J. Phys. Chem. 1993,97, 5024-5031
5024
Interpolation Formulas for Unimolecular Rate Coefficients in the Falloff Region Z. Pawlowska,t
W.C. Gardiner,'*$and I. Oref
J
Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel, and Department of Chemistry and Biochemistry, University of Texas at Austin, Austin, Texas 78712 Received: November 5, 1992; In Final Form: February 24, I993
Three algebraic formulas accounting for the pressure and temperature dependence of unimolecular reaction rate coefficients are developed and compared with one another and with rate coefficients computed with the kOp) [(k, kOp)2 4(J3/2 master equation-RRKM method. The three equations are k = {-(k, l)kmkOp]O.5]/2 P3p and some show better agreement at P < P3/2. Better agreement could hardly be expected. The deviations appear to be independent of the bath gas. At 1300 K (Figure 5) the situation is different. The fit is better above P3/2 than below it. It is better for the J and FZ equations than the a and FI equations. The a equation is good at valuesjust below P3/2,and at lower pressure they all deteriorate to give deviations larger than 30%. Cyclobutane fission showsdifferentbehavior. At 800 K (Figure 6) the errors above P3/2 are at the most f20%, while below 9 1 2 , the errors are larger and depend on which formula is used. The effect of the weak collider on the deviation is minor while its effect on the value of P312is of course major. The stronger the collider the lower the value of P3/2, i.e., the falloff curve moves to lower pressures. At 1500 K (Figure 7) the deviations span a 150% to -30% range. At pressures above P3/2 the errors range from +60% to -30% and the J and F2 models give the best fits. At pressures below P3/2 the a and F2 equations give the best fits. Again, the effect of the weak collision is there but small. Temperature Dependence of 83/2. The quadratic eq 37 is used to represent the ratio as a function of temperature for various colliders. Figure shows a sample calculation of a;',/a?, for cyclobutane fission for a very weak (250 cm-I) and a #airly strong (1200cm-I) collider. It can clearly be seen that the analytical expression (eq 37) gives reasonable agreement of aY[Ja;cl2with values calculated from master equation solutions. It IS thus possible to calculate ayT2/aT2at any temperature once the constants in eq 37 are known. hince a is a function of a (=( m ) d o w n ) the constants in eq 37 are a function of the value of a. The dependence on a of the three constants A, B, and C in eq 37 is well represented by the polynomial expressions eqs 3840. Sample calculations for cyclobutane fission are shown in Figure 9. The smooth function through the calculated points indicates that by obtaining the coefficients in eqs 3840,a772/aF2can be calculated for every temperature and pressure. The set of coefficients resulting from calculations on cyclobutane and cyclobutene is given in Table I. TemperatureDependence of 5312. The temperature dependence of J was discussed at length in ref 1 1 in terms of eqs 33-35. The
+
"i
c
---
t 0 500
loo0
MEC
.
0 1500
2000
TEMPERATURE, K Figure 14. Dependence of Fcen,values for cyclobutane fission upon temperature from eq 42 and from master equation calculations.
values of A ( a ) and B ( a ) in eq 33, calculated using eqs 34 and 35 with the coefficients given in Table 11, are shown in Figure 10 for cyclobutane fission." These values are used then in eq 33 to obtain the ratio In Figure 11 the values of J;f,/Jsc12 calculated from eqs 33-35 are compared with the master equation results for the three colliders. The agreement is seen to be very good. It is thus possible to obtain J;fi at any temperature, pressure, and type of collider. Temperature Dependence of F. The F,,, dependence of the weak collision shape factor Pcis given by low-temperature F;' (eq 12)and high-temperature (eq 13) expressions. The temperature-dependent Fcent value (eq 29)is represented in Figure 12 for cyclobutane fission. Values of P cas functions of pressure calculated from eqs 12, 13, and 29 are compared to master equation calculations in Figure 13. The discrepancy is large at 800 K and small at 1500K. Moresatisfactoryresults are actually obtained using just a quadratic fit:
ct;lcz/c12.
F,", =f1+ f i T + f 3 P
(42) where t h e 5 are constants. An example (Table I11 and Figure 14) shows that eq 42is a very good representation of Fcent. Figure 15 shows the good fit obtained by using eq 42 instead of eq 33 for obtaining F,,,,.
Conclusions Three interpolation formulasfor the calculation of unimolecular rate coefficients in the falloff region were studied. Each has
The Journal of Physicul Chemistry, Vol. 97, No. 19, 1993 5031
Unimolecular Rate Coefficients
F, (empirical)
C Y CLOBUTANE FISSION
parameters
I
T*1500K
'Y -10
-15
-5
-exact .-)--.
-----
fit to F2 Fl fit to FI
cIOCt
0
5
log ( k ,PI ka 1 Figure 15. Dependence of F Cupon pressure for cyclobutane fission from eq 42 and from master equation calculations.
advantages and disadvantages with respect to various criteria such as pressure range of interest, simplicity, empiricality or semiempiricality, and the total number of parameters needed to calculate the rate coefficient for a given combination of temperature, pressureand bath gas. To summarize the three models:
k,, k,, a,,
a,, a,,
bo, b,, b2
The eight parameters define k completely, Le., at any pressure or temperature and for any collider. For strong colliders, five parameters are needed. u (empirical)
parameters
k,, k,, a,,
a,, 02,
bo, b,, 4,CO, c1, c2
For strong colliders, five parameters are needed.
F,(empirical) parameters
or
For strong colliders, six parameters are needed. Interpolation formulas are a simple and computationally efficient way of obtainingrate coefficients as functionsof pressure and temperature. The deviatiorlsof the values of the interpolated coefficients from values calculated by master equation solutions, however, do depend on the model and on the temperature. While deviations as large as a factor of 2 arise in the worst situations, interpolation formulas are still the only practical way of using falloff corrections in large scale dynamic modeling. There is a way to suppress errors while still taking advantage of the computational efficiency of interpolation formulas. It is not necessary, as we have done here, to derive parameter values for the interpolation formulas from the behavior of the functions at the P3/2measure of the center of the falloff region, which may be far removed from the pressure range of actual interest. One can instead readily derive parameter values to give exact agreement with the full unimolecular. reaction rate theory (or with an experiment) at a pressure in the middle of a range of practical interest, then use an interpolation formula to connect that value smoothly to the correct low- and high-pressurelimits.
Acknowledgment. This research was supported by the U.S.Israel Binational Science Foundation, the Technion VPR Fund (to I.O.), the Gas Research Institute, and the Robert A. Welch Foundation (to W.C.G.).
References and Notes
J (semiempirical) parameters
ko,k,, u, P,P*, P** c, N,d, k,, L A , f 2 , f 3 , C, N,d
k,, k,, P,P*, P** or ko, k,, ~ , f , , f p f 3
For strong colliders, six parameters are needed.
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