Unimolecular Rate Expression for Cyclohexene Decomposition and Its

Jan 13, 2015 - Unimolecular Rate Expression for Cyclohexene Decomposition and Its Use in Chemical Thermometry under Shock Tube Conditions...
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Unimolecular Rate Expression for Cyclohexene Decomposition and Its Use in Chemical Thermometry under Shock Tube Conditions Wing Tsang and Claudette M. Rosado-Reyes* National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States ABSTRACT: The methods used in deriving the rate expressions from comparative rate single-pulse shock tube studies, recent direct shock tube studies, and high-pressure flow experiments bearing on the data for the reverse Diels−Alder decomposition of cyclohexene to form ethylene and 1,3-butadiene are reviewed. This current interest is due to the increasing need for accurate kinetics and physical data (particularly the temperature) for realistic simulations in practical areas such as combustion. The rate constants derived from the direct shock tube studies and high-pressure flow experiments are somewhat larger than those used in comparative rate single-pulse shock tube experiments. For the latter, it is shown that they have been derived from a variety of independent experiments that include rate constants for unimolecular decomposition and isomerization processes that are considered to be well understood. The possibility of non-Arrhenius behavior in the unimolecular rate constants as a consequence of the large range covered in rate constants (as much as 12 orders of magnitude) for the comparative rate experiments has been examined and ruled out as a source of the discrepancy. Our analysis shows that there is the need to consider the possibility of radical-induced decompositions for verifying the correctness of the reaction mechanisms in studying unimolecular reactions. In the case of cyclohexene decomposition, recent experiments demonstrating the presence of residual amounts of H atoms in shock tube experiments suggest that addition to the double bond can also lead to the formation of ethylene and 1,3-butadiene and hence to rate constants larger than the true values. This possibility is even more likely to occur in high-pressure flow experiments. As a result, the internal standard method must be used with care and a radical inhibitor should always be present in sufficiently large quantities to suppress possible chain reactions. The present analysis results have important implications for the determination of temperatures in shock tubes.

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INTRODUCTION Unimolecular reactions, at the high-pressure limit, are the simplest of kinetic processes. It is a fundamental property of a molecule and describes the changes it undergoes as a function of time in the absence of any chemical interactions with other molecules at a given temperature. Quantitatively, it is characterized as the rate expression describing the chemical changes as a function of time and involves the first-order decomposition of a polyatomic molecule at a given temperature. For present purposes, we consider only molecules with a thermal distribution. The more general case involving molecules at lower pressures with nonthermal distributions are not of present applicability. All molecules can undergo unimolecular decomposition at an appropriate temperature. We are interested in the temperature range in which larger polyatomic organic molecules decompose. Because of their relative complexity, other pathways can lead to products that are similar to those from the unimolecular pathway. It is only when certain precautions are taken that a true unimolecular rate process can be studied experimentally. Indeed, many of the first reactions that have been characterized as unimolecular processes subsequently have been found to be in error. The general problems are well-known and described in standard texts.1 A particular aspect of kinetic measurements is the contrast with thermodynamic properties. Because of the dynamic nature of the phenomena, uncertainties are much more poorly defined and difficult to quantify. This will become clear in the subsequent discussion. © XXXX American Chemical Society

The rate constants for a true unimolecular reaction can be expressed in the Arrhenius format, k = A exp(−E/RT), or over extensive range of temperatures in the modified Arrhenius format, k = ATn exp(−E/RT), where n is a small number. The exponential dependence means that if the rate is known then determination of the rate constant can lead directly to a measure of the reaction temperature. The exponential dependence of the rate constant on the reaction temperature also means that the former can be an extremely sensitive measure of temperature. In shock tubes, the conservation relations permit the determination of the reaction temperature from the shock velocity. The growth of boundary layer makes this relation subject to errors, especially when mixtures containing even trace amounts of larger polyatomic molecules are shocked and if studies are carried out in the reflected shock region. A direct consequence from these errors is the large scatter in shock tube results. Because of the dependence of the incident shock temperature on the square of the shock velocity, an accuracy of 1 degree in reaction temperature at 1000 K requires an accuracy of 0.05% in the shock velocity. This is a severe limitation and Special Issue: 100 Years of Combustion Kinetics at Argonne: A Festschrift for Lawrence B. Harding, Joe V. Michael, and Albert F. Wagner Received: September 25, 2014 Revised: December 11, 2014

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The initial step in the unimolecular decomposition of any molecule can be obscured by contributions from the subsequent steps if they produce reactive species. To prevent induced decompositions and suppress chain processes in singlepulse shock tube experiments, two prerequisites have to be fulfilled. First, the reaction in question has to lead to the formation of stable products. Second, radical-induced reactions cannot lead to the same products as those formed from the purely unimolecular reaction of interest. Both are achieved by the addition of methylated benzene or similar compounds where loss of hydrogen leads to a resonance-stabilized radical in large amounts. As a result, reactive radicals such as H atom or methyl radicals are removed from the system and much less reactive species, such as benzyl type radicals, are formed. The general process is

may have never been achieved. The comparative rate technique that we developed many years ago removes the temperature as a measured quantity and substitutes rate constants. If one of these rate expressions has been determined from an independent measurement, then a reaction temperature can be deduced. The technique has the added advantage that there is large-scale cancellation of uncertainties. Comparative rate single-pulse shock tube studies are the basis of a great deal of the existing kinetics work on the quantitative details of the stability of organic molecules. Much of the accepted bond dissociation energy (BDE) and heat of formation values of the radicals that are formed upon bond cleavage are deduced from the activation energies. These chemical properties have largely been confirmed by other types of kinetics experiments. The rate expressions are the basis for the rate rules that allow the prediction of relevant quantities for unstudied molecules. In contrast to the direct determination of the temperature from the shock velocity, the comparative rate method relies on the relative rate constant or the relative concentrations. For example, at 1000 K, a 3.5% change in the rate constant is equivalent to a change of 1 degree in the temperature if the activation energy is in the range of the reactions discussed here. The determination of temperature in this fashion is defined as chemical thermometry and is particularly useful for the case of pulse heating (as in shock tubes) or where the heating time is short and hence thermocouples or other steady state methods cannot be utilized. We discuss, summarize, and analyze the discordancy among rate expressions and the conclusions that can be deduced from the various experiments. We will begin by giving a synopsis of the past work. We will then explore possible routes for consolidating the varying results. Results, particularly those involving the comparative rate single-pulse shock tube experiments, are unlikely to be greatly in error because of the multiplicity of data, whereas others, particularly those from recent direct measurements, must be subject to more studies to validate postulated mechanisms upon which the reported rate expressions are based. Current interest in the determination of highly accurate rate constants and expressions is due to the use of these values in the simulation of complex phenomena to reproduce global experiments such as ignition delays. Obviously the predictions can be only as good as the input data. From a more basic viewpoint, the strength of chemical bonds is of fundamental interest and accurate experimental values are the ultimate test of theory. In addition to the rate constants, the possibility of inverting the process and the making of highly accurate temperature measurements have also been of interest. We discuss the necessary conditions for the determination of correct unimolecular rate constants and the associated reaction temperatures.

The reduced reactivity of phenyl CH2 is due to the benzyl resonance energy of the order of 45 kJ/mol. With the short residence time (500 μs) arising from shock heating, the benzyl radical does not react with the decomposing system. Instead, it reacts rapidly with the reactive radicals that may be present in the system, further inhibiting chain reactions, and isolates a particular stable product for analysis. Thus, chain-induced decompositions cannot occur. Active radicals are converted to benzyl type radicals which will ultimately recombine and are hence removed from the system. The addition of the methylated benzenes thus inhibits chain decomposition. The unique feature of the comparative rate experiments is the overwhelming concentrations of the inhibitors that are used and thus reduce the possible contribution from chain processes to the overall reaction to a minimum. The concentration of the product that is formed from the unimolecular decomposition can then be used as a measure of the extent of unimolecular reaction. The use of radical traps is a well-established technique developed by Szwarc.3,4 Such isolation is a general technique that can be used to study the rates of decomposition of any compound that is substantially less stable than the methylated benzenes that are added as radical traps. The use of the smallest concentration of the reactant in the shortest possible reaction times in the presence of the largest excess of a radical trap is necessary. These requirements are most easily met in singlepulse shock tube studies. The variation of inhibitor-to-reactant concentrations makes it possible to determine the smallest possible rate constant. This must then be the unimolecular rate constant of interest.1,2,5 Once the individual reactions for study are isolated, it is possible in principle to study a whole host of reactions at once (if they lead to different products) and test the correctness of any particular rate constant and expression against a whole set of “standard reactions”. A standard reaction is a unimolecular reaction whose mechanism and rate expression for decomposition has been unambiguously established from earlier studies. From past work,1 there are a large number of “standard” reactions that can be used for this purpose. The unimolecular reaction and rate expression for the reverse Diels−Alder decomposition of cyclohexene to form ethylene and 1,3-butadiene has been used as an “internal standard” in many single-pulse shock tube studies for the characterization of

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SHOCK TUBE EXPERIMENTAL AND COMPARATIVE RATE METHODOLOGIES We will begin with a discussion of the experimental measurement of unimolecular reactions and then focus on the comparative rate single-pulse shock tube technique. A description of the method can be found in previous communications.2 The present treatment will be focused on the likely uncertainties in the rate constants and rate parameters. We will then consider the more direct measurements of rate constants where internal standards are not used. B

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Table 1. Summary of Rate Expressions for Cyclohexene Decomposition (Cyclohexene → Ethylene + 1,3-Butadiene) Based on Comparative Rate Single-Pulse Shock Tube Resultsa rate expression internal standard at lower temperatures derived from static systems

experimental rate constant relations from comparative rate single-pulse shock tube experiments

k(ECB) = 1015.56 exp(−31 219/T), (693− 733 K) [15] k(1122TMCP) = 1014.75 exp(−31 370/T), (699−750 K) [16] k(ISPCl) = 1013.4 exp(−25 428/T), (640− 679 K) [8]

log k(cyclohexene) = (1.08 ± 0.004) log k(ECB) − (1.498 ± 0.004) [6] log k(cyclohexene) = (1.071 ± 0.006) log k(1122TMCP) − (0.726 ± 0.008) [7] log k(cyclohexene)= (1.304 ± 0.009) log k(SPCl) − (2.816 ± 0.002) [9]

cyclohexene decomposition rate expression (sec−1); derived from columns 1 and 2

log(rate constant) at T (K)

k(cyclohexene) = 1015.3 exp(−33 690/T)

1.37 (1050) 0.5 (987) 1.20 (1050) 0.32 (987) 1.17 (1050) 0.26 (987)

15.1

k(cyclohexene) = 10

exp(−33 600/T)

k(cyclohexene) = 1015.02 exp(−33 552/T)

a

The uncertainties in the third column can be deduced from the data in column 2, assuming the correctness of the internal standard reaction. bECB = ethylcyclobutane → ethylene + 1-butene; 1122TMCP = tetramethylcyclopropane → 2,4-dimethylpentene-2; ISPCl = isopropyl chloride → C3H6 + HCl; and cyclohexene = cyclohexene → ethylene + 1,3-butadiene.

reflected shock region, and the simple relation between shock velocity and temperature no longer holds exactly. All the experiments in the comparative rate shock tube studies are carried out in mixtures that contain significant quantities of large polyatomic molecules. There should therefore be important boundary layer effects. However, such deviations should have minimal results because the test and reference gases that are undergoing reaction suffer the same temperature. If their activation energies are the same, differences in the temperature histories will have no effects. Because it is difficult to carry out studies with widely varying activation energies, we have shown that for the relatively small activation energy differences encountered here, this also is minimal. This has been demonstrated in one of the first papers in this series.9 It can also be inferred from the vanishingly small deviation in the linear relations between the rate constants shown in Table 1. The situation in the comparative rate single-pulse shock tube experiments is easily demonstrated as follows. Consider two rate constants, k and k′; then, from the Arrhenius dependence of the rate constant on temperature, one obtains

the unimolecular stability of a large number of polyatomic organic molecules2,5 and has been validated by testing against other standards. At this point it is important to establish the criteria for using a particular internal standard. Particularly important is the carrying out of studies under the reaction conditions where the true unimolecular mechanism is operative. Much of the subsequent discussion will deal with these issues. Here again the reader is reminded of the fundamental nature of this data. It is intrinsic to the molecule in question in the same way as is a boiling point. A summary of the various rate expressions and constants from comparative rate single-pulse shock tube experiments using three completely independent internal standards can be found in Table 1. The similarity in the rate expressions is notable. The three derived rate expressions are practically identical. Note the very small uncertainties of the experimental rate relationship in the second column. For shock tube experiments,2,5−10 the scatter (in terms of the uncertainties at the one sigma level) shows the data is of the highest precision. There is, however, the scatter in the derived rate constants is greater than that in the rate parameters. This is in contrast to the usual situation in rate determinations for which the precision of the rate constants is greater than that of the rate parameters. Before describing these experiments in more detail, we discuss the justification for the use of the cyclohexene reaction and consider possible uncertainties. A unimolecular reaction can be represented as C → D where −dC/dt = dD/dt = kuniC. The unimolecular rate constant kuni can be expressed in the Arrhenius form as A exp(−E/RT) where A is the preexponential factor and E is the activation energy. The key factor in comparative rate experiments is the removal of the temperature dependency. Because of the exponential dependence, temperature is a key variable in any attempt at the determination of rate constants. The overall scatter of the three results in Table 1 at an appropriate temperature (1050 K) is 60%. The deviation from the average value is 25%. Note the practically identical activation energies. This is of utmost importance because it is from these activation energies that important information on radical heats of formation are derived. Indeed, it is the differences in the activation energies of the direct shock tube and comparative rate shock tube results (of the order of 13 kJ/mol) that should be of the utmost concern. This is a particularly serious problem in shock tube experiments because aside from uncertainties in the shock velocity there are many nonideal effects (growth of boundary layer)2 when larger polyatomic molecules are used (because of their larger heat capacity), studies are carried out in the

1/T = R(log k′ − log A′)/E′ and 1/T = R(log k − log A)/E ; then (log k′ − log A′)/E′ = (log k − log A)/E

This clearly demonstrates the linearity of the relationship between the logarithm of the two rate constants and the relationship between the slope and the intercepts. Once this relationship has been established, it is straightforward to convert the standard rate constant to a 1/T value so as to obtain the usual Arrhenius dependence. The uncertainties in the comparative rate experiments are reduced to the uncertainties in the extents of reaction of the target and the selected internal standard, and the rate expression for the latter process. We have discussed the uncertainties arising from these and possible errors in the residence time and nonisothermicity in one of the earliest papers on the present single-pulse shock tube experiments.9 An accurate temperature measurement is achieved only for the time scale of about 500 μs. The time resolution could be modified by changing the length and dimensions of the shock tube. This is inconvenient, and the range so covered cannot be large. In no case can it be substituted for direct, accurate measurements in real time. Although the cyclohexene decomposition reaction is used as the internal standard for most of the comparative rate singlepulse shock tube studies, this choice is really a matter of convenience. Ethylene and 1,3-butadiene are both extremely C

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constant. The estimation of accurate uncertainty limits in kinetic experiments is extremely difficult because of the introduction of time and the strong temperature dependence of the rate constants. It is a problem that has been treated extensively in atmospheric kinetics.11 In the present case the uncertainties in the reaction time and temperature are largely eliminated by the use of the internal standard.9 It is suspected that the major uncertainty is coming from the rate constants of the internal standard. Our results in cyclohexene decomposition6,7,9 are based on three internal standards (see Table 1) and note the small scatter of the results for each set of experiments. These standards have been selected on the basis of experimental evidence that indicates the lack of mechanistic artifacts.1 We believe that tests with more internal standards will reduce the spread of results. The comparative rate method assumes an Arrhenius dependence of the rate constants over a large range of temperatures. Overall, this range can be as large as ten or more orders of magnitude. Curvature in the standard Arrhenius plot will reveal itself in terms of differences in the rate constants. It may be that the similarities in the activation energies from the comparative rate experiments are purely coincidental and that the spread of results in terms of the rate constants may be a true measure of the uncertainties. Nevertheless, the fact that three completely independent measurements should lead to virtually the same rate expression and a spread of rate constants is significant and justifies the assigned uncertainties for cyclohexene decomposition as suggested in Table 1. This provides justification for our use of the cyclohexene unimolecular decomposition reaction as the internal standard. Because of their fundamental nature many other unimolecular reactions can be used as the internal standard. However, for the type of accuracy it is necessary to carry out similar types of verification experiments.

stable to thermal decomposition; thus, it is possible to cover a large temperature range. There is, however, no guarantee that the gas-phase reverse Diels−Alder reaction is the only plausible reaction. Indeed the possibility of a hydrogen atom-induced decomposition H + cyclohexene → cyclohexyl → hexen‐5‐yl‐1 → ethylene + buten3‐yl‐1 → 1,3‐butadiene + H

will also lead to the expected products. The likelihood for this channel to take place illustrates the importance of the addition of radical inhibitors to the system. As mentioned earlier, the addition of large quantities of polyatomic molecules to a shock tube system is incompatible with maintaining the correct relation between shock velocity and temperature because of the growth of the boundary layer.2 In high-pressure flow tube experiments this option should certainly be tested. There is of course always the possibility of surface processes in the latter experiments because of the manner in which the sample is heated and the longer residence time. Figure 1 contains Arrhenius plots of the results from the comparative rate single-pulse shock tube experiments. Because

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DIRECT STUDIES ON CYCLOHEXENE DECOMPOSITION Recently, Stranic et al.12 carried out shock tube studies on the pyrolysis of cyclohexene. From the rate of formation of ethylene determined from laser absorption in real time they found the rate expression for the decyclization reaction to be 4.8 × 1014 exp(−31 900/T) s−1, somewhat different from the expression 1.45 × 1015 exp(−33 500/T) s−1 (this is derived from the data in Table 1, third column and the work of Uchiyama et al.,13 which we have used in many experiments as an internal standard in comparative rate single-pulse shock tube experiments). Heyne and Dryer14 have carried out studies of cyclohexene decomposition in a high-pressure reactor and concluded that the temperature determined by the rate constant for cyclohexene decomposition is considerably different than that measured by thermocouples. The results are equivalent to an even larger rate constant for cyclohexene decomposition. Particularly significant was their detection of scores of other organic compounds that signify the presence of other decomposition processes. It will therefore be not

Figure 1. Arrhenius plot for the decomposition of cyclohexene over the temperature range of 70−1100 K. ECB15 refers to studies in which ethylcyclobutane is the internal standard; 1122TMCP16 are studies using 1,1,2,2-tetramethylcyclobutane, and ISPCl8 indicates those using isopropyl chloride.

of the different stability of the internal standard, the data covers 6 orders of magnitude. This is far larger than most studies on the rate constants for unimolecular reactions. All of these results are the basis for the rate expression that we have been using. Note the similarities in the slope or the activation energies of the results. From the data in Figure 1 and Table 1 we estimate an uncertainty of about a factor of 1.25 in the rate

Table 2. Rate Expressions and Constants Derived from Earlier and More Recent Direct Studies on Cyclohexene Decomposition methods 12

direct shock tube study high-pressure flow experiments14 low-pressure flow experiments13

rate expressions, sec−1

log rate constants (at T (K))

k(cyclohexene) = 1014.68exp(−31 900/T) (950−1300) not given k(cyclohexene) = 1015.15exp(−33 233/T) (814−902 K)

1.49 (1050) 0.65 (987) 0.34 (957), 0.61 (977), 0.74 (987) 1.409 (1050) 0.55 (987)

D

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The Journal of Physical Chemistry A surprising that there will also be contributions from nonunimolecular cyclohexene decomposition processes. The earlier work by Uchiyama et al.13 was in excellent agreement with the comparative rate single-pulse shock tube studies. This work was included in our recommended rate expression for this reaction and played an important role in our analysis. Table 2 contains rate expressions and constants determined on an absolute basis. Figure 2 contains Arrhenius plots of all the

Table 3. Summary of Rate Expressions and Constants of Cyclohexene Decomposition k(cyclohexene → ethylene + 1,3-butadiene)

rate relation used

rate expression

log rate constant, 1050 K

comparative rate experiments comparative rate experiments and low-pressure flow experiments comparative rate experiments and low-pressure flow experiments and direct shock tube study

1015.14±0.2 exp(−33 621 ± 90/T) 1015.15±0.16 exp(−33 500 ± 360/T)

1.24 1.29

1015.02±0.35 exp(−33 170 ± 1000/T)

1.49

The second includes the data from the comparative rate experiments and the low-pressure flow experiments of Uchiyama,13 used as the standard rate expression in past reports.2,5 The last is a summary of all the data (including the recent direct studies).13 We have not used the results of Heyne and Dryer14 because rate expressions were not derived in their paper. However, a rate constant was derived from the fractional conversion and reaction temperature that was measured by thermocouples. The rate constant is also larger than the comparative rate single-pulse shock tube experiments. It is probably significant that all the direct measurements lead to rate constants that are larger than the comparative rate experiments. This is consistent with the addition of chain inhibitors in the single-pulse shock tube experiments and the absence of surface process in general. This is completely expected on the basis of the present analysis. Specifically, the directly measured rate constants from Stranic et al.12 and Heynes and Dryer14 are all larger than those determined by us. This is expected because we removed all parallel chemical processes with the use of inhibitors, and no attention is paid to the mechanistic aspects in the more recent determinations. In the context of unimolecular rate studies in general,1 the deviations noted here are not particularly large. These are based on studies carried out many years ago, and it should certainly be possible to do better today. The reader is reminded again of the greater experimental difficulties in carrying out kinetic measurements in contrast to the situation for static properties. Also, in terms of the current interest in using unimolecular reaction for thermometry, higher accuracy is needed. All of the rate expressions that have been used in deriving cyclohexene decomposition from comparative rate single-pulse shock tube work are based on measurements at temperatures considerably lower than those from the shock tube measurements. The lower-temperature results used as standards have rate constants that are lower by about 4 orders of magnitude in rate constants (Figure 1). A key assumption made is that the lower-temperature determinations expressed in the Arrhenius form can be directly extrapolated to the higher-temperature region. Transition state theory does not justify an Arrhenius extrapolation over all temperature ranges. However, the only definitive evidence for strong curvature in rate constants exists for bond-breaking reactions, from room to single-pulse shock tube temperatures, which covers a range of rate constants far greater than those displayed here. If one assumes that all the experimental measurements are correct and that the observed deviations are purely due to the non-Arrhenius behavior of the unimolecular reactions, a rough estimate can be made from the data in Tables 1 and 2. Assume that the lower-temperature rate expression for cyclohexene

Figure 2. Rate constants for cyclohexene decomposition from direct measurements, direct shock tube study;12 high-pressure flow experiments;14 and low-pressure flow experiments.13 For comparison, data are from the comparative rate experiments and extrapolation of lowtemperature results assuming modified Arrhenius behavior based on direct measurements at 1000 K and low-temperature results are included (see text). The dark line marked at comparative rate refers to the average of the three comparative rate experiments listed in Table 1.

direct measurements on the rate constants for the unimolecular decomposition of cyclohexene. Also included are our best estimates of the rate constants for this reaction from the comparative rate measurements. These have been derived from the data in Figure 1 and Table 1. It can be seen that at the highest temperatures all the results are in good agreement. As the temperature is lowered, discrepancies increase. It is interesting that all the direct measurements lead to rate constants that are larger than or equal to the comparative rate values. The actual magnitude of these differences can be seen in Figure 3. It is of the order of 2 (at the lowest temperatures) and 2.5 for the direct shock tube and high-pressure experiments, respectively. This is completely expected on the basis of inhibition of radical-induced decomposition in the comparative rate studies and its absence in the more recent experiments. In the case of high-pressure flow experiments there is also the possibility of surface processes. The decrease of such effects at the higher temperatures from the comparative rate experiments is a direct consequence of the relative importance of the two processes.

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PUTTING IT ALL TOGETHER From the results in Tables 1 and 2, we derive the average values for the rate expressions and rate constants, found in Table 3. We have simply taken the average value of the logarithm of the A-factors, the activation energies, and the logarithm of the rate constants, and the uncertainties are derived and expressed as one standard deviation. The rate constants and expressions are divided into three separate categories. The first is based on results from comparative rate single-pulse shock tube alone. E

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The Journal of Physical Chemistry A decomposition is correct. Instead of a standard Arrhenius format for the rate expression, assume that the appropriate relation is k(cyclohexene) = 1015.15(T/650)α exp(−33 500/T) s−1 where an appropriate value of α = 2 will fit the discrepancy in the 900−1000 K region. Physically this means that the transition state for cyclohexene decomposition has a nonArrhenius behavior larger than that of the reference standards. The results can be seen in Figure 2. However, fitting the data in the region of 900−1000 K leads to larger discrepancies in the higher-temperature region. The required activation energy is now too large, which as a simple “fix” to the problem does not work. The present analysis leads to the conclusion that the recent studies on the cyclohexene decomposition reaction are contaminated by ancillary reactions that lead to increased reaction rate constants. In a recent paper from Urzay et al.,17 hydrogen atoms appear to be present in an oxygen system in their shock tube studies. In systems with organics (such as cyclohexene) present, it would not be surprising that hydrogen atoms may be present as a trace impurity. This of course illustrates the importance of adding radical inhibitors into the reaction system. Unfortunately, the addition of a large amount of a compound such as 1,3,5-trimethylbenzene as used in the comparative rate experiments to a shock tube system is incompatible with maintaining the correct relation between the shock velocity and the temperature. Thus, a correct reaction temperature cannot be calculated. In high-pressure flow experiments, the heating of the reaction sample may introduce parallel decomposition channels. Indeed, the unsaturated compounds that they detected are suggestive of this possibility. It may well be that the addition of radical inhibitors may reduce the contributions from additional reaction channels. The data in Figures 1 and 2 show that the deviations in the rate constants are not very large. Indeed, the rate constant differences are so small that it is difficult to discern the actual magnitude of the discrepancies. In Figure 3 we have plotted the ratio of the various rate constants against those derived from the rate expression that we have recommended and used a considerably expanded scale. If all the measurements are of equal validity, it would be straightforward to simply minimize the ratios. Unfortunately, as we have shown in the recent direct measurements, the presence of H atoms in direct shock tube studies17 and of additional unexplained products in the study of Heyne and Dryer14 are all flawed on a mechanistic basis. The comparative rate studies suffer from uncertainties in the standard rate. It is suspected that it will be simpler to increase the accuracy of the latter because the measurements used here were all based on studies carried out many years ago and modern techniques should certainly provide better accuracy. In addition, there are a whole host of internal standards that can be used. Hence, replicate measurements can be made and proper statistical treatment of the data can lead to extremely accurate rate expressions. Note that all the direct measurements have rate constants that are larger than the comparative rate determinations. This is in accord with our suggestion that there are extraneous contributions to the rate constants from the direct determinations (see earlier observations on the presence of H atom in the direct shock tube determination and the presence of additional products in the high-pressure flow work). Finally, it is interesting to speculate on the nature of the observed divergences. At the highest temperatures, cyclohexene is decomposing so fast so that contributions from the induced decomposition is minimized. The low-pressure flow experi-

Figure 3. Ratio of measured rate constants for the unimolecular decomposition of cyclohexene versus our recommended rate constants (see Table 3, second column). The high-pressure flow refers to the experiments of Heyne and Dryer;14 direct shock tube is from the experiments of Stranic et al.;12 low-pressure flow refers to the work of Uchiyama et al.;13 comparative rate studies are from refs 6, 7, and 9. The thin dark line is from the average of the values of rate constants derived from the low-pressure flow experiments and the comparative rate study.

ments are carried out under conditions in which H atoms make minimal contributions.

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GENERAL CONSIDERATIONS AND THE USE OF UNIMOLECULAR REACTIONS FOR THERMOMETRY IN SHOCK TUBE STUDIES All of the above leads to the conclusion that there is no particular property of the cyclohexene decomposition reaction that makes it especially suitable as an internal standard in hightemperature shock tube systems. As noted earlier, it was really selected because of the thermal stability of ethylene and 1,3butadiene. All the usual precautions must still be taken to ensure the integrity of the reaction process. Actually, small cyclic compounds may be more suitable internal standards. We have also used ethylcyclobutane and 1,1,2,2-tetramethylcyclobutane as internal standard to calibrate the cyclohexene decomposition reaction. Both compounds are completely saturated, and therefore radical attack cannot lead to products with the same number of hydrogens as the starting reactant. This class of three- and four-membered ring compounds has been extensively studied in static systems; they are extremely clean and lead to reproducible rate constants. In carrying out the comparative rate experiments, we have made use of work performed many years ago. Undoubted the accuracy can be improved. Thus, it is suspected that the small discrepancies exhibited in Figure 1 may well be experimental artifacts that can be reduced by more careful, modern, lowertemperature experiments. Of particular interest will be the testing of internal standards against each other in the lowtemperature regime. We note that uncertainties in classical kinetic experiments of the magnitude of concern to the comparative rate experiments are endemic.1 For example, ethylcyclobutane could be tested against isopropyl iodide or 1,1,2,2-tetramethylcyclopropane. In combination with modern theory this will permit the proper extrapolation to the higher F

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contributions from parallel reactions that are not considered. When these chemical processes are taken into account, then only the addition of inhibitors and hence the use of an apparently more complex chemical system can ensure correct answers. A corollary to this observation is that in direct measurements with the shock tube or in high-pressure flow reactors there is the crucial need to establish correct mechanisms before proper rate constants can be obtained. We emphasize that the high-pressure rate constant for a unimolecular reaction of a polyatomic molecule is a fundamental property of the molecule. As such the determination of the proper values must be carried out with careful attention to the purity of the chemical process in the same way, as an example, as the purity of a sample in a melting point determination is achieved. Finally, the present discussion of the influence of parallel reactions in direct studies may also have impact on uncertainty estimates for the large volume of measurements that now exist on ignition delays.18

shock tube temperatures and a correct derived value. We note that at the present time in terms of the possible uncertainties mentioned earlier the accuracy limits of temperature derived from comparative rate unimolecular kinetics may be as much as an order of magnitude smaller than that derived from the shock velocity. For a variety of tested standards replicate measurements can be made and the accuracy increased so as to reflect modern practices. The present estimated uncertainties in the rate constants are a factor of 1.25. This implies an uncertainty in the derived temperature of about 7 degrees at 1000 K. From direct shock wave velocity measurement this is equivalent to accuracy in the shock velocity measurement of 0.35%. This is just about the limit of shock velocity measurements. A more careful determination of the rate constant of the standard may reduce the uncertainty to 1−2 degrees at 1000 K.

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CONCLUSIONS It is clear that the cyclohexene decomposition rate expression can be derived from a wide variety of starting points in comparative rate single-pulse shock tube experiments. Each of these rate expressions is in striking agreement with one another. The small discrepancies that have been observed are due to errors in the rate expressions of the standard reactions measured many years ago. There is no particular reason that the cyclohexene decomposition reaction will not suffer from ancillary reactions that will distort the measured rate constant unless steps are taken to minimize these contributions. The direct measurement by Stranic et al.12 and the temperature determination of Heyne and Dryer14 are subject to errors arising from neglecting of possible contributions from additional reactions that lead to the same products as those from the unimolecular decomposition. Evidence for this is provided by subsequent work from the same shock tube laboratory17 of the presence of H atom impurities and the detection in the high-pressure flow experiments of products that can arise from ancillary reaction processes. The use of a unimolecular reaction as a chemical thermometer would appear to be a method that is probably superior to the use of the shock velocity relation for temperature determinations. Here, the focus must be on the selection of a proper internal standard. The ideal situation is to carry out studies with reactions that are not enhanced by hydrogen atoms. A combination of conventional (at the lower temperatures) and shock tube experiments in the presence of a chemical inhibitor for both types of experiments should lead to state-of-the-art results. We believe that uncertainties in the 1−2 degree range at 1000 K can be achieved. The present conclusions highlight the importance of adding chemical inhibitors to the reaction systems when the aim is to obtain high-accuracy rate expressions. For results of the highest accuracy it is important to establish the integrity of the reaction system. This is an issue that has been neglected in the most recent direct experiments and in the early days of studies on unimolecular processes. To a large extent the latter is due to the lack of any incentive to obtain data of the highest accuracy. The comparative rate and related single-pulse shock tube methods do offer a means of obtaining data of the highest accuracy and consistency and thus form a basis of obtaining the rate rules for further prediction and as tests of theory. Although the newer measurements are more direct, they cannot lead necessarily to more accurate numbers unless attention is paid to reaction mechanisms due to possible

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AUTHOR INFORMATION

Corresponding Author

*100 Bureau Drive, Gaithersburg, MD 20899. Tel.: 240-8886547. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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