J. Phys. Chem. 1996, 100, 7541-7549
7541
Ab Initio Calculations and Three Different Applications of Unimolecular Rate Theory for the Dissociations of CCl4, CFCl3, CF2Cl2, and CF3Cl S. S. Kumaran, M.-C. Su,† K. P. Lim, J. V. Michael,* A. F. Wagner,* and L. B. Harding* Chemistry DiVision, Argonne National Laboratory, Argonne, Illinois 60439
D. A. Dixon*,‡ DuPont Central Research and DeVelopment Experimental Station, Wilmington, Delaware 19880-0328 ReceiVed: January 3, 1996X
Previously measured Cl dissociation rate constants for CCl4 and CFCl3 were analyzed with three different kinetics modeling calculations. The three models differ in detail but primarily are distinguished by the manner in which the high-pressure limiting rate constant is determined: model 1 involves a calibration to transport properties of the dissociated fragments, model 2 uses a Gorin model with a hindrance parameter, and model 3 requires variational transition state theory on an ab initio reaction path where all low-frequency motion off the path is presumed to be a free rotation. All three models have two adjustable parameters: the dissociation energy E0 and the average energy transferred to the buffer gas 〈∆E〉down. All three models are found to give comparable fits to the experiment and produce quite similar values for the adjustable parameters. For CCl4, the values are E0 ) (68.2 ( 1.2) kcal mol-1 and 〈∆E〉down ) (750 ( 125) cm-1. For CFCl3, the values are E0 ) (76.5 ( 0.5) kcal mol-1 with 〈∆E〉down ) (800 ( 215) cm-1. These values are compared to those obtained in similar studies for CF2Cl2 and CF3Cl. The results indicate a substantial and consistent decrease in the C-Cl bond energy with each additional chlorine substitution in the chlorofluoromethanes. Isodesmic electronic structure calculations at the MP2 level confirm this effect but find it to be a little smaller than the experimental results indicate. Extended electronic structure calculations provide heats of formation for all nine CHxFyClz methyl radicals.
Introduction In the preceding paper, we documented the thermal decomposition kinetics of the chlorofluoromethane, CFCl3.1 This complements earlier studies from this laboratory on CCl4,2 CF3Cl,3 CF2Cl2,4 CH3Cl,5 CH2Cl2,6 and COCl2.7 All of these were shock tube studies and utilized primarily Cl-atom atomic resonance absorption spectrometry (ARAS) as the detection technique. In CCl4 and CF3Cl, the database was extended by combining the ARAS with laser schlieren (LS) results,2,3 and the combined rate constants were then rationalized with one version of Rice-Ramsperger-Kassel-Marcus (RRKM) theory. However, for CFCl31 and CF2Cl2,4 with the same data input, the results were explained by applying three different formulations of unimolecular reaction rate theory. The final conclusions from the three versions of theory are quite similar even though they differ significantly in complexity. Here the other two cases, CCl42 and CF3Cl,3 are considered in order to further explore the comparison of methods. Validating theoretical predictions of unimolecular rate data with less rigorous theoretical methods therefore supplies a motivation for the present work. For both CFCl3 and CF2Cl2,1,4 the three versions of theory yield nearly identical threshold energies. This suggests that these derived threshold energies are reliable estimates of the C-Cl bond strengths in these two molecules. One conclusion from this analysis was that the C-Cl bond strength of CF2Cl2 is ∼9 kcal/mol stronger than that of CFCl3, suggesting a large Cl-atom substituent effect on the C-Cl bond strengths for the CFnCl4-n series. Another motivation for this study then is to †
On sabbatical leave from Butler University, Indianapolis, IN 46208. Present address: Pacific Northwest Laboratory, P.O. Box 999, K1-83, Richland, WA 99352. X Abstract published in AdVance ACS Abstracts, April 1, 1996. ‡
S0022-3654(96)00047-0 CCC: $12.00
use the existing data on CCl4 and CF3Cl with these same three unimolecular rate theory analyses to determine the C-Cl bond strengths for these molecules and thereby fill in the remaining gaps in this CFnCl4-n bond strength series. Also, we have carried out isodesmic electronic structure calculations at the MP2 level, and the theoretical Cl substituent effect is compared to experiment for the four molecules. The idea of a Cl substituent effect on C-Cl bond energies may also be applicable in the CHnCl4-n cases; unfortunately, possible molecular HCl elimination can increase the complexity of unimolecular theoretical analyses. However, additional electronic structure calculations for the heats of formation of all nine CHxFyClz methyl radicals are also presented, and these can provide estimates of this effect in the CHnCl4-n methanes. Unimolecular Theory on CCl4 In this study, as in the studies on CFCl3 and CF2Cl2, three versions of unimolecular reaction rate theory are used to model the rate behavior for CCl4 and CF3Cl. These are a semiempirical Troe calculation and two distinct RRKM calculations.8-11 All three approaches include the average energy transfer parameter, 〈∆E〉down, which determines the weak collision efficiency factor, βc. Theory depends strongly on the two highly coupled quantities, the threshold energy, E0, and 〈∆E〉down. All three calculations have therefore been iteratively carried out for various mutual combinations of E0 and 〈∆E〉down until the best fit with the measured data is obtained. From a consideration of the 12 calculations for the four molecules, final conclusions can be made on the same basis regarding threshold energies and energy transfer parameters. For CCl4, ARAS rate constants were measured as a function of temperature at approximately four constant densities: 1.3 × © 1996 American Chemical Society
7542 J. Phys. Chem., Vol. 100, No. 18, 1996
Kumaran et al.
TABLE 1: Best Fit Values for the High- and Low-Pressure Limiting Rate Constants Used in the Theoretical Calculations for the Dissociation Reaction: CCl4 (+M) f CCl3 + Cl (+M)a k0 ) A0T-n0e-T0/T calculation type Troe/Lennard-Jones RRKM/Gorin RRKM/ab initio
A0
/(1032
cm3
molecule-1 s-1)
0.2881 0.2404 0.9013
k∞ ) A∞T-n∞e-T∞/T n0
T0/K
10.744 10.750 10.884
38 199 37 486 38 946
A∞
/1016 s-1
1.400 18 323.0 49 885
n∞
T∞/K
0 1.30 1.18
32 462 34 727 35 448
a The derived back reaction rate constant values for each of the models over the temperature range of the experiment are 5.85, 1.75, and (8.6 ( 0.6), respectively, from top to bottom, all in units of 10-11 cm3 molecule-1 s-1.
1018, 1.9 × 1018, 2.8 × 1018, and 5.3 × 1018 molecules cm-3. The LS rate constants measurements as a function of temperature can also be grouped around four approximately constant densities: 0.75 × 1018, 1.1 × 1018, 2.4 × 1018, and 2.9 × 1018 molecules cm-3. This grouping by densities was not made explicit in ref 2, but it is convenient here, and displays of the comparison between theory and experiment will label the measurements by the above densities. For both techniques, measurements at the highest and lowest densities were fewer in number and more prone to experimental uncertainties. Furthermore the scatter of the data with the ARAS method is noticeably larger than that of the LS method, complicating an even-handed treatment of the two sets of data. In recognition of these features of the experimental data sets, the “best fit” between theory and experiment was defined as follows. The measured rate constants of the two middle data sets of each experimental technique were fit to an Arrhenius expression, C exp(-D/T), leading to four pairs of (C,D) values. The root-mean-square (rms) relative error of the model calculations with these Arrhenius fits were then determined for each of the four densities. (This negates the difference in scatter between the ARAS and LS measurements.) These four rms relative errors for each model calculation were then combined into one grand rms relative error. Minimization of this grand rms relative error will produce a good fit. However, for one or two data sets, the relative error is not evenly balanced over the temperature range (i.e., the mean relative error is not close to zero). Small variations in parameters will slightly increase the grand rms relative error but significantly improve the balance of the error across the temperature range of each data set. This compromise defines the “best fit” for each model. This approach ensures a balanced representation for the most reliable measurements from both experimental techniques. Comparison with the measurements at the highest and lowest densities of each experimental technique show that model calculations optimized in this way are in good agreement with experiment even though the measurements were not explicitly included in the best fit definition. Then, in a manner similar to that illustrated previously,1,4 the second-order dissociation rate constants were theoretically modeled in three ways. The first method (model 1) utilizes the semiempirical method of Troe8-11 to calculate the pressure and temperature behavior for CCl4 (+M) f CCl3 + Cl (+M). The molecular parameters necessary for this determination are given in Table II of ref 2. The calculation uses the Whitten-Rabinovitch method for calculating the density of states. The collision rate between CCl4 and the buffer gas is calculated with the Lennard-Jones expression, eq 8 of ref 1, from potential parameters derived from polarizabilities12 as suggested by Cambi et al.13 This effectively replaces the combining rules that are commonly used.12 The strong collision low-pressure rate constant can be determined from two quantities: (1) density of states and (2) collision rate.8 We have carried out the calculations with variations in E0 between 65 and 69 kcal mol-1. In an earlier calculation where only the ARAS data were considered,14 E0 ) 66.7 kcal mol-1
and 〈∆E〉down ) 735 cm-1 provided the best fit; however, including the LS data in the analysis requires a slight adjustment to E0 ) 67.2 kcal mol-1 and 〈∆E〉down ) 621 cm-1. The predicted rate constants for 1100 e T e 2200 K in the middlepressure range are between ∼0.25 and 0.65 of the low-pressure limit, βcksc 0 , indicating some sensitivity to both pressure and the rate constant for the high-pressure limit. Since the data are slightly in the falloff region, the Troe fit is influenced by the method for calculating the limiting highpressure rate constant, k∞. With this method this rate constant is evaluated by assuming that the transition state is a LennardJones complex. Hence, E0 ) ∆H°0. The rate constant for the back reaction is calculated from the collision rate constant with electronic degeneracy corrections giving a nearly constant value of kLJ ) (5.85 ( 0.05) × 10-11 cm3 molecule-1 s-1 between 1100 and 2200 K. This can be compared to a recent measurement15 of (6.5 ( 1.4) × 10-11 cm3 molecule-1 s-1 at 298 K for the recombination of CCl3 + Cl. The agreement is excellent. Equilibrium constants, K, are directly calculated from the molecular constants in Table II of ref 2 for the best fit value, E0 ) ∆H°0 ) 67.2 kcal mol-1, giving
K ) 2.37 × 1026 exp(-32450 K/T) molecules cm-3
(1)
to within (5% for 1100 e T e 2200 K. The value for the high-pressure Cl-atom dissociation rate constant is then calculated as k∞ ) kLJK. The resulting k∞ and an analogous fit to the low-pressure limit, k0, are listed in Table 1. All fits listed in this table have an rms relative error over the 1100-2200 K range of
