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.

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.

On the interpretation of vibrational relaxation times

1983 ◽  
Vol 61 (6) ◽  
pp. 1253-1266 ◽  
Author(s):  
Heshel Teitelbaum
Keyword(s):  
Shock Tube ◽  
Thermal Effects ◽  
Vibrational Energy ◽  
Chemical Activation ◽  
Relaxation Times ◽  
Boltzmann Distribution ◽  
Vibrational Energy Level ◽  
Rate Laws ◽  
Transfer Processes ◽  
Special Case

Rate laws for the evolution of vibrational energy level populations are derived when the Bethe–Teller law is obeyed. It is assumed that a Boltzmann distribution is maintained via rapid V–V processes. A variety of different rate laws result depending on the size and direction of the perturbation, the extent from equilibrium, and how classical the oscillator is at the initial and final conditions. An earlier analysis by Breshears is shown to be a special case. A prescription is given for procedures to compare relaxation times obtained from shock tube experiments and from laser-induced fluorescence experiments, when T–V energy transfer processes are rate-determining. Corrections for thermal effects are included. Shock tube, fluorescence, and chemical activation experiments are proposed which provide meaningful conditions for testing the Bethe–Teller law and for testing the existence of a Boltzmann distribution.

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.

scholarly journals Linking gas and particle ejection dynamics to boundary conditions in scaled shock-tube experiments

2021 ◽  
Vol 83 (8) ◽  
Author(s):  
Valeria Cigala ◽  
Ulrich Kueppers ◽  
Juan José Peña Fernández ◽  
Donald B. Dingwell
Keyword(s):  
Particle Size ◽  
Shock Tube ◽  
Initial Conditions ◽  
Volcanic Eruptions ◽  
Tube Length ◽  
Experimental Investigations ◽  
Clear Correlation ◽  
Angular Deviation ◽  
Dynamic Nature ◽  
Experimental Conditions

AbstractPredicting the onset, style and duration of explosive volcanic eruptions remains a great challenge. While the fundamental underlying processes are thought to be known, a clear correlation between eruptive features observable above Earth’s surface and conditions and properties in the immediate subsurface is far from complete. Furthermore, the highly dynamic nature and inaccessibility of explosive events means that progress in the field investigation of such events remains slow. Scaled experimental investigations represent an opportunity to study individual volcanic processes separately and, despite their highly dynamic nature, to quantify them systematically. Here, impulsively generated vertical gas-particle jets were generated using rapid decompression shock-tube experiments. The angular deviation from the vertical, defined as the “spreading angle”, has been quantified for gas and particles on both sides of the jets at different time steps using high-speed video analysis. The experimental variables investigated are 1) vent geometry, 2) tube length, 3) particle load, 4) particle size, and 5) temperature. Immediately prior to the first above-vent observations, gas expansion accommodates the initial gas overpressure. All experimental jets inevitably start with a particle-free gas phase (gas-only), which is typically clearly visible due to expansion-induced cooling and condensation. We record that the gas spreading angle is directly influenced by 1) vent geometry and 2) the duration of the initial gas-only phase. After some delay, whose length depends on the experimental conditions, the jet incorporates particles becoming a gas-particle jet. Below we quantify how our experimental conditions affect the temporal evolution of these two phases (gas-only and gas-particle) of each jet. As expected, the gas spreading angle is always at least as large as the particle spreading angle. The latter is positively correlated with particle load and negatively correlated with particle size. Such empirical experimentally derived relationships between the observable features of the gas-particle jets and known initial conditions can serve as input for the parameterisation of equivalent observations at active volcanoes, alleviating the circumstances where an a priori knowledge of magma textures and ascent rate, temperature and gas overpressure and/or the geometry of the shallow plumbing system is typically chronically lacking. The generation of experimental parameterisations raises the possibility that detailed field investigations on gas-particle jets at frequently erupting volcanoes might be used for elucidating subsurface parameters and their temporal variability, with all the implications that may have for better defining hazard assessment.

PLATELET AGGREGATION DOES NOT CONFORM TO SIMPLE PARTICLE COLLISION THEORY

1987 ◽  
Author(s):  
Moideen P Jamaluddin
Keyword(s):  
Platelet Aggregation ◽  
Rate Equation ◽  
Rate Constants ◽  
Kinetic Scheme ◽  
Order Type ◽  
Second Order ◽  
Particle Collision ◽  
Collision Theory ◽  
Rate Law ◽  
First Order

Platelet aggregation kinetics, according to the particle collision theory, generally assumed to apply, ought to conform to a second order type of rate law. But published data on the time-course of ADP-induced single platelet recruitment into aggregates were found not to do so and to lead to abnormal second order rate constants much larger than even their theoretical upper bounds. The data were, instead, found to fit a first order type of rate law rather well with rate constants in the range of 0.04 - 0.27 s-1. These results were confirmed in our laboratory employing gelfiltered calf platelets. Thus a mechanism much more complex than hithertofore recognized, is operative. The following kinetic scheme was formulated on the basis of information gleaned from the literature.where P is the nonaggregable, discoid platelet, A the agonist, P* an aggregable platelet form with membranous protrusions, and P** another aggregable platelet form with pseudopods. Taking into account the relative magnitudes of the k*s and assuming aggregation to be driven by hydrophobic interaction between complementary surfaces of P* and P** species, a rate equation was derived for aggregation. The kinetic scheme and the rate equation could account for the apparent first order rate law and other empirical observations in the literature.

PARTON RECOMBINATION IN CLASSICAL CASCADE

1990 ◽  
Vol 05 (28) ◽  
pp. 2377-2383 ◽  
Author(s):  
A. V. BATUNIN ◽  
O. P. YUSHCHENKO
Keyword(s):  
Explicit Form ◽  
Heavy Ion Collisions ◽  
Charged Particles ◽  
Heavy Ion ◽  
Considerable Decrease ◽  
High Energies ◽  
Ion Collisions ◽  
Parton Cascade ◽  
Energy Formula ◽  
The Mean

An equation for parton multiplicity in cascade with the recombination 1 → 2 ⊕ 2 → 1 is derived from a Kolmogorov-Chapman equation and solved. An evolution parameter τ of the cascade depends on the c.m. energy [Formula: see text]; an explicit form of the dependence is obtained from the condition that the mean multiplicity of charged particles in pp, [Formula: see text] collisions be reproduced. A considerable decrease in the mean multiplicity in heavy-ion collisions per pair of the colliding nucleons at high energies is predicted and compared to the parton cascade with no recombination.

Phenobarbitone modulation of postsynaptic GABA receptor function on cultured mammalian neurons

1979 ◽  
Vol 206 (1164) ◽  
pp. 319-327 ◽  
Keyword(s):  
Time Constant ◽  
Gaba Receptor ◽  
Aminobutyric Acid ◽  
Current Fluctuations ◽  
Receptor Function ◽  
Spinal Neurons ◽  
Synaptic Currents ◽  
Open Time ◽  
Postsynaptic Action ◽  
The Mean

The anticonvulsant barbiturate phenobarbitone increases membrane current and conductance responses to γ-aminobutyric acid (GABA) in cultured mouse spinal neurons. Analyses of GABA current fluctuations under control conditions and in the presence of phenobarbitone show that the principle action is to increase the average time during which GABA- activated channels remain open. The duration of miniature synaptic currents with a time constant of decay similar to the mean open-time of GABA-activated channels is prolonged by the drug. The results suggest that (1) the synaptic events are GABA-mediated and (2) the enhancement of these events by barbiturate is due to the postsynaptic action of the drug.

Ferricyanide Oxidation of Butanol Homogeneously Catalysed by Chlororuthenium Complexes

1979 ◽  
Vol 32 (9) ◽  
pp. 1905 ◽  
Author(s):  
AF Godfrey ◽  
JK Beattie
Keyword(s):  
Aqueous Solution ◽  
Complex Formation ◽  
Linear Dependence ◽  
Rate Constants ◽  
Homogeneous Catalyst ◽  
Basic Solution ◽  
Rate Law ◽  
Kinetics Of ◽  
Solution Estimates ◽  
Butanol Concentration

The oxidation of butan-1-ol by ferricyanide ion in alkaline aqueous solution is catalysed by solutions of ruthenium trichloride hydrate. The kinetics of the reaction has been reinvestigated and the data are consistent with the rate law -d[FeIII]/dt = [Ru](2k1k2 [BuOH] [FeIII])/(2k1 [BuOH]+k2 [FeIII]) This rate law is interpreted by a mechanism involving oxidation of butanol by the catalyst (k1) followed by reoxidation of the catalyst by ferricyanide (k2). The non-linear dependence of the rate on the butanol concentration is ascribed to the rate-determining, butanol-independent reoxidation of the catalyst, rather than to the saturation of complex formation between butanol and the catalyst as previously claimed. Absolute values of the rate constants could not be determined, because some of the ruthenium precipitates from basic solution. With K3RuCl6 as the source of a homogeneous catalyst solution, estimates were obtained at 30�0�C of k1 = 191. mol-1 s-1 and k2 = 1�4 × 103 l. mol-1 s-1.

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