Influence of Molecular Parameters on Rate Constants of Thermal Dissociation/Recombination Reactions: The Reaction System CF4 ⇄ CF3 + F

The possibilities to extract incompletely characterized molecular parameters from experimental thermal rate constants for dissociation and recombination reactions are explored. The reaction system CF4 (+M) ⇄ CF3 + F (+M) is chosen as a representative example. A set of falloff curves is constructed and compared with the available experimental database. Agreement is achieved by minor (unfortunately not separable) adjustments of reaction enthalpy and collisional energy transfer parameters.


INTRODUCTION
Statistical theories of unimolecular reactions have reached a mature state such that they can be used to predict rate constants of thermal dissociation/recombination reactions over a wide range of conditions (see, e.g., refs 1 and 2). Obviously, this is of considerable practical importance. Nevertheless, there are often molecular parameters contributing to the rate constants that are not well known and that require detailed considerations, either by theoretical modeling or by educated guessing. In the following, we demonstrate the influence of selected molecular parameters on the rate constants. Such parameters are the average energy ⟨ΔE⟩ total transferred per collision between the excited molecules and the surrounding bath gas species, characteristic parameters of the potential energy surface (PES) of the reaction system, and, for dissociation reactions in particular, the enthalpy ΔH o of the reaction. These quantities all influence the rate constants, but they do it in different ways. It may not be easy to unravel their respective contributions. This is the issue of the present article.
The dissociation reaction . We illustrate the influence of the mentioned molecular parameters on the respective rate constants k 1 and k 2 , and we try to specify their values by combining experimental and modeling results. Regardless of the precision of the derived parameters, the described approach allows one to construct an internally consistent set of falloff curves and to identify uncertainties of the analysis. In this way, it may help improve recommendations of rate constants (see, e.g., the difference between various experimental results reported in ref 8).

MODELING APPROACH
The falloff curves of the rate constants k 1 and k 2 are dominated by limiting low-pressure (subscript 0) and high-pressure (subscript ∞) rate constants. For this reason, attention is first focused on these limiting rate constants. Approximate expressions for the transition of k 1 from k 1,0 to k 1,∞ (or k 2 from k 2,0 to k 2,∞ ) are available (see, e.g., refs 9−11) and are used when the limiting rate constants have been fixed. Here, we employ the expressions recommended in ref 10, see below.
High-pressure recombination rate constants k 2,∞ correspond to rate constants k cap for capture between the combining species. As long as the anisotropy of the PES can be neglected, with the parameters α 0 = −15.7706, α 1 = −8.6364, and α 2 = 0.9975. If recombination from a single electronic level of the reactants into a single electronic state of the adduct is considered, the electronic weight factor f el (given by the relevant ratio of electronic partition functions) has to be multiplied with k cap PST in addition. Table 1 shows results for k cap PST = k 2,∞ PST as a function of the temperature T. Accounting for the anisotropy of the PES reduces k 2,∞ to values below k 2,∞ PST . This is accounted for by a rigidity factor f rigid , which is smaller than unity, that is, Eq 5 provides a quick access to f rigid and shows its dependence on molecular parameters (B e here denotes the relevant rotational constant of the forming adduct, see below). It should be noted that the derivation of eq 5 in ref 12 was based on the validity of an approximate relation between α e and β e . Deviations from eq 5 of less than about ±10% were observed when α e /β e differed from 0.5 by less than ±10%. Anisotropies characterized by eq 6 initially were suggested by the analysis of experimental results, see ref 14.
Quantum chemical calculations, meanwhile, often confirmed the validity of this relationship (see also the results for the present reaction system shown in the Supporting Information). While the dependence of k 2,∞ on details of the PES is only weak, this is different for the limiting high-pressure rate constant k 1,∞ . As k 1,∞ follows from k 2,∞ through the equilibrium constant K eq = k 1,∞ /k 2,∞ and as K eq and k 1,∞ both contain the strongly temperature-dependent Boltzmann factor exp(−ΔH 0 o /kT), the analysis of k 1,∞ can be used to determine ΔH 0 o . Table 1 compares calculated temperature dependences of k 1,∞ and k 2,∞ , k 1,∞ PST and k 2,∞ PST , as well as the corresponding thermal rigidity factors f rigid (the used molecular parameters of the calculations are given in the Supporting Information).
The calculation of limiting low-pressure rate constants k 1,0 and k 2,0 has been described in detail in ref 11 and is not reproduced here. k 1,0 has been expressed in the form with various contributions characterized by individual factors. Table 2 shows calculations of such factors (the critical energy E 0 here is identified with ΔH 0 o ; temperature-dependent quantities are given for 300 K. While the calculation of most of the factors is straightforward and shows the dependence on molecular parameters in a transparent manner, the rotational factor F rot requires attention. It depends on the centrifugal barriers E 0 (J) of the system and, hence, on properties of the PES like the rotational constant of the system along the dissociating bond (see the Supporting Information). Properties of collisional energy transfer enter through the overall Lennard-Jones collision frequency Z LJ and a collision efficiency β c . Information on the latter quantity can be derived from the analysis of experimental values of k 1,0 and k 2,0 and is of particular importance. Through the solution of simplified master equations, a relation between β c and the average (total) energy ⟨ΔE⟩ total of the form was derived (see refs 11 and 15; F E corrects for the used expression of the energy dependence of the vibrational density a k 1,∞ in s -1 , k 2,∞ in cm 3 mol -1 s -1 , and equilibrium constant K eq = k 1,∞ /k 2,∞ in mol cm -3 ; superscript PST: phase space theory; rigidity factor f rigid = k 2,∞ /k 2,∞ PST , see the text.  Table 2 includes values of β c estimated with a value of −⟨ΔE⟩ total /hc = 100 cm −1 ; later on, ⟨ΔE⟩ total will be used as an empirical fit parameter to be extracted from those experiments that are sensitive to k 1,0 and k 2,0 .
Falloff curves connecting the limiting low-and high-pressure rate constants primarily are expressed in the Lindemann− Hinshelwood form; however, additionally accounting for a broadening of the falloff curves where x = k 1,0 /k 1,∞ or x = k 2,0 /k 2,∞ are proportional to [M] such that x represents a reduced pressure scale. Broadening factors F(x) (being smaller than unity) here are approximated in the form recommended in refs 9−11. The deviations of the falloff curves from the Lindemann−Hinshelwood form are most pronounced near the center of the curves, that is, close to those values of [M] where k 1,0 is equal to k 1,∞ (or where k 2,0 is equal to k 2,∞ ). The corresponding center broadening factors F cent can be estimated using the methods outlined in refs 9−11 and 16. They can be related to effective numbers of oscillators of the system, but they also contain energy transfer contributions. Symmetric broadening factors with N = 0.75−1.27 log F cent from ref 11 often are sufficient for applications, but refined asymmetric expressions with n = [ln 2/ln(2/F cent )] [0.8 + 0.2x q ] and q = (F cent − 1)/ ln(F cent /10) have also been tested (see refs 9 and 10). F cent has strong collision (for β c = 1) and weak collision (β c < 1) contributions, that is, F cent = F cent sc F cent wc , with F cent sc as given above (from refs 9−11 and 16) and F cent wc = max (β c 0.14 , 0.64) from ref 9. F cent first increases with temperature, and then, it decreases again. Representative values for the present system (with M = Ar) are F cent ≈ 0.74, 0.17, 0.095, 0.084, and 0.089 for T/K = 300, 1000, 2000, 3000, and 4000, respectively. Obviously, the values of the limiting rate constants are more important for the construction of the falloff curves than the precise values of F cent . Table 3 summarizes the derived values of k 1,0 , k 1,∞ , k 2,0 , and k 2,∞ , together with the corresponding K eq and approximate values for F cent .

EVALUATION OF EXPERIMENTAL RESULTS
The comparison of experimental results with modeled falloff curves such as that described in Section 2, on one hand, validates the used molecular parameters of the system entering the given analysis. On the other hand, it allows one to locate the experimental rate constants at their proper position along the falloff curves. We illustrate this with the available experimental data for reactions 1 and 2. The rate constant k 2 in the bath gas Ar was first measured in ref 7 at room temperature and at [Ar] = 1.1 × 10 −7 and 3.7 × 10 −7 mol cm −3 . As an increase of the measured k 2 by about a factor of 4 has been observed, the recombination reaction was assumed to be near its third-order range. The comparison of the data with modeled falloff curves from the present work in Figure 1, however, casts doubt on this interpretation of the measurements. Nevertheless, the measured absolute value of k 2 = 2.6 (±0.7) × 10 13 cm 3 mol −1 s −1 at [Ar] = 3.7 × 10 −8 mol cm −3 appears roughly reconcilable with the present calculated value of k 2,∞ = 1.3 × 10 13 cm 3 mol −1 s −1 . Meanwhile, the modeling predicts a weaker dependence on [M] than that claimed for the experiments. Indeed, the later measurements of refs 6 and 8 (in He between 3.9 × 10 −8 and 3.8 × 10 −7 mol cm −3 ) showed weaker pressure dependences than that found in ref 7. Figure 2 compares the corresponding results with a modeled falloff curve for M = He. The remaining differences between the results from refs 6 and 8 cannot be explained by uncertainties in the used molecular parameters, in particular in the used value for ⟨ΔE⟩ total . Therefore, the modeled falloff curves of Figures 1 and 2 based on assumed values of −⟨ΔE⟩ total /hc = 100 or of 500 cm −1 for M = He or of Ar appear to provide a more reasonable representation of the pressure dependence of k 2 . The agreement with the experiments from ref 6 appears quite satisfactory, while the difference from the results from ref 8 calls for a reinterpretation of the assumed mechanism of the experiments. Table 3. Modeled Limiting High-pressure (Subscript ∞) and Low-pressure (Subscript 0) Rate Constants for the Dissociation Reaction CF 4 (+Ar) → CF 3 + F (+Ar) (k 1 , for Temperatures 1000−4000 K) and the Recombination Reaction CF 3 + F (+Ar) → CF 4 (+Ar) (k 2 , for Temperatures 300−1000 K) a k 1,∞ = 4.0 × 10 16   The question arises how well measured dissociation rate constants k 1 agree with modeling results and whether they provide a further access to molecular parameters. Figure 3 compares measurements of k 1 at T = 2500 K and [Ar] between 10 −5 and 10 −4 mol cm −3 with modeling results for a series of values of −⟨ΔE⟩ total /hc. The best agreement seems to be obtained with a value of −⟨ΔE⟩ total /hc of the order of 500 cm −1 . Modeling of the temperature dependence of the corresponding falloff curves in Figure 4 clearly confirms that the dissociation measurements have been made far below the high pressure limit of the reaction, while recombination measurements at 300 K were made close to the high pressure limit of the reaction. As the latter thus practically cannot provide an access to ⟨ΔE⟩ total , one might be tempted to extract ⟨ΔE⟩ total from Figure 3. However, there is an alternative interpretation. Besides ⟨ΔE⟩ total , k 1 in Figure 3 also depends on the precise value of the reaction enthalpy ΔH 0 o . Decreasing −⟨ΔE⟩ total /hc from 500 to 100 cm −1 in Figure 3 would decrease the modeled k 1 at [Ar] = 10 −5 mol cm −3 by about a factor of 2.9. This decrease could be compensated by a decrease of ΔH 0 o of reaction 1 by about 22 kJ mol −1 . As the uncertainty of ΔH 0 o apparently is markedly smaller today (see the Supporting Information), it appears more probable that −⟨ΔE⟩ total /hc in the present case is of the order of 500 cm −1 , such as that fitted in Figure 3, although this value is larger than the value of 100 cm −1 generally assumed in our modeling. In any case, this example illustrates the sensitivity of the modeled rate constants on molecular parameters. Unfortunately, the thermochemical parameter ΔH 0 o and the energy transfer parameter ⟨ΔE⟩ total cannot be separated in the analysis. In particular, no conclusion on the temperature dependence of ⟨ΔE⟩ total for the reaction CF 4 (+Ar) ⇄ CF 3 + F (+Ar) can be drawn. In other cases [e.g., the CH 4 (+Ar) ⇄ CH 3 + H (+Ar) system analyzed in ref 17], only a weak temperature dependence of ⟨ΔE⟩ total was postulated.

CONCLUSIONS
Regardless of whether correct molecular parameters have been used in the modeling or not, fixing ⟨ΔE⟩ total and ΔH 0 o to the values used for the interpretation of Figure 3 and assuming only a weak temperature dependence of ⟨ΔE⟩ total , the corresponding set of modeled falloff curves shown in Figure  4 provides an internally consistent picture of the temperature and pressure dependence of reactions 1 and 2. The outlined procedure illustrates the influence of various molecular parameters on the rate constants, and it may be used to identify problems of the interpretation of experimental results.