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.

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.

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.

A simple reason for non-linear mixture rules in chemical kinetics. Part 1. Vibrational relaxation of diatomic molecules

1985 ◽  
Vol 63 (2) ◽  
pp. 381-393 ◽  
Author(s):  
Chris Carruthers ◽  
Heshel Teitelbaum
Keyword(s):  
Vibrational Relaxation ◽  
Thermal Effects ◽  
Diatomic Molecules ◽  
Boltzmann Distribution ◽  
Initial Distribution ◽  
Rate Law ◽  
Initial Deviation ◽  
Transfer Processes ◽  
Non Linear ◽  
Morse Oscillators

The generalized rate law for the vibrational relaxation of diatomic molecules is extended to include inert collision partners. V–V energy transfer processes are accounted for explicitly as are thermal effects. The molecules are treated as Morse oscillators as far as energetics are concerned; however, the microscopic rate constants are Landau–Teller type. It is found that the phenomenon of non-linear mixture rules arises when experimental data are forced to fit a first-order rate law. The persistence of V–V processes at times well-advanced into the relaxation zone is responsible for deviations from linearity. The non-linearities are most pronounced at high temperatures, and can be avoided only by using extremely dilute mixtures. Several sources of ambiguity are pointed out. The type of excitation method influences the initial deviation from a Boltzmann distribution and plays a crucial role in determining the importance of V–V processes and hence the degree of non-linearity. Thus, when the initial distribution is Boltzmann as in shock waves, the mixture rule is found to be absolutely linear for the vibrational relaxation of diatomic molecules.Several examples, heretofore not recognized as such, are pointed out in the literature.

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.

Substituent effects of rate constants for thermal [4π + 2π] cycloreversion of spiro-fused β-lactam oxadiazolines

1993 ◽  
Vol 71 (6) ◽  
pp. 907-911 ◽  
Author(s):  
Michel Zoghbi ◽  
John Warkentin
Keyword(s):  
Thermal Decomposition ◽  
Rate Constant ◽  
Rate Constants ◽  
Transition States ◽  
Substituent Effects ◽  
Ground States ◽  
Phenyl Substituent ◽  
First Order ◽  
Average Value ◽  
Carbonyl Ylide

Twelve Δ3-1,3,4-oxadiazolines in which C-2 is also C-4 of a β-lactam moiety (spiro-fused β-lactam oxadiazoline system) were thermolyzed as solutions in benzene. Substituents in the β-lactam portion affect the rate constant for thermal decomposition of the oxadiazolines to N2, acetone, and a β-lactam-4-ylidene. The total spread of first-order rate constants at 100 °C was 47-fold and the average value was 6.7 × 10−4 s−1. A phenyl substituent at N-1 or at C-3 was found to be rate enhancing, relative to methyl. At C-3, H and Cl were also rate enhancing, relative to methyl. The data are interpreted in terms of the differential effects of substituents on the stabilities of the ground states, and on the stabilities of corresponding transition states for concerted, suprafacial, [4π + 2π] cycloreversion. The first products, presumably formed irreversibly, are N2 and a carbonyl ylide. The latter subsequently fragments to form acetone (quantitative) and a β-lactam-4-ylidene.

Effects of oxidant concentration and temperature on decolorization of azo dye: comparisons of UV/Fenton and UV/Fenton-like systems

2012 ◽  
Vol 65 (11) ◽  
pp. 1970-1974 ◽  
Author(s):  
C. Y. Kuo ◽  
C. Y. Pai ◽  
C. H. Wu ◽  
M. Y. Jian
Keyword(s):  
Rate Constant ◽  
Rate Constants ◽  
Azo Dye ◽  
Activation Energies ◽  
First Order ◽  
First Order Kinetics ◽  
Oxidant Concentration ◽  
Decolorization Efficiency ◽  
Pseudo First Order ◽  
Reactive Red 2

This study applies photo-Fenton and photo-Fenton-like systems to decolorize C.I. Reactive Red 2 (RR2). The oxidants were H2O2 and Na2S2O8; Fe2+, Fe3+, and Co2+ were used to activate these two oxidants. The effects of oxidant concentration (0.3–2 mmol/L) and temperature (25–55 °C) on decolorization efficiency of the photo-Fenton and photo-Fenton-like systems were determined. The decolorization rate constants (k) of RR2 in the tested systems are consistent with pseudo-first-order kinetics. The rate constant increased as oxidant concentration and temperature increased. Activation energies of RR2 decolorization in the UV/H2O2/Fe2+, UV/H2O2/Fe3+, UV/Na2S2O8/Fe2+ and UV/Na2S2O8/Fe3+ systems were 32.20, 39.54, 35.54, and 51.75 kJ/mol, respectively.

The Kinetics of Isotope Effects in a Model Solvolysis System Involving One Optically Active Ion Pair Intermediate

1974 ◽  
Vol 52 (10) ◽  
pp. 1937-1941 ◽  
Author(s):  
P. Christian Vogel
Keyword(s):  
Rate Constant ◽  
Rate Constants ◽  
Isotope Effects ◽  
Kinetic Scheme ◽  
Ion Pair ◽  
Rate Equations ◽  
Differential Analysis ◽  
First Order ◽  
Order Rate ◽  
Special Cases

The derivation of the observed first-order rate constants from the "exact" integrated rate equations for the kinetic scheme of reaction 1 is presented. It is shown that the solvolytic exponential first-order rate constant is a special case of the polarimetric rate constant and that the optical activity of the product is determined by a multiplicative ratio of rate constants for the optically important reactions of the ion pair intermediate. A form of the integrated first-order polarimetric rate equation with a linearly independent parameter set is presented. The functions for the first-order rate constants derived using the steady state approximation are special cases of the functions derived from the exact equations, as are the functions for the first-order rate constants for two systems which involve pre-equilibria followed by a slow product forming step. These functions cannot all be derived one from the other. A differential analysis of observed isotope effects as functions of isotope effects on the rate constants for reactions involving the intermediates is presented.

An analytical method for determining rate constants using a periodical irradiation technique

1969 ◽  
Vol 47 (24) ◽  
pp. 4531-4535 ◽  
Author(s):  
W. Kawakami ◽  
H. S. Isbin
Keyword(s):  
Rate Constants ◽  
Analytical Method ◽  
Apparent Rate Constant ◽  
Chloral Hydrate ◽  
Life Time ◽  
First Order ◽  
Radical Concentration ◽  
Γ Rays ◽  
The Mean

A new analytical method is presented for the determination of rate constants using a periodical irradiation technique. This method is applicable to reactions which have both a first and a second order termination for the chain radical concentration, and is illustrated for the radiolysis of aqueous solutions of chloral hydrate by 137Cs γ rays. The apparent rate constant of first order termination, and the mean chain life time of the chain radical were determined to be 1.10 s−1, and 0.25 s for 400 rad/min, at 22 °C, respectively.

Reversible ring opening in the hydrolysis of spiro ortho esters

1985 ◽  
Vol 63 (10) ◽  
pp. 2673-2678 ◽  
Author(s):  
Robert A. McClelland ◽  
Claude Moreau
Keyword(s):  
Rate Constant ◽  
Rate Constants ◽  
Ring Opening ◽  
Lone Pair ◽  
Order Rate Constant ◽  
Product Analysis ◽  
Ortho Ester ◽  
First Order ◽  
Order Rate ◽  
Ortho Esters

Hydrolysis kinetics are reported for four spiro ortho esters: 3,4-dihydro-6-methoxy-1H-2-benzopyran-1-spiro-2′-1′,3′-dioxolane (13), its 1′,3′-dioxane analog (14), and the 6-unsubstituted versions of each (11 and 12). For comparison, also included are the diethoxy analogs: 1,1-diethoxy-3,4-dihydro-6-methoxy-1H-2-benzopyran (10) and the 6-unsubstituted compound (9). Product analysis implicates an initial opening of the dioxolane or dioxane ring in the spiro ortho esters, as expected on the basis of stereoelectronic considerations. The intermediate dialkoxycarbocations can be observed in HCl solutions. A detailed analysis has been carried out for the 6-methoxy systems to provide the rate constants k1, the second-order rate constant for H+-catalyzed formation of the cation from the ortho ester, k2, the first-order rate constant for water addition to the cation, and k−1, the first-order rate constant for ring closing of the cation to reform the ortho ester. The two spiro ortho esters are shown in this analysis to undergo reversible ring opening in their hydrolysis, in that values of k−1, are greater than k2. The differences, however, are not large, k−1/k2 being 1.2 (dioxolane, 13) and 3.8 (dioxane, 14). Comparison with the diethoxy ortho ester also reveals that the ring opening process (k1, rate constants) is inherently more difficult with the dioxolane, although not with the dioxane. An argument involving lone pair orientation is advanced to explain this.

ABSOLUTE RATE CONSTANTS FOR HYDROCARBON AUTOXIDATION: III. α-METHYLSTYRENE, β-METHYLSTYRENE, AND INDENE

1966 ◽  
Vol 44 (10) ◽  
pp. 1113-1118 ◽  
Author(s):  
J. A. Howard ◽  
K. U. Ingold
Keyword(s):  
Isotope Effect ◽  
Rate Constant ◽  
Rate Constants ◽  
Propagation Rate ◽  
Absolute Rate ◽  
Order Process ◽  
Deuterium Isotope Effect ◽  
First Order ◽  
Process Rate Constant ◽  
Propagation Rate Constant

Absolute rate constants for the copolymerization of α-methylstyrene and oxygen have been measured from 13 to 50 °C. The propagation and termination rate constants can be represented by[Formula: see text]Experiments with 2,6-di-t-butyl-4-methylphenol at 65 °C have shown that C6H5C(CH3):CH2 and C6H5C(CD3):CD2 have the same propagation rate constant but that chain termination involves a deuterium isotope effect (kt)H/(kt)D ≈ 1.5.Absolute rate constants for the copolymerization of oxygen with β-methylstyrene and with indene at 30 °C showed that a significant fraction of the oxidation chains were terminated by a kinetically first order process (rate constant kx). The rate constants for β-methylstyrene and indene at 30 °C are kp = 51 and 142 l mole−1 s−1, kt = 1.6 × 107 and 2.5 × 107 l mole−1 s−1, and kx = 0.61 and 1.2 s−1, respectively. The propagation rate constant for indene can be separated into a rate constant for the copolymerization with oxygen (kadd = 128 l mole−1 s−1) and a rate constant for hydrogen atom abstraction (kabstr = 14 l mole−1 s−1). In the presence of heavy water the first order process for indene had a deuterium isotope effect (kx)/(kx)D2O ≈ 3.

Sign in / Sign up

Export Citation Format

Share Document