Two-dimensional Master Equation Research Articles

A Two Transition State Model for Radical−Molecule Reactions: A Case Study of the Addition of OH to C2H4

A two transition state model is applied to the study of the addition of hydroxyl radical to ethylene. This reaction serves as a prototypical example of a radical-molecule reaction with a negative activation energy in the high-pressure limit. The model incorporates variational treatments of both inner and outer transition states. The outer transition state is treated with a recently derived long-range transition state theory approach focusing on the longest-ranged term in the potential. High-level quantum chemical estimates are incorporated in a variational transition state theory treatment of the inner transition state. Anharmonic effects in the inner transition state region are explored with direct phase space integration. A two-dimensional master equation is employed in treating the pressure dependence of the addition process. An accurate treatment of the two separate transition state regions at the energy and angular momentum resolved level is essential to the prediction of the temperature dependence of the addition rate. The transition from a dominant outer transition state to a dominant inner transition state is predicted to occur at about 130 K, with significant effects from both transition states over the 10 to 400 K temperature range. Modest adjustment in the ab initio predicted inner saddle point energy yields theoretical predictions which are in quantitative agreement with the available experimental observations. The theoretically predicted capture rate is reproduced to within 10% by the expression [4.93 x 10(-12) (T/298)(-2.488) exp(-107.9/RT) + 3.33 x 10(-12) (T/298)(0.451) exp(117.6/RT); with R = 1.987 and T in K] cm3 molecules(-1) s(-1) over the 10-600 K range.

The H + C2H2(+M) ⇄ C2H3(+M) and H + C2H2(+M) ⇄ C2H5(+M) reactions: Electronic structure, variational transition-state theory, and solutions to a two-dimensional master equation

In this article we investigate the kinetics of the H + C2H2 and H + C2H4 reactions, as well as their reverse dissociations, in some detail. High level electronic structure calculations are used to characterize the potential energy surfaces, and these results are not adjusted to obtain good agreement with experiment in the subsequent kinetic analysis. An approximate two-dimensional master equation is used to determine phenomenological rate coefficients, k(T,p). The effects of angular momentum conservation, tunneling, and the use of variational transition-state theory (as opposed to conventional transition-state theory) to compute microcanonical rate coefficients are investigated in detail. For both reactions, the low-pressure limit is approached very slowly, because reaction just above threshold must occur strictly by tunneling. Assuming a single-exponential-down model for P(E,E′), we deduce from experiment values of 〈ΔEd〉, the average energy transferred in a deactivating collision, as a function of temperature for both C2H3 and C2H5 in baths of He, Ar, and N2. Our results support the idea that 〈ΔEd〉 increases roughly linearly with temperature, at least for weak colliders. The agreement between theory and experiment is remarkably good for both reactions. Values of k(T,p) for the two reactions are given in the Troe format for use in modeling.

Solution of Some One- and Two-Dimensional Master Equation Models for Thermal Dissociation: The Dissociation of Methane in the Low-Pressure Limit

Using three formulations of the master equation (ME), we have investigated theoretically the dissociation of methane in the low-pressure limit. The three forms of the ME are as follows: (1) A one-dimensional model in which E, the total energy, is the independent variable (the E model). (2) The two-dimensional strong-collision-in-J model of Smith and Gilbert (Int. J. Chem. Kinet. 1988, 20, 307−329) in which ε, the energy in the active degrees of freedom, and J, the total angular momentum quantum number, are the independent variables (the ε,J model). (3) A two-dimensional variant of the ε,J model in which E and J are the independent variables (the E,J model). The third form of the ME is the most physically realistic, and for this model we investigate the dependence of values of the energy transfer moments (〈ΔEd〉, −〈ΔE〉, and 〈ΔE2〉1/2) deduced from experiment on assumed forms of the energy transfer function, P(E,E‘), and on temperature. All three moments increase as the temperature rises; −〈ΔE〉 increases from 20−25 cm-1 at 300 K to 110−120 cm-1 at 4000 K. The variation in the energy transfer moments with the form of P(E,E‘) depends on the particular moment and the temperature, but generally the variation is not greater than 25%. For the same input to the models, the E and E,J models give similar values of the rate coefficient at high temperature, implying that the rotational degrees of freedom behave increasingly as if they are active as temperature is increased. For T > 3000 K, the dissociation perturbs the equilibrium energy distribution of the molecule so much that the detailed-balance condition begins to fail; i.e., k0(T)/kr(T) ≠ Keq(T), where k0(T) and kr(T) are the dissociation and recombination rate coefficients and Keq(T) is the equilibrium constant.

Diffusion approximations of the two-dimensional master equation

Diffusion approximations of the two-dimensional master equation

Diffusion approximations of the two dimensional master equation

The effects of rotational energy transfer on unimolecular dissociation are assessed using a two-dimensional master equation approach. The collision operator for such a system can be very large, and approximations are sought that reduce the size of the matrix that has to be stored. The density of states of the vibrational energy is large for most molecules and can be treated as virtually continuous. As a consequence, motion in vibrational energy space can be treated using a diffusion equation approach. This reduces the storage requirements greatly and increases the speed at which diagonalization occurs. Results are presented for the dissociation of ethane and methane.

Diffusion approximations of the two-dimensional master equation

Diffusion approximations of the two-dimensional master equation

Full Iterative Solution of the Two-Dimensional Master Equation for Thermal Unimolecular Reactions

The application of the Nesbet algorithm to the full iterative solution of the two-dimensional master equation for thermal unimolecular reactions is presented. This leads to an efficient algorithm for computing thermal rate coefficients in the falloff regime for reactions such as simple fission dissociations, radical−radical recombinations, and ion−molecule reactions wherein microcanonical dissociation rate coefficients are sensitive to both the vibrational energy and the angular momentum of the excited species.

Reduction of the two-dimensional master equation to a Smoluchowsky type differential equation with application to CH4→CH3+H

A master equation (ME) approximation describing the effect of vibrational/rotational (V/R) relaxation on unimolecular reactions in the gas phase is presented. It is demonstrated that some forms of ME can be transformed exactly to an effective two-dimensional Smoluchowsky type differential equation (SE) in the V/R energy variables, V and R. The SE allows interpretation in a unified way of both weak and strong collision limits. Analytical expressions for the unimolecular reaction rate coefficient ku(T) are derived for simple models of the energy dependence of the microscopic reaction rate k(V,R) and V/R density of states. For realistic k(V,R) (obtained from flexible transition state theory for the reaction CH4→ArCH3+H) the SE is solved numerically. Both numerical and analytical calculations show that the collisional rotational energy transfer can, in principle, affect the rate coefficient ku(T). However, for the particular reaction considered, ku is found to be less sensitive to changes in the average rotational energy transfer than to changes in the average vibrational energy transfer. Furthermore, the strong collision limit is achieved only for unrealistically large values of the average rotational energy transfer. This means that it is a good approximation in this case one can treat V/R energy transfer and dissociation by a one-dimensional master equation in total energy.