RRKM Theory Of Unimolecular Reactions Research Articles

A Short Account of RRKM Theory of Unimolecular Reactions and of Marcus Theory of Electron Transfer in a Historical Perspective

The RRKM Theory of Unimolecular Reactions and Marcus Theory of Electron Transfer are here briefly discussed in a historical perspective. In the final section, after a general discussion on the educational usefulness of teaching chemistry in a historical framework, hints are given on how some characteristics of Marcus’ work could be introduced in courses of physical chemistry or chemical kinetics to show how experiments drive the formulation of theories and, on the other hand, how the predictions of theories may suggest experiments and even predate their results.

Role of tunneling of hydrogen in the Norrish type II processes in a thione and a ketone

Intra-molecular hydrogen abstraction reactions were examined in pentane-2-thione in its lowest triplet state using the AM-1 semiempirical molecular orbital method with the geometry optimisation in the unrestricted Hartree-Fock frame. Results were compared with those of pentane-2-one. The role of tunneling of hydrogen was examined in both the conventional transition-state approach and in the RRKM theory of unimolecular reactions. Results reveal that tunneling of hydrogen is less significant to sulfur than to oxygen.

An overview of the statistical theory of unimolecular reactions (RRKM) and recent developments

After a brief historical introduction, the RRKM theory of unimolecular reactions is reviewed with emphasis on the general concepts that have influenced the subject over the years. Discussed are recent developments concerning both the microcanonical and canonical unimolecular rate constant.

Statistical calculation of unimolecular rate constant

Over the years, the RRKM theory of unimolecular reactions has continued to represent a useful statistical approach to the microcanonical unimolecular rate constant which measures the rate of decomposition of molecules with specified total energy E.

The estimation of specific rate constants by RRKM theory

It is noted that the usual aposteriori quantization of the classical result for the specific rate constants k a ( E) in the RRKM theory of unimolecular reactions leads to numerically ill-defined rate constants when the density of states is taken to be discrete, i.e. as if the energy levels corresponded to fully quantized bound states. This situation arises when exact count schemes are used for density of states evaluation and only vibrations are included. In the absence of tractable quantum dynamical methods smoothing of the density of reactant states is recommended to produce a well-defined if not ultimately correct set of rate constants. Similar problems arise in general in RRKM theory but are usually hidden by some simplification which implies smoothing.

Kinetic study of the gas-phase decomposition of the trifluoroacetyl radical. Effects of temperature and pressure upon the rate constants

The kinetics and pressure dependence of the decomposition of the trifluoroacetyl radical: CF3CO + M→CF3+CO + M (5′), have been studied by generating the radicals from the selective photolysis of azomethane in the presence of 1,1,1-trifluoroacetaldehyde over the temperature range 338–417 K. The rate constants of the decomposition reaction (k5′) have been measured relative to the radical combination reactions: CH3+CF3CO→CF3COCH3(6), 2CH3→C2H6. (2)The rate constants (k5′), were found to be pressure-dependent over the pressure range 50–740 Torr, with SF6 as a bath gas.The A factor of the radical decomposition reaction has been calculated from transition-state theory to be A∞5′≈ 2.2 × 1013 s–1. Treatment of the pressure dependence of the rate constants (k5′), according to the RRKM theory of unimolecular reactions, yielded a high-pressure limiting activation energy E∞5′= 83.1 kJ mol–1.Thus the present data are consistent with a high-pressure limiting Arrhenius expression: log(k∞5′/s–1)=(13.34±0.8)–[(10 000±1000)/2.303T] where the error limits are estimates of the accuracies of the Arrhenius parameters.The results are discussed in relation to existing data on the CF3CO radical and are compared with kinetic data on the decomposition reactions of the CH3CO and C2H5CO radicals.

Dynamical treatment of unimolecular decomposition reactions. II. Short-range interfragment coupling and incomplete randomization

We investigate the dynamics of an isolated polyatomic molecule undergoing unimolecular dissociation. The intramolecular vibrational energy transfer step is treated by applying a theory of vibrational relaxation previously developed for permanently bound molecules. In contrast to an earlier study, the decomposition step is treated here subject to the assumption that, when sufficient energy is concentrated in the reaction coordinate, transitions between internal states of the molecule occur only while the incipient fragments are near their minimum classically allowed separation. Under such circumstances, it is generally not possible to divide configuration space cleanly into a randomized and a nonrandomized region, as required by the RRKM theory of unimolecular reactions. In spite of this, it is shown that, under certain conditions, the RRKM specific rate constant expression may remain valid. More generally, it is shown that the accurate specific rate constant assumes a form predicted by the transition state treatment of unimolecular reactions. The explicit rate expression derived here reveals factors which determine the accuracy of the transition state approximation for unimolecular reactions—the basic, simplifying assumption that a transition state exists which coincides with a ’’configuration of no return’’ for both decomposition and association reactions. Estimates based on our rate formulas suggest that the transition state approximation may often be sufficiently accurate to justify use of the RRKM rate expression. As part of our analysis of randomization and decomposition dynamics, we examine the time-dependent behavior of a molecule which has fragment–fragment interactions as described above but which is constrained to remain bound by the presence of an artificial ’’wall’’ which prevents the fragments from separating completely. It is found that statistical equilibrium may be achieved in this system—a significant result because the system violates some rather stringent assumptions previously invoked to prove randomization in bound molecules. This motivates a future search for a more general theory of intramolecular relaxation.

An extension of the RRKM theory of unimolecular reactions - the effective strong collision frequency

It is proposed that the accuracy of the RRKM theory can be improved by replacing the usual collision frequency by an effective collision frequency defi

Dynamical treatment of unimolecular decomposition reactions. The RRKM formula

To explore the range of validity of the RRKM theory of unimolecular reactions, we present a completely dynamical derivation of the RRKM expression for the decomposition rate of isolated molecules. This derivation avoids the usual equilibrium statistical assumptions and expresses the conditions for validity of the RRKM theory in terms of fundamental, static, molecular properties. To carry out this derivation we apply a treatment of energy transfer and decomposition which combines the Wigner–Eisenbud R-matrix approach to scattering with a technique we previously developed for studying the internal dynamics of nonreactive molecules. We obtain a molecular dissociation rate which agrees with the predictions of microcanonical transition state theory by introducing conditions which ensure statistical equilibration of all states describing the molecular fragments in close proximity of each other. We verify that, under the conditions of our derivation, the distribution of product states is statistical, i.e., in agreement with the implications of simple phase space theory, when the RRKM unimolecular rate formula is valid. Our treatment relies on a number of assumptions which may be regarded as sufficient (although not necessary) conditions for RRKM behavior. These include assumptions concerning the relative magnitudes of various interaction matrix elements (and thus the relative rates of intramolecular relaxation, passage along the reaction coordinate, and decomposition), the properties of intramolecular potential energy surfaces, and the energy dependence of certain quantities. One assumption belonging to the last category may not be entirely valid in all relevant cases but can probably be relaxed considerably. Other assumptions seem plausible but ultimately remain somewhat uncertain due to our ignorance about detailed features of potential energy surfaces for polyatomic molecules.

Some difficulties in the application of the concept of density of states

The problem of defining N( E), the density of states, for molecular systems is discussed. It is shown that a numeric evaluation of N( E) is often only approximate, even for continuous data, and is not well defined for quantized systems. The application of the concept of density of states is discussed, particularly with regard to the RRKM theory of unimolecular reactions. The sum of states, W( E), and density of states curves are evaluated for several harmonic and anharmonic model systems and the results discussed in order to illustrate the foregoing considerations.

Application of RRKM theory to the chemical and thermal activation of ethyl radicals

The quantum statistical RRKM theory of unimolecular reactions has been applied to the decomposition of excited ethyl radicals. These radicals can be formed either by collisional activation in a thermal system or chemically by the addition of hydrogen atoms to ethylene molecules. The assessment of pertinent parameters has been based on an appraisal of theory and experiment. The pressure dependence of the rate constant for the chemical activation reaction was used to assist in the assignment of activated complex parameters. These parameters were varied until good agreement was obtained between the calculated pressure falloff and data from this laboratory for the H+C2H4 reaction. The final parametric assignments were then used in calculations at higher temperatures for comparison with the thermal decomposition data for ethyl radicals from other laboratories.

Application of the RRKM theory of unimolecular reactions to the thermal reactions of spiropentane

The RRKM theory of unimolecular reactions is applied to the pyrolysis of spiropentane. The product distribution as a function of pressure, and the overall fall-off curves, are calculated on the basis of two possible detailed mechanisms. Either mechanism is found to fit the experimental data which, with new data presented here, cover the pressure range from 10–3 Torr to 1 atm.

The kinetics of the thermal gas phase reactions of methylspiro[2,2]-pentane at low pressure

A study of the thermal unimolecular reactions of methylspiro[2,2]pentane has been carried out at three temperatures in the range 347·6 to 387·1 °C and at pressures between 3·8 and 0·004 Torr. As the pressure is reduced de-composition products, allene, ethylene, propene, and buta-1,2-diene, are formed in increasing amounts at the expense of the isomerization products ethylidenecyclobutane, 2-methyl(methylene) cyclobutane, and 3-methyl-(methylene)cyclobutane. A similar increase in the proportion of 2-ethylbuta-1,3-diene and 2-methylpenta-1,4-diene relative to the methylenecyclobutanes is also observed. Calculations using the RRKM theory of unimolecular reactions are carried out in an attempt to confirm the detailed reaction mechanism.

Application of the RRKM theory to the unimolecular decomposition of 1,2-difluoroethane

The rate constants for the unimolecular elimination of HF from chemically-activated CH2FCH2F, produced in the photolysis of 1,3-difluoroacetone alone and in the presence of silanes, have been interpreted in terms of the RRKM theory of unimolecular reactions, and a critical energy of 63 kcal mol–1* obtained for the reaction. The rate constants are compared with previous data, and the dependence of the results on the thermochemistry of the system, and on the transition state model adopted are discussed.

Thermal isomerization of cyclobutenes. Part 11.—Perdeuteriocyclobutene

The thermal isomerization of perdeuteriocyclobutene has been studied in the temperature range 140–180°C. The high-pressure rate constants for the Arrhenius equation k∞D= 1013.66 ± 0.16 exp (–33,760 ± 200/RT)sec–1. The rate constant decreases with pressure below ∼20 mm. Rate constants for the isomerization have been measured down to 0.004 mm at 150.5°C. The “fall-off” curve obtained at this temperature has been compared with that obtained for cyclobutene and with various theoretical curves computed on the basis of the RRKM theory of unimolecular reactions.

RRKM Theory Calculation of the Unimolecular Decomposition of Hexafluoroethane: Thermal Activation

The thermal decomposition of hexafluoroethane is discussed in terms of the quantum-statistical RRKM theory of unimolecular reactions. The falloff behavior is calculated at several temperatures using the approximate rotation–vibration energy-level sums of Whitten and Rabinovitch. A very loose, Gorin-type activated complex model with five active internal rotations is found to be consistent with earlier shock-tube results. Centrifugal effects, associated with a significant extension of the critical bond length in the activated complex, are found to be important and must be taken into account. The approximate method is also applied to C2H6 and satisfactory agreement is found with previous exact computations.

RRKM Calculated Unimolecular Reaction Rates for Chemically and Thermally Activated C2H5Cl, 1,1-C2H4Cl2, and 1,2-C2H4Cl2

The RRKM theory of unimolecular reactions has been applied to the HCl elimination reactions of C2H5Cl, 1,1-C2H4Cl2, and 1,2-C2H4Cl2, in both chemical and thermal activation systems. The experimental data are adequately represented by a model which employs a four-center transition state. For example, the model for C2H5Cl gives a pre-exponential factor of 1.14×1013 compared to the experimental value of 0.97×1013. The calculated and experimental nonequilibrium rate constants for C2H5Cl formed by combination of CH3 and CH2Cl radicals are 1.7×109 and 2.6×109 sec−1, respectively. Various models for the molecule were investigated in an attempt to represent the torsional degree of freedom of the molecule and its interaction with over-all rotation. Also several models for the activated complex were studied. For the present data, the conclusions presented are as follows: (1) RRKM theory with all internal degrees of freedom taken as active, satisfactorily reproduces the observed data for these four-center elimination reactions; (2) the torsional degree of freedom can be adequately represented as a low-frequency vibration and the over-all rotational degree of freedom about the ``figure axis'' need not be considered active. The activated complex models developed in this work are significantly different from the models developed by Benson and Haugen for the empirical calculation of activation energies for the H—X+olefin reactions.

Theoretical Rate Constant for Thermal Unimolecular Reactions in a Multilevel System

A macroscopic unimolecular rate constant is given here in terms of the detailed microscopic rates of transport and relaxation in a many-quantum-level system. A rate equation for each quantum level is written considering all input and output processes. The resulting set of equations constitutes a master equation. The conventional unimolecular rate constant is then the lowest eigenvalue to the relaxation problem defined by the master equation. This is then a general prescription for the theoretical treatment of unimolecular behavior based only on the assumed mechanism and explicitly including the multilevel behavior of real systems undergoing chemical reaction. General methods for solving this eigenvalue problem are discussed. These methods are illustrated with the master equation for a generalized Lindemann mechanism. The lowest eigenvalue for this mechanism is obtained by an iteration procedure. The zero-order result of the iteration is shown to be equivalent to the rate constant derived from a steady-state approximation. Both the zero-order and exact expressions for the lowest eigenvalue are given in terms of a computationally useful many-shot expansion. The lowest eigenvalue result then constitutes an exact and computationally simple version of the unimolecular rate constant which is an unambiguous consequence of the microscopic parameters inserted into the chosen relaxation mechanism. Since the particular mechanism used here is also the point of departure in the derivation of the well-known RRKM theory of unimolecular reactions, the lowest eigenvalue solution is directly compared with the RRKM rate constant. The additional restrictive conditions for agreement between the two rate constants are enumerated. Some generalized expressions for mean first passage times in unimolecular rate theory are also derived and expressed in terms of many-shot solutions. These latter expressions are computationally useful for extremely rapid reactions.