J. Phys. Chem. 1995,99, 14738-14741
14738
Energy Dependence of (AE)down and the Shape of Falloff Curves: Implications for Modeling of Experimental Data Vadim D. Knyazev Department of Chemistry, The Catholic University of America, Washington, D.C. 20064 Received: May 18, 1995; In Final Form: July 21, 1995@
The influence of the energy dependence of ( a d o w n on the shape of falloff curves in unimolecular reactions is investigated using two model systems: the decomposition of C2H5 and the isomerization of cycloheptatriene (CHT). An exponential-down model of collisional energy transfer is used with ( a d o w n ( @ = E', where i = 1 for the C2H5 decomposition and i = [/2 for the CHT isomerization. It is demonstrated that neglect of the energy dependence of ( a d o w n in the fitting of experimental falloff data may yield significant discrepancies between low-pressure and high-pressure results.
Introduction The unimolecular decomposition of molecules and free radicals, as well as the related reverse reaction of recombination, plays an important role in the chemical kinetics of many hightemperature processes, including the pyrolysis and oxidation of hydrocarbon^.'-^ Many of these reactions are in the falloff region under combustion conditions. On the other hand, knowledge of the rate parameters of these reactions is usually derived from experiments conducted over a limited range of conditions (temperature and pressure) which are frequently far from the actual conditions of combustion. Therefore, there exists a need for reliable and easily implemented tools for extrapolating rate constants of decomposition and association reactions to conditions other than those of laboratory experiments. During recent years, modeling techniques based on the formulation of the master e q ~ a t i o n have ~ . ~ been increasingly used for treatment of experimental data on the rates of decomposition and association reactiom6 Typically, in this treatment one or several parameters of a reaction model are adjusted to fit the observed falloff behavior of the reaction under study. The reaction model necessarily includes an assumption about the form of the function describing the collisional energy transfer. The most frequently used form is the exponentialdown model7 characterized by the expression
where P(F,E)is the probability of transition from energy E to E' on collision, A is a normalization constant, and (&?)down is the average energy lost in deactivating collisions. Although the exact real form of the P(E',E) function is not known, the fact of its dependence on the total energy of the excited molecule is well established.8-'0 Generally, the average energy transferred in both upward and downward collisions, (AE), increases with energy. The functional form of this increase depends on the particular system of excited molecule and collider. However, some general trends can be seen8s9 For small molecules (CS2, S 0 2 ) (AE) was found to increase proportional to ?I at E = 2000-36 000 cm1-1.13*12 For large molecules, such as azulene and benzene, a directly proportional dependence of (AE) on E was observed at energies up to %25 000 cm-' with a leveling~~
'Abstract published In Advance ACS Abstracts, September 1, 1995.
off above this e n e r g ~ . ~ O , ' ~In - ' ~the case of molecules of intermediate size (CnFzn+2,n = 3-8, and CnHzn+lF,n = 6-8), it was foundi6that (AE) is directly proportional to E in the range 15 000-40 000 cm-I. This same type of directly proportional dependence was observed" for CF2Cl2 and CF2HCl at E = 10 000-30 000 cm-I with a stronger dependence (approximately (AE) at E below 8000-10 000 cm-l. This dependence of (AE) (and, consequently, of ( W d o w n ) on excitation energy is generally ignored in the modeling of decomposition and association reactions. A study of the influence of the energy dependence of (AE)down on the shape of the falloff is presented here. The effect of this dependence on the results of the fitting of experimental data in different parts of the falloff curve is also investigated.
Method Two model unimolecular reactions are considered. One is the thermal decomposition of ethyl radical in He; the other is the isomerization of cycloheptatriene (CHT) in Ar. In both cases the values of &E), energy-specificrate constants, are calculated by the RRKM method.I8 Models of the molecules and transition states involved are taken from Feng et aLi9 for the C2H5 decomposition and from Astholz et aLZofor the CHT isomerization (Model A1 of ref 20). The exponential-down model of energy transfer described by eq I was applied in both cases. The energy dependence of (AE)down was expressed in terms of a power function: (AE)down(E) = aEi with i = 1 for the C2Hs decomposition and i = l/2 for the CHT isomerization.Use of these functions results in approximate dependencies of (AE) = E for the former (which is appropriate for small molecules) and (AE) = E for the latter in the vicinity of the reaction threshold EO. The coefficients a were chosen to satisfy the requirements that (AE)dOwn(E=Eo) match the values of (@down used in refs 19 and 20. For the ethyl radical decomposition this requirement yields a temperature-dependent a = 1.971 x 10-5(TIK) (to obtain (AE)down(Eo)= 0.255 T cm-I K-I, as recommended by Feng et al.I9). In the case of CHT isomerization we use a = 3.768 cm-1'2 to obtain (AE)down(Eo)= 500 cm-'. The method of Gaynor, Gilbert, and of solving the steady-state master equation was employed. An energy step size of 50 cm-l was used. Recently, Bemshtein and 0reP2 demonstrated that the steady-state approximation
0022-365419512099-14738$09.0010 0 1995 American Chemical Society
(-down
J. Phys. Chem., Vol. 99, No. 40,1995 14739
Energy Dependence and Falloff Curve Shape
cannot be used for calculation of unimolecular reaction rates by the master equation solution at high temperatures and low pressures, where relaxation to the steady-state population occurs on time scales comparable to the reaction time. The examples given by these authors indicate that reasonable agreement between the exact values of rate constants and those obtained under the steady-state assumption is reached after several hundred collisions with the bath gas. The calculations presented here are limited to conditions where the ratio of collision frequency to the rate constant is higher than 400.
-3 1.6
-2
t
-1
1
1
0
8
0
0
K
2
1
n
1.4
0 Y
L.
1.2
1 .o
Results and Discussion The qualitative effect of the energy dependence of (AE)down on the shape of falloff can be understood from simple considerations. In the low pressure limit, the rate constant of the unimolecular reaction is determiqed by the rate of excitation of molecules from lower energy levels to the reaction threshold energy (Eo) and, hence, an “effective” (AE)down corresponds to (AE)down(E) at some energy E lower than EO. Under the conditions where the reaction is close to its high-pressure limit, the deviation of the population from the Boltzmann distribution (and the falloff from the high-pressure limit) is determined by the competition between energy transfer and reaction at energies above the threshold. Hence, an effective (-down corresponds to (AE‘)down(E) at some energy E higher than EO. Thus, if a falloff curve calculated assuming a constant (AE)down is compared with one calculated assuming that (AE)down increases with energy (with the values of (&!?)down of both models equal at the reaction threshold), in the latter case one can expect lower values of rate constant at low pressures and higher values at higher pressures. The effect of the energy dependence of (AE)down on the shape of the falloff can be expressed in terms of an “energy dependence factor”, FED, which is defined as the ratio of the rate constant value obtained assuming an energy-dependent (AE)down(E) to that calculated with constant (AE)z$ The latter is chosen in such a way that the low-pressure-limit rate constants for these two cases coincide:
0.01
1
0.001 -3
-2
I
I
I
L
-1
0
1
2
10 9 ( k,
[W k - 1
Figure 1. (a) Dependence of FED,the energy dependence factor, on the reduced pressure, k~[M]/k“,for the decomposition of C& in He at T = 800-1800 K. (AE)T;A (see text) = 195, 237, 274, 313, and 362 cm-l at 800, 1O00, 1200, 1500, and 1800 K. (b) Shapes of the falloff curves for the same system at 1800 K with (solid line) and without (dashed line) accounting for the energy dependence of (AE)down.
1.4
I
I
I
I
1.3 1200K
n w
LL
1.2
1.1
This factor is, therefore, a measure of the difference between two falloff curves with identical ka and k” calculated with and without accounting for the energy dependence of (&?)down. As discussed above, when one goes from the low-pressure limit to higher pressures, the effective (-down increases and, hence, FED increases, too. As one approaches the high-pressure limit (which is the same for both curves), FED necessarily decreases to 1. The dependence of FED on the reduced pressure, h[M]l k“, is shown in Figures l a and 2 for the two systems considered here. The difference between the falloff curves representing the energy-dependent and energy-independent P(F,E)can be as high as 58% (C2H5 decomposition in He). The maximum values of FED for these two systems are comparable at equal temperatures. Although the dependence of ( W d o w n on energy is weaker in the case of CHT isomerization, the population distribution of reacting molecules in this system is broader due to a higher number of degrees of freedom, and the effective values of (AE)down at low and high pressures correspond to more widely separated values of active intemal energy. Figure l b also illustrates the difference in the shape of falloff curves (kl k“ vs log(l~[M]/k”))calculated with and without accounting for the energy dependence of ( W d o w n .
1 .o
0.9
Figure 2. Dependence of PD, the energy dependence factor, on the reduced pressure, b[M]/k“, for the isomerization of cycloheptatriene in Ar at T = 800- 1200 K. Dashed line indicates the conditions when the ratio of collision frequency to the decomposition rate constant is lower than 400 and the steady-state master equation is inapplicable. (AE)TdA (see text) = 490,479, and 474 cm-’ at 800, 1000, and 1200
K.
Implications for Fitting of Experimental Data. Typically, in experiments measuring falloff in unimolecular reactions or reactions of association, only a limited range of pressures is covered and only a portion of the falloff curve is obtained. If the experimental results are used in a modeling procedure with
14740 J. Phys. Chem., Vol. 99, No. 40, 1995 8oo
700
r
E \ c)
m
e
0 U
P 0
Knyazev
5
c
1800K
/
A
I 300
I
I
I
I
I
L
i
200
W
:loot E
500
a V
I
400 0
300 200
-t u1
100 -4
-2
0
2
I og ( ko[ M I/ km) Figure 3. Values of (AE)ZiAobtained from fitting the falloff curves for CzH5 decomposition in He calculated using (AE)down= 1.971 x lO+ET (see text).
the purpose of extracting quantitative information on weak collision effects (such as the values and temperature dependence of (&??)down), the outcome can be significantly affected by neglect of the energy dependence of (&??)down. Figure 3 presents the results of fitting of the calculated (with (AE)down(E) = E, r ) values of the rate constants of the C2H5 decomposition in He using the model with constant ( A E ) Z i A . The fitted values of (AE)Fi:display a pronounced pressure dependence which increases with temperature. At the highest temperature considered, 1800 K, the difference between values of (AE)ZiA obtained from the fitting of calculated rate constants close to the high-pressure limit and in the low-pressure limit is almost a factor of 2. One of the model reactions considered here, C Z H ~ C2& H (1, - l), presents an exceptional case of a system for which experimental falloff data are available over a relatively wide range of temperatures and conditions. Addition of H atom to the double bond of ethylene was directly studied in the falloff region relatively close to the high-pressure limit at temperatures 285-800 K.23-26 The decomposition reaction 1 was directly studied in the low-pressure limitI9 at 876-1094 K and close to the high-pressure limit at 800 K.23 These experimental data and the fitting procedure employed by Feng et al.I9 illustrate the noticeable effect of the inclusion of the energy dependence of the collisional energy transfer parameters on the fitted values of (AE)down. The fitting exercise described by Feng et al. was repeated twice, once assuming that (&??)down(E) = E (here the fitting parameter is (AE)down(Eo), the value of (&??)down(E) at E equal to the reaction threshold), and again assuming independent of energy. The results in the latter case practically coincide with those of Feng et al., as expected. The fitted values of (AE)down(Eo) obtained from the low-pressure measurements lie above the (AE)FiA values. The treatment of the highpressure results, however, yields (LLE)down(EO) lower than the corresponding (&??)F$ If these values of (AE)Fii and (A.f?)down(&) are plotted together versus temperature (Figure 4), one can see that including the energy dependence in the model
+
I
**
!!i J
600
c 1 W
-
I
I
I
I
I
200
400
600
800
1000
1200
T / K
Figure 4. Plot of values of (AE)down(Eo)and :Z:)E'A.( vs temperature obtained from fitting the experimental data on reaction 1 (small circles, low-pressure limit, ref 19) and on reaction -1 (close to the high-pressure limit, large circles, refs 24 and 25; large squares, ref 23; large triangles, ref 26). Open symbols: values obtained assuming (AE)ZiAindependent of energy. Closed symbols: values of (AE)down(Eo)obtained assuming (AQdowndirectly proportional to energy. The line represents the temperature dependence of (AE)downrecommended in ref 19.
noticeably (although not completely) reduces the disagreement between the results of the treatment of low- and high-pressure data. The examples in Figures 3 and 4 indicate the importance of accounting for the energy dependence of (-down in the modeling of experimental falloff data at high temperatures. Use of energy-independentmodels of collisional relaxation can result in disagreement between collisional efficiencies obtained at low pressures and at pressures at which reaction is significantly in the falloff or relatively close to the high-pressure limit. If the results of fitting (such as absolute values of (&??)down and its temperature dependence) are used for extrapolation and prediction of rate constants at other temperatures and pressures, errors associated with this extrapolation may be considerably enhanced.
Acknowledgment. The author thanks Professor Irene R. Slagle for her support of this work. References and Notes (1) Warnatz, J. In Combustion Chemistry; Gardiner, W. C., Jr., Ed.; Springer-Verlag: New York, 1984. (2) Hucknall, D. J. Chemistry of Hydrocarbon Combustion; Chapman and Hall: New York, 1985. (3) Westbrook, C. K.; Dryer, F. L. frog. Energy Combust. Sci. 1984, IO, 1. (4) Tardy, D. C.; Rabinovitch, B. S. Chem. Rev. 1977, 77, 369. (5) Gilbert, R. G . ; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell: Oxford, U.K.,1990. (6) Pilling, M. J.; Robertson, S. H.; Green, N. J. B. In Vibrational Energy Transfer Involving Small and Large Molecules, Vol. 2; Advances in Chemical Kinetics and Dynamics; Barker, J. R., Ed.; JAI Press: Greenwich, CT, in press. (7) Rabinovitch, B. S.;Tardy, D. C. J. Chem. Phys., 1966, 45, 3720. (8) Hippler, H.; Troe, J. In Bimolecular Reactions; Ashfold, M. N. R., Baggott, J. E., Eds.; The Royal Society of Chemistry: London, 1989. (9) Oref, I.; Tardy, D. C. Chem. Rev. 1990, 90, 1407. (10) Barker, J. R.; Toselli, B. M. Znt. Rev. fhys. Chem. 1993, 12, 305. (1 1) Dove, J. E.; Hippler, H.; Troe, J. J. Chem. Phys. 1985, 82, 1907. (12) Heymann, M.; Hippler, H.; Nahr, D.; Plach, H. J.; Troe, J. J. Phys. Chem. 1988, 92, 5507. (13) Hippler, H.; Otto, B.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 428. (14) Toselli, B. M.; Barker, J. R. J. Chem. Phys. 1992, 97, 1809. (15) Shi, J. H.; Barker, J. R. J. Chem. Phys. 1988, 88, 6219. (16) Tardy, D. C.; Song, B. H. J. Phys. Chem. 1993, 97, 5628. (17) Tardy, D. C. J. fhys. Chem. 1993, 97, 5624.
(&??)r;;
(hE)down Energy Dependence and Falloff Curve Shape (18) Robinson, P. J.; Holbrook, K. A. Unimoleculnr Renctions; WileyInterscience: New York, 1972. (19) Feng, Y.; Niiranen, J. T.; Bencsura, A.; Knyazev, V. D.; Gutman, D.; Tsang, W. J. Phys. Chem. 1993, 97, 871. (20) Astholz, D. C.;Troe, J.; Wieters, W. J. Chem. Phys. 1979, 70, 5107. (21) Gaynor, B. J.; Gilbert, R. G.: King, K. D. Chem. Phys. Lett. 1978, 55,40. (22) Bemshtein, V.; Oref, I. J. Phys. Chem. 1993, 97, 6830.
J. Phys. Chem., Vol. 99,No. 40,1995 14741 (23) Hanning-Lee, M. A.; Green, N. J. B.; Pilling, M. J.; Robertson, S. H. J. Phys. Chem. 1993, 97, 860. (24) Kurylo, M. J.; Peterson, N. C.; Braun, W. J. Chem. Phys. 1970, 53,2776. (25) Michael, J. V.; Osbome, D. T.; Suess, G. N. J. Chem. Phys. 1973, 58,2800. (26) Lightfoot, P. D.; Pilling, M. J. J. Phys. Chem. 1987, 91, 3373.
JP951376F
