J . Phys. Chem. 1990, 94, 4941-4945
4941
Rate Constants for Some Radical-Radical Cross-Reactions and the Geometric Mean Rule Leslie J. Garlandt and Kyle D. Bayes* Department of Chemistry & Biochemistry, University of California, Los Angeles, Los Angeles, California 90024- I569 (Received: October 30, 1989)
Rate constants for four radical-radical cross-reactions have been measured at 300 K. A laser pulse formed two different radicals simultaneously, and their decays were followed with a photoionization mass spectrometer. The major radical, formed at high concentration, decayed predominantly by reaction with itself. The minor radical decay was determined primarily by reaction with the major radical. Computer modeling was used to simultaneously fit the decays of both radicals. The measured rate constants, in units of IO-" cm3 molecule-' s-I, are 6.5 f 2.0 for CH3 + C3H5,9.3 f 4.2 for CH3 + C2HS. 10.5 f 4.0 for C3H5+ C2H5, and C0.6for CH, CCI,. The slow rate observed for CH, CCI, is not caused by a falloff effect. These results, and others in the literature, are compared to the predictions of the geometric mean rule.
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Introduction
Gas-phase radical-radical reactions have been actively studied for more than 50 years.' These reactions are the reverse of unimolecular decompositions, and so their rate constants can be used to quantitatively test theories of unimolecular reactions. In addition, the rate constants are valuable in understanding the chemistry of combustion,2 hydrocarbon cracking, and air polluti~n.~ Most experimental work has focused on the rates of radical self-reaction, with CH, + C H 3 receiving the most a t t e n t i ~ n . ~ Reactions between unlike radicals (cross-reactions) have been studied less thoroughly, in part due to the additional experimental complexities involved. Kerr and Trotman-Dickenson5 observed that cross-reaction rate constants were related to the rate constants for the self-reactions of the radicals by what was called the "cross combination ratio" 4'
+' =
[AB1 ([AA] [BB])'/2
--
kEm ( k E r mk B B
(1)
where [AB], [AA], and [BB] are the concentrations of various recombination products formed by reaction of radicals A and B, and km refers to the rate constant for the recombination pathway only (i.e., disproportionation is not included). The experimental values of +'were found to be approximately 2 for a variety of alkyl
radical^.^ Further work6 showed that, according to the theory that Kerr and Trotman-Dickenson had used to explain their findings, a total cross-reaction ratio 4, defined by should be equal to 2.0. In eq 2, kAB,k A A , and kBBrepresent total rate constants, in which all reaction pathways are included. This then led to the "geometric mean rule"
(3) which has been used to estimate k A B when only the self-reaction rate constants are known.' It is important to know for what conditions this rule is valid, so that it is not used in inappropriate situations. While the geometric mean rule appears to be valid for many cross-reactions, some combinations, especially involving polar radicals ( C H 3 0 and CH2COCH3),show significant deviations.* Anastasi and Arthur9 recently used molecular modulation spectrometry to determine the rate constants for several alkyl radical cross-reactions. They found that for CH, t-C,H, was significantly less than 2.
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'Present address: Loker Hydrocarbon Research Institute, University of Southern California, Los Angeles, CA 90089-1661.
0022-3654/90/2094-494l$02.50/0
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TABLE I: Radical Precursors, Typical Partial Pressures, and Resonance Lines Used To Photoionize the Radicals partial resonance radical precursor pressure. mTorr line, nm CH, acetone 1-35 121.6 (H) CIHS 3-pentanone 5 147 (Xe) C,H, 1,Shexadiene I-so 147 (Xe) CCI, hexachloroacetone 17 121.6 (H) carbon tetrachloride 50 121.6 (H) chloropicrin 50 121.6 (H)
Photoionization mass spectrometry is well suited for studying hydrocarbon free radicals, as well as other radicals that have low ionization potentials. Radicals can be identified by their parent mass, and the signals can be followed in real time. This technique has been used'O to measure a variety of rate constants for alkyl radicals reacting with 02,O,, and CI2. In the present paper, photoionization mass spectrometry is used to determine four cross-reaction rate constants. Experimental Section
Two different radicals were made simultaneously in the same laser flash. By controlling the precursor concentrations, it was possible to form one radical (the major radical) with a much higher concentration than the other (the minor radical). Consequently, the decay of the major radical was determined primarily by the rate of reaction with itself, with only minor contributions from other reactions. The decay of the minor radical was dominated (1) Steacie, E. W. R. Atomic and Free Radical Reactions; 2nd ed.; Reinhold: New York, 1954. (2) Gardner, W. C., Jr., Ed. Combustion Chemistry; Springer-Verlag: New York, 1984. (3) Finlayson-Pitts, B. J.; Pitts, J. N., Jr., Atmospheric Chemistry; Wiley-lnterscience: New York, 1986. (4) (a) Macpherson, M. T.;Pilling, M. J.; Smith, M. J. C. J . Phys. Chem. 1985, 89, 2268. (b) Slagle, I . R.; Gutman, D.; Davies, J. W.; Pilling, M. J. J . Phys. Chem. 1988, 92,2455. (c) Wagner, A. F.; Wardlaw, D. M. J. Phys. Chem. 1988, 92, 2462. ( 5 ) Kerr, J. A.; Trotman-Dickenson, A. F. frog. React. Kine!. 1961, I , 105. (6) Blake, A. R.; Henderson, J. F.; Kutschke, K. 0. Can. J . Chem. 1961, 39, 1920. (7) (a) Tsang, W.; Hampson, R. F. J . Phys. Chem. Ref. Data 1986, 15, 1087. (b) Tsang, W. J . Phys. Chem. Ref. Data 1988, 17, 887. (8) (a) Calvert, J. A. Annu. Reu. Phys. Chem. 1960, ! I , 41. (b) Benson, S. W.; DeMore, W. B. Annu. Reo. Phys. Chem. 1965, 16, 391. (9) Anastasi, C.; Arthur, N . L. J . Chem. Soc., Faraday Trans. 2 1987,83, 277. (IO) (a) Xi, 2.;Han, W.-J.; Bayes, K . D. J . Phys. Chem. 1988, 92, 3450. (b) Ruiz, R.; Bayes, K. D. J. Phys. Chem. 1984,88, 2592. (c) Paltenghi, R. N.; Ogryzlo, E. A,; Bayes, K. D. J. Phys. Chem. 1984,88,2595. ( d ) Timonen, R. S.; Gutman, D. J . Phys. Chem. 1986, 90, 2987.
0 I990 American Chemical Society
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Garland and Bayes
The Journal of Physical Chemistry. Vol. 94, No. 12, I990
by reaction with the major radical. The cross-reaction rate constant was determined by fitting a computer model to the two time-dependent radical signals. Small amounts of radical precursor gases were mixed with a bath gas, usually argon, in a 12-L bulb attached to a gas handling manifold. The mixture of gases flowed continuously through the cylindrical reaction cell (volume 100 cm3), where the radicals were generated photolytically by the pulsed radiation from an ArF excimer laser. The total pressure in the cell was approximately 4 Torr. T o achieve a uniform initial distribution of radicals, the laser radiation ( 1 93 nm) was defocused before entering the reaction cell, giving a uniform intensity of approximately 5 mJ cm-* a t the front window. A small portion of the reaction mixture continuously exited the reaction cell via a pinhole and entered the high-vacuum region of the mass spectrometer where the radicals were selectively photoionized by ultraviolet radiation from microwave-powered resonance lamps. The precursors used to generate the various radicals, their partial pressures in the cell, and the resonance lines used for photoionization are given in Table I . More experimental details are available elsewhere." There was no evidence that the radicals reacted significantly with precursor molecules or contaminants. When a single precursor was used at low concentration, the radical signal decayed exponentially with a time constant only slightly larger than the pump-out rate. The latter was calculated by using measured flows and treating the system as a stirred-flow reactor. Increasing the precursor concentration, while also attenuating the laser intensity so that the radical signal stayed the same, did not alter the decay rate, indicating no significant reaction of radicals with precursor for the conditions used. These exponential decay rates (10-19 S I ) were slightly larger than the calculated pump-out rates (5-9 s-l) probably due to slow reaction of the radicals on the walls; cleaning the reactor would sometimes reduce these exponential decay rates. In experiments with high precursor concentrations, the radical signal was well described by a mixed first- and second-order fit with the first-order losses equivalent to pump-out losses. For some radicals, different precursors and different bath gases were used, with no significant effects on the decay rates. The radical precursors were all degassed by four or five freeze-pump-thaw cycles. The helium (99.998%). argon (99.995%1), and 1% hydrogen-in-argon mixture, used for the hydrogen resonance lamp, were used without further purification. Computer Fitting and Simulations
Because of the mixed first- and second-order decay of the major radical, and the dependence of the rate of the minor radical decay on the major radical concentration, it was necessary to perform numerical simulations of the reaction system in order to extract the cross-reaction rate constant. The coupled differential equations were numerically integrated on a computer (DEC 1 1 /23). The kinetic model used is quite simple, with only five basic reactions involving radicals A and B. A
+A
+B A + B B
A
B
+
-
-
+
products
(kAA)
products
(kB8)
products
(kAB)
removal
(k,)
removal
(k,)
Since only the radical signals were measured, the nature of the products need not be specified in the model (provided none of the reactions re-form radicals A or B; as mentioned below, radicals may have been regenerated for some conditions). The two first-order loss processes represent the sum of the rates of pump-out from the cell and reaction on the walls. Reactions of the radicals with precursor molecules are not included in the model as there ( I I ) Garland, L. J . Ph.D. Dissertation, University of California, Los Angeles. 1989
TABLE 11: Values Used for Radical Self-Reaction Rate Constantsu radical k,,, lo-" cm3 molecule-' s-I ref CH3 5.98 f 0.67 4b 4.8 (in 4 Torr of Ar) 3.7 (in 4 Torr of He)
C2HS C3HS CCI,
2.03 f 0.61 2.66 i 0.20 0.33 f 0.08
7a b. c 18
"These represent high-pressure limiting values at 298 K, unless noted otherwise. *Tulloch, J . M.; Macpherson, M. T.;Morgan, C. A,; Pilling, M . J. J . Phys. Chem. 1982, 86, 3812. CSears.T. S.; Volman, D. H. J . Photochem. 1984, 26. 85.
was no experimental evidence for these reactions for the conditions used. I n order to carry out the numerical integrations, values for the initial radical concentrations and the various rate constants were needed as input. The rate constants for the self-reaction of the radicals (kAA and kee) were taken from the literature; the values used are collected in Table 11. Values for k , and kB were usually just the calculated pump-out rate or, sometimes, the values measured in single-radical decay experiments; the final fits were not critically dependent on these values. The initial radical concentrations were calculated by using the spectrometer sensitivities as measured in separate experiments. The radical signals are related to the radical concentrations by the mass spectrometer sensitivity, SAor S;, for example signal, = S,[A] (4) In practice, there is also a detector background signal observed at long times after the laser flash. This background signal is not indicated in eq 4, although it was included when doing the data analysis. The sensitivities were measured in experiments having a large concentration of a single radical. For these conditions, the radical decay was primarily second order. The most critical fitting parameter is the ratio of rate constant to sensitivity. By substituting eq 4 into eq 5 , rearranging, and adding in the detector background, B,, we obtain
The experimental signals as a function of time could be fitted by eq 6 using three fitting parameters, which were interpreted as ( ~ ~ A A S A and - ' ) ,B,. Using the literature value for kAA, we could then calculate the sensitivity of the system to radical A. Once an SAvalue was available, the full signal as a function of time could be converted to an absolute concentration of A as a function of time. In actual practice, the model also included the possibility of a first-order decay process for A as well as the self-reaction. For a single radical, the working equation is then signal, = (signal,),((I
+ C ) exp(k,t)
--
C1-l
+ Bg
(7)
where C is a collection of terms In the limit as k A approaches zero, eq 7 reduces to eq 6. In fitting the experimental signals to eq 7, it was possible to impose a value for k , and search for the best fit to the remaining three parameters or to search for all four parameters. The former procedure was usually used, with a value for k A equal to the pump-out rate. When all four parameters were fit, the resulting values of k , were comparable to the pump-out rate. In practice, the critical value of (2kA,SA-') was not strongly dependent on the value of k , imposed. Due to slow changes of lamp intensity with time, the radical sensitivities were redetermined daily. By use of these sensitivities, the measured signals of A and B in a mixed experiment could be converted to concentrations and the initial concentrations estimated by extrapolating back to zero
Radical-Radical Cross-Reactions 7000
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990 4943
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Figure 1. Methyl plus allyl reaction with allyl as the minor radical. The experiments were performed with 1.2 mTorr of 1,5-hexadieneand 34 mTorr of acetone in He at a total pressure of 4.006 Torr and a pumpout rate of 5.7 s-I. The simulations shown used [C3HS],= 4.25 X 1O’O and [CH,],= 2.93 X 10l2cN3; kAB(in lo-” cm3 molecule-’ s-I) varied by &50% around the best-fit value: solid line, kAB= 5.83; long dashed line, kAB= 8.70 short dashed line, k A B = 3.91. (a) Methyl radical decay,
signal averaged over 1463 laser pulses. (b) Allyl radical decay, signal averaged over 2568 laser pulses. time. Inserting literature values for kAA and kBB, and an initial guess for kAB,the numerical integration could be done. Then goodness-of-fit parameters, x2, were calculated for both the concentrations of [A] and [B]:12 x2 = N-’ E.Cvi(expt) - yi(model)12(a?)-’ (9) Here N is the number of data points, usually 255, and the ai values were calculated from the square root of the number of counts in each channel divided by the radical sensitivity. A minimum in x2 was used as a criterion for a good fit of the model to the experimental measurements. Usually the major radical could be fit by minor variations in its initial concentration; variations in kABor the initial concentration of the minor radical had little effect on the calculated X2(major). Once the major radical signal was well fitted by the model, the X2(minor) was minimized by varying its initial concentration and the value of k A Bby means of a grid search.I2 Clearly, the most important input parameter for X2(minor) was k A B , especially when the initial major radical concentration was much larger than that of the minor radical. Both the major radical and the minor radical signals were compared with the same computer simulation. Even though the model showed that the major radical decay was insensitive to the initial minor radical concentration and to kAB,the major radical fit was checked again once the best conditions for the minor radical were determined. Examples of these fits are shown in Figures 1 and 2. Figure I represents an experiment in which the initial concentration of the major radical, CH3,was larger than that of the minor radical, ( I 2) kvington, P. R.Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969.
TIME/ms
Figure 2. Methyl plus allyl reaction with methyl as the minor radical. The experimental conditions were 1.2 mTorr of acetone and 17 mTorr of 1,s-hexadienein He with a total pressure of 4.005 Torr and a p u m p out rate of 6.1 s-l. The simulations shown used [CH,], = 3.12 X 10” cm-) and [C3H5lo= 1.76 x 1 O l 2 kAB(in lo-” cm3 molecule-I s-l) varied by &50% around the best-fit value: solid line, kAB = 8.97; long dashed h e , kAB= 13.38; short dashed line, k A B = 6.01. (a) Allyl radical decay, signal averaged over 746 laser pulses. (b) Methyl radical decay, signal averaged over 2849 laser pulses.
C3H5,by a factor of 73. Figure 2 shows an experiment in which the initial allyl radical concentration was 5.6 times greater than the initial methyl radical concentration. In each of these figures, the experimental data points are shown along with a solid line representing the computer simulation best fit. In addition, two computer simulation runs are shown as dashed lines in which the value of kAB was varied by f50%. As can be seen, the fits are quite sensitive to the value of kABused in the simulation. It was observed that some of the high-concentration radical decays are not fit well by the simple model described above. The radicals are initially lost more quickly and later more slowly than the model predicted. Such an effect might be caused by the radicals absorbing reversibly onto the walls. At early times some radicals would be lost to the walls, while at long times desorption from the walls would replenish the radical concentration in the gas phase. This interpretation is supported by the observation that the effect was not entirely reproducible; cleaning the cell tended to eliminate the problem for a period of time. If this interpretation is correct, the rate of radical adsorption on the walls can be estimated from the low-concentration single-radical experiments. The first-order radical loss was usually greater than the rate of pump-out; the difference should be the rate of adsorption on the cell walls. kadsorption
=
kA
- kpumpout
(10)
The amount of radicals adsorbed onto the walls would be small in all cases; a coverage of only monolayer would be enough to account for the observed effect. When such an absorptiondesorption process was included in the model, the major radical fit was improved significantly in those cases which were previously
The Journal of Physical Chemistry, Val. 94, No. 12, 1990
4944
TABLE 111: Experimental and Calculated Cross-Reaction Rate Constantso radicals
1.4-
kAB
A
B
CH, CH, C2H5
C*HS C3H5 C,H, CCI,
CH,
Garland and Bayes
experimental 9.3 f 4.2 6.5 f 2.0
10.5 f 4.0
