RRKM calculations

Variational RRKM calculations, as described above, show that a imimolecular dissociation reaction may have two variational transition states [32, 31, 34, 31 and 36], i.e. one that is a tight vibrator type and another that is a loose rotator type. Wliether a particular reaction has both of these variational transition states, at a particular energy, depends on the properties of the reaction s potential energy surface [33, 34 and 31]- For many dissociation reactions there is only one variational transition state, which smoothly changes from a loose rotator type to a tight vibrator type as the energy is increased [26],... 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

Zhu L, Chen W, Hase W L and Kaiser E W 1993 Comparison of models for treating angular momentum in RRKM calculations with vibrator transition states. Pressure and temperature dependence of CI+C2H2 association J. Phys. Chem. 97 311-22... 

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

A much more detailed and time-dependent study of complex hydrocarbon and carbon cluster formation has been prepared by Bettens and Herbst,83 84 who considered the detailed growth of unsaturated hydrocarbons and clusters via ion-molecule and neutral-neutral processes under the conditions of both dense and diffuse interstellar clouds. In order to include molecules up to 64 carbon atoms in size, these authors increased the size of their gas-phase model to include approximately 10,000reactions. The products of many of the unstudied reactions have been estimated via simplified statistical (RRKM) calculations coupled with ab initio and semiempirical energy calculations. The simplified RRKM approach posits a transition state between complex and products even when no obvious potential barrier... 

Ab initio and RRKM calculations indicate that the reactions of C, CH, and (H2C ) with acetylene occur with no barrier." Laser flash photolysis of the cyclopropanes (69) and (70) was used to generate the corresponding dihalocarbenes. The absolute rate constant for the formation of a pyridine ylide from Br2C was (4-11) x 10 lmoP s. The rates of additions of these carbenes to alkenes were measured by competition with pyridine ylide formation and the reactivity of BrClC was found to resemble that of Br2C rather than CI2C . 

Figure 2. RRKM calculations of the kinetic shift for model hydrocarbon ion dissociations as a function of ion size. Calculations are shown both for a fairly weakly bonded ion (1.86 eV) and a fairly strongly bonded one (3.10 eV), and in each case both the conventional and the intrinsic kinetic shifts are plotted.

Schindler and coworkers verified the formation of hydroxyl radicals kinetically and further RRKM calculations by Cremer and coworkers placed the overall concept on a more quantitative basis by verifying the measured amount of OH radical. An extensive series of calculations on substituted alkenes placed this overall decomposition mechanism and the involvement of carbonyl oxides in the ozonolysis of alkenes on a firm theoretical basis. The prodnction of OH radicals in solution phase was also snggested on the basis of a series of DFT calculations . Interestingly, both experiment and theory support a concerted [4 4- 2] cycloaddition for the ozone-acetylene reaction rather than a nonconcerted reaction involving biradical intermediates . 

It has been predicted from an RRKM calculation that 3-oxetanone would photolyze readily, forming formaldehyde and ketene (75JPC1990). This is in accord with the observation that 2,2,4,4-tetramethyl-3-oxetanone is transformed in high yield to dimethylketene and acetone on irradiation with 3130 A light in polar solvents, such as acetone and ethanol. In nonpolar solvents, however, photo-decarbonylation to tetramethyloxirane becomes a competing process (equation 5) (66JA1242). 

The thermodynamic stability of unsubstituted silacyclopropane to fragmentation has been studied by ab initio quantum mechanical methods and the enthalpy of decomposition to H2Si + CH2=CH2 was predicted to be 44.878 and 43.279 kcalmol-1. There is indirect experimental support for these theoretical estimates. When these values were employed in RRKM calculations on silirane decomposition, the pressure dependence of the bimolecular rate constant for addition of H2Si to ethylene could be accurately modeled80. 

Some of the initial work dealt with the formation of proton-bound dimers in simple amines. Those systems were chosen because the only reaction that occurs is clustering. A simple energy transfer mechanism was proposed by Moet-Ner and Field (1975), and RRKM calculations performed by Olmstead et al. (1977) and Jasinski et al. (1979) seemed to fit the data well. Later, phase space theory was applied (Bass et al. 1979). In applying phase space theory, it is usually assumed that the energy transfer mechanism of reaction (2 ) is valid and that the collisional rate coefficients kx and fc can be calculated from Langevin or ADO theory and equilibrium constants. 

To complete the RRKM calculations for the cluster dissociation rates and final bare 4EA molecule product distributions, the cluster binding energy E0 and the energy v of the chromophore vibrational state to be populated must be found. These can be estimated from selected fits to the experimental rates and intensities (Hineman et al. 1993a). The results of the rate and product distribution calculations are presented in Table 5-4. The predictions of the model are quite good—less than 30% error for all observations for the 4EA(N2)1 and 4EA(CH4), clusters. 

A possible alternative decomposition as described in equation 25 was not observed. A four-membered cyclic transition state and an Arrhenius factor similar to that of the HC1 elimination from chlorocyclobutane was assumed for the RRKM calculations. The experimental unimolecular rate coefficients are consistent with the Arrhenius equation log kx... 

In general, the structure and frequencies of the transition complex are not known for unimolecular reactions and, consequently, neither transition state theory nor detailed RRKM calculations can be tested. However, provided a physically plausible choice is made which will match the koc over the range of measured temperatures, the derived ft (e) are only slightly dependent on the particular model selected. Details of these procedures are available [11—13] and an excellent discussion is given by Robinson and Holbrook [11]. Readers should also refer to the detailed methods used by Schneider and Rabinovitch [14] for the CH3NC isomerisation. The following brief comments are intended to complete this introductory outline of the basic theory and to show how it may be applied. 

Recently, some additional low-pressure rate constants for the syn-anti process in MeONO have been obtained [770]. Figure 5 shows pressure-dependent rate constants (k/k J for syne anti exchange in pure MeONO at 262.2 K. The solid line corresponds to RRKM calculated values using E0 = 49.8 kJ mol-1 and a = 5.4 A. 

Figure 5. Pressure dependent rate constants (k/kx) of methyl nitrite gas. Solid line represents RRKM calculated rate constants as described in the text.

Nitrogen inversion in aziridine and 2-methylaziridine was studied using H NMR by Bauer et al. [775], This study observed no dependence of the nitrogen inversion rate constant of aziridine on pressure down to a pressure of 5 torr, in sharp contrast to RRKM calculations which indicated that this system should be in the bimolecular... 

For most of the molecules discussed above, the experimentally determined pressure-dependent gas-phase rate constants can be modeled adequately with RRKM theory. SF4 and possibly aziridine are the exceptions. Due to uncertainties in the model parameters used in RRKM calculations, and the sensitivity of the calculations to these parameters, only qualitative conclusions can be drawn from the observed agreement. Since major departures are not observed, it can be... 

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