The influence of non-equilibrium transient effects on measurements of vibrational relaxation times and of dissociation rate constants

1982 ◽  
Vol 60 (23) ◽  
pp. 2927-2942 ◽  
Author(s):  
Heshel Teitelbaum
Keyword(s):  
Rate Constants ◽  
Vibrational Relaxation ◽  
Vibrational Energy ◽  
Relaxation Times ◽  
Dissociation Rate ◽  
Harmonic Oscillators ◽  
Dissociation Limit ◽  
Transient Effects ◽  
Rate Law ◽  
Non Equilibrium

A semi-empirical analysis based on a rate law for vibrational relaxation of dissociating simple harmonic oscillators allows for a detailed study of measurements of vibrational relaxation times τ and of steady dissociation rate coefficients k0. It is shown that non-equilibrium populations of vibrational energy levels prevent attainment of the thermodynamically expected equilibrium energy. Even under near-isothermal and mild conditions, [Formula: see text], serious experimental errors result when the Bethe–Teller relaxation rate law is used. Closed form expressions are given which permit evaluation of these errors. Measurements should be analyzed using the rate law[Formula: see text]where ε is the vibrational energy per molecule, τ the relaxation time, kd the non-equilibrium rate coefficient, ετ the thermodynamically expected vibrational energy at temperature T, and (ε* + hv) the energy just above the dissociation limit. It is also shown that if[Formula: see text]a local minimum and maximum are predicted for measured density gradients in shock tube dissociations of diatomic molecules, where tine is the incubation time, D′ the effective dissociation energy, and x0 the mole fraction of dissociating molecules in an inert diluent. Expressions are given for extracting incubation times and rate constants from such studies when [Formula: see text]. Analysis of experimental data actually showing such phenomena (J. Chem Phys. 55, 4017 (1971)) is carried out. There are indications that any analysis which does not explicitly account for transient effects could result in errors in measured k0's of factors of 2 or more.

Generalized rate law for vibrational relaxation of a pure diatomic gas

1983 ◽  
Vol 61 (6) ◽  
pp. 1267-1275 ◽  
Author(s):  
Heshel Teitelbaum
Keyword(s):  
Rate Constant ◽  
Rate Constants ◽  
Vibrational Relaxation ◽  
Vibrational Energy ◽  
Boltzmann Distribution ◽  
Rate Law ◽  
First Order ◽  
Non Linear ◽  
Energy Formula ◽  
The Mean

The master equation for the vibrational relaxation of a pure gas of diatomic molecules AB is reduced to a simple analytical rate law. Anharmonicity is accounted to first order, and both T–V and near-resonant V–V energy transfer processes are included with the limitation that Δν = ± 1. L and au–Teller type transition probabilities are used to scale the rate constants. The rate law consists of a pair of simultaneous first order non-linear differential equations — one for the mean vibrational energy, [Formula: see text], and one for the mean squared vibrational energy [Formula: see text]; or equivalently a non-linear second order differential equation for [Formula: see text], with respect to time, t, plus an algebraic equation for [Formula: see text] These lead to[Formula: see text]where χe is the anharmonicity factor, N the molecular concentration, νe,. the spectroscopic vibrational frequency; ν′ = νe (1 − χe); ν″ = νe. (1 − 3χe); [Formula: see text]; 1/τ = Nk1.0(1 − e−hν″/KT); k1.0 the rate constant for the process AB(ν = 1) + AB(ν) → AB(ν = 0) + AB(ν); and [Formula: see text] the rate constant for the process 2AB(ν = 1) → AB(ν = 0) + AB(ν = 2). It is shown that the Bethe–Teller law, [Formula: see text], is valid only in the limit of zero anharmonicity or slow V–V processes, or when the initial population is Boltzmann, such as in shock tube experiments. Furthermore, a population distribution which is initially Boltzmann will remain so; whereas a non-Boltzmann distribution rapidly becomes a Boltzmann distribution on a time scale determined by the sum of T–V and V–V rate constants. The present study allows one to gauge the importance of two common assumptions: the validity of the Bethe–Teller law and the existence of a Boltzmann distribution or vibrational temperature during the relaxation.

scholarly journals Picosecond-Duration Vibrational Relaxation Kinetics in the Excited S1 State of Perylene and Anthracene Molecules

1983 ◽  
Vol 3 (1-6) ◽  
pp. 249-261 ◽  
Author(s):  
A. Freiberg ◽  
T. Tamm ◽  
K. Timpmann
Keyword(s):  
Steady State ◽  
Luminescence Spectrum ◽  
Vibrational Relaxation ◽  
Vibrational Energy ◽  
Streak Camera ◽  
Relaxation Times ◽  
Temporal Behavior ◽  
Vibrational Energy Relaxation ◽  
The Matrix ◽  
Measured Relaxation

We present and discuss the results of a direct observation of the picosecond range temporal behavior of vibronic lines in the luminescence spectrum of the matrix-isolated perylene and anthracene molecules. A novel subtractive dispersion mount of monochromators in conjunction with the synchroscan streak camera has been used. From spectrochronograms measured at different excitation wavelengths the vibrational energy relaxation times have been obtained. These are in the range of 20–30 ps and are most probably determined by the existence of the phonon bath of the matrix. A comparison of the measured relaxation constants with those estimated from the steady-state hot luminescence spectrum has been made.

Analytic solution of relaxation and dissociation of ozone and sulphur dioxide at low pressures

1981 ◽  
Vol 59 (17) ◽  
pp. 2569-2574 ◽  
Author(s):  
Wendell Forst
Keyword(s):  
Rate Constants ◽  
Sulphur Dioxide ◽  
Vibrational Relaxation ◽  
Analytic Solution ◽  
Thermal Dissociation ◽  
Relaxation Times ◽  
Average Energy ◽  
Collision Rate ◽  
Low Pressures ◽  
Good Agreement

The analytic solution of vibrational relaxation in a low-pressure gas is applied to the thermal dissociation of O3 in helium and of SO2 in argon. Use is made of experimental relaxation times to obtain average energy lost per collision. Calculated weak-collision rate constants are in very good agreement with experiment in the case of SO2, but only in fair agreement in the case of ozone. Several curious aspects of the ozone system, both experimental and theoretical, are discussed.

Solutions of the generalized rate law for vibrational relaxation of a pure diatomic gas

1983 ◽  
Vol 61 (6) ◽  
pp. 1276-1287 ◽  
Author(s):  
Heshel Teitelbaum
Keyword(s):  
Shock Tube ◽  
Time Constant ◽  
Rate Constants ◽  
Vibrational Energy ◽  
Initial Conditions ◽  
Chemical Activation ◽  
Rate Law ◽  
Diatomic Gas ◽  
Energy Formula ◽  
The Mean

The generalized rate law for the relaxation of the vibrational energy of a pure diatomic gas, AB, derived earlier, is solved analytically for a variety of initial conditions corresponding to shock tube, laser-excited fluorescence, and chemical activation experiments. The resulting expressions can be used to easily predict whether a given system will relax according to a V–V or a T–V mechanism or both. The initial conditions and the molecular anharmonicity are shown to be as important, if not more important, for this purpose than the ratio of T–V and V–V rate constants. Behind shock waves the energy relaxes exponentially with a T–V time constant. The initial distribution remains Boltzmann. In laser or chemical activation experiments the energy does not relax exponentially, leading to phenomenological time "constants" [Formula: see text] or [Formula: see text] which are not constant in time and prevent direct comparisons with shock tube data. It is only after an incubation period during which the vibrational energy is redistributed via V–V processes that the energy then exchanges with translational energy and decays. Prescriptions are given to extract T–V and V–V rate constants from such data. The initial degree of laser excitation, a, and the time regime probed, t/τ, must be known for this purpose. However, when direct overtone excitation is used, a careful choice of α can lead to extraction of the T–V constant directly. Even though the vibrational energy itself does not relax exponentially, it is shown that the mean energy, [Formula: see text], and the mean squared energy, [Formula: see text], relax in such a way that the quantity [Formula: see text] does decrease exponentially with a time constant very closely related to the V–V rate constant for 2AB(ν = 1) → AB(ν = 2). A short survey of various laser and chemical excitations in the literature is presented and analyzed in terms of initial conditions. In general, the larger the degree of excitation and the higher the quantum numbers of the excited levels, the more V–V character does the energy relaxation have.

Statistically steady states of forced isotropic turbulence in thermal equilibrium and non-equilibrium

2016 ◽  
Vol 797 ◽  
pp. 181-200 ◽  
Author(s):  
Diego A. Donzis ◽  
Agustin F. Maqui
Keyword(s):  
Turbulent Flows ◽  
Thermal Equilibrium ◽  
Degrees Of Freedom ◽  
Isotropic Turbulence ◽  
Vibrational Energy ◽  
Relaxation Times ◽  
Steady States ◽  
Laminar Flows ◽  
Vibrational Modes ◽  
Non Equilibrium

We investigate statistically steady states of turbulent flows when molecular degrees of freedom, in particular vibration, are taken into account. Unlike laminar flows initially in thermal non-equilibrium which asymptotically relax towards thermal equilibrium, turbulent flows present persistent departures from thermal equilibrium. This is due to fluctuations in temperature and other thermodynamic variables, which are known to increase with turbulent Mach number. Analytical results are compared to direct numerical simulations at a range of Reynolds and Mach numbers as well as molecular parameters such as relaxation times. Turbulent fluctuations are also shown to disrupt the distribution of energy between translational–rotational–vibrational modes even if thermal equilibrium is attained instantaneously relative to turbulence time scales, an effect that increases with characteristic relaxation times. Because of the nonlinear relation between temperature and vibrational energy in equilibrium, the fluctuation of the latter could be strongly positively skewed with long tails in its probability density function. This effect is stronger in flows with strong temperature fluctuations and when vibrational modes are partially excited. Because of the finite-time relaxation of vibration, departures from equilibrium result in very strong transfers of energy from the translational–rotational mode to the vibrational mode. A simple spectral model can explain the stronger departures from thermal equilibrium observed at the small scales. The spectral behaviour of the instantaneous vibrational energy can be described by classical phenomenology for passive scalars.

Oxygen vibrational and dissociation relaxation behind regular reflected shocks

1976 ◽  
Vol 74 (3) ◽  
pp. 477-495 ◽  
Author(s):  
H. Oertel
Keyword(s):  
Theoretical Model ◽  
Vibrational Relaxation ◽  
Relaxation Times ◽  
Dissociation Rate ◽  
Numerical Calculations ◽  
Rate Coefficients ◽  
Double Exposure ◽  
Experimental Conditions ◽  
Density Profiles ◽  
Best Fit

The oxygen vibrational and dissociation relaxation behind regular reflected shocks has been calculated and measured. Numerical calculations using published rate coefficients supplied the relaxation-zone data needed to estimate the range of most useful experimental conditions. Then photographs of the shock reflexion were taken using a complementary double-exposure interferometer. The density profiles in the relaxation zones behind the reflected shocks were measured by means of a multibeam laser-differential interferometer. The results of these experiments confirmed the theoretical model adopted for the calculations within a certain range of experimental conditions, but clearly revealed the need for revising the rate coefficients. New calculations with different vibrational relaxation times and dissociation rate coefficients then had the result that the best fit of calculated to measured profiles was obtained when the following values were inserted.Vibration\begin{eqnarray*} & p\tau_v = A_v\exp(B_vT^{-\frac{1}{3}}),\\ & A_v = (2.1\pm 0.2)\times 10^{-5}\,{\rm kg/ms},\quad B_v = 129\,{}^{\circ}{\rm K}^{\frac{1}{3}}. \end{eqnarray*}Dissociation: O2+ O_2[rlarr ] 2O + O_2\begin{eqnarray*} & {\mathop {k_1}\limits^{\rightharpoonup}} = A_1T^{-2.5}\exp (-\theta_D/T),\\ & A_1 = (6.2 \pm 0.5)\times 10^{18}\,{\rm m}^3\,{}^{\circ}{\rm K}^{2.5}/{\rm mol}\,{\rm s},\quad\theta_D = 59\,136\,{}^{\circ}{\rm K}. \end{eqnarray*}Dissociation: O2+ O[rlarr ]3O\begin{eqnarray*} & {\mathop {k_1}\limits^{\rightharpoonup}} = A_2T^{-1.0}\exp (-\theta_D/T),\\ & A_2 = (4.0 \mp 0.5)\times 10^{-13}\,{\rm m}^3\,{}^{\circ}{\rm K}/{\rm mol}\,{\rm s}. \end{eqnarray*}

Theory of vibrational relaxation in mixtures of ortho- and para-hydrogen

1981 ◽  
Vol 59 (8) ◽  
pp. 1277-1283 ◽  
Author(s):  
Avygdor Moise ◽  
Huw O. Pritchard
Keyword(s):  
Rate Constants ◽  
Vibrational Relaxation ◽  
Numerical Study ◽  
Relaxation Times ◽  
High Dilutions ◽  
State Rate ◽  
The Mean ◽  
The Individual ◽  
Individual Rate ◽  
Very High

A numerical study of the vibrational relaxation at 500 K of a mixture of ortho-H2 and para-H2 is described. The required state-to-state rate constants were calculated from the quantum results of Rabitz and co-workers, and missing pieces of data were estimated by interpolation.It is concluded that only one relaxation time will be observed in any mixture of ortho-H2 and para-H2 and that (except at very high dilutions in a third inert gas) the relaxation rate constant will be close to the mean of the individual rate constants for relaxation, weighted according to the respective mole fractions of ortho-H2 and para-H2 present in the mixture.We find that the relaxation process can be modelled very accurately as an electrical RC network, whose time constants can be written down quite easily as sums of the appropriate microscopic rate constants, and by using this model, it is a simple matter to explore the conditions required for a mixture of two gases to exhibit two distinct vibrational relaxation times.

Vibrational Energy Transfer in CHF3 -Mixtures

1976 ◽  
Vol 31 (10) ◽  
pp. 1268-1270 ◽  
Author(s):  
K. Frank ◽  
P. Hess
Keyword(s):  
Energy Transfer ◽  
Vibrational Relaxation ◽  
Vibrational Energy ◽  
Relaxation Times ◽  
Rare Gas ◽  
Vibrational Energy Transfer

Abstract The vibrational relaxation times for pure CHF, and CHF3 diluted in H2, D2, Ar, Kr and Xe are 0.55; 0.01, 0.025, 2.6, 4.8, and 5.6 /μsec atm at 298 K. These measurements complete previous results obtained for the systems CHF3-He, Ne, Ar. Correlation of the rare-gas results according to SSH-theory shows that relatively small rotational contributions may be expected for the heavy collision partners Kr and Xe.

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