2• h C•H2C(CH3)HO2H 98 C3H6 + HO2 kr can be estimated,12-14 and it can be shown that kf > k-r so that kr is rate determining. We estimate kr as
kr ∼ 1012.8-28/θ s-1 where θ ) 2.303RT in kcal/mol. kr has uncertainties of a factor of 2 in the A-factor and about 1 kcal in the activation energy.6 At 500 K where the test of step r was made, its value is about 5 s-1, too small to be significant in the 0.03 s of the test run. However in the range 592-692 K where the runs on iPrO2 were made its value ranges from 102.7 to 104.3 s-1, putting it in the range of m1 and k-1. An estimate of k-1 at its high-pressure limit yields
k-1(iPrO2•) ∼ 1015.5-31/θ s-1
k-1/kr ) 102.7-3/θ which varies from a value of 40 at 592 K to 50 at 692 K. These are precisely in the range to make kr of major importance and far more important than the wall reaction rate constant kw (eq 25) when k-1 is much below its high-pressure limit. If we were to use the author’s value of ∆H1,7
Positive values of k1′ are possible from eq 26 in two cases: (1) kr > kw and m2 > kw or (2) kr < kw and m2 < kw. However, in case 1, we obtain a positive value for k-1 (eq 27) only when kr > m2 and m1 > kr. In case 2 with kr and m2 both less than kw we obtain a positive value for k-1 (eq 27) only if kr < m2. Summarizing the two cases,
case I:
Again uncertainties are about a factor of 2 in the A-factor and 1 kcal in the activation energy. We see that
k-1 = 1015.5-39/θ s-1 and
k-1 = 102.7-9/θ kr which has the value of 0.3 at 592 K and about 0.9 at 692 K, making kr faster than k-1 over the entire range. Uncertainties in our estimates are increased in the ratio k-1/ kr so that it is not useful to try to further examine the data, particularly in the light of uncertainties about whether k1 or k-1 is in pressure dependent regimes. We are forced to conclude that there are unreconcilable discrepancies between the data reported in these papers and the mechanisms used to interpret them. Until these can be resolved, thermochemical data derived from these measurements must be placed in abeyance. References and Notes (1) Benson, S. W. J. Am. Chem. Soc. 1965, 87, 972. (2) Heneghan, S. P.; Benson, S. W. Int. J. Chem. Kinet. 1983, 15, 815. (3) Kondo, O.; Benson, S. W. J. Phys. Chem. l984, 88, 6675. (4) Shum, L. G. S.; Benson, S. W. J. Phys. Chem. 1983, 87, 3479. (5) Benson, S. W., Cohen, N., Alfassi, Z., Eds. John Wiley & Sons, in press. (6) Benson, S. W. Thermochemical Kinetics, 2nd ed.; John Wiley and Sons: New York. (7) Slagle, I. R..; Ratajczak, E.; Heaven, M. C.; Gutman, D.; Wagner, A. J. Am. Chem. Soc. 1985, 107, 1838. (8) Slagle, I. R.; Gutman, D. J. Am. Chem. Soc. 1985, 107, 5342. (9) Slagle, I. R.; Ratajczak, E.; Gutman, D. J. Phys. Chem. Soc. 1986, 90, 402. (10) Russell, J. J.; Seetula, J. A.; Gutman, D.; Melius, C. F.; Senkan, S. M. 23rd Symposium on Combustion, 1900; Combustion Inst. Pittsburgh: PA, 1990. (11) Ruiz, R. P.; Bayes, K. D.; McPherson; M. T.; Pilling, M. J. J. Phys. Chem. 1981, 85, 1622. (12) Tullouch, J. M.; McPherson, M. T.; Morgan, C. A.; Pilling, M. J. J. Phys. Chem. 1982, 86, 3812. (13) Some Current Problems in Oxidation Kinetics. Benson, S. W. The Mechanisms of Pyrolysis, Oxidation Burning Materials; NBS Spec. Publ. 357; Wall, L. Ed.; 1972. Invited Lecture at 4th Materials Research Symposium, NBS, 1970. (14) The Kinetics and Thermochemistry of Chemical Oxidation with Application to Combustion and Flames. Benson, S. W. Prog. Energy Combust. Sci. 1981, 7, 125. (15) Khachatryn, L. A.; Niazin, O. M.; Mantashyan, A. A.; Vedeneev V. I.; Teital’boim, M. A. Int. J. Chem. Kinet. 1982, 14, 1231. (16) Su, F.; Calvert, F. G.; Shaw, F. H. J. Phys. Chem. 1979, 83, 3185. (17) Wagner, A. F.; Slagle, I. R.; Sarzynski D.; Gutman, D. J. Phy. Chem. 1990, 94, 1853. (18) Burrows, J. P.; Moortgat, G. K.; Tynall, G. S.; Cox, R. A.; Jenkin, M. E.; Hayman, G. D.; Veyret, B. J. Phys. Chem. 1989, 93, 2375.
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