Breakdown of the local linearization of the rate equation at high [CH 3 ]

5. Discussion

5.5 Breakdown of the local linearization of the rate equation at high [CH 3 ]

The master equation methodology used to fit the experimental data relies on the local linearization developed in Section 3. As discussed in that Section, because reaction 3 is reversible and both CH3 and C2H6 are recognised in the ME analysis, there are two chemically significant eigenvalues, 1 and 2; the former is zero and relates to mass balance. CH and the two rate coefficients can be separated using the equilibrium constant.

There is a limitation imposed by the truncation of the Taylor series and it is that the equation of motion is only valid for very short intervals of time. The Jacobian, and therefore 2,depend on the methyl concentration, which necessarily changes as the system evolves. Therefore the expansion limited to linear terms will only be valid for that period of time for which the change in methyl radical concentration is very small, and this time period will vary depending on where the system is on the evolution trajectory. At the end of this period the Jacobian and therefore  need to

be re-evaluated. As discussed above, the approach accounts for the reversibility of the process. The master equation analysis refers to an infinitesimally short period of time in which [CH3] is invariant.

Note that even though the eigenvalue of smallest magnitude is changing as [CH3] changes, the second order rate coefficient is invariant.

The application of the master equation to association / dissociation reactions, with particular reference to CH3 + CH3, has been discussed by Davis and Klippenstein.

They showed that problems arise, especially at high [CH3]. These derive from a number of potential sources, but there is a specific problem when the timescale of the association process becomes comparable with the timescales of collisional relaxation in C2H6*. Davis and Klippenstein provided a particularly clear analysis of this aspect of the problem by reference to the Lindemann model for association. Problems arising from the overlap of chemically significant eigenvalues (CSEs) with eigenvalues for internal energy relaxations (IEREs) are widely recognised in master equation analyses of multiple well systems.71,72 Eigenvalue overlap was investigated for CH3 + CH3 in order to establish that such problems had been avoided in the data analysis. Figure 7 shows a plot at 900 K and a bath gas concentration of 1018 cm-3, of 2, 3 and4 vs [CH3]a; 3 and4 are the two IEREs of smallest magnitude. Figure 7 also shows the rate coefficient, kr, calculated from 2; kf is negligible at 900 K.

At the lower values of [CH3]a, 2 increases linearly with [CH3]aand kris constant. For [CH3]a 5×1016 cm-3, however, 2 increases more slowly with [CH3]a, there is a clear mixing between the three eigenvalues 2 - 4 and the apparent kr decreases. These changes reflect those discussed in

more detail by Davis and Klippenstein.48 The deviation from linearity in 2, and the decrease in kr,

occur for a greater separation of the CSEs and IEREs than is generally found in studies of linear multiwell reactions, reflecting the complexity in the dynamical behaviour, as discussed by Davis and Klippenstein. Since the IEREs scale linearly with the bath gas concentration, the values for [CH3]a at which the apparent rate coefficient begins to diverge from the true value increases with bath gas concentration. Figure 7b shows a plot of kr/ kr limit vs [CH3], where kr limit is the true value for the

association rate coefficient, obtained at the limit low [CH3] where the dynamical complexities are eliminated. The plot was used to establish conditions under which accurate rate coefficients could

be obtained: broadly speaking, the results are accurate provided [CH3]a is at least a factor of 100 less than the bath gas concentration.

Figure 7. (a)The non-zero chemically significant eigenvalue, 2, and the two internal energy relaxation eigenvalues of smallest magnitude, 3 and 4, as a function of [CH3]a, for T = 900 K and a total bath gas concentration of 1018 cm-3. The rate coefficient for association, calculated from 2, is also shown; under these conditions, kCH3+CH3 = 1.60 × 10

-11

cm3 molecule-1 s-1 =klimit..

(b) A plot of kr / kr, limit, vs [CH3] , for T = 900 K and a total bath gas concentration of 1018 cm-3. kr, limit

is the limiting value as [CH3]a tends to zero (see Figure 7a and its caption); the plot shows the convergence of the rate coefficient under the conditions used in the data fitting, where [CH3]a = 10

1015 cm-3 was used. The slight positive curvature at the highest [CH3] is a result of the eigenvalue mixing shown in Figure 7a by the onset of non-linearity in 2.

a

39 6. Conclusions

We have demonstrated the fitting of p and T dependent rate data for an association reaction using a master equation analysis in which the microcanonical dissociation rate constants, kd(E), are linked to

the high pressure limiting rate coefficient for association, , via inverse Laplace transformation (ILT). The fits have been made over an unprecedentedly wide range of temperature, so that the ILT parameters, A and n, are well-defined and compare well with the most accurate theoretical values. The final parameters obtained were primarily influenced by the absorption spectroscopy data obtained below 1000 K, which lie closest to the high pressure limit.

Jasper et al.58 argued that it is necessary to use a 2D master equation, allowing collisional changes in both E and J, in order to accurately predict fall-off behaviour in association reactions. The use of the present fitting methodology is not compatible with such an approach, because the ILT procedure generates only kd(E) not kd(E,J), although it does contain the correct averaging over J in

the high pressure limit; in addition the large number of ME calculations needed in a data-fitting procedure are not currently feasible when using a 2-D master equation model. An investigation of the relationship between energy transfer parameters obtained in 1-D and in 2-D ME models would be of value, especially in relation to the use of experimental data to obtain representations of rate coefficients for use in complex kinetic models.

The recommended expression for is 5.76 × 10-11 (T/298 K)-0.34 cm3 molecule-1 s-1 with an uncertainty of ±10% at 300 K increasing to ±20% at 2000 K. Chebyshev polynomials for calculating the rate coefficient as a function of pressure and temperature in Ar and He are given in the Supporting Information, together with a parameterization using the Troe and Ushakov procedure.70

Acknowledgements. We are grateful to Ahren Jasper for providing the energy transfer parameters from reference67 and Lev Krasnoperov for providing the high pressure limiting rate coefficients included in Figure 6 prior to publication. We are grateful to EPSRC for funding for RJS under grant EP/J010871/1.

40 Supporting Information Available

Three files of Supporting Information are available:

Supporting information I contains:

1. Log file from PGOPHER fit to the available ground state data. This also gives the constants for all the states used in Model I. The log file can be found in Supplementary Information III.

2. The correction factors used to correct the rate coefficients of Slagle et al. for the absorption coefficients calculated using absorption coefficients from Models I and II. 3. The results of fitting the data used in Fit 5, but with different energy transfer parameters

above and below 1000 K.

4. The results of fitting the data used in Fit 5 plus the experimental data of Du et al.18The results of fitting the data used in Fit 5 plus the experimental data of Yang et al.56

5. Calculated values for k0(T)_using the data from Fits 5 and 6, together with a comparison

of the recommendation of Baulch et al.4 and parameterization of the temperature dependence. Broadening factors for calculating k(p,T) using the method of Troe and Ushakov.70

6. Chebyshev polynomials for calculating k(p,T) for CH3 + CH3. A spreadsheet is provided as a separate SI file to facilitate use of the Chebyshev polynomials. The instructions for using the spreadsheet to calculate k(p,T) under any conditions within the fitted range are given in the spreadsheet in Supporting Information II.

7. The MESMER input file, that includes the moleculardata used in the master equation calculations.

Supporting Information II is a spreadsheet which can be used to calculated k(p,T) from the Chebyshev polynomials under any conditions within the fitted range. Instructions for use are contained in the spreadsheet

Supporting Information III is the PGOPHER log file.

This information is available free of charge via the Internet at http://pubs.acs.org.

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