4047
J . Phys. Chem. 1989, 93, 4047-4059
+ XH,,
where X = C, N, 0, CI, S, Br, and I is given in Figure
3. Since the H-Br bond length in HBr and the H-P bond length in PH3 are nearly identical at 1.40 and 1.42 A, the activation energy for reaction Ib should be very close to that for O(3P) + HBr. The activation energy for O(,P) + HBr may be taken as 3.08 k ~ a l / m o l . Adopting ~~ an approximate value of A exp(-1550/T) cm3 cm3 s-l then gives k l b = 1 X =1X s-’. This is shown as line klb in Figure 2. It is clear that, on the basis of a reasonable expression for k l b ,the abstraction channel for the reaction O(,P) + PH3 can be expected to make a measurable contribution to the experimental overall rate constant kl only at T 2 -1000 K ( h e k l , k l b in Figure 2). A lower value for A (5 X lo-” cm3 s-I would be a reasonable lower limit) does not alter this conclusion. Thus, the lack of evidence for products of the abstraction channel at 298 K2’s2’ is understandable. Figure 2 further shows that, even at the highest temperature in the present experiments (423 K), the abstraction channel ( I b ) is still very minor ( 3 X I O i 2 ~ m - (b) ~ , 2 C (IO1’k2/cm3
+
-
(33) Lightfoot, P. D.; Kirwan, S . P.; Pilling, M. J. J . Phys. Chem. 1988, 92, 4938.
(34) Baggott, J. E.; Brouard, M.; Coles, M. A,; Davis, A,; Lightfoot, P. D.: Macpherson, M. T.; Pilling, M. J. J . Phys. Chem. 1987. 9 / . 317
Modeling Study of C H 3
+ H and C H 3 + D Reactions
The Journal of Physical Chemistry, Vol. 93, No. IO, 1989 4051
I
4
T
-
l.oo
"i, d
00
25
50
400 600 Helium pressurelTorr
Figure 4. Dependence of k,(D) on pressure at 400 K. The rate coefficients have been determined from a full numerical data analysis. The error bars refer to flu, and the full line refers to a pressure-independent weighted mean.
..-"'
P
200
75
Timelms
SCHEME I
Typical decay profiles for (a) H and (b) CH3 [CH&OCHJ = 1.06 X 10" cm-3,percentage photolysis = 0.53, total pressure = 100 2 Torr of He, and temperature = 504 K Figure 2.
CH3
+
=CHaD* JJklE(D)
D
kat€)
k (E)
CH2D
+
H
MJko
CH3D
1
I
200
400
I
+
I
600 800 Helium p r e s s u r e / T o r r
Rate coefficients for CHI + H, k,(H), as a function of helium pressure. A, 300 K; 0,400 K 0 , 500 K; A, 600 K. Error bars refer to 95.5% confidence limits. -, Troe factorization fits (Table VI). Figure 3.
molecule-' s-') Q 50, (c) [HIo < 0.2 [CH,],, and (d) atom signal to noise ratio Z 30. In addition, because acetone photolysis produces both CH, and H , it was necessary to keep [CH,], C 3 X lOI3 cm-, so as to avoid generation of H atoms in excess of -3 x 10l2 ~ m - ~ . Typical decay curves (and fits to the data) obtained under the above conditions are illustrated in Figure 2. Having characterized the radical decays (via the parameters A. and k 2 / a ) and, in separate experiments, determined the diffusion rate coefficients, kj, the H atom decays were fitted by use of nonlinear least squares via (6) with [HI, (relative units) and kl/u as variable parameters. Good fits were invariably obtained, as demonstrated by the random distribution of the residuals (see, for example, Figure 2), thus confirming the validity of the data analysis procedure employed. A potential complication arises from the generation of ,CHI in a two-photon process and its subsequent reaction with CH3, which generates H: ,CH2
+ CH3
-
supplementary material over the range of conditions 25 Q P/Torr C 600, in a helium buffer gas, and 300 d T/K C 600. The pressure range was restricted a t low pressures ( P 5 25 Torr) by H atom diffusion and the low CH3 + H rate coefficients and a t high pressures by quenching of the H(2P) atoms. Temperatures in excess of 600 K would be feasible if a detailed study of the H + CH3COCH3reaction were undertaken. Of more interest would be experiments below 300 K, where k , would more rapidly approach its limiting high-pressure value; a minor modification to the cell design should enable such measurements to be made. A discussion of error analysis is given in the supplementary material. d . CH,+ D. i. Introduction. It is evident from Figure 3 that the CH3 H reaction is well into the falloff regime and that quite long extrapolations are required to obtain k; at all temperatures. Measurements of the rate coefficient for CH3 + D, k,(D), present, in principle, a means of resolving this difficulty. The reaction may be represented by Scheme I. Zero point energy differences in channels a and b in Scheme I suggest that kb(E) >> k,(E) a t accessible energies, so that whenever CH3D* is formed it is either stabilized or decomposes via channel b, regardless of the pressure. Thus measurements of the rate of removal of D should generate pressure-independent rate coefficients that correspond to the high-pressure limit, kT(D), from which k;(H) could be calculated by using transition-state theory. The photolysis system was outlined in section 111 and is further discussed in the supplementary material. Typical reactant/precursor concentrations were varied within the following ranges: N,O:D2:CH3COCH3 = (2-11) X 1014:(1-15) X 10'6:(3-10) X 1014~ m - ~All. experiments were performed in a buffer gas of He. With laser fluences in the photolysis region of =12 mJ cm-*, the following typical reactant concentrations were generated: [D]o:[CH,]o:[H]o:[CH2]o = 0.02-0.04:1:> k , ( E ) (Scheme I). Table I 1 shows the rate coefficients, k,(D), averaged over the range of deuterium concentrations employed a t each temperature and pressure. The error bars (which are 95% confidence limits including errors in a) are discussed fully in the supplementary material. Within experimental error, k,(D) is independent of temperature over the range 300-400 K, with a mean value of (1.75 f 0.045) X cm3 molecule-I s-I, where the error limits refer to the 95.5% confidence limits, as returned from the error analysis. To account for unforeseen systematic errors, it would seem wise to widen these limits somewhat, giving k,(D) = (1.75 A 0.15) X IO-1o cm3 molecule-’ s-I. V. Discussion As discussed in the Introduction, most theoretical studies have concentrated on kf and there has been very little work directed a t a description of the reaction in the falloff region. It is clear from Figure 3 that the reaction is far from the high-pressure limit and some form of extrapolation is required in order to determine kf and hence compare the experimental results with theory. Alternatively, we must develop, if possible, a means of calculating k ; ( H ) from k;“(D). I n section Va, we present an analysis of the
TABLE 11: Rate Coefficients for CH, and TemDerature
T/K 289 291 300 300 315 325 350 351 376 401 40 I
He press./Torr 100 100 50 200
100 100 100 100 100 100 50-60
+ D as a Function of Pressure no. of
lOlok1(D)/
cm3 molecule-I
s-l
1.75 f 0.0866 1.72 f 0.100 1.81 f 0.076 1.88 f 0.110 1.75 f 0.086 1.69 f 0.079 1.65 f 0.097 1.69 f 0.103 1.78 f 0.140 1.80 f 0.088 1.78 f 0.130
expts
S(kl(D))”
20 17 17 12 20 20 14 15 15 20 21
0.030 0.036 0.021 0.039 0.03 1 0.028 0.038 0.041 0.060 0.036 0.058
Osingle standard deviations about the mean, excluding the contribution from u. *95.5% confidence limits, including uncertainties in k2, as returned from consideration of A,, u, and the rate coefficients employed in the full FACSIMILE analysis. See the supplementary material for discussion.
data using the Troe factorization procedure, a technique that leads, in the present case, to a strong negative temperature dependence of kT( H), in contrast to the temperature-invariant experimental data for kf(D). In section Vb we present a simple transition-state model of ( R l ) , which enables k f ( H ) to be predicted from k;“(D). This treatment demonstrates that the values of kf(H) obtained from the Troe analysis significantly exceed the values predicted from kf(D) in the 300-400 K range over which the latter were measured. These calculations are supplemented by microcanonical variational
Modeling Study of C H 3 + H and C H 3 + D Reactions
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4053 b. Determination of kf(H)f r o m k y ( D ) . i. Transition-State Theory. Applying transition-state theory to reactions R1 (H) and R 1(D) gives
TABLE 111: Limiting Rate Coefficients Obtained from Fitting the Data for k ,(H)Using the Troe Factorization 1OZ9k%H) /
300 400 500 600
7.6 f 1.6 4.7 i 1.2 3.1 f 0.6 1.8 f 0.6
2.7 f 0.4 3.1 f 0.6 3.0 f 0.5 2.5 f 0.8
0.659 0.627 0.599 0.566
kT(H)/kf(D) = qCH‘ qD exp(
0.45 0.39 0.34 0.30
qcHgD*
+
(7) where F,, is a broadening factor correcting for the energy dependence of the microcanonical dissociation rate coefficient and F,, corrects for weak collision effects. Fscdepends primarily on a parameter ST,which is the effective number of oscillators in the activated complex, on whose structure it therefore depends. F,, depends on STand on the average energy transferred per collision in a downward direction, ( AE)down.Fits to experimental data and the value of kf obtained by extrapolation depend most sensitively on ST. The aim of the fitting procedure is to obtain an analytic representation of t h e data and to determine kf( T ) . It is important that the values of STemployed are compatible with the structure of the activated complex and, therefore, with kT( T),since STis defined by
ST= 1
-(7)) EH - E D
(8)
*
R R K M (pVRRKM) calculations, designed both to compare k f ( H ) and kf(D) and to confirm the validity of the assumption made in section IVd that the experimental k,(D) values correspond to the high-pressure limit. These calculations lead to the same conclusion that the values of k f ( H ) predicted on the basis of kf(D) are significantly less than those obtained from a Troe analysis of the experimental k l ( H ) data. This discrepancy could, in principle, derive from the inadequacy of the extrapolation procedure or from scatter in the experimental data. The aim of section Vc is to demonstrate that this is not the case. We present a Troe analysis of the experimental k,(H) data with kf(H) constrained to the values predicted from kT(D) and very poor fits are obtained. More detailed fits, based on a full energy grained master equation (EGME)/pVRRKM analysis, are also presented which lead to the same conclusion, that the CH3 H rate coefficients cannot be predicted from the C H 3 D values. Finally in section Vd we compare our experimental data with previous measurements and provide an analytic representation over our experimental range. a. Falloff Analysis of the Experimental Data f o r CH, + H. T r ~ has e ~developed ~ a procedure for analyzing and for representing rate data in the falloff region. It is based on a modified Lindemann-Hinshelwood expression:
+
kH
+ T(d In q v * / d T )
where qv* is the vibrational partition function of the activated complex. Accordingly, STwas related to kf via a pVRRKM treatment (see below) and optimized iteratively until the kf value obtained from a nonlinear least-squares analysis of the experimental data at a specific temperature coincided with that calculated from the pVRRKM model. The optimal fits are shown in Figure 3, while the rate parameters are given in Table 111. It should be noted that the limiting rate coefficients were not sensitive to (AE)down. The uncertainties in k;” given in Table 111 are 95.5% confidence limits and are determined simply from the larger of the external and internal standard deviations of the mean. They do not include contributions from STnor errors inherent in the model. Even so the limits are quite wide, and it is clear that the reaction is so far into the falloff range that precise extrapolation cannot be made. (36) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 161. (37) Gilbert, R. G.; Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 169.
where qx is the total partition function for species X, refers to the activated complex, and E x is the threshold energy for reaction CH3 + X. Assuming (a) that the threshold energies, which arise from small maxima in the zero point energy surfaces, are zero (they are very small, provided there is no maximum in the electronic potential energy surface), (b) that the vibrational partition functions of the activated complex are unchanged on isotopic substitution, and (c) that the average geometries of the complex are the same for reactions R l ( H ) and Rl(D), then only the masses in the rotational and translational partition functions in (8) are significant and
Thus ky(H) = 21/2ky(D) N (2.5 f 0.2) X cm3 molecule-] s-I over the range 300-400 K. This value is significantly smaller than those obtained by fitting the experimental data (Table 111). ii. Microcanonical Variational Transition State Theory. It is possible that the origin of this discrepancy might reside in oversimplifications implicit in the transition-state model described in the last paragraph. Accordingly we have performed detailed pVRRKM calculations which we shall also employ, in section Vc, to reexamine the falloff behavior of reaction R l ( H ) . The pVRRKM procedure we employed is described in more detail elsewhere.38 In outline, a radial CH3-X potential was adopted and the rovibrational sum of states, N(E,,), determined as a function of r, the CH3-X bond distance, with the microcanonical transition state being located at the minimum in the sum of states. The distance-dependent rotational constants were evaluated explicitly and those vibrational frequencies that change significantly on dissociation described by a parameter a (see below), following the procedure adopted by Quack and Troe;’s2a is related, albeit in a complex way, to the angular potential. The approach employed was to vary a until the value calculated for kT(D) agreed with experiment and then to employ the same value of a and of the electronic potential energy function, in the calculation of kT(H). The pVRRKM method has been criticized39and it is important, in order to justify our calculation, that our approach is described in more detail. A Morse potential energy function
Ve(r) = Det1 - exp(-p(r - req))l2 was employed in the majority of the calculations, with @ = 1.88 and Truhlar,6 and Duchovic et a].@ have published ab initio surfaces that are significantly steeper than the Morse surface. Accordingly, to test the effect of such a surface on kT(H)/k;(D), we performed some calculations using the variable p representation proposed by Duchovic et aL4O
k’. H i r ~ t Brown ,~
p’ = a ( r - r e J 3
+ b(r - r,q)2+ c(r - rQ) + d
The values of a-d are given in the Appendix. The rotational coefficients, A and B, of CH3-X were evaluated explicitly as a function of r by solving the inertial tensor matrix equation,“’ assuming a constant C-H bond length and the variation in H C X bond angle calculated by Duchovic et aL40 (see the Appendix). -
~~~~
(38) Legg, A. F.; Pilling, M. J.; Whitham, C. J. Unpublished work. (39) Glanzer, K.; Quack, M.; Troe, J. Symp. ( I n t . ) Combust. 1976, Jdth, 959. (40) Duchovic, R. J.; Hase, W. L.; Schlegel, H. B. J . Phys. Chem. 1984, 88, 1339. (41) Thompson, H. B. J . Chem. Phys. 1967, 47, 3407.
4054
The Journal of Physical Chemistry, Vol. 93, No. 10, I989
Following Quack and Troe,',2 the vibretional modes were divided into two classes, (a) conserved modes, in which there is little change in frequency on dissociation, and (b) transitional modes. For CH,X, the latter category includes one of the deformations that correlates with the out-of-plane bending vibration in CH, and the rocking modes that correlate with product rotations; e.g., for CH3D u4(CH3D) (1306 cm-I)
-
u2(CH3) (606 cm-I)
v,(CH3D) (1477 cm-', doubly degenerate)
-
product rotations
Two models were employed to accommodate, in a tractable manner, these variations. I n the first, following the approach employed by H a ~ e , a~ *single parameter, a , was employed and an exponential variation in u assumed: udr) = u4(CH3X) exp(-a(r -
+ u2(CH3)[I - exp(-a(r - req))l
V2(r) = V2(CH3X) exp(-a(r This simple approach leads to overestimates of the densities of states for loose complexes, since the frequencies of the rocking m, whereas, in reality, they modes are forced to zero as r correlate with quantized rotations. To test the effects of this approximation, some more detailed calculations were made in which the reactant states were correlated with product states. This correlation, which is described in more detail in the Appendix, follows the approach used by Quack and Troel in their statistical adiabatic channel model (SACM). Unlike Quack and Troe,' however, who used the r-dependent correlated state energies to determine the number of "open" channels at a given reactant energy, we have simply employed the correlations to generate expressions for r-dependent rovibrational energies that have then been incorporated in a conventional yVRRKM procedure via direct state counting routines. Angular momentum conservation was accommodated in both correlated and uncorrelated models by ensuring that the total angular momentum quantum number, J , only takes values which are energetically accessible to both reactant, R, and complex, C, i.e.
-
B,J(J
+ 1) + E,*
6 E,,*;
BcJ(J
+ I ) + E,* < E,,*
*
where * and refer to the energized reactant and to the complex, respectively, and E, and E,, to vibrational and rovibrational energies. Angular momentum about the main symmetry axis was assumed to be active, so that K was not conserved. An effective potential curve Veff(r) was generated by adding the r-dependent zero-point energies of all modes except the reaction coordinate. The r-dependent sum of rovibrational states and the reactant density of states were then evaluated by a direct count p r o c e d ~ r e . first ~ ~ *counting ~~ the rotational states and then convoluting these sums with the vibrational states, which were assumed harmonic. Detailed rotational statistical weights were not evaluated, the sums simply being divided by the relevant symmetry numbers. Explicit calculations demonstrated that this procedure is valid at the temperatures of this investigation. Once the minimum in the sum of states N(E,,*) had been determined at r*, the microcanonical rate coefficient for dissociation was evaluated in the usual way: k(Evr*) = N(Evr*)/hp(Evr*)
where p(E,,*) is the density of states in the molecule, R, and E,' = (E,,* - Veff(r*)). These calculations were also employed to generate a Boltzmann averaged value for r* and hence of qv* for use in the iterative Troe fitting procedure described in section Va. cy was the only variable parameter in these calculations and k(Ev,*) and k;"(X), the high-pressure rate coefficient for CH, + X, given by (42) Hase, W . L . J. Chem. Phys. 1976, 64, 2442. (43) Stein. S. E.: Rabinovitch, B. S. J . Chem. Phys. 1973, 58, 2438.
Brouard et al. Kc(X) k:(X)
=
C b(Evr*)
k(Evr*) exp(--Evr*/kT)JCH,X
E,*
(qvr) CHlX
where KJX) is the equilibrium constant for the reaction CH3
+X
2
CH3X
were determined for 0.5 < a/.&-' < 1.0. kf(X) increases with a , because the activated complex becomes more "productlike". For large values of a, the assumed form for the rocking motions may be a poor representation, because the frequencies are assumed to tend to zero. To test the validity of this approximation, a limited number of calculations were made with a full rovibrational correlation (see above and the Appendix). The sums of correlated states were evaluated at a total energy E,,* and convoluted with the states from the remaining vibrations. &Evr*) and k T ( X ) were then evaluated as before. For a ;5 0.9 k', there was good agreement between both the microcanonical and canonical rate coefficients as calculated by the two techniques. The values for kT(X) agreed to better than 20% for all a within this range, and the forms of the dependence on a were very similar. At higher values of a, k;"(X)was found to increase much more strongly with CY for the uncorrelated model, because of the poor representation of the rocking frequencies for "productlike" complexes. The good agreement a t lower a values, however, demonstrates the general validity of the uncorrelated model, within the general framework of the present approach. Because of its computational simplicity, the uncorrelated model has been employed in the bulk of the calculations described below, in which optimal fits are obtained within the range of validity of a . As a final check on the models employed, the canonical rate coefficients were compared with values obtained using Quack and Troe's SACM.' For the range of a values studied here (0.5-1 .O the canonical rate coefficients calculated by SACM and by the correlated model agreed to within IO%, with the correlated model giving the higher rate coefficients. We emphasize, once again, the aim of these calculations. They are intended primarily to give a reasonably realistic description of k(EVr*)for comparison with experimental data on k,(H) and k,(D). I n particular they are required to test extrapolation of k,(H) to the high-pressure limit and to aid the comparison between k , ( H ) and k;"(D). There are obvious limitations in a rigid rotor-harmonic oscillator model and the simple representation of effects arising from the angular potential. Thus the calculations are unlikely to be as realistic as the detailed analysis by Wardlaw and Marcusx of CH3 + CH, or, indeed, as the k T ( X ) calculations of Hase et aL4 Nevertheless, in view of the good agreement with SACM calculation^, they are likely to be adequate for our present purpose. iii. yVRRKM Fitting to k;"(D).Using a Morse potential with p = 1.88 k', the uncorrelated model gave agreement with experiment at 300 K (kT(D) = 1.72 X cm3 molecule-'s-') with cy = 0.801 A-' (Le., within the range of validity as demonstrated by comparison with the correlated model and SACM.) The model did not quite reproduce the experimental temperature independent but gave instead a slight increase in k;"(D) to 1.96 X cm3 molecule-' s-' at 400 K. The stiff potential of Duchovie et al.39 gave much smaller rate coefficients (e.g., 1.0 X lo-" cm3 molecule-' s-' at 300 K for a = 1.0 A-') and required much larger values of a to produce rate coefficients in agreement with experiment. ic. Predicted Values for k;"(H).The simple transition-state model described above gives R = kT(H)/kT(D) = 2'12, but it contains several assumptions. The uncorrelated yVRRKM model was employed with a = 0.801 A-' and the molecular parameters given in the Appendix to determine R and k;"(H)as a function of temperature; values of R N 1.34 were obtained over the temperature range 300-600 K. Calculations based on the correlated model gave an even smaller ratio ( R N 1.25), while incorporating the stiff potential in the uncorrelated model gave R = 1.36 at 300 K . These calculations give k;(H) N (2.3-2.6) X lo-'' cm3 molecule-' s-' at 300 K, confirming the conclusions made above
Modeling Study of C H 3 + H and C H 3 + D Reactions
The Journal of Physical Chemistry, Vol. 93, No. IO, 1989 4055
TABLE 1V: Troe Factorization Fits to Experimental k,(H) Data with k,(H) Constrained to pVRRKM Values 101°k;(H)/cm3 (AE),,, = 300 Cm-' molecule-I s-l 1029k:(H)/ o ~ ~ ~ : ( Hs-I) / T IK (fixed) cm6 molecule-* SKI PC cm61molecule-2
e' fixed
e'
300 400 500 600 a
2.31 2.63 2.90 3.13
2.48 2.69 2.87 3.67
1.25 0.91 0.63 0.35
0.45 0.39 0.34 0.30
pent
2.44 2.66 2.85
a
SC
Pc
0.66 0.63 0.60 0.57
220 26 0.89
a
Fit did not converge.
that the values of kT(H), predicted from the experimental values of k;(D), are much lower than those obtained by using the Troe extrapolation on experimental k,(H) data, at least at lower temperatures. u. Falloff Behauior in CH3 + D. The C H 3 + D experiments were performed because it was anticipated that no pressure dependence would be found and that k,(D) would correspond to the high-pressure limit. This prediction was made because kb(E) is expected to be substantially larger than ka(E) at all accessible energies (Scheme I), so that (k,[M] 4- kb(E)) >> ka(E), for all E, and that the total rate of removal of D corresponds to the total rate of formation of CH3D*, Le., to the high-pressure limiting rate. This conclusion was supported by experiment which showed k,(D) to be pressure independent (Figure 4). The purpose of this section is to provide further confirmatory evidence. The simplest way to provide this evidence is to examine the reaction under limiting zero-pressure conditions: Figure 5. Comparison of experimental falloff data for kl(H) at 300 K and falloff curves generated by the EGME/rVRRKM procedure. 0, experimental data. -, best fit with a and (AE)down varied; best fit parameters were CY = 0.91 .&-I (kf(H) = 4.68 X cm3 molecule-' s-I) and (AE)down = 208 cm-l. - --,best fit with kf(H) fixed at the value calculated for kT(D) and (AE)downvaried; best fit parameter was (AE)down= 394 cm-I. -.-, k ; ( H ) calculated from k;(D).
where f(E) =
pa(E) ka(E) e x ~ ( - E / k T ) ~
L o P a ( E ) ka(E) ~ x P ( - E / ~ T d~ ) and p,(E) is the density of vibrational states in CH,D a t energy E, subject to angular momentum conservation in channel a. ka(E) and k b ( E ) were calculated by the uncorrelated pVRRKM technique with a = 0.801 A-' and @ = 1.88 A-', giving kY(D) = 1.69 X cm3 molecule-l s-l at 300 K and 1.91 X cm3 molecule-' s-I a t 400 K, values differing by less than 2% and 3%, respectively, from the high-pressure limits (see Section Vb(iii)). These calculations demonstrate that the overall C H 3 D rate coefficient is negligibly different at the high- and low-pressure limits; Le., it is pressure independent. vi. Revised Falloff Curves with k y ( H ) Constrained. In order to test the compatibility of the predicted values of kf(H) with the experimental falloff data, fits were performed with k;(H) constrained to the values required by the pVRRKM model with a = 0.801 A-l, which was required to generate the experimental value for kT(D) a t 300 K. Two sets of calculations were performed; in the first, ( AE)downwas fixed at 300 cm-' and kY(H) and varied to obtain the best nonlinear least-squares fit. is the value of F, in the center of the falloff curve.37 In the second, was calculated from the pVRRKM values of qv* (see above) and kY(H) and @, were varied. @, is the collisional energy transfer The results are shown in efficiency and is related to ( ilE)down.36 Table IV. The first set of calculations require an unusually strong dependence of on temperature and, most significantly, generate a value at 300 K that is greater than unity. Thus a negative broadening factor is needed at this temperature if the predicted kT(H) is to be approached a t the high-pressure limit. This conclusion is further illustrated by a simple, linear Lindemann plot of the 300 K data, which gives kf(H) = 3.3 X lo-'' cm3 molecule-' s-I, which is greater than the predicted value. Similarly unrealistic behavior is required of p, in the second set of calculations with, once again, an unrealistically strong temperature dependence and impossibly high values (>>I) at the lower temperatures. We conclude that a Troe factorization treatment of the experimental data gives extrapolated values of k f ( H ) that are significantly
+
e'
et
e?'
e'
higher than those obtained from k;(D), a t least a t 300 and 400
K. c. Energy Grained Master Equation (EGME) Calculations. The poor agreement between the predicted k;(H) values and those obtained via the Troe factorization procedure could conceivably arise from inadequacies in the Troe model. A more fundamental approach is to incorporate microcanonical rate coefficients into an energy grained master equation, where the population of the ith energy grain, n,, is obtained by solution of the coupled differential equations of the general form44
dn,/dt = gi - wni
+ wXPijnj - kini j
where g, is the rate of production of molecules in the ith grain from CH3 + H , w is the collision frequency, k, is the microcanonical rate coefficient for dissociation from level i, and PIJis the transition probability for collisional transfer from level j to level i. g, is related to k, via detailed balance,4s and an exponential down model was adopted for PIJ.The proper discretization of the master equation and its method of solution are described elsewhere.& In the present context, only solutions in the steady state (dn,/dt = 0) are required. Typical grain sizes of 100-200 cm-I were employed, but careful checks were made with smaller grain sizes to ensure convergence of the solutions. Attempts to fit the CH, H falloff data were initially made with a = 0.801 k', the value required to fit k;(D) a t 300 K. Figure 5 shows the experimental rate data at 300 K and the best fit theoretical falloff curve generated from the EGME model with (AE)dow,, employed as a variable parameter. The fit is poor, with too high a curvature, demonstrating, once again, that the value
+
(44) Tardy, D. C.; Rabinovitch, B. S . J . Chem. Phys. 1966, 45, 3720. (45) Robinson, P. J.; Holbrook, K. A. Cnimolecular Reactions; Wiley: London, 1972. (46) Clifford, P.; Green, N. J. B.; Bezant, M.; Pilling, M. J. Chem. Phys. Lett. 1987, 135, 477.
4056
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989
Brouard et al. TABLE VI: Ratio of Quantum to Classical Dissociation Rate Coefficients, $, for Specific Rovibration States and a = 0.801 (See Text)''
A-1
- -__-_------_---___ -------__
A
-v1
c
-1ZJ0
I
I
I
I
1
2
3
4 LgIPITorri
+
.,
Figure 7. Comparison of rate coefficient data for CH3 H, k,(H), at 300 K. 0, Patrick et al. (Ar);" 0, Cheng and Yeh (C2H6);51952Teng and Jones (H2);540, Michael et al. (H2);55A, Brown et al. (Ar);56e , Dodonov et al. (Ar);s7+, Halstead et a!. (Ar);" A, Sworski et al. (H2,N2);53 0 , this work. High-pressure limit: - - - Cheng , and Yeh; Patrick et al.; -, this work. Low-pressurelimit: -., Pratt and Veltman (He):48349 -, this work.
-
-e-,
200 400 600 Helium pressurelTorr Figure 6. Comparison of experimental falloff data for k,(H) and falloff curves generated by EGME/FVRRKM procedure at (a) 400, (b) 500, and (c) 600 K. ---,fit with k;(H) fixed at -4.7 X lo-'' molecule-' SKI and varied (see Table V). -, optimal fits at 500 and 600 K with both k;"(H) and (riE)down varied. The best fit k;"(H) values were 3 X IO-" cm3 molecule-' s-I at 500 K and 2 X cm3 molecule-' s-' at 600 K.
0'
Values Obtained with an Uncorrelated uVRRKM-EGME Analvsis
TABLE V: Optimum k; and (AE
T IK 300 400 500
600
CY/A-'
0.9 I 0.89 0.88 0.86
(AE)down/Cm-' 208 285 331 315
1 O''kY(H)/ cm3 molecule-' s-'
d . Quantum Mechanical Tunneling. One possible explanation of the discrepancy between the CH, H and CH, D rate coefficients is the presence of a large tunneling contribution in the former case. Because there is no maximum in the purely electronic potential energy surface, tunneling can only take place through the barrier generated by centrifugal effects and from variations with r of the transitional vibrational frequencies. T o examine this possibility, calculations were performed using representative examples of the channel energy profiles employed in the correlated pVRRKM calculations. The channel maximum was located, the curvuature of the channel energy, (d2V/dr2),*, calculated, and and approximate transmission coefficient determined for a parabolic approximation to the potential surface4'
+
+
4.7
4.5 4.6
4.4
required for k;"(H) significantly exceeds that predicted from k f ( D). Similar results were obtained a t higher temperatures, although the fits were not quite so poor. Fits were also performed in which both a and ( ilE)down were varied. Figure 5 shows the fit a t 300 K, with a = 0.91 A-I and ( u ) d o m = 208 cm-I, corresponding to kf(H) = 4.68 X cm3 molecule-' SKI.This is the lowest value of a for which a satisfactory fit could be obtained and corresponds, therefore, to a lower limit for kY(H) which is a factor of -2 greater than that expected from an R R K M analysis of CH, + D. At higher temperatures fits were obtained by constraining kY(H) to a temperature-invariant value of 4.7 X IO-'' cm3 molecule-' s-l, by varying both a and ( AE)down.The values requires are shown in Table V and the fits in Figure 6. At 500 and 600 K, better fits were obtained with a lower value of a and, therefore, of k;"(H), suggesting that the latter may decrease with temperature. However, the reaction is so far into the falloff region at these higher temperatures that this conclusion should be treated with caution. I n addition, a t these higher temperatures, the experimental data show a larger curvature a t low pressures than is required by the model.
where o* = [{d2V/dr21/w]1/2
and x is the translational energy along the reaction coordinate. The ratio of the quantum to classical rate coefficients, 4, is therefore given by
4 =
[ 1,P(x) exp(-x/kT)
dx]/kT
Representative results are shown in Table VI, which demonstrates that the tunneling contribution is comparatively small, even for states with high quantum numbers in the transitional vibrational modes and in rotation, where curvature in the channel energies is likely to be most pronounced. Thus, although these calculations are comparatively crude, it seems safe to conclude that the discrepancy between the CH3 + H and CH, + D rate data cannot be attributed to tunneling. e . Comparsion with Experiment. i. Recombination. Figure 7 compares the present measurements of k , with previous studies (47) Nikitin, E. E. Theory of Elementary Atom and Molecular Processes in Gases: Clarendon Press: Oxford, 1974.
Modeling Study of C H 3
+ H and C H 3 + D Reactions
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4051 The shock-tube data all lie below a temperature-independent
extrapolation of the kT(H) data presented in this paper. It seems 9 7 unlikely, however, that k;(H) shows a strong negative temperature
-
9
m
I E
r 5
-11
%- -I\
dependence. A more probable explanation is that the high-temperature data were obtained in the falloff region with inadequate or even no extrapolation to the high-pressure limit being made. RRKM-EGME calculations, with kT(H) = 4.7 X IO-'' cm3 molecule-' s-' and (AE)down= 900 cm-I, show that the halfpressure is 2 2 0 atm a t 2000 K, while the majority of the data shown in Figure 8 were obtained a t pressures in the range 1-10 atm. It is of interest to note that Cobos and Troe6' claim that the data of Hartig et al.,64obtained a t pressures of up to 1 atm, are consistent with kT(H) = 4.4 X IO-" cm3 molecule-' s-'. iii. Theory. The most recent calculations on k; are those by Hase et al.,4 who employed a canonical variational approach on an a b initio surface (multireference double excitation with a 6-31G** basis) calculated by This paper also summarizes earlier calculations by Hase and co-workers. In agreement with calculations by Duchovic et a1.@ and by Brown and Truhlar,6 the C-H stretching potential was found to be steeper than a Morse potential; the r dependences of the switching functions for the transitional modes were determined from the curvature of the surface with respect to the angular coordinates. For CH, + D, 1.8 X IO-'' cm3 molecule-' s-' with Hase et aL4 obtained k;"(D) no adjustable parameters, in excellent agreement with the experimental data. Less good agreement was found for C H 3 + H, where values of k T ( H ) increase from 2.5 X cm3 molecule-' SKI a t 300 K to 3.2 X lo-'' cm3 molcule-' s-l a t 1000 K. Their kT(H)/kT(D) ratio is in good agreement with that calculated in the present paper. Hase et aL4 showed that the rocking modes were adequately represented by an exponential correlation function over the most significant range of r*, with a = 1.65 A-'. A much smaller a value was required in our pVRRKM calculations because a Morse potential was employed. The steeper potential calculated by H i r d requires the larger a value if it is to reproduce the experimental data. In this respect it is interesting to note the results of Lewerenz and who determined the rovibrational spectrum of CHD, over the range 900-12000 cm-' and interpreted their spectra in terms of a Fermi resonance Hamiltonian for the coupled C-H stretching and bending vibrations. The Hamiltonian was also derived from a b inito full CI potential surface calculations. They used both the experimental and a b initio potentials to obtain an effective a value of 0.84 ,&-I. Quack69 earlier employed a = 1 .O k'together with a Morse potential ( p = 1.88 %.-I) to calculate k;(H) = 4.7 X lo-'' cm3 molecule-' s-I in excellent agreement with the present experimental results. The most detailed calculations on the CH, + H reaction have recently been reported by Aubanel and Wardlaw.'' They used a microcanonical variational R R K M technique, specifically designed to model loose transition states, in which the state counting for the interacting transitional modes is achieved by using a Monte Carlo method.8 Aubanel and Wardlaw also incorporated anharmonicity in the umbrella motion and in the C-H stretches. They found that the calculated values for kT(H) increased somewhat with temperature, in contrast to the dependence found for CH, + CH39. Calculations on k;"(D) generated an isotope ratio of 1.4, in agreement with that determined from the less detailed calculations presented here.
EL -12
23
3.0
3.5
Ig(TIK) Figure 8. Comparison of the values of k;(H) obtained in the present
work and those calculated from K, (Appendix) and high-temperature dissociation data. -, present data; A, Back et al.;" B, Kevorkian et al.? C, Kondrat'ey6* D, Skinner et E, Hartig et al.;" F, Kozlov et al.;65 G . Palmer et a1.66 a t 300 K. The data have not been corrected for different bath gas efficiencies. The experimental technique described in this paper represents the most direct approach to date to the measurement of k , . Good agreement is obtained with the most recent low-pressure measurements of Pratt and V e l t n ~ a n n . ~ *The * ~ ~remaining lowpressure data, which are rather scattered, were obtained with a variety of third bodies. Of the higher pressure studies, the data of Patrick et aLS0are significantly lower than the present measurements and give a limiting high-pressure rate coefficient of 1.5 X cm3 molecule-' s-l, a factor of 3 lower than that reported here. They employed flash photolysis coupled with end-product analysis; the disagreement can largely be accounted for by the ~ from their kinetic omission of the H a ~ o m e t h a n e ,reaction scheme and their use of a low value for kZ(CH3 + CH3).25 The data of Cheng and Yeh,5',52obtained in an end-product-analysis study of the mercury photosensitized decomposition of ethane, gives k f ( H ) = 3.8 X cm3 molecule-' s-l, in better agreement with our value. The previous most direct study, by Sworski et al.,53 in which CH3 was monitored following flash photolysis of a C H 4 / H 2 0 / N 2 or H, mixture, is in good agreement with the present data. ii. Dissociation. There have been no previous measurements of k , a t elevated temperatures, and the most useful comparison of the temperature dependence is made with dissociation data, usually obtained from shock-tube measurements. Figure 8 compares the present values of k;(H) with values calculated from dissociation data via K,, using the spectroscopic and thermodynamic parameters given in the Appendix. K , was calculated by using a rigid rotor, harmonic oscillator approximation, which is a reasonable approximation over the required temperature range.59
+
(48) Pratt, G.; Veltmann, I. J . Chem. Soc., Faraday Trans. I 1974, 70, 1840. (49) Pratt, G.; Veltmann, 1. J . Chem. SOC.,Faraday Trans. I 1976, 72, 1733. (50) Patrick, R.; Pilling, M. J.; Rogers, G. J. Chem. Phys. 1980, 53, 279. (51) Cheng, J. T.;Yeh, C. T.J. Phys. Chem. 1977, 81, 1982. (52) Cheng, J. T.; Lee, Y.-S.; Yeh, C. T.J. Phys. Chem. 1977, 81, 687. (53) Sworski, T. J.; Hochanadel, C. T.; Ogren, P. J. J . Phys. Chem. 1980, 84, 129. (54) Teng, L.; Jones, W. E. J. Chem. SOC.,Faraday Trans. I 1972, 68,
1267.
(55) Michael, J. V.; Osborne, D. T.; Suess, G. N. J . Chem. Phys. 1973,
58, 2800.
(56) Brown, J. M.; Coates, P. B.; Thrush, B. A. Chem. Commun. 1966, 843. (57) Dodonov, A. T.; Lavroskaya, G. K.; Tal'rose, V L. Kinet. Cutal. 1969, 10. 391.
(58) Halstead, M. P.; Leathard, D A,; Marshall, R. M.; Purnell, J. H. Proc. R. Soc. 1970, A316, 575. (59) Frurip, D. J.; Syverud, A. N.; Chase, M . W. J . Nucl. Mater. 1985, 130, 189.
(60) Chen, C. J.; Back, M. H.; Back, R. A. Can. J . Chem. 1975.53, 3580. (61) Kevorkian, V . ; Heath, C. E.; Boudart, M. J . Phys. Chem. 1960, 64, 964. (62) Kondrat'ev, V . N. Symp. (Int.) Combust. 1965, 10th. 319. (63) Skinner, G. B.; Ruehrwein, R. A. J . Phys. Chem. 1959, 63, 1736. (64) Hartig, R.; Troe, J.; Wagner, H. Gg. Symp. (In!.) Combust. 1971, 13th, 147. (65) Kozlov, G. 1.; Knorre, V. G. Combust. Flame 1962, 6, 253. (66) Palmer. H. B.; Hirt. T. J. J . Phys. Chem. 1963, 67, 709. (67) Cobos, C. J.; Troe, J. Chem. Phys. Lett. 1985, 113, 419. (68) Lewerenz, M.; Quack, M. J . Chem. Phys., in press. (69) Quack, M. J . Phys. Chem. 1979, 83, 150. (70) Aubanel, E. E.; Wardlaw, D. M. Submitted for publication in J . Phys. Chem.
4058
Brouard et al.
The Journal of Physical Chemistry, Vol. 93, No. IO, 1989
TABLE VI1 CH,
reactants" products CHID CH,- + H ,/ D Conserved Modes
3020 (3) 1306 (2)
2982 ( I ) 3030 (2) 1156 (2)
I306 ( I ) 1534 (2) K
I306 (1) 1477 (2) K
5.29 (3)
5.29 (1) 3.91 (2)
2917
2205
3044 ( 1 ) 3162 (2) 1396 (2)
ref 76
Transitional Modesb 606 (1)
product rot. x, J.' (j, k , L, M,. m d
76 77
Rotational Constants 9.58 (2) 4.74 (1)
77 41
Relative Translationb transl, z
"All energies in cm-'. bReaction coordinate is defined here as the z axis.
TABLE VI11 ref Dissociation Energy DJcm-'
39 1 7 2 -
AHoo/cm-'
36072 (CH,)
36646 (CH3-D) 78
Morse Parameters 1.88
LVA-' 8'
a = 0.3039
(see section Vb) c = -0.1156
A-2 1.094
req/A
79 b = 0.09847 k3 40 d = 1.9216 A-] 76, 79
l g I T / K)
Figure 9. Comparison of experimental and theoretical data for k;(H). -, experimental data from this work; - - - ,H, Hase et al., canonical VTST;4 - - -, C, Cobos, SSACM? L, Leblanc and Pacey, canonical VTST;', ---, D. Hase and Duchovic, canonical VTST;" A,A, Aubanel
-..
and Wardlaw.70 The numbers refer to the type of potential employed: 1, Duchovic et al. surface40with stiff Morse potential; 2, Duchovic et al. surface with standard Morse potential; 3, Hirst s ~ r f a c e4,; ~Brown and Truhlar surface;65, Schlegel surface. -, B, present pVRRKM calcuCobos and Troe. SACM. CY = 0.94 .k';67 lations with a = 0.801 A; 0, Quack, canonical VTST, a = 1 A-'.69 -.e-,
Lennard-Jones Parameters aHc/A
ms/A ( ECHI/ k ) / ( EHe/ k ) / K
2.551 3.758 148.6 10.22
80
Cobos3 has recently applied the simplified version of Quack and Troe's statistical adiabatic channel model (SSACM) to the CH3 H reaction, employing, once again, the a b initio surfaces of ~ Duchovic et al.,40 Brown and Truhlar,6 and H i r ~ t . Morse, stiff-Morse, or a b initio C-H stretching potentials were used, with transitional mode switching functions calculated in a variety of ways, including directly from a b initio surfaces. The values for k f ( H ) at 300 K ranged from 1.12 X 1O-Io to 5.8 X IO-" cm3 molecule-' s-' depending on the potential function employed. All the surfaces resulted in calculated rate coefficients that increase with temperature. Cobos3 presents a useful comparison between the results of his own calculations and those of Hase and cow o r k e r ~ and ~ ~ of - ~Leblanc ~ and P a ~ e y . Figure ~ ~ 9 compares the extrapolated experimental values for k f ( H ) with a range of theoretical estimates. All of the calculations predict an increase in kf with T .
+
VI. Conclusions a . Pressure and Temperature Dependence of k , ( H ) . Table 1 and Figure 3 summarize the experimental rate data for C H 3 + H in ;Ihelium diluent over the temperature range 300-600 K. The reaction is i n the falloff region under all conditions, particularly at higher temperatures, and the extrapolation to k f ( H ) is unrealiable. The recombination of methyl radicals, reaction R 2 shows a slight decrease in k; over the same temperature range, a result that has been ascribed by Wardlaw and Marcus* to a reduction of the phase space available to the rocking motions as the energy of the C2H,* adduct increases and the complex tightens. The decrease in k ; is comparatively small for C H , + C H , over
J=3 CH,
Duchovic, R. J.; Hase, W . L . Chem. Phys. Lerr. 1984, 110, 474. Duchovic, R . J.; Hase, W . L. J . Chem. Phys. 1985, 82, 3599. Hase, W .1.:Duchovic, R. J . J . Chem. Phys. 1985, 83, 3448. LeBlanc. J . F.: Pacey. P. D. J . Chent. P h j s 1985. 83, 4511
CH,
+
H
Figure 10. Correlation diagram for the correlated pVRRKM program for J = 3. The degeneracy of each level is given in brackets.
the range 300 < T / K < 600 and is likely to be even less marked for CH, + H . Indeed all of the theoretical models, including that of Aubanel and Wardlaw,'O show a slight increase in k f ( H ) over the experimental range. Given the temperature independence of kT(D) over the range 300-400 K , and the reasonable fits of the master equation model shown in Figures 5 and 6, we propose that. in the absence of further data at higher pressures, a temperacm3 molecule-' s-' is adopted ture-invariant value of 4.7 X for kT(H). The error limits are difficult to define, but logarithmic uncertainties, A log k f ( H ) of -+0.2 to -0.1 at 300 K , rising to -0.4 at 600 K , are probably reasonable. With this value for k,( H ) , the experimental data are adequately reproduced with the following parameters, using the Troe factorization procedure described by Gilbert et al.:37 k: = 4.0
X
F,,,, (71) (72) (73) (74)
-
cm6 molecule-' s-I
(helium diluent)
= (0.902 - 1.03) X 1 0-3 ( T / K )
These parameters produce fits to the experimental data with a mean absolute deviation of 9%. Despite the large uncertainties in the high-pressure limiting values. these parameters are coupled
Modeling Study of CH,
+ H and C H 3 + D Reactions
with smaller uncertainties over the experimental range (300-600 K, 25-600 Torr of He) of around *l5% at 300 K rising to *25% a t 600 K. b. Temperature Dependence of k ; ( D ) . The rate coefficient for C H 3 D was shown experimentally to be independent of pressure and temperature, with a mean value of 1.75 X lo-'' cm3 molecule-' 8. The lack of a pressure dependence arises because k , ( E ) 3.137 A
The number of states correlated by using eq A I rapidly increases with J and soon becomes prohibitively large for a system as complex as CH, H/D. However, eq A1 can be approximated without significant loss of accuracy by assuming that the orbital angular momentum -takes an averagevalue of J (with degeneracy 2J 1 or 2j 1, whichever is smaller). This procedure avoids the detailed summing over the L quantum number, and in this approximation eq A1 can be simplified to
+
+
+
E," = E: exp[-a(r - r,)]
+ &{I
+
- exp[-a(r - r e q ) ] ] B,J(J 1) (A2)
+
The energy level ladders and the correlation between them can best be illustrated by reference to Figure I O , which shows the correlation scheme for J = 3 . Numerous such schemes are constructed for J = 0 to J,,,, where the latter limit is imposed by the available energy in the complex and reactant, as discussed in section 5b. Having calculated the transitional rovibrational sums of states as a function of reaction coordinate, r, the t a a l , obtained by convolution with sum of complex states, N ( E v r * ) is the remaining conserved vibrations and approximate correction for the statistical weights using the activated complex symmetry numbers. Registry No. CH,, 2229-07-4; H, 12385-13-6.
Supplementary Material Acailable: Tables S I-S3, Figures 1S-5S, text discussing simulations of the basic scheme, weighting factors for fluorescence decays, H / D atom diffusion, additional reactions in the CH, H system, additional reactions in the CH, + D system, error analysis (23 pages). Ordering information is given on any current masthead page.
+
