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Temperature and Pressure Dependence of the Reaction 2CF3 (+ M) S C2F6 (+ M)† C. J. Cobos,‡,§ A. E. Croce,‡,§ K. Luther,§,⊥ and J. Troe*,§,⊥ INIFTA, Facultad de Ciencias Exactas, UniVersidad Nacional de La Plata, Argentina, Max-Planck-Institut fu¨r Biophysikalische Chemie, Am Fassberg 11, D-37077 Go¨ttingen, Germany, and Institut fu¨r Physikalische Chemie, UniVersita¨t Go¨ttingen, Tammannstrassse 6, D-37077 Go¨ttingen, Germany ReceiVed: September 23, 2009; ReVised Manuscript ReceiVed: December 3, 2009
Limiting low- and high-pressure rate coefficients as well as full falloff curves have been modeled by unimolecular rate theory for the recombination reaction 2CF3 (+ M) f C2F6 (+ M) and the reverse dissociation of C2F6. The results are compared with experimental data from the literature. Although there are considerable discrepancies (up to a factor of 5) between various experimental data near 300 K and the database for high temperatures is still limited, we try to conclude on the temperature dependence of the high-pressure rate coefficient. We suggest that there is only a small and probably positive temperature coefficient of the latter quantity. The present theoretical modeling seems to be in agreement with this experimental result, but it is in disagreement with conclusions from earlier theoretical work. The difference is attributed to different empirical assumptions about the anisotropy of the potential. It is shown that nearly all previous experiments (except high-temperature shock wave and very low pressure pyrolysis/photolysis experiments) correspond to nearly limiting high-pressure conditions. 1. Introduction Despite their importance for the chemistry of fluorocarbons in various applications, the thermal dissociation of hexafluoroethane
(1)
C2F6 (+M) f 2CF3 (+M) and the reverse recombination of perfluoromethyl radicals
2CF3 (+M) f C2F6 (+M)
(-1)
are not yet well-characterized. Rate coefficients k1, to our knowledge, so far have only been estimated indirectly; see below. On the other hand, rate coefficients k-1 have been determined more frequently,1-21 mostly near room temperature. Measurements markedly above 300 K are scarce, the broadest range of bath gas pressures being studied in shock waves in ref 8 near 1300 K. More recently, the isochoric isothermal pyrolysis of trifluoroiodomethane at around 700 K from ref 21 has provided important additional information for intermediate temperatures. Despite these efforts, the experimental data for k-1 still look scattered, and neither the temperature nor the pressure dependences show a clear picture. Unfortunately, the situation does not look much better on the theoretical side. Simplified RRKM (Rice-Ramsperger-Kassel-Marcus) calculations from ref 6, depending on the transition-state model, led to temperatureindependent values or to positive temperature coefficients of k-1,∞. Simplified SACM (statistical adiabatic channel model) calculations in ref 8 led to nearly temperature-independent k-1,∞, as well, while canonical flexible transition-state theory in ref †
Part of the special section “30th Free Radical Symposium”. * To whom correspondence should be addressed. E-mail:
[email protected]. ‡ Universidad Nacional de La Plata. § Max-Planck-Institut fu¨r Biophysikalische Chemie. ⊥ Universita¨t Go¨ttingen.
TABLE 1: Modeled Equilibrium Constants Kc ) k1/k-1 for the Reaction C2F6 S 2CF3 (see text) Kc/molecules cm-3 T/K
ref 23
refs 21, 24
this work
300 700 1000 1300 1500 2000
3.1 × 10-41 1.3 × 10-1 6.9 × 107 2.9 × 1012 3.0 × 1014 5.0 × 1017
1.0 × 10-40 2.4 × 100 6.7 × 107 3.0 × 1012 3.4 × 1014 7.1 × 1017
5.4 × 10-43 1.9 × 10-2 1.7 × 107 9.1 × 1011 1.1 × 1014 2.2 × 1017
22 predicted a markedly negative temperature coefficient of k-1,∞. It is the intention of the present article to attempt a synthesis of the available data by modeling temperature and pressure dependences of k-1 and k1 in terms of unimolecular rate theory and by comparing these results with experimental data from the literature. The result is by no means satisfactory, but it will provide useful guidelines for future work. 2. Equilibrium Constants k1 and k-1 are related by the equilibrium constant
Kc ) k1 /k-1
(2)
As different expressions for Kc were obtained in different evaluations, we recalculated Kc on the basis of today’s input data, and we compare our results with literature values. We used a harmonic oscillator/hindered rotor approach with molecular parameters such as those summarized in the Appendix. Our results in Table 1 are shown and compared with the data from refs 23 and 24 (as evaluated in ref 21). The differences primarily are due to different reaction enthalpies (∆H°0 ) 400.6, 397.3, and 410.0 kJ mol-1 used in refs 23 and 21 and in the present work, respectively). The analysis of the present work relies on the most recent determination of the enthalpy of formation of CF3 from ref 25 and, therefore, is preferred. Minor additional
10.1021/jp9091464 2010 American Chemical Society Published on Web 01/05/2010
2CF3 (+ M) S C2F6 (+ M) Temperature and Pressure Dependence differences between the three evaluations are due to somewhat different frequencies and treatments of the hindered internal rotor of C2F6; see the Appendix. So far, the equilibrium constants Kc have not been measured directly, although this would have been possible. For example, the thermal decomposition of C2F6, followed by the more rapid decomposition of CF3 radicals26 to CF2 + F and monitored by means of the UV absorption of CF2, could be studied in shock waves under conditions of incomplete dissociation of C2F6. However, there are27 shock wave measurements of C2F6 decomposition in the presence of H2 which provided values of the ratio Kc1/2k3, where k3 is the rate constant of the reaction
CF3 + H2 f CF3H + H
(3)
Combined with direct measurements of k3 in shock waves from ref 28, this leads to values of Kc of 5.9 × 1011 and 3.3 × 1013 molecules cm-3 for 1300 and 1500 K, respectively, which roughly (within a factor of 2-3) agree with the results of the present calculations such as given in Table 1. As we have used the most recent reaction enthalpy, we conclude that the analytical representation of our numerical results from Table 1, such as that given by
Kc ) 2.32 × 1030(T/300 K)-2.59 exp[-(50174 ( 500) K/T) molecules cm-3
(4a)
over the range of 300-2000 K, provides more accurate values of Kc than the data from refs 23 and 24 (the latter represented in ref 21). The given uncertainty reflects the estimated uncertainty of the reaction enthalpy. An alternative representation of Kc would be
Kc ) 1.69 × 1029(T/300 K)-1.29 exp(-∆H°/kT) molecules cm-3 0
(4b)
) (49316 ( 500) K. By means of eqs 2 and 4, data with ∆H°/k 0 for k-1 later on will be converted to values of k1. Table 1 documents the considerable differences between the various values of Kc, which then also determine the uncertainty of k1 when it is derived from k-1 and Kc. 3. Experimental Rate Coefficients A closer inspection of the experimental results1-21 for k-1 leads to a series of observations. First, some rate measurements were absolute, and some were relative; second, falloff effects had to be taken into account in measurements at higher temperatures; third, there were clearly some internal inconsistencies between various studies. The modeling of the falloff curves by unimolecular rate theory, such as shown below, indicates that measurements at 300 K in all bath gases at pressures above about 0.1 Torr all practically correspond to the high-pressure limit, that is, k-1 ≈ k-1,∞. Discrepancies between measurements near 1 and 100 Torr, therefore, cannot be attributed to falloff effects. A summary of experimental results is given in Table 2a. There is one group of data with k-1,∞ ≈ 10-11 cm3 molecule-1 s-1 and another group suggesting markedly lower values, in the range of (1.8-3.9) × 10-12 cm3 molecule-1 s-1. We have carefully inspected the various studies but were unable to identify the reasons for the discrepancies. However, we tend to favor the higher values
J. Phys. Chem. A, Vol. 114, No. 14, 2010 4749 TABLE 2: Experimental Rate Coefficients k-1 for the Reaction 2CF3 (+ M) f C2F6 (+ M) k-1/cm3 molecule-1 s-1 reference
T/K
M
P/Torr
(a) 298 298 298 298 298 296 290
Ar, N2, CO2 Ar (CF3CO)2O SF6 (CF3)2CO, Ar He He
100-400 100 20 20 0.6-1.0 600 1-6
1.3 × 10-11 1.3 × 10-11 9.1 × 10-12 1.0 × 10-11 2.2 × 10-12 3.9 × 10-12 1.8 × 10-12
2 5, 15, 21 7 19 13 18 20
(b) 700 700 1300 1300
CF3I CF3I Ar Ar
29 298 250 19000
7.1 × 10-12 1.3 × 10-11 3.3 × 10-12 1.7 × 10-11
21 8
because our theoretical modeling, with standard potential parameters such as those given below, appears more compatible with the higher values of the room-temperature data and the measurements at higher temperatures from refs 8 and 21. A number of experimental results from refs 1-21 have not been included in the table. For example, k-1 in ref 1 was obtained with the rotating sector method and extracted from a mechanism of secondary reactions. The initially derived value of k-1 ) 3.8 × 10-11 cm3 molecule-1 s-1 in the re-evaluation of ref 17 was lowered to k-1 ) 2.5 × 10-11 cm3 molecule-1 s-1. Because of the complexity of the system, there still may be uncertainties. Besides other early and very uncertain results, very low pressure pyrolysis/photolysis (VLPP/φ) measurements of refs 6, 11 and 14 are not considered directly because effective pressure and heterogeneous effects, to some extent, remain uncertain. These data, however, later on are shown to be consistent with our general analysis of the falloff curves. For the shock wave experiments of ref 8 at 1300 K, both experiments and modeling indicate the presence of marked falloff effects. Representative data are given in Table 2b. This table also includes data from ref 21 for temperatures near 700 K. We emphasize that in the present article, we cannot analyze the sources of error which obviously must have been present in several of the experimental studies. In particular, we cannot explain why there was such a discrepancy between the data (obtained between about 300 and 400 K) of the one group of values near k-1,∞ ≈ 10-11 cm3 molecule-1 s-1 and the other group of lower values such as those given in Table 2a. However, by combining these data with the two high-temperature determination of refs 8 and 21, which independently led to k-1,∞ ≈ 1.5 × 10-11 cm3 molecule-1 s-1 (within a factor of 2), one at least can conclude that k-1,∞ can have only a weak negative or a more or less strong positive temperature coefficient. The experimental results for 300, 700, and 1300 K in Figures 1-3 are compared with modeled falloff curves such as those described in the following. The figures illustrate the described problem of quite diverse experimental results. The modeling given in the next section only in part can improve this situation. 4. Modeled Rate Coefficients 4.1. High-Pressure Rate Coefficients. The conclusion about a positive or, at most, only weakly negative temperature coefficient of k-1,∞, such as that described in section 3, is in contrast to the predictions of a treatment employing canonical flexible transition-state theory (CFTST) from ref 22, which suggested k-1,∞ ∝ T -1.07 between 300 and 1000 K. In discussing the apparent discrepancy with the experiments, one might
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consider the possibility of problems with the used CFTST approach and/or with the employed potential energy surface. The latter in ref 22 was constructed on an elaborate but nevertheless empirical level. The validity of the approach of ref 22 was demonstrated for CH3 recombination, where a T -0.43 dependence between 300 and 1000 K was obtained, while the variable reaction coordinate transition-state theory (VRC-TST) on a high-level quantum chemical calculation of the potential in refs 29 and 30 led to a T -0.31 dependence. Although experimental high-temperature falloff curves in this case are not complete, experiments and theoretical modeling apparently fit to each other. One then asks why the experimental results for CF3 recombination from the present analysis and the theoretical modeling by CFTST from ref 22 apparently disagree. In order to assess the relevance of theoretical modeling of k-1,∞, in the present work we used a different theoretical approach, doing statistical adiabatic channel model/classical trajectory (SACM/CT) calculations on a partly theoretical, partly empirical potential energy surface such as that described in ref 31. Using ab initio potentials, for H + O2 f HO2 and HO + HO f H2O2, the SACM/CT approach led to excellent agreement with the experiments; see refs 32 and 33, respectively. The application to CF3 recombination suffers from the lack of an ab initio potential for the C2F6 system. Therefore, a “standard” potential energy surface was constructed. This potential here was characterized by a Morse stretching potential between the CF3 groups and an overall anisotropy represented by a standard value of the effective ratio R/β (R ) Pauling parameter;34 β ) Morse parameter). A useful starting point for the guess of the ratio R/β is the “standard value” of R/β ≈ 0.5 (see ref 35); the ratio then is slightly modified in order to fit the resulting k-1,∞ to experimental data. Although this method fits the ratio R/β by comparing k-1,∞ with the absolute value of experimental k-1,∞ at a single temperature, it predicts the temperature dependence of k-1,∞. In the absence of complete ab initio calculations of the potential, at least this procedure may provide a valuable interim analysis of the results. While we use the ratio R/β as a fit parameter, we employ Morse parameters β which are either estimated from experimental data or from ab initio calculations. A rough guess of β is obtained by identifying the reaction coordinate with the C-C stretching mode (νRC ) 807 cm-1) using a dissociation energy of De ) 422 kJ mol-1 and employing the simple relation36 β ≈ (2π2µ/Deh2)1/2hcνRC. This leads to β ≈ 3.07 Å-1. Fitting ab initio G3(MP2)B3 results for the C-C stretching potential (following the method described in ref 37) to a Morse potential on the other hand gives β ≈ 1.84 Å-1. Employing B3LYP/6311+G(3df) calculations for estimating the force constant fC-C for the C-C bond finally gives fC-C ≈ 6.9 mdyn/Å and, through β ≈ (fC-C/2De)1/2 from ref 36, results in β ≈ 2.22 Å-1. Neglecting the anisotropy of the potential (R/β > 1), first one obtains the PST (phase space theory) result given by35 PST k-1,∞ ) (kT/h)(h2 /2πµkT)3/2
∏ Qel,iQ*cent
(5)
with the product of the electronic partition functions ∏ Qel,i ) 1/4 and the centrifugal partition function Q*cent depending on the value of the Morse parameter. For different values of β, one obtains different results. For example, for T ) 300, 700, -11 cm3 molecule-1 s-1 ≈ 4.2, 5.7, and 1300 K, one has kPST -1,∞/10 -1 and 7.0 for β ) 3.07 Å and 7.5, 9.8, and 11.7 for β ) 1.84 Å-1, respectively These values illustrate the uncertainty of k-1,∞ on the PST level. However, as is well known, PST only defines
TABLE 3: Modeled Limiting High-Pressure Rate Coefficients k-1a k-1,∞/10-11 cm3 molecule-1 s-1 T/K
R/β ) 0.45
0.475
0.5
0.45
0.475
0.5
300 700 1300
0.75 1.2 1.6
1.3 1.8 2.4
2.1 2.8 3.4
1.3 1.8 2.3
2.1 2.8 3.4
3.5 4.3 4.8
a Left values: β ) 3.07 Å-1; right values: β ) 1.84 Å-1 (see text).
the upper limit for k-1,∞. The “thermal rigidity factors” frigid ) PST express the effects of the anisotropy of the potential k-1,∞/k-1,∞ PST . In our approach, these and reduce k1,∞ to values below k-1,∞ rigidity factors for various R/β were estimated with the representation of SACM/CT results given in ref 31. In the absence of more detailed information on the potential, the anisotropy of the potential was represented by four bending modes of two CF3 relative to each other. That is, the anisotropy of the two CF3 tops relative to each other was replaced by that of two quasi-linear species approaching each other and forming a linear adduct. This is a system which was systematically evaluated by SACM/CT calculations in section IV of ref 31 (eqs 4.1-4.10 of ref 31). As long as we did not have better information on the potential, we followed the described method, which obviously can only be of semiquantitative value. Table 3 shows representative results. No matter which combination of β and R/β is chosen, one always obtains positive temperature coefficients of k-1,∞. With the combinations β ≈ 3.07 Å-1 and R/β ≈ 0.475, as well as with β ≈ 1.84 Å-1 and R/β ≈ 0.45, for example, one would obtain a value at 300 K of k-1,∞ ≈ 1.3 × 10-11 cm3 molecule-1 s-1 and a temperature dependence following
k-1,∞ ≈ 1.3 × 10-11(T/300 K)0.42 cm3 molecule-1 s-1
(6) Comparing the experimental data from Table 2 and Figures 1-3 between 300 and 1300 K, one would have an even larger positive temperature coefficient of k-1,∞; see below. With a standard value of the ratio R/β of 0.5, the modeling would agree within about a factor of 2-3 with the group of higher values for k-1,∞ shown in Table 2. The group of lower values at 300 K would be reproduced only with R/β of the order of 0.4, which would appear unusually small. At the same time, the hightemperature data would be markedly underestimated. For this reason, we favor eq 6, possibly with an even higher temperature coefficient; see below. Although the presented modeling at least appears to be in semiquantitative agreement with the scattered experiments, one would ask why ref 22 led to different results. Inspecting the representative SACM/CT calculations of ref 31, one concludes that negative temperature coefficients of frigid and, hence, also those of k-1,∞ are obtained when the effective ratio R/β varies along the reaction path. Such effects have been observed in the ab initio potential of the HO2 system in ref 38. Without ab initio calculations of the potential, such as is the case here, one cannot account for such effects. Most probably, the differences between the results of the present modeling and ref 22 are due to such differences in the empirical potentials and not to differences in the CVTST and SACM/CT approaches to the dynamics. The question why such effects would be stronger for the CF3 system than for the CH3 system cannot be answered at this stage.
2CF3 (+ M) S C2F6 (+ M) Temperature and Pressure Dependence
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4.2. Low-Pressure Rate Coefficients. Limiting low-pressure rate coefficients k1,0 are given by the product of strong collision SC and collision efficiencies βc, both contriburate coefficients k-1,0 tions depending on the bath gas M. Following the formalism SC and converting the result outlined in refs 39 and 40 for k-1,0 with Kc in an intrinsically consistent manner, for M ) Ar, we obtain SC k-1,0 ≈ [Ar]7.96 × 10-19(T/300 K)-11.46
exp(-2847 K/T) cm6 molecule-2 s-1
(7a)
between 300 and 2000 K; a second, slightly less accurate, fit to the results is given by SC k-1,0 ≈ [Ar]1.35 × 10-22(T/300 K)-7.16 cm6 molecule-2 s-1 (7b)
Details of the used parameters are given in the Appendix. We SC estimate this calculation of k-1,0 to be accurate within about a SC , factor of 2. For example, two independent calculations of k-1,0 using different methods to characterize the reaction F + CF2 (+ M) f CF3 (+ M), in refs 26 and 41 agreed within about a factor of 2. The ratio of the experimental extrapolated limiting lowSC is identified pressure rate coefficient k-1,0 and the modeled k-1,0 with an experimental collision efficiency βc, that is, k-1,0 ) SC . The collision efficiency through the solution of the βck-1,0 master equation40,42 is related to the average energy 〈∆E〉 transferred per collision. As only parts of the falloff curves so far were explored experimentally and limiting low-pressure rate coefficients k-1,0 were difficult to obtain by extrapolation, we can base our determination of βc only on a comparison of modeled and fitted full falloff curves for 1300 K and/or assume standard values for 〈∆E〉 following from the tabulations given in refs 39 and 40. For example, we employ 〈∆E〉 ≈ hc 200 cm-1 for the bath gas M ) Ar and we assume a temperatureindependent 〈∆E〉. The corresponding βc then is given by the relation -〈∆E〉/FEkT ) βc/(1 - βc1/2) from refs 39 and 40. 4.3. Modeled Falloff Curves. Without further calculations of specific rate constants k(E,J) of C2F6 dissociation on ab initio potentials, such as those demonstrated, for example, for the H2O2 system in ref 33, it appears most reasonable to model falloff curves by the method of refs 40, 43, and 44. This approach SC ) leads to strong collision center broadening factors40,43,44 Fcent 0.35, 0.10, and 0.059 at T/K ) 300, 700, 1300, respectively (see Appendix). There are additional weak collision broadening 40 factors FWC cent estimated from the collision efficiencies βc through WC log Fcent ≈ 0.14 log βc and using βc obtained with temperatureindependent 〈∆E〉; see section 4.2. SC WC Fcent are compaThe derived broadening factors Fcent ) Fcent rably small, which is due to the low vibrational frequencies of C2F6. As a consequence, the modeled falloff curves differ from the simplest form given by [1 + k-1 /k-1,∞ ) [x/(1 + x)]Fcent
(log x)2]-1
(8)
where x ) k-1,0/k-1,∞ and considerable additional broadening and asymmetries of the falloff curves arise. Here, we employed eqs 6.3-6.6 from ref 44 (together with eqs 5.1-5.8 from ref 43) to represent these effects. As Fcent in the present case is calculated to be below 0.3, marked deviations from the simple
Figure 1. Rate coefficients for 2CF3 (+ M) f C2F6 (+ M) near 300 K in various gases M (full line ) modeled falloff curve for M ) Ar from this work (see text); experimental points: 1 (ref 2), 4 (refs 5, 15, and 21), 3 (ref 7), 9 (ref 19), 2 (ref 13), [ (ref 18), and b (ref 20)).
expression of eq 8 are predicted. It should be emphasized that this approach to the full falloff curves is still preliminary. However, it is believed to be realistic within the experimental uncertainties, and presently, one cannot do better. The present calculation of falloff curves supersedes earlier attempts to characterize the pressure dependence of k-1. The representation of ref 21 employed in the Lindemann model, on the basis of the apparent pressure dependence of the experiments in the bath gas M ) CF3I, extrapolates to values of k-1,0 which are orders of magnitude below those from the present modeling for M ) Ar, that is, it predicts much more falloff than suggested now, although CF3I should be a stronger collider than Ar. The experimental falloff curves for 1300 K from ref 8 for M ) Ar are roughly in line with the present modeling but were empirical. The modeling of the falloff curves for M ) He in ref 18 used the same method as that applied in the present work and led to essentially the same results (see Figure 8 of ref 18); however, it did not go into much detail. 5. Comparison of Experimental and Modeled Rate Coefficients The modeled falloff curve for 300 K and M ) Ar, such as that included in Figure 1, clearly indicates that the shown experiments at pressures above about 0.1 Torr (3.2 × 1015 molecules cm-3) within experimental accuracy all should correspond to the high-pressure limit (this conclusion is independent of the used bath gases). The absence of a pressure dependence, for example, in refs 13 and 20, conforms with this statement. However, the reasons for the discrepancy up to a factor of 5 between the group of high and low values of k-1,∞ remain unknown. In order to observe falloff reductions of the second-order rate coefficient k-1 at 300 K, much lower pressures than 0.1 Torr would have to be employed such as illustrated in Figure 4. This conclusion is supported by the VLPφ study of ref 11. In this case, wall collision frequencies of CF3 radicals of ω ) 1.28 × 104 s-1 were calculated. A similar gas-phase collision frequency of ω ) ZLJ[Ar] between C2F6 and Ar would be realized at [Ar] ) 3.7 × 1013 molecules cm-3, corresponding to a pressure of about 10-3 Torr. According to Figure 4, k-1/ k-1,∞ then should be about 0.77, which within a factor of 2-3 agrees with the measured k-1 ) 3.0 × 10-12 cm3 molecule-1 s-1 of ref 11. Earlier VLPP measurements near 1000 K from
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Figure 2. Rate coefficients for 2CF3 (+ M) f C2F6 (+ M) near 700 K (full line ) modeled falloff curve for M ) Ar from this work (see text); experimental points from ref 21 at pressures of 29 (O) and 298 Torr (b) in M ) CF3I at 700 ((50) K).
Figure 3. Rate coefficients for 2CF3 (+ M) f C2F6 (+ M) near 1300 K in M ) Ar (full line ) modeled falloff curve for M ) Ar from this work (see text); experimental points (b) from ref 8).
ref 6 are similarly consistent with the falloff curves of Figure 3. The wall collision frequency of ω ) 1.90 × 104 s-1 here would correspond to the same gas-phase collision frequency of ω ) ZLJ[Ar] at [Ar] ) 4.1 × 1013 molecules cm-3, corresponding to a pressure of about 1.3 × 10-3 Torr. According to Figure 4, now, one should be deep in the falloff range with k-1 ≈ k-1,0. For the bath gas Ar at [Ar] ) 4.1 × 1013 molecules cm-3, SC according to eq 7a, k-1,0 should be near 1.0 × 10-12, and k-1,0 SC ) βc k-1,0 should be near 1.0 × 10-13 cm3 molecule-1 s-1, if a collision efficiency of βc ≈ 0.1 is estimated, like shown below. The measured 1.3 × 10-13 cm3 molecule-1 s-1 from ref 6 is close to this value, although the efficiency of wall collisions remains somewhat uncertain. The VLPφ and VLPP results from refs 6 and 11 thus appear to be fully consistent with the falloff curves in Figure 4, although they could only be used for quantitative conclusions when the efficiency of wall collisions would be known more precisely. The modeling of the falloff curves at 700 K for M ) Ar included in Figure 2 again indicates that a pressure dependence of k-1 should not have been observed in the experiments of ref 21, which were performed at pressures of M ) CF3I above about 30 Torr. However, for unknown reasons, the data show some pressure dependence and also fall somewhat below the modeled curve for M ) Ar (and even more below a curve for M ) CF3I). Nevertheless, these data appear to be closer to the group of high values of k-1,∞ than to the low values shown in Figure 1. In addition, they clearly support the conclusion that k-1,∞ has either a positive or only a small negative temperature coefficient. The falloff curve of Figure 2 has been modeled with k-1,∞ ) 2.5 × 10-11 cm3 molecule-1 s-1, in contrast to the SACM/CT value from eq 6 of 1.9 × 10-11 cm3 molecule-1 s-1. In this way, a better consistency with Figure 3 is obtained. The deviations between the measured points from ref 21 and the modeled falloff curve in Figure 2 may be attributed to the fact that the experiments of ref 21 were evaluated using a Lindemann expression for the pressure dependence of k-1 instead of the more realistic falloff expressions from unimolecular rate theory. More pronounced pressure dependences of k-1,∞ could only be detected near 1300 K in ref 8, such as illustrated in Figure 3. Because of the large width of the falloff curves and the indicated experimental uncertainties, even the pressure variation over 2 orders of magnitude of Figure 3 does not allow one to extrapolate reliably to k-1,∞ and k-1,0. For this reason,
we only compare one possibility of a modeled falloff curve with the experiments. This was chosen to be a curve with the following parameters: k-1,0 ) [Ar] 4.6 × 10-28 cm6 molecule-2 s-1 (βc ) 0.097, -〈∆E〉 ) hc 200 cm-1), k-1,∞ ) SC 4.0 × 10-11 cm3 molecule-1 s-1, and Fcent ) 0.043 (Fcent ) WC ) 0.72). Falloff curves with -〈∆E〉 ) hc 100 0.059, Fcent cm-1 (βc ) 0.054, k-1,0 ) [Ar] 2.5 × 10-28 cm6 molecule-2 s-1) and k-1,∞ ) 5.0 × 10-11 cm3 molecule-1 s-1 would fit the experiments equally well. In any case, one realizes that k-1,∞ has a markedly positive temperature coefficient. The preferred value for 〈∆E〉 and the assumption of an only weak temperature dependence is consistent with the analysis of several other reaction systems; see, for example, refs 39, 40, and 45. 6. Representation of Rate Coefficients for Dissociation and Recombination According to the foregoing discussion, a recommendation for k-1,∞ at this stage can only be made in two preliminary ways, (i) there is the discrepancy up to a factor of 5 near 300 K between two groups of values and (ii) by preferring the higher values (k-1,∞ ≈ 1.3 × 10-11 cm3 molecule-1 s-1), the temperature coefficient of k-1,∞ could be either T+0.42 (modeled with SACM/CT on a simplified potential) or T+0.77 (based on the experiments from refs 8 and 21; see Figures 2 and 3). Tentatively, here, we select the latter and recommend
k-1,∞ ≈ 1.3 × 10-11(T/300 K)0.77 cm3 molecule-1 s-1
(9) The modeled k-1,0 for M ) Ar is approximated roughly by a power law
k-1,0 ≈ [Ar]3.0 × 10-23(T/300 K)-7.2 cm6 molecule-2 s-1 (10a) However, a simple power law gives only fair agreement with the modeled values. For exmaple, the calculated values with this expression at 300, 700, and 1300 K are 3.0 × 10-23, 6.7 × 10-26, and 7.8 × 10-28, while the modeled values in Figures 1-3 are 2.1 × 10-23, 1.5 × 10-25, and 4.6 × 10-28 cm6
2CF3 (+ M) S C2F6 (+ M) Temperature and Pressure Dependence
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k1,∞ ≈ 2.2 × 1018(T/300 K)-0.52 exp(-∆H°/kT) s-1 0
(12) with ∆H°/k ) 49316 K. Together with eq 10a, one furthermore 0 obtains
k1,0 ≈ [Ar]5.07 × 106(T/300 K)-8.49 exp(-∆H°/kT) cm3 molecule-1 s-1 0
(13a)
If eq 10b is adopted, the more accurate expression
k1,0 ≈ [Ar]7.13 × 1010(T/300 K)-13.8 Figure 4. Rate coefficients for 2CF3 (+ M) f C2F6 (+ M) in M ) Ar (modeled falloff curves from this work; see text).
molecule-2 s-1. A more accurate fit to the modeling is obtained using
k-1,0 ≈ [Ar]4.22 × 10-19(T/300 K)-12.51 exp(-2973 K/T) cm6 molecule-2 s-1
(10b)
Finally, the modeled Fcent (for M ) Ar) can be represented by
Fcent ≈ (1 - a) exp(-T/T*) + a exp(-T**/T)
(11) where a ) 0.069, T* ) 260 K, and T** ) 880 K. The falloff curves only to a rough approximation can be represented by eq 8 because the low values of Fcent are accompanied by broadening and asymmetries. At this stage, it appears sufficient to account for this43 by replacing the exponent [1 + (log x)2]-1 in eq 8 by {1 + [(log x + a)/(N ( ∆N)]2}-1 where N ≈ 0.75 - 1.27 log Fcent, a ≈ -0.12, and ∆N ≈ 0.1 + 0.6 log Fcent (the positive sign for ∆N applies to x > 1 and the negative sign to x < 1). Figure 4 shows a complete set of falloff curves such as those modeled for M ) Ar in the present work. On the basis of the recommended rate coefficients k-1 of eqs 9-11 and the equilibrium constants Kc of section 2, one may also arrive at a recommendation for the so far not directly measured dissociation rate coefficients k1. Using the representation of Kc from eq 4b, with eq 9, one has
exp(-52289 K/T) cm3 molecule-1 s-1
(13b)
results. The shapes of the falloff curves for dissociation and the values of Fcent are the same for dissociation and for recombination. Previous estimates of k1 are superseded by the present determination. In earlier work for high-temperature conditions (above 1500 K), either second-order low-pressure behavior46 or first-order high-pressure behavior27 was assumed. In the work of ref 46, the measured value of k-1 from ref 1 was assumed to correspond to k-1,0 and to increase with T1/2 and then was converted to k1,0 with JANAF Tables of 1966. As this procedure appears inadequate today, clearly, this approach to k1 has to be rejected. In the work of ref 27 at 1300-1600 K, for [M] ) [Ar] ) 2((0.5) × 1019 molecules cm-3, the ratio k3Kc1/2 was determined. It then was combined with a ratio k3/k-11/2 from ref 47, which was measured at 490-620 K in H2/D2 mixtures at [M] near 4 × 1017 molecules cm-3 (in a relative rate study based on CF3 + C2H6 as a standard reaction48). The resulting value of k1 ) 4.3 × 1017 exp(-47504 K/T) s-1 for the applied [Ar] according to Figure 4 should not correspond to the limiting high-pressure first-order range but be about a factor of 4 below k1,∞. Nevertheless, it is a factor of 1.5 higher than the present result for k1,∞ from eq 12. The reasons for this inconsistency can clearly be attributed to the neglect of pressure dependences in extrapolating the ratio k3/k-11/2 to higher temperatures over a wide range. The determination of k1 in ref 27 therefore cannot be considered to be of quantitative value. 7. Conclusions Although still a number of unexplained discrepancies among groups of experiments exists, and the theoretical modeling cannot yet be carried out on an ab initio level, a number of conclusions can be drawn. Recombination experiments below 1000 K within experimental uncertainty so far were all performed very close to the high-pressure limit (except the
TABLE A1: Contributing Factors in the Modeling of k-1,0 and Fcent (see text and refs 39-41)a T/K
300
FE Frot Frotint SC k-1,0 βc WC k-1,0 SC Fcent Fcent
1.10 5.65 14.72 6.09 × 10-23 0.35 2.13 × 10-23 0.35 0.30
a
700 1.25 3.66 5.15 7.67 × 10-25 0.19 1.46 × 10-25 0.10 0.079
1000
1300
1500
2000
1.40 2.83 3.31 4.87 × 10-26 0.13 6.33 × 10-27 0.069 0.052
1.58 2.33 2.42 4.74 × 10-27 0.097 4.60 × 10-28 0.059 0.043
1.73 2.07 2.05 1.20 × 10-27 0.079 9.48 × 10-29 0.058 0.041
2.21 1.60 1.49 6.53 × 10-29 0.051 3.33 × 10-30 0.067 0.044
SC WC Fvib,h(E0) ≈ 4.2 × 1014 (kJ mol-1)-1, Fanh ≈ 1.43, -〈∆E〉 ≈ hc 200 cm-1, k-1,0 , and k-1,0 in cm3 molecule-1 s-1.
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J. Phys. Chem. A, Vol. 114, No. 14, 2010
VLPP/φ experiments of refs 6, 11, and 14, whose effective pressures are difficult to establish). The high-pressure rate coefficients k-1,∞ for CF3 recombination have either a positive or only a small negative temperature coefficient. This conclusion is based both on experiments and on SACM/CT modeling with “standard” potential parameters, but it is in contrast to the predictions from the canonical flexible transition-state theory of ref 22. We suggest that the differences in the modeling results are due to different properties of the empirically expressed anisotropies of the potential energy surfaces. However, final answers can only be given when better ab initio results for the potential become available. Then also, the apparent differences between CH3 and CF3 recombinations will become understandable. A preliminary representation of full falloff curves for dissociation and recombination is given in section 6. These results from the present work should be useful for designing new experiments. Such experiments obviously are needed to resolve the unsatisfactory discrepancies between the various recombination data near 300 K. Acknowledgment. The authors thank D. M. Golden, M. J. Pilling, M. Rossi, and G. Skorobogatov for helpful discussions of this reaction system. This work also profited from funding within the Max-Planck partner group La Plata s Go¨ttingen. A.E.C. also thanks the Alexander von Humboldt Foundation for support. Appendix Molecular Parameters Used for Modeling Rate Coefficients and Equilibrium Constants. Frequencies (in cm-1). C2F6: 1251 (2), 1243 (2), 1228, 1117, 807 (reaction coordinate), 714, 619 (2), 523 (2), 376 (2, transitional modes), 348, 220 (2, transitional modes), 68 (torsion); from ref 23. CF3: 1253.8 (2), 1086, 701.4, 508.7 (2); from ref 49. Rotational constants (in cm-1). C2F6: A ) 0.095, B ) C ) 0.062, σext ) 6, σrot,int ) 3. Barrier of hindered internal rotor Vo ) 16.57 kJ mol-1, Bred ) 0.382; from ref 23; hindered rotor partition function approximated by Qhind.rot. ≈ Qtors [exp(-kT/ Vo)]1.2 + (Qrot.int.free/σrot.int.[1 - exp(-kT/Vo)]1.2 following ref 39. CF3: A ) B ) 0.364, σ ) 3; from ref 50; C ) 0.189; from ref 51. Enthalpies of formation at 0 K (in kJ mol-1). C2F6: -1335.55 ( 4; from ref 23. CF3: -462.75 ( 2; from ref 25. Lennard-Jones parameters (σ in Å, ε/k in K). C2F6: σ ) 5.19, ε/k ) 201; Ar: σ ) 3.47, ε/k ) 114; from ref 52. For contributing factors in the modeling of k-1,0 and Fcent, see Table A1. References and Notes (1) Ayscough, P. B. J. Chem. Phys. 1956, 24, 944. re-evaluated in ref 17. (2) Ogawa, T.; Carlson, G. A.; Pimentel, G. C. J. Phys. Chem. 1970, 74, 2090. (3) Basco, N.; Hathorn, F. G. M. Chem. Phys. Lett. 1971, 8, 291. (4) Hiatt, R.; Benson, S. W. Int. J. Chem. Kinet. 1972, 4, 479. (5) (a) Skorobogatov, G. A.; Seleznev, V. G.; Slesar’, O. N. Dokl. Akad. Nauk SSSR 1976, 231, 1407. (b) Translation: Skorobogatov, G. A.; Seleznev, V. G.; Slesar’, O. N. Dokl. Phys. Chem. 1976, 231, 1292). (6) Rossi, M.; Golden, D. M. Int. J. Chem. Kinet. 1979, 11, 775. (7) Fagarash, M. B.; Moin, F. B. React. Kinet. Catal. Lett. 1979, 11, 265. (8) Glaenzer, K.; Maier, M.; Troe, J. J. Phys. Chem. 1980, 84, 1681.
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