J. Phys. Chem. 1996, 100, 2141-2144
2141
Global Fits of Methyl-Methyl Recombinational Data to Prezhdo’s New Interpolation Formula Jan P. Hessler Chemistry DiVision, Argonne National Laboratory, 9700 South Cass AVenue, Argonne, Illinois 60439 ReceiVed: August 24, 1995; In Final Form: October 30, 1995X
Prezhdo has suggested (J. Phys. Chem. 1995, 99, 8633) a new interpolation formula for the pressure-dependent behavior of dissociative and recombinational reactions. This formula, k(Pr) ) k∞[Pr/{1 + Pr + (Pr/B)A}], introduces two phenomenological parameters, A and B. Fits to isothermal methyl-methyl recombinational data are used to determine their temperature dependence. Global fits to an extensive set of data give highpressure rates coefficients which are compared to values derived by other formulations.
I. Introduction Recently, Prezhdo suggested a new interpolation formula for the pressure-dependent behavior of dissociative and recombinational reactions.1 This formula and others were compared to RRKM and master-equation calculations of methyl-methyl recombination to form ethane. Prior to the publication of Prezhdo’s suggestion, we had submitted the results of global fits to an extensive data set for this reaction.2 In this article, similar global fits of this data to Prezhdo’s interpolation formula are reported. The temperature dependence of the limiting rate coefficients and the two phenomenological parameters introduced by Prezhdo has been obtained. Three parameters are needed to describe the temperature-dependent behavior in the falloff region. The best-fit results for the high-pressure rate coefficient agree with global fits based on Oref’s “J-equation”3 and differ significantly from fits based upon broadening functions4,5 and Gardiner’s “a-equation”.6 Prezhdo has suggested that the Lindemann-Hinshelwood formula be modified by the addition of a term in the denominator which contains two phenomenological parameters, A and B. His modified formula for the pressure-dependent rate coefficient, k(Pr), is
Pr k(Pr) ) k∞ 1 + Pr + (Pr/B)A
(1)
where A is constrained by 0 < A < 1, the reduced pressure Pr ) k0P/k∞RT, the high-pressure rate coefficient is k∞, and the low-pressure rate coefficient is k0. Of course, both the highand low-pressure rate coefficients and the parameters A and B may depend upon temperature. Equation 1 may be reformulated in several different ways, but because of its simplicity, this formulation is preferred. The same procedures and data discussed previously2 are used to systematically determine the temperature dependence of the high-pressure rate coefficient and the parameters A and B and to perform the global fits. II. Analysis of Isothermal Data Previously, we showed it is very difficult to extract a lowpressure rate coefficient from the current set of methyl-methyl recombinational data. Furthermore, its value depends upon the formula used to reduce the data. On the basis of these results, this discussion is started by setting the low-pressure rate coefficient equal to 5.0 × 10-26 cm6 s-1 and assuming it is X
Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-2141$12.00/0
independent of temperature. Later, the global fits will show this is reasonable. The isothermal data at 577 K may be considered representative. The results of a least-squares analysis of this data (k∞, A, and B varied but k0 fixed) are shown in Figure 1. The best-fit result is the solid line, and the dashed lines represent the 68.3% confidence envelopes7 calculated from a Monte Carlo simulation.8 The correlations of the parameters A and B and the nonlinear nature of the fit are shown in Figure 2. Here, δA ) (A - A′)/A′ and δB ) (B - B′)/B′, where A′ and B′ are the best-fit values. From Figure 2, we note that a 31% change in A is correlated with a 1272% change in B and a -34% change in A is correlated with a -99.7% change in B. At every temperature the isothermal data sets provide more information on the high-pressure rate coefficient than the parameters A and B. As we showed previously, the temperature dependence of the high-pressure rate coefficient is given by
k∞(T) ) A∞ exp(-T(K)/T∞)
(2)
Four isothermal data sets were used to establish the applicability of eq 2 and to estimate its coefficients, A∞ ≈ 9.0 × 10-11 cm3 s-1 and T∞ ≈ 800 K. With these coefficients nine isothermal sets (296 e T(K) e 1525) were used to determine the temperature dependence of the parameters A and B. These results are shown in Figures 3 and 4. The error bars represent the 68.3% confidence limits determined by Monte Carlo simulations. From Figure 3 one can see that A ≈ 0.5 and is independent of temperature. From Figure 4 the temperature dependence of B may be approximated by
B(T) ) B0 exp(-T/TB)
(3)
where B0 ≈ 1 and TB ≈ 100 K. Equations 1, 2, and 3 and the low-pressure rate coefficient, k0, define the model used to perform the global fits. III. Global Fits The results of six different nonlinear least-squares global fits to data which span 296 e T(K) e 1750 and 0.15 e P(Torr) e 1.7 × 105 are given in Tables 1 and 2. The uncertainities shown in these tables are 68.3% confidence limits calculated from the values of χ2 and the diagonal elements of the error matrix at the best-fit values of the parameters.9 For the first fit only three parameters were varied, A∞, T∞, and TB. The fact that χ2/ndf (number of degrees of freedom) is less than unity indicates that the interpolation formula reproduces the data to within their experimental uncertainities. Examination of the normalized © 1996 American Chemical Society
2142 J. Phys. Chem., Vol. 100, No. 6, 1996
Figure 1. Results of least-squares analysis of 577 K isothermal data. The solid line is the best fit, and the dashed lines represent the 68.3% confidence envelope determined by a Monte Carlo simulation. See the text for details and references.
Hessler
Figure 3. Best-fit values of the parameters A of eq 1 determined by analysis of isothermal data sets. The error bars represent 68.3% confidence limits which were determined by Monte Carlo simulations.
Figure 2. Scatter plot of 68.3% of the best-fit values of A and B determined by the Monte Carlo simulations. The axes δA and δB refer to fractional deviations from the best-fit values shown in Figure 1.
residuals and their distribution indicates there is no systematic deviation between the data and the formula. The results of three four-variable fits are also shown in Table 1. Of these, the two where A was held fixed at 0.5 appear to be slightly better. Finally, fits with five and six variable parameters are reported in Table 2. The quality of these fits is better than those in Table 1, but it is questionable whether this improvement is significant. The relative weights and correlation coefficients9 for the sixvariable fit are shown in Table 3. It is interesting to observe that for this fit the parameter A has the largest relative weight, 0.704, the weights for the coefficients of the high-pressure rate coefficient sum to 0.182, and the sum of the weights of the coefficients of the parameter B is 0.111. The relative weight for the low-pressure rate coefficient is only 0.004. Perhaps of more significance are the correlation coefficients. The correlation coefficients for A with TB, B0, and k0 are 0.978, 0.972, and 0.971, respectively. The coefficients for TB with B0 and k0 are both -0.909. The coefficient of B0 with k0 is -1.000. These relatively large coefficients indicate that the above parameters are inexorably linked. Furthermore, because of these correlations it is possible to reduce the number of varied parameters and yet obtain a reasonable fit to the data. This is consistent with the fact that the root-mean-squared deviations of the threeand six-variable fits are not significantly different.
Figure 4. Best-fit values of the parameter B of eq 1 determined by analysis of isothermal data sets. The error bars represent 68.3% confidence limits which were determined by Monte Carlo simulations.
To demonstrate that the initial value of the low-pressure rate coefficient does not affect these calculations, consider the results of the six-variable fit. The crucial point is that the uncertainities for both the low-pressure rate coefficient and the phenomenological parameter B0 are both larger than their values. Of course, neither of these parameters may be less than or equal to 0. This comparison demonstrates the low significance of these two parameters. Conversely, the data does not contain enough information to determine them accurately. IV. Discussion Prezhdo compared the fractional deviations of his formula and others to RRKM and master-equation calculations of the methyl-methyl recombinational rate and concluded his formula is significantly better. Here, a different problem has been addressed, i.e. the extraction of limiting rate coefficients from a set of experimental data. There are three significant differences between comparisons of interpolation formulas to model calculations and the use of such formulas in global fits of experimental data. First, the pressure range of calculations may
Global Fits of Methyl-Methyl Recombinational Data
J. Phys. Chem., Vol. 100, No. 6, 1996 2143
TABLE 1: Best-Fit Values of the Three- and Four-Variable Global Fits
a
parameter
3 variables
4 variables
4 variables
4 variables
A∞ (cm3 s-1) T∞ (K) k0 (cm6 s-1) A B0 TB (K) χ∞/ndf rmsd (cm3 s-1)
8.86 ( 0.29E-11a 850.0 ( 61.0 5.0E-26 0.5 1.0 112.2 ( 2.1 0.961 9.77E-13
8.87 ( 0.30E-11 815.0 ( 71.0 5.0E-26 0.471 ( 0.036 1.0 102.0 ( 13.0 0.952 9.71E-13
9.03 ( 0.38E-11 838.0 ( 69.0 5.0E-26 0.5 0.80 ( 0.18 115.0 ( 4.0 0.940 9.64E-13
9.02 ( 0.38E-11 837.0 ( 69.0 4.07 ( 0.81E-26 0.5 1.0 115.0 ( 4.0 0.941 9.65E-13
Read E-11 as × 10-11.
TABLE 2: Best-Fit Values of the Five- and Six-Variable Global Fits parameter
5 variables
(cm3 s-1)
9.19 ( 743.0 ( 63.0 5.0E-26 0.409 ( 0.053 0.51 ( 0.22 86.0 ( 18.0 0.875 9.28E-13
0.35E-11a
A∞ T∞ (K) k0 (cm6 s-1) A B0 TB (K) χ2/ndf rmsd (cm3 s-1) a
6 variables
9.16 ( 0.41E-11 749.0 ( 55.0 3.30 ( 0.70E-26 0.411 ( 0.063 1.0 87.0 ( 21.0 0.880 9.31E-13
9.19 ( 0.43E-11 743.0 ( 79.0 4.8E-25 ( 2.5E-23 0.406 ( 0.082 0.0178 ( 1.37 86.0 ( 24.0 0.872 9.24E-13
Read E-11 as × 10-11.
TABLE 3: Relative Weights and Correlation Coefficients B0 k0 T∞ TB A∞ A
5 variables
B0
k0
T∞
TB
A∞
A
0.002
-1.000 0.004
-0.824 -0.823 0.057
-0.909 -0.909 -0.752 0.109
-0.750 -0.749 -0.938 -0.596 0.125
0.972 0.971 0.838 0.978 0.710 0.704
TABLE 4: Comparison of the High-Pressure Coefficients from Various Approximations approximation
χ2
asym. Lorentz sym. Gauss J-equation a-equation Prezhdo
170 172 172 207 185
χ2/ndf rmsd (cm3 s-1) A∞ (cm3 s-1) T∞ (K) 0.80 0.81 0.81 0.97 0.88
8.9 × 10-13 8.9 × 10-13 8.9 × 10-13 9.8 × 10-13 9.3 × 10-13
7.71 × 10-11 7.68 × 10-11 8.78 × 10-11 6.44 × 10-11 9.19 × 10-11
1298 1095 723 3709 743
be as large as needed. For example in the calculations performed by Prezhdo the pressure extends from 10-5 to 105 Torr for a maximum-to-minimum ratio of 1010. On the other hand, the largest range in pressure for the experimental data is at 296 K, where the pressure extends from 1.5 × 10-1 to 7.5 × 103 Torr for a maximum-to-minimum ratio of only 5 × 104. Second, the limiting rate coefficients are known in model calculations, whereas they are never known in the experimental case. In fact, one of the more important applications of interpolation formulas is their use to extract limiting rate coefficients from data. Third, each experimental datum has an uncertainty, and these uncertainties are not necessarily constant or a constant fraction of the experimental values. Previously, we performed global fits of the methyl-methyl recombinational data with the asymmetric Lorentzian broadening function recommended by Gilbert, Luther, and Troe;4 the symmetric Gaussian broadening function suggested by Wang and Frenklach;5 Gardiner’s “a-equation”;6 and Oref’s “J-equation”.3 One test of these different formulations is to compare the coefficients of the high-pressure rate coefficient deduced by each. This comparison is given in Table 4. The results fall into three groups. One is associated with approximations which use broadening functions and gives A∞ ≈ 7.7 × 10-11 cm3 s- and T∞ ≈ 1200 K. Another is associated with Oref’s and Prezhdo’s modifications of the Lindemann-Hinshelwood formula and gives A∞ ≈ 9.0 × 10-11 cm3 s-1 and T∞ ≈ 730 K, and the third
is Gardiner’s “a-equation”, which gives A∞ ≈ 6.4 × 10-11 cm3 s-1 and T∞ ≈ 3700 K. These differences are statistically significant. When values of χ2 are compared, Prezhdo’s formula is only 8% higher than the three lower values and Gardiner’s “a-equation” is 21% higher. Since the values of the χ2/ndf are all less than unity, these differences may not be statistically significant. Finally, when comparing interpolation formulas, one must consider the number of parameters which must be extracted from the data to describe the temperature-dependent behavior in the falloff region. For example, the asymmetric Lorentzian broadening formulation requires between one and four parameters (a, T***, T**, and T*). The symmetrical Gaussian formulation requires at least two (R the shift and σ the width) and possibly more if either R or σ depends upon temperature. Oref’s “Jequation” may require only one or several. Here, the number depends upon the complexity of the temperature dependence of the limiting rate coefficients. Gardiner’s “a-equation” may require one, two, or more. For Prezhdo’s formula, we needed three parameters (A, B0, and TB), but as Prezhdo has shown, five or six may be required. For all of these approximations, we have found significant correlations between these parameters. Perhaps one of the roles future calculations might perform would be to identify the physical significance of some of the parameters or the phenomenological relationships between them. This would reduce the number of free parameters and, thereby, lower the uncertainty in the values of the limiting rate coefficients. Another approach may be to force the RRKM and/or masterequation calculations to reproduce the experimental results. Realistic experimental uncertainities and scatter may be added to these results to produce a “fictitious” set of data. With this data set a systematic evaluation of the various empirical formulas could be performed. Parts of this approach have been discussed by Robertson, Pilling, Baulch, and Green.10 Acknowledgment. William C. Gardiner, Jr., Joe V. Michael, and Albert F. Wagner are thanked for their helpful comments. S. H. Robertson, M. J. Pilling, D. L. Baulch, and N. J. B. Green kindly provided a preprint of their work. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences.
2144 J. Phys. Chem., Vol. 100, No. 6, 1996 References and Notes (1) Prezhdo, O. J. Phys. Chem. 1995, 99, 8633-8637. (2) Hessler, J. P.; Ogren, P. J. J. Phys. Chem., in press. (3) Oref, I. J. Phys. Chem. 1989, 93, 3465. (4) Gilbert, R. G.; Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 169. (5) Wang, H.; Frenklach, M. Chem. Phys. Lett. 1993, 205, 271. (6) Gardiner, W. C., Jr. In 12th IMACS World Congress on Scientific Computation, Proceedings; Vichnevetsky, R., Borne, P., Vignes, J., Eds.; IDN, Grande EÄ cole d’inge´nieurs: Villeneuve d’Asq, France, 1988; p 582.
Hessler (7) Meyer, S. L. Data Analysis for Scientists and Engineers; John Wiley and Sons, Inc.: New York, 1975. (8) Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences, 2nd ed; McGraw-Hill Book Co.: New York, 1993. (9) Hessler, J. P.; Ogren, P. J.; Current, D. H. Comput. Phys., in press. (10) Robertson, S. H.; Pilling, M. J.; Baulch, D. L.; Green, N. J. B. J. Phys. Chem. 1995, 99 (36), 13452-13460.
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