T H E
J O U R N A L
OF
PHYSICAL CHEMISTRY Registered in C: S. Patent Office 0 Copyright, 1974, by t h e American Chemical SocieQ
VOLUME 78, NUMBER23 NOVEMBER 7, 1974
Rice-Ramsperger-Kassel-Marcus Theory Applied to Decomposition of Hot Atom Substitution Products. c=C4H7Tand c=C4D7T C. C. Chou Department of Chemistry, University of California, Irvine, California 92664
and William L. Hase*’ Department of Chemistry, Wayne State University, Detroit, Michigan 48202 (Received January 3 1, 1974; Revised Manuscript Received July 25, 1974) Publication costs assisted by the Petroleum Research Fund
The RRKM theory has been used to interpret experimental measurements of the decomposition of excited c-C4H7T and c- C4DjT following T-for-H and T-for-D substitution in recoil tritium experiments. An expression is obtained for calculating the pressure dependence of decomposition to stabilization. It is found that (1)rotational effects are unimportant for either c- C4H;T or c- C4D;T decomposition; (2) when the excitation function for T-for-H substitution in c- C4H8 is approximated by one derived from molecular beam experiments, the displaced H atoms carry on the average 40% or more of the incident T translational energy; (3) for similar T-for-H and T-for-D excitation functions, displaced H and D atoms have nearly the same average kinetic energies.
Introduction
The decomposition of excited molecules and radicals formed by hot atom substitution and addition reactions has been widely observed and has provided valuable information about the preceding hot atom reactions.* Measurements of the pressure dependence of the decomposition to stabilization ratios for hot atom substitution products have been combined with the RRKM (Rice-Ramsperger-KasselMarcus) theory to yield values for the median excitation energies of the hot substitution product^.^,^ It has been found that more than 250 kcal/mol may be deposited on the substitution product and the median excitation energies for the different systems are not identical. A detailed explanation of those observations requires an understanding of the dynamics of the substitution process. For the molecular dynamics of T-for-H and T-for-D substitution reactions of recoil tritium atoms with c-C4Hs and cC4D8, Rowland and coworker^^^^ have made the following conclusions and hypotheses from the pressure dependence of product distributions and RRKM calculations: (a) a broad excitation with a median energy of 115 f 23 kcal/mol for both C- C4HjT and C- C ~ D T T(b) , the kinetic energies of
the displaced H and D atoms are nearly equivalent, and (c) the displaced and D atoms emerge from the substitution reaction with small amounts of energy. Bunker6 has shown hot atom substitution may produce a species with large amounts of angular momentum, and the RRKM expression appropriate for thermal systems must be modified for treating the decomposition of hot atom substitution products. He found inclusion of rotational effects (angular momentum) in the RRKM calculations could in some cases greatly change the calculated rates and explain some seemingly paradoxical experimental observations. In particular he found the secondary decomposition of CH2TNCi and CH3CF218F,3which appeared to display non-RRKM behavior, is in accord with the RRKM theory if rotational effects are included. Bunker also found under certain circumstances the RRKM calculations may be a useful source of information about the dynamics of primary substitution and addition reactions. In this paper we have extended Bunker’s formulation so the decomposition to stabilization ratio of the substitution product may be calculated as a function of pressure. An analysis is then made of experimental measurements of cC4H;T decomposition. This molecule has provided one of 2309
231 0
C. C. Chou and William L. Hase
the best and most important examples of unimolecular reaction following hot atom substitution. The original motivation for this study was to see if a more complete analysis, one which includes rotational effects, would alter any of the earlier conclusions reached regarding the cyclobutane systems. However, it was also found that specific information about the substitution reaction, gained from molecular beam experiments,8 could be used to correlate the experimental data for c- C4H7T decomposition with several dynamical quantities.
Theory The theoretical formulation we use is an extension of the one presented by Bunker.6 Parts of it are described below for completeness and clarity. In a hot atom substitution reaction 'the average angular momentum and resulting rotational energy retained on each molecular axis of the polyatomic product are Li =
E , ~= L ~ ~ i/ =~x, Ly , ~o r z
where b is the impact parameter, E , is the translational energy of the hot atom, m ~mA, , and m p are the masses of the hot atom, displaced atom, and product, respectively, and f is the fraction of E , left behind on the product. If there is no A-P relative translation, f = fmax = 1 - m R / ( m ~ mp). The vibrational-internal rotational energy of the energized molecule (*) and critical configuration (+) are
+
E,*
= .E,
+
Evrth - x E r i * i
E,*
= fE,
(21
+ Eyrth- ETr{* - E, i
with Eo the threshold energy for the unimolecular decomposition of the energized molecule and EWththe thermal vibrational energy of the product molecule. Substituting eq 1 into 2 yields
E,'=
f E , + EIrth - b2E,(1/Z,
*
+
1/Z,
*
+ 1/Z,
*
)a/4
These energies were inserted into standard RRKM express i o n ~to~calculate unimolecular rate constants. The semiclassical Whitten-Rabinovitch approximation was employed to calculate the necessary sum and density terms.1° The total decomposition (D)and stabilization (S) of the product molecule (P*) depends on the competition between the processes
P*
k(flb,ES)
decomposition (D) fragments (4 )
0:
* P (S) where w is the collision deactivation probability. For a fixed value off, the ratio of decomposition to stabilization is The Journal of Physical Chemistry, Vol. 78, No. 23, 1974
D/S =
where P,(b ) and P ( b ) are the reaction probability (opacity function) and collision probability, and P(E,), u ( E , ) , and u ( E , ) are the relative translational energy distribution, excitation function, and velocity, respectively. This expression was used to calculate the theoretical D/S values. Lin and Laidlerl' have performed an extensive RRKM analysis of the thermal decomposition of cyclobutane. Using energetic and mechanistic arguments they hypothesized that cyclobutane decomposes via a tetramethylene diradical. A recent ab initio calculation12also suggests that this is the reaction path. There is some question as to the location of the critical configuration along the reaction coordinate. Thermodynamic arguments have been presented which suggest that the critical configuration is located between the biradical and the two ethylene product molecules. l3 However, earlier thermodynamic argumentsI4 and extended Hiickel calculation^'^ indicate that the location of the critical configuration is between the molecule and biradical. The structure for the critical configuration used in our calculations is intermediate of those of the molecule and biradical. The four C atoms lie in a plane with one C-C bond extended; i.e., a trapezoidal structure similar to that of eclipsed n-butane. The three unextended C-C bonds have a length of 1.55 A. The two expanded C-C-C angles are 110'. Skeletal deformations and CH2 motions were lowered in the critical configuration to yield Arrhenius parameters in agreement with the measured ones.16 Therefore, the RRKM model is calibrated to the thermal decomposition rates. A ring deformation was taken as the reaction coordinate. The frequencies for c - C4H7T and c- C4D7T were derived from those of c- C4H8 and c- C4D8I7 by an approximate application of the Teller-Redlich product rule.18 Frequencies for c-C4H8 and c-C4D8 were grouped according to the type of mode (stretch, bend, wag, twist, etc.) so the geometric mean of the frequencies remained the same. The C-T stretching frequency was then derived from the C-H (or C-D) stretch group by the square root of the C-T to C-H (or C-D) reduced mass ratios. A CH2 (or C D 3 wag, twist, rock, and deformation were then varied proportionally to account for the remaining frequency difference specified by the Teller-Redlich product rule. In Table I are presented the molecular and critical configuration frequencies and other pertinent structural and energy parameters. Cyclobutane may also decompose by the simultaneous cleavage of two C-C bonds to produce two ethylene molecules. The importance of this process was estimated by using the critical configuration structures in Table I and the activation energy of 156 kcal/mol, which was determined by ab initio calculations.12It was found that decomposition by the simultaneous cleavage of two C-C bonds is slower than decomposition uia the tetramethylene diradical by 10, 7, 4,and 3 orders of magnitude at 200, 250, 300, and 350 kcal/mol of vibrational excitation of the cyclobutane molecules, respectively. These numbers should be representative since there is little reason to expect a significantly looser critical configuration structure (larger A factor) for the mechanism involving simultaneous cleavage of the two C-C bonds. In fact, one might expect a tighter crit-
231 1
Decomposition of Excited c-C4H7Tand c-C4D7T TABLE I: Energy and Structural P a r a m e t e r s for the Decomposition of c-C,H;T and c-C,D;T
1
Frequenciesa __
Molecule
Critical configuration 16 0
c-CaH;T 2935 (7) 2935(7) 1695(1) 1695 (1) 1450(3) 1450(3) 1245(3) 1245(1), 628(2) 1230(3) 1230(1), 624(2) 1150 (1) 1150(1) lOlO(1) 508(1) 990(1) 990 (1) 975(1) 975(1) 926 (4) 926(3), rc* 720(3) 720(1), 364(2) 570(1) 570(1) 193(1) 9W) I,* = 4 8 . 6 ~ 1 , *= 5 4 . 5 1 2 * = 8 7 . 2 I,- = 8 1 . 3 I U L = 4 8 . 9 1 , - = 113.8 Eo = 5 9 . 5 kca1,'mol E , = 6 3 . 2 kcal 'mold A = 7 . 0 X 1015sec-l
Ida-
b=2\
,-
//
J50
33.0
, ;2 0
c-C~D~T 2170(7) 2170(7) 1770(1) 1770(1) 1070(3) 1070(3) 1060 (3) 1060(1), 521 (2) 980 (1) 980 (1) 970 (1) 970(1) 900 (3) 900(1), 443(2) 873 (1) 428(1) 825 (1) 825 (1) 790(4) 790(3), rcb 545(3) 545(1), 267(2) 500 (1) 500(1) 151(1) 74V) lz* = 6 6 . 3 ~ 1 , *= 6 9 . 2 I , * = 1 0 7 . 6 I,' = 9 8 . 8 I , + = 6 6 . 1 I,+ = 1 3 6 . 3
Eo = 6 0 . 6 kcal/mol E , A = 9 . 4 X 1015sec-l
=
6 4 . 1 kcallmold
"Frequencies are in cm-'. Those that change in the critical coniiguration are italicized. brc = rea$ion COordinate. c Moments of inertia in units of amu h2. The Arrhenius parameters are calculated for 722°K. ical configuration structure due to the simultaneous formation of two C=C bonds and the above factors would then represent lower limits. Therefore, since the largest degree of internal excitation of the cyclobutane molecules considered in this paper is less than 350 kcal/mol only the biradical mechanism was included in the calculations. In the derivation of eq 5 it was assumed the collision deactivation probabilities can be calculated from gas kinetic cross sections. Data collected from thermallg and chemical activation experiments2c24 indicate that this assumption is valid for molecules as large as cyclobutane. The chemical activation studies show that C4 hydrocarbons and halocarbons transfer on the average -10 kcal/mol from molecules which contain -90-115 kcal/mo12c22 of vibrational excitation. Though this amount of energy transfer only reduced the unimolecular rate constants by an order of magnitude, it was sufficient to yield a unit gas kinetic collisional deactivation efficiency at pressures where stabilization competes with decomposition. Only a t lower pressures when decomposition overwhelmed stabilization (D/S > 10) did collisional inefficiency become evident.
Nevertheless, there remains an uncertainty in using collision frequencies in eq 5 even if comparisons are made with experiments where decomposition and stabilization are competitive which is the case for this study. As the degree of internal excitation is increased the decrease in the unimolecular rate constant is smaller for equal amounts of energy transfer. For cyclobutane the rate constant changes by a factor of 8 a t 100 kcal/mol but by only a factor of 1.5 a t 200 kcal/mol, for an energy change of 10 kcal/mol (Figure 1).Though a factor of 8 is sufficient for a unit deactivation efficiency 1.5 is not. However, there is little justification for assuming the average amount of energy transferred remains constant for all excitation energies. Since vibrational levels become denser a t greater degrees of excitation the probability of energy transfer should increase. This effect is seen when the chemical activation s t ~ d i e s 2 6are ~ ~compared with photochemical experiment^^^,^^ which produce molecules with much smaller amounts of vibrational excitation. This result combined with that of the chemical activation experiments suggests the unit deactivation efficiency may be physically realistic for this cyclobutane study. However, even if not completely valid, it provides the only tractable approach, since very little is known about intermolecular energy transfer at energies as high as those attained in hot atom substitution reactions. The effect of using other deactivation models on the calculational results is discussed later. That hot atom substitution products can possess significant amounts of rotational energy may add an additional complication for some studies since less is known about rotational energy transfer than vibrational energy transfer a t large degrees of excitation. This problem has recently been discussed in reference to the isomerization of CH2TNC.27 However, for cyclobutane decomposition this complication is not important since the substitution reaction produces only a small amount of rotational excitation (see the following section). In calculating the collision deactivation probabilities a diameter of 5.0 A was used for all isotopic cyclobutanes. The oxygen and helium present in the experiments (less The Journal 01 Physical Chemistry. Voi. 78.
No. 23. 1974
2312
C.C.Chou and William L. Hase
than 20%) were assumed to make no contribution to the collision deactivation probability. The resulting values at 300 K are W ~ - C ~ = H ~1.18 T X lo7 Torr-' sec-' and U,-C~D,T = 1.14 X loi Torr-' sec-'. -6 0 Results and Discussion A. Unimolecular Rate Constants. Calculated rate constants for the decomposition of c- C4H7T for f = fmax and f = 0.5 are displayed in Figures 1 and 2, respectively. Also shown are the ratios of the decomposition rate constants for c- C4HjT and c- C4DjT, k H/k D. In contrast to the calculations for CH2TNC isomerization,6 the rate constants for c- C4H7T and c- CdD7T are only slightly dependent on the impact parameter. Experimental values of D/S for c-CdHjT and c - C ~ D ~have T been measured in the 50-850-Torr pressure range.5 Over this range, rate constants of 6 X loE-1 X 1O1O sec-l will be competitive with collisional stabilization. For this large variance in rate constants, k ( b = O)/k(b = 2 A) for c - C ~ H ~only T varies from 5 to 3 for f = fmax and 3.5 to 2.5 for f = 0.5. In the liquid phase where only molecules with rate constants >10l2 '05/.40 220 300 380 460 540 sec-' decompose, rotational effects are minimal; k ( b = 0 ) / E, ( k col/rnole) k ( b = 2 A) is
