Phenomenological Rate Coefficients Research Articles

An experimental and computational study of the unimolecular-decay reaction of diethyl-substituted Criegee intermediate (C2H5)2COO.

We have performed direct kinetic measurements to determine the thermal unimolecular-decay rate coefficient of (C2H5)2COO as a function of temperature (223-296 K) and pressure (4-100 torr) using time-resolved UV-absorption spectroscopy. The stabilised (C2H5)2COO Criegee intermediate was produced by photolysing 3-bromo-3-iodopentane ((C2H5)2CIBr) with 213 nm radiation in the presence of O2. We performed quantum-chemistry calculations and master-equation simulations to complement the experimental work. At 296 K and 100 torr, we measure 1530 ± 440 s-1 (2σ) for the unimolecular-decay rate coefficient, and both the experiments and simulations indicate that the reaction is effectively at the high-pressure limit under these conditions. Key parameters in the master-equation model were optimised using the experimental results, and phenomenological rate coefficients were then computed to facilitate the use of the present results in modelling. A roaming channel that forms (R/S)-2-hydroxypentan-3-one was included in the master-equation model. We also performed similar calculations for the unimolecular-decay reaction of (CH3)2COO to improve the kinetic modelling of our previous work [J. Peltola et al., Phys. Chem. Chem. Phys., 2022, 24, 5211-5219].

Automated reaction kinetics and network exploration (Arkane): A statistical mechanics, thermodynamics, transition state theory, and master equation software

AbstractThe open‐source statistical mechanics software described here, Arkane–Automated Reaction Kinetics and Network Exploration–facilitates computations of thermodynamic properties of chemical species, high‐pressure limit reaction rate coefficients, and pressure‐dependent rate coefficient over multi‐well molecular potential energy surfaces (PES) including the effects of collisional energy transfer on phenomenological kinetics. Arkane can use estimates to fill in information for molecules or reactions where quantum chemistry information is missing. The software solves the internal energy master equation for complex unimolecular reaction systems. Inputs to the software include converged electronic structure computations performed by the user using a variety of supported software packages (Gaussian, Molpro, Orca, TeraChem, Q‐Chem, Psi4). The software outputs high‐pressure limit rate coefficients and pressure‐dependent phenomenological rate coefficients, as well as computed thermodynamic properties (enthalpy, entropy, and constant pressure heat capacity) with added energy corrections. Some of the key features of Arkane include treatment of 1D, 2D or ND hindered internal rotation modes, treatment of free internal rotation modes, quantum tunneling effect consideration, transition state theory (TST) and Rice‐Ramsperger‐Kassel‐Marcus (RRKM) rate coefficient computations, master equation solution with four implemented methods, inverse‐Laplace transform of high‐pressure limit rate coefficients into the energy domain, energy corrections based on bond‐additivity or isodesmic reactions, automated and efficient PES exploration, and PES sensitivity analysis. The present work describes the design of Arkane, how it should be used, and refers to the theory that it employs. Arkane is distributed via the RMG‐Py software suite (https://github.com/ReactionMechanismGenerator/RMG‐Py).

Examining the accuracy of methods for obtaining pressure dependent rate coefficients.

The full energy-grained master equation (ME) is too large to be conveniently used in kinetic modeling, so almost always it is replaced by a reduced model using phenomenological rate coefficients. The accuracy of several methods for obtaining these pressure-dependent phenomenological rate coefficients, and so for constructing a reduced model, is tested against direct numerical solutions of the full ME, and the deviations are sometimes quite large. An algebraic expression for the error between the popular chemically-significant eigenvalue (CSE) method and the exact ME solution is derived. An alternative way to compute phenomenological rate coefficients, simulation least-squares (SLS), is presented. SLS is often about as accurate as CSE, and sometimes has significant advantages over CSE. One particular variant of SLS, using the matrix exponential, is as fast as CSE, and seems to be more robust. However, all of the existing methods for constructing reduced models to approximate the ME, including CSE and SLS, are inaccurate under some conditions, and sometimes they fail dramatically due to numerical problems. The challenge of constructing useful reduced models that more reliably emulate the full ME solution is discussed.

Theoretical Investigations on the Possibility of Prebiotic HCN Formation via O-Addition Reactions.

Until now, reactions between methane photolysis products (CH3•, CH2) and active N atom or reactive NO radical are proposed as routes of HCN formation in the prebiotic Earth. Scientists think that the reducing atmosphere of primitive Earth was made of H2, He, N2, NO, CH4, H2O, CO2, etc., and there was no molecular oxygen. However, it has been evident from experiments that the vacuum ultraviolet (VUV) photolysis of CO2 can produce atomic oxygen. Therefore, it can be presumed that atomic oxygen was likely present in early Earth's atmosphere. Was there any impact of atomic oxygen in production of early atmospheric HCN for the emergence of life? To hunt for the answer, we have employed computational methods to study the mechanism and kinetics of CH3NO + O(1D) and CH2NO• + O(3P) addition reactions. Current study suggests that the addition of O(1D) into nitrosomethane (CH3NO) and the addition of O(3P) into nitrosomethylene radical (CH2NO•) can efficiently produce HCN through an effectively barrierless pathway. At STP, Bartis-Widom phenomenological loss rate coefficients of O(1D) and O(3P) are obtained as 2.47 × 10-12 and 4.67 × 10-11 cm3 molecule-1 s-1, respectively. We propose that addition reactions of atomic oxygen with CH3NO and CH2NO• might act as a potential source for early atmospheric HCN.

Pragmatic Solution for a Fully E,J-Resolved Master Equation.

Including the effects of total angular momentum is essential to determine highly accurate pressure-dependent phenomenological rate coefficients when there are significant changes of rotational constants along the reaction coordinate. In this work, a deterministic (matrix) method has been used to solve a completely E,J-resolved two-dimensional master equation (2DME) for reaction systems that have many intermediates and many products. The practicality of the method is due to the need to obtain just a few eigenvalues and corresponding eigenvectors. Three examples are provided in order to test the performance of the implementation. It is found that the impact of rotational energy transfer via collisions on a loose transition state (TS) is more noticeable than a tight TS. The calculated results are then compared with those obtained from a recent fixed-J 2DME model.

Global uncertainty analysis for the RRKM/master equation modeling of a typical multi-well and multi-channel reaction system

Theoretically predicted rate coefficient parameters play an increasing role in combustion kinetic models. Understanding the uncertainty propagation during RRKM/master equation modeling and quantitatively evaluating the uncertainties of the phenomenological rate coefficients would be a great boon to chemical modeling. In the present work, the global uncertainty and sensitivity analysis were performed for a prototypical multiwell and multichannel reaction system (on the C4H7 potential energy surface) over the temperature and pressure range of 300–2300 K and 0.01–100 atm. Six reactions including chemical activation reactions and unimolecular reactions were studied, focusing on the uncertainty of both absolute rate coefficients and their branching ratios. To reveal the nature of uncertainty propagation from input parameters at different theoretical levels, five scenarios were designed to mimic the uncertainty from high-, moderate- and low- levels of theories. For absolute rate coefficients, the temperature dependence of uncertainties for all the reactions is found to be much stronger than the pressure dependence. The pressure dependence is mainly determined by the reaction type (chemical activation or thermal dissociation, directly-connected or well-skipping). For the branching ratio, sensitivity analysis shows that the input parameters that are very important in determining the absolute rate coefficients do not necessarily have high sensitivity coefficients in predicting the branching ratio. As a consequence, the uncertainty of the theoretically calculated branching ratio can be smaller than that of the rate coefficients.

Direct Kinetics and Product Measurement of Phenyl Radical + Ethylene.

The phenyl + ethylene (C6H5 + C2H4) reaction network was explored experimentally and theoretically to understand the temperature dependence of the reaction kinetics and product distribution under various temperature and pressure conditions. The flash photolysis apparatus combining laser absorbance spectroscopy (LAS) and time-resolved molecular beam mass spectrometry (MBMS) was used to study reactions on the C8H9 potential energy surface (PES). In LAS experiments, 505.3 nm laser light selectively probed C6H5 decay, and we measured the total C6H5 consumption rate coefficients in the intermediate temperature region (400–800 K), which connects previous experiments performed in high-temperature (pyrolysis) and low-temperature (cavity-ring-down methods) regions. From the quantum chemistry calculations by Tokmakov and Lin using the G2M(RCC5)//B3LYP method, we constructed a kinetic model and estimated phenomenological pressure-dependent rate coefficients, k(T, P), with the Arkane package in the reaction mechanism generator. The MBMS experiments, performed at 600–800 K and 10–50 Torr, revealed three major product peaks: m/z = 105 (adducts, mostly 2-phenylethyl radical, but also 1-phenylethyl radical, ortho-ethyl phenyl radical, and a spiro-fused ring radical), 104 (styrene, co-product with a H atom), and 78 (benzene, co-product with C2H3 radical). Product branching ratios were predicted by the model and validated by experiments for the first time. At 600 K and 10 Torr, the yield ratio of the H-abstraction reaction (forming benzene + C2H3) is measured to be 1.1% and the H-loss channel (styrene + H) has a 2.5% yield ratio. The model predicts 1.0% for H-abstraction and 2.3% for H-loss, which is within the experimental error bars. The branching ratio and formation of styrene increase at high temperature due to the favored formally direct channel (1.0% at 600 K and 10 Torr, 5.8% at 800 K and 10 Torr in the model prediction) and the faster β-scission reactions of C8H9 isomers. The importance of pressure dependence in kinetics is verified by the increase in the yield of the stabilized adduct from radical addition from 80.2% (800 K, 10 Torr) to 88.9% (800 K, 50 Torr), at the expense of styrene + H. The pressure-dependent model developed in this work is well validated by the LAS and MBMS measurements and gives a complete picture of the C6H5 + C2H4 reaction.

Three-Dimensional Master Equation (3DME) Approach.

The master equation technique is a standard tool to interpret gas-phase experimental kinetic results as well as to provide phenomenological rate coefficients for modeling. When there are significant changes of rotational constants along the reaction coordinate from a reactant through a transition state (TS) to product(s), including effects of angular momentum explicitly in a master equation model becomes vitally important. In this work, assuming that the K quantum number is adiabatic for both the TS and reactant, we developed an algorithm for pragmatic solutions of a three-dimensional master equation (3DME) that involves internal energy, total angular momentum ( J), and its projection K. Two examples (one is for a thermally activated isomerization of CH3NC to CH3CN via a tight TS, and the other is for a thermally activated dissociation of NH3 to H + NH2 via a loose, variational TS) are given. In addition, comparison of 3DME results with experiment as well as with those of 1DME and 2DME are documented.

Frequency Variation of the AC Order Parameter Susceptibility of the Metamagnetic Ising Model

The extensive investigation of the absorptive and reactive parts of the AC order parameter susceptibility spectra of iron group dihalides, which is obtained on the basis of Onsager theory of irreversible processes, revealed the fact that the diagonal phenomenological rate coefficients γs and γm have an important impact on the nature of the order parameter relaxation process. The number of the relaxation peaks appearing in the double logarithmic plots of χs′′ versus field frequency ω and the number of plateau regions in χs′ spectrum depends on the values of γs and γm. Only for γs≫γm does the relaxation evolve from a simple Debye exponential at high temperatures to a two-step process at lower temperatures in which there exist two long relaxation times characterizing the relaxation of staggered magnetization. In parallel with these characteristics of the order parameter relaxation, Cole-Cole plots (χs′′-χs′) are shown to consist of two arcs in the metamagnetic phase and of a semicircle in the paramagnetic phase.

A theoretical study of the thermal stability of the FS(O2)OSO2 radical and the recombination kinetics with the FSO3 radical

The kinetics of the thermal reaction of FS(O2)OO(O2)SF with SO2 have been theoretically studied. Experimental investigations performed at 293–323 K indicate that the FSO3 radical, in equilibrium with the peroxide FS(O2)OO(O2)SF ⇄ 2 FSO3 (1, -1), initially attacks the SO2 forming the FS(O2)OSO2 radical which afterwards may dissociate back, FSO3 + SO2 ⇄ FS(O2)OSO2 (2, -2), or recombine with FSO3 generating the final product, FSO3 + FS(O2)OSO2 → (FS(O2)O)2SO2 (3). Several DFT formulations and composite ab initio models were employed to characterize FS(O2)OSO2 molecular properties and to determine relevant potential energy surfaces features of reactions (2), (-2) and (3). Transition state theory calculations lead to the high pressure rate coefficients k∞,2=9.1×10-14exp(-5.2kcal mol-1/RT)cm3molecule-1s-1 and k∞,-2=4.9×1015exp(-13.9kcal mol-1/RT)s-1 while statistical adiabatic channel model (SACM/CT) calculations predict for the barrierless reaction (3) the expression k∞,3=2.9×10-11(T/300)0.4cm3molecule-1s-1. The experimental phenomenological rate coefficients are very well reproduced by these rate coefficients.

Initiation Reactions in Acetylene Pyrolysis.

In gas-phase combustion systems the interest in acetylene stems largely from its role in molecular weight growth processes. The consensus is that above 1500 K acetylene pyrolysis starts mainly with the homolytic fission of the C-H bond creating an ethynyl radical and an H atom. However, below ∼1500 K this reaction is too slow to initiate the chain reaction. It has been hypothesized that instead of dissociation, self-reaction initiates this process. Nevertheless, rigorous theoretical or direct experimental evidence is lacking, to an extent that even the molecular mechanism is debated in the literature. In this work we use rigorous ab initio transition-state theory master equation methods to calculate pressure- and temperature-dependent rate coefficients for the association of two acetylene molecules and related reactions. We establish the role of vinylidene, the high-energy isomer of acetylene in this process, compare our results with available experimental data, and assess the competition between the first-order and second-order initiation steps. We also show the effect of the rapid isomerization among the participating wells and highlight the need for time-scale analysis when phenomenological rate coefficients are compared to observed time scales in certain experiments.

Experimentally Determined Site-Specific Reactivity of the Gas-Phase OH and Cl + i-Butanol Reactions Between 251 and 340 K.

Product branching ratios for the gas-phase reactions of i-butanol, (CH3)2CHCH2OH, with OH radicals (251, 294, and 340 K) and Cl atoms (294 K) were quantified in an environmental chamber study and used to interpret i-butanol site-specific reactivity. i-Butyraldehyde, acetone, acetaldehyde, and formaldehyde were observed as major stable end products in both reaction systems with carbon mass balance indistinguishable from unity. Product branching ratios for OH oxidation were found to be temperature-dependent with the α, β, and γ channels changing from 34 ± 6 to 47 ± 1%, from 58 ± 6 to 37 ± 9%, and from 8 ± 1 to 16 ± 4%, respectively, between 251 and 340 K. Recommended temperature-dependent site-specific modified Arrhenius expressions for the OH reaction rate coefficient are (cm3 molecule-1 s-1): kα(T) = 8.64 × 10-18 × T1.91exp(666/T); kβ(T) = 5.15 × 10-19 × T2.04exp(1304/T); kγ(T) = 3.20 × 10-17 × T1.78exp(107/T); kOH(T) = 2.10 × 10-18 × T2exp(-23/T), where kTotal(T) = kα(T) + kβ(T) + kγ(T) + kOH(T). The expressions were constrained using the product branching ratios measured in this study and previous total phenomenological rate coefficient measurements. The site-specific expressions compare reasonably well with recent theoretical work. It is shown that use of i-butanol would result in acetone as the dominant degradation product under most atmospheric conditions.

Communication: Thermal unimolecular decomposition of syn-CH3CHOO: A kinetic study.

The thermal decomposition of syn-ethanal-oxide (syn-CH3CHOO) through vinyl hydrogen peroxide (VHP) leading to hydroxyl radical is characterized using a modification of the HEAT thermochemical protocol. The isomerization step of syn-CH3CHOO to VHP via a 1,4 H-shift, which involves a moderate barrier of 72 kJ/mol, is found to be rate determining. A two-dimensional master equation approach, in combination with semi-classical transition state theory, is employed to calculate the time evolution of various species as well as to obtain phenomenological rate coefficients. This work suggests that, under boundary layer conditions in the atmosphere, thermal unimolecular decomposition is the most important sink of syn-CH3CHOO. Thus, the title reaction should be included into atmospheric modeling. The fate of cold VHP, the intermediate stabilized by collisions with a third body, has also been investigated.

Unimolecular HO2 Loss from Peroxy Radicals Formed in Autoxidation Is Unlikely under Atmospheric Conditions.

A concerted HO2 loss reaction from a peroxy radical (RO2), formed from the addition of O2 to an alkyl radical, has been proposed as a mechanism to form closed-shell products in the atmospheric oxidation of organic molecules. We investigate this reaction computationally with four progressively oxidized radicals. Potential energy surfaces of the O2 addition and HO2 loss reactions were calculated at ROHF-RCCSD(T)-F12a/VDZ-F12//ωB97xD/aug-cc-pVTZ level of theory and the master equation solver for multienergy well reactions (MESMER) was used to calculate Bartis-Widom phenomenological rate coefficients. The rate coefficients were also compared with the unimolecular rate coefficients of the HO2 loss reaction calculated with transition state theory at atmospheric temperature and pressure. On the basis of our calculations, the unimolecular concerted HO2 loss is unlikely to be a major pathway in the formation of highly oxidized closed-shell molecules in the atmosphere.

A Theoretical Study of the Photoisomerization of Glycolaldehyde and Subsequent OH Radical-Initiated Oxidation of 1,2-Ethenediol.

It has recently been discovered that carbonyl compounds can undergo UV-induced isomerization to their enol counterparts under atmospheric conditions. This study investigates the photoisomerization of glycolaldehyde (HOCH2CHO) to 1,2-ethenediol (HOCH═CHOH) and the subsequent (•)OH-initiated oxidation chemistry of the latter using quantum chemical calculations and stochastic master equation simulations. The keto-enol tautomerization of glycolaldehyde to 1,2-ethenediol is associated with a barrier of 66 kcal mol(-1) and involves a double-hydrogen shift mechanism to give the lower-energy Z isomer. This barrier lies below the energy of the UV/vis absorption band of glycolaldehyde and is also considerably below the energy of the products resulting from photolytic decomposition. The subsequent atmospheric oxidation of 1,2-ethenediol by (•)OH is initiated by addition of the radical to the π system to give the (•)CH(OH)CH(OH)2 radical, which is subsequently trapped by O2 to form the peroxyl radical (•)O2CH(OH)CH(OH)2. According to kinetic simulations, collisional deactivation of the latter is negligible and cannot compete with rapid fragmentation reactions, which lead to (i) formation of glyoxal hydrate [CH(OH)2CHO] and HO2(•) through an α-hydroxyl mechanism (96%) and (ii) two molecules of formic acid with release of (•)OH through a β-hydroxyl pathway (4%). Phenomenological rate coefficients for these two reaction channels were obtained for use in atmospheric chemical modeling. At tropospheric (•)OH concentrations, the lifetime of 1,2-ethenediol toward reaction with (•)OH is calculated to be 68 h.

Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method.

A rovibrational collisional model is developed to study the internal energy excitation and dissociation processes behind a strong shock wave in a nitrogen flow. The reaction rate coefficients are obtained from the ab initio database of the NASA Ames Research Center. The master equation is coupled with a one-dimensional flow solver to study the nonequilibrium phenomena encountered in the gas during a hyperbolic reentry into Earth's atmosphere. The analysis of the populations of the rovibrational levels demonstrates how rotational and vibrational relaxation proceed at the same rate. This contrasts with the common misconception that translational and rotational relaxation occur concurrently. A significant part of the relaxation process occurs in non-quasi-steady-state conditions. Exchange processes are found to have a significant impact on the relaxation of the gas, while predissociation has a negligible effect. The results obtained by means of the full rovibrational collisional model are used to assess the validity of reduced order models (vibrational collisional and multitemperature) which are based on the same kinetic database. It is found that thermalization and dissociation are drastically overestimated by the reduced order models. The reasons of the failure differ in the two cases. In the vibrational collisional model the overestimation of the dissociation is a consequence of the assumption of equilibrium between the rotational energy and the translational energy. The multitemperature model fails to predict the correct thermochemical relaxation due to the failure of the quasi-steady-state assumption, used to derive the phenomenological rate coefficient for dissociation.

Adventures on the C3H5O potential energy surface: OH + propyne, OH + allene and related reactions

We mapped out the stationary points and the corresponding conformational space on the C3H5O potential energy surface relevant for the OH+allene and OH+propyne reactions systematically and automatically using the KinBot software at the UCCSD(T)-F12b/cc-pVQZ-F12//M06-2X/6-311++G(d,p) level of theory. We used RRKM-based 1-D master equations to calculate pressure- and temperature-dependent, channel-specific phenomenological rate coefficients for the bimolecular reactions propyne+OH and allene+OH, and for the unimolecular decomposition of the CH3CCHOH, CH3C(OH)CH, CH2CCH2OH, CH2C(OH)CH2 primary adducts, and also for the related acetonyl, propionyl, 2-methylvinoxy, and 3-oxo-1-propyl radicals. The major channel of the bimolecular reactions at high temperatures is the formation propargyl+H2O, which makes the title reactions important players in soot formation at high temperatures. However, below ∼1000K the chemistry is more complex, involving the competition of stabilization, isomerization and dissociation processes. We found that the OH addition to the central carbon of allene has a particularly interesting and complex pressure dependence, caused by the low-lying exit channel to form ketene+CH3 bimolecular products. We compared our results to a wide range of experimental data and assessed possible uncertainties arising from certain aspects of the theoretical framework.

A theoretical kinetics study of the reactions of methylbutanoate with hydrogen and hydroxyl radicals

The chemical kinetics for the reactions of methylbutanoate (MB) with hydrogen and hydroxyl radicals were studied theoretically with the ab initio transition state theory. In addition to the hydrogen abstraction reactions of MB by the radicals, the potential energy surfaces of MB+H and MB+OH were further investigated to search for additional significant hydrogen addition channels, which are followed by β-scission reactions to produce non-hydrogen and non-water products, respectively. Stationary points on the potential energy surfaces were calculated at the QCISD(T)/CBS//B3LYP/6-311++G(d,p) level. Phenomenological rate coefficients for temperature- and pressure-dependent reactions were calculated over broad ranges of temperature (200–2500K) and pressure (1.3×10−3–102atm) by solving the time-dependent multiple-well master equation. The theoretical rate coefficients were compared with the available experimental and theoretical data and observed discrepancies were analyzed. The predicted rate coefficients are represented in the forms that may readily be used in combustion modeling of MB.

Atmospheric chemistry of enols: a theoretical study of the vinyl alcohol + OH + O(2) reaction mechanism.

Enols are emerging as trace atmospheric components that may play a significant role in the formation of organic acids in the atmosphere. We have investigated the hydroxyl radical ((•)OH) initiated oxidation chemistry of the simplest enol, vinyl alcohol (ethenol, CH2═CHOH), using quantum chemical calculations and energy-grained master equation simulations. A lifetime of around 4 h was determined for vinyl alcohol in the presence of tropospheric levels of (•)OH. The reaction proceeds by (•)OH addition at both the α (66%) and β (33%) carbons of the π-system, yielding the C-centered radicals (•)CH2CH(OH)2, and HOCH2C(•)HOH, respectively. Subsequent trapping by O2 leads to the respective peroxyl radicals. About 90% of the chemically activated population of the major peroxyl radical adduct (•)O2CH2CH(OH)2 is predicted to undergo fragmentation to produce formic acid and formaldehyde, with regeneration of (•)OH. The minor peroxyl radical HOCH2C(OO(•))HOH is even less stable and undergoes almost exclusive HO2(•) elimination to form glycolaldehyde (HOCH2CHO). Formation of the latter has not been proposed before in the oxidation of vinyl alcohol. A kinetic mechanism for use in atmospheric modeling is provided, featuring phenomenological rate coefficients for formation of the three main product channels ((•)O2CH2CH(OH)2 [8%]; HC(O)OH + HCHO + (•)OH [56%]; HOCH2CHO + HO2(•) [37%]). Our study supports previous findings that vinyl alcohol should be rapidly removed from the atmosphere by reaction with (•)OH and O2 with glycolaldehyde being identified as a previously unconsidered product. Most importantly, it is shown that direct chemically activated reactions can lead to (•)OH and HO2(•) (HOx) recycling.

Reformulation and Solution of the Master Equation for Multiple-Well Chemical Reactions

We consider an alternative formulation of the master equation for complex-forming chemical reactions with multiple wells and bimolecular products. Within this formulation the dynamical phase space consists of only the microscopic populations of the various isomers making up the reactive complex, while the bimolecular reactants and products are treated equally as sources and sinks. This reformulation yields compact expressions for the phenomenological rate coefficients describing all chemical processes, i.e., internal isomerization reactions, bimolecular-to-bimolecular reactions, isomer-to-bimolecular reactions, and bimolecular-to-isomer reactions. The applicability of the detailed balance condition is discussed and confirmed. We also consider the situation where some of the chemical eigenvalues approach the energy relaxation time scale and show how to modify the phenomenological rate coefficients so that they retain their validity.