Showing papers on "Rate equation published in 1995"

Kinetics Model for the Growth of Silicon Carbide by the Reaction of Liquid Silicon with Carbon

Abstract: The kinetics and mechanism of reaction of glassy carbon with a pure silicon melt or a Si + Mo melt were investigated. The results showed that the growth of a continuous reaction-formed SiC layer followed a fourth-power rate law in the temperature range of 1430° to 1510°C. A model that could explain the fourth-power rate law was developed. In this model, an internal electric field was assumed to be set up over the reaction-formed SiC layer through a negative space charge, and then the diffusion of the carbon-ion vacancies across this layer, driven predominately by this electric field, was considered as the rate-limiting step for the SiC growth. Neither an increase in the processing temperature nor an addition of 10 wt% Mo into the silicon melt had a significant influence on the reaction kinetics. X-ray diffraction analysis revealed that the reaction products were β-SiC, and β-SiC + MoSi2 for the Si-C and Si-C-Mo reactions, respectively.

A continuous and mechanistic representation of calcite reaction-controlled kinetics in dilute solutions at 25oc and 1 atm total pressure

Abstract: Calcite dissolution rates were measured using a free-drift technique at 25°C, 1 atm total pressure, and various $$P_{CO_2 } $$ in deionized water. The data were corrected for gas phase disequilibrium and fitted to a kinetic expression derived by coupling the mechanistic models of Plummeret al. (1987a) and Chouet al. (1989) to the surface complexation model of Van Cappellenet al. (1993). Corrected dissolution and precipitation rate measurements from previous investigations were combined to our data set and fitted to the same expression. The following reactions provide an adequate description of the calcite dissolution and precipitation mechanism in dilute solutions: for which the overall reaction rate is given by where >i are the densities of surface complexes (mol/m2),a i are the activities of dissolved species and,k i are the rate constants corresponding to the above reactions. This rate equation satisfies the principle of microscopic reversibility and applies to both dissolution and precipitation reactions over a wide range of $$P_{CO_2 } $$ , pH and saturation states. The rate constants obtained from fitting the data set to Equation (3) are compatible with values reported by Plummeret al. and Chouet al., as well as yielding a very good estimate of the thermodynamic solubility constant of calcite, K 0 sp .

Single-Event Rate Parameters for Paraffin Hydrocracking on a Pt/US-Y Zeolite

Abstract: A single-event kinetic model is applied to octane hydrocracking on a Pt/US-Y zeolite over a wide range of experimental conditions (T = 200--260 C; P = 10--50 bar). The single-event kinetic approach used in this paper draws on the chemical knowledge of the elementary steps occurring on the catalyst surface, retaining the full detail of the reaction network. The hundreds of rate coefficients of the elementary steps in the reaction network are expressed in terms of a practical number of single-event rate coefficients. Rate equations are developed for the paraffins and the olefin and carbenium ion surface intermediates based on this network. Finally, single-event rate coefficients are estimated for both isomerization and cracking reactions.

Geochemical controls on the kinetics of quartz fracture at subcritical tensile stresses

Abstract: A new kinetic model links physical and chemical controls on the subcritical fracture kinetics of quartz from the assumption that molecular level reactions governing fracture and dissolution proceed by similar pathways. The model formulation combines fracture theory with a mechanistically based description of chemical, thermal, and tensile stress effects on reactivity in aqueous environments. Water, as a vapor or liquid, promotes rupture of Si-O bonds by end-member processes : (1) reaction of a protonated surface with molecular water and (2) reaction of hydroxyl ions at an ionized surface. In humid environments, reaction frequency is determined by water accessibility to the crack tip. In wetted environments, the relative contributions of these mechanisms are determined by bulk solution composition which affects surface ionization and solvent-surface interactions. The macroscopic fracture rate law is given in meters per second by the fractional sum of these end-member reaction mechanisms per a first-order equation. Agreement of this empirical rate expression with reported measurements of quartz fracture rates suggests the model is robust. It gives a good fit to fracture rates over 6 orders of magnitude and explains increasing rates with increasing solution pH, the dependence of rate upon crystallographic direction, and thermal dependence of rate over 20° to 80°C. Findings in this study suggest that (1) fracture models based upon changes in surface free energy with solution composition are macroscopic descriptions of solvent-surface interactions and parallel the mechanistic model presented here ; (2) faster fracture rates observed in basic solutions are not facilitated by decreases in the activation barrier but are due to a transition in the solvent-surface reaction to give a higher reaction frequency and (3) power law expressions applied to fracture rate versus stress intensity measurements may not have direct mechanistic significance since log-linear relations describe rate data equally well. Results of the model indicate that common chemical constituents in groundwater could affect mineral reactivity in near-surface and deeper earth environments. This quartz fracture rate model may lead to a better understanding of the time-dependent integrity of rock and glassy materials in contact with the fluids of natural systems.

Cooperative effects in the nonlinearly driven spin-boson system.

Abstract: We study the quantum dynamics of a periodically driven biased two-state system which additionally is subject to either Ohmic or to frequency-dependent damping. Within the noninteracting-blip approximation for the stochastic forces, the solution is given in terms of a series which is defined by recursion relations. Avoiding rotating wave and Markovian approximation, we deduce the solution in closed form for the important cases of high- and low-frequency driving. For low-frequency driving, the dynamics is governed by a rate equation with a time-dependent rate and with a time-dependent adiabatic equilibrium state. In the high-frequency case, we find novel cooperative effects, such as resonances in the spectral distribution near fractional values of the static detuning energy.

Eyring transition-state theory and kinetics in catalysis

Abstract: It is shown that for catalysis in general the kinetics of reactions can be discussed in terms of an extended Eyring rate equation which takes account of equilibria between free reactants and catalyst-reactant complexes as well as the equilibrium between complexed reactants and transition state. This extended equation, k exp = kT h·K ads ·K ≠ , where Kads incorporates all the complexation or Langmuir adsorption coefficients necessary is the basis of a novel understanding of the well-known compensation effect.

Modeling of quantum-well lasers with electro-opto-thermal interaction

Abstract: An equivalent circuit model has been developed for quantum-well (QW) lasers from the electrical and optical rate equations and thermal conduction equation. This model and its SPICE implementation allow self-consistent calculation of the electrical, optical, and thermal interactions using lumped elements, and make possible the simulation of multiple components in optoelectronic integrated circuits in the presence of these interactions. Physical effects which are strongly temperature dependent, such as gain variation, leakage current, and Auger recombination, have been incorporated. The model has been validated with measured laser characteristics. >

Kinetics of pyrite oxidation in sodium hydroxide solutions

Abstract: The kinetics of pyrite oxidation in sodium hydroxide solution were investigated in a stirred reactor, under temperatures ranging from 50 °C to 85 °C, oxygen partial pressures of up to 1 atm, particle size fractions from -150 + 106 to -38 + 10µm (-100 + 150 mesh to -400 mesh + 10 µ), and pH values of up to 12.5. The surface reaction is represented by the rate equation:-dN/dt = Sbk″pO0.5 2[oH- 0.25/(1 +k‴ pO2 0.5) where N represents moles of pyrite, S is the surface area of the solid particles,k″ andk″ are constants,b is a stoichiometric factor, pO2 is the oxygen partial pressure, and [OH-] is the hydroxyl ion concentration. The corresponding fractional conversion (X) vs time behavior follows the shrinking particle model for chemical reaction control: 1 - (1 -X)1/3 =k ct The rate increases with the reciprocal of particle size and has an activation energy of 55.6 kJ/mol (13.6 kcal/mol). The relationship between reaction rate and oxygen partial pressure resembles a Langmuir-type equation and thus suggests that the reaction involves adsorption or desorption of oxygen at the interface. The square-root rate law may be due to the adsorption of a dissociated oxygen molecule. The observed apparent reaction order with respect to the hydroxyl ion concentration is a result of a complex combination of processes involving the oxidation and nydrolysis of iron, oxidation and hydrolysis of sulfur, and the oxygen reduction.

Theoretical modeling of relaxation oscillations in Er-doped waveguide lasers

Abstract: An analysis of relaxation oscillations (/spl lambda//sub s//spl sim/1.5 /spl mu/m) in locally Er-doped optically pumped (/spl lambda//sub p//spl sim/1.48 /spl mu/m) waveguide lasers is reported. The theoretical model is based on time dependent rate equations for a quasi-two-level-system and on the equation of continuity for a gain medium. For the first time a numerically reliable simulation of the elementary properties of the laser oscillations was possible: the build-up time and decay of the relaxation oscillations, the time-dependent repetition period, the steady state signal output power and the evolution of the pump power versus time. Mathematically the problem can be characterized as a large boundary value problem, which can approximately be replaced by a stiff initial value problem of ordinary differential equations. In this report, pump- and signal evolution versus time are presented for planar Er-diffused Ti:LiNbO/sub 3/ waveguide lasers. The numerically obtained results show a good quantitatively agreement with experimental investigations. >

Reaction-Diffusion Model for A + A Reaction

Abstract: We formulate an approach to the A + A {yields} products reaction that is based on a reaction-diffusion equation frequently used for the A + B problem but requires an appropriate generalization for the A + A problem. This approach leads naturally to the proportionality of the rate of the reaction to the pair correlation function evaluated at r = a, where a is the diameter of the reacting particles. In other words, the reaction rate is proportional to the probability that two A particles are sufficiently close. We also present numerical simulations in one and two dimensions in order to check our predictions. We confirm the well-known anomalous rate law in one dimension (the anomalies are marginal in two dimensions) and the proportionality of the reaction rate to the two-particle correlation function. Our simulations show that the rate of the reaction is indeed determined entirely by the spatial distribution of a very small shell of particles around a given reactant particle. Anomalous kinetics is a direct reflection of the deviation of the spatial distribution of this small shell from a random configuration. We also present simulation results that confirm the predicted distance and time scaling of the pair correlationmore » function in one dimension. 92 refs., 3 figs.« less

Modeling of Adsorption and Kinetics in Catalysis over Induced Nonuniform Surfaces - Surface Electronic Gas-Model

Abstract: Several aspects of adsorption and reaction kinetics in catalysis over induced nonuniform surfaces are considered. They are discussed in terms of the surface electronic gas model, which accounts for the case of inhomogeneous surface. The adsorption equilibrium of gas mixtures was described, and adsorption isotherms were obtained. The reaction rate for the two-step sequence was deduced, and for that scheme the problems of optimum catalyst and metal-support interactions were considered. Structure sensitive and insensitive reactions with nonuniformly reactive adsorbates are also discussed in terms of the surface electronic gas model using the Polanyi parameter. The procedure for the rate law computation is given. Examples include a reaction graph with a pendent vertex and irreversible deactivation and reaction mechanisms with nonlinear elementary steps

Development and Evaluation of a Monte Carlo Technique for the Simulation of Multifunctional Polymerizations

Abstract: A stochastic simulation technique was developed to describe the kinetic behavior of bulk multifunctional photopolymerizations and the structural evolution of the polymer networks. Rate equations were written for each of the reacting species. In cross-linking polymerizations, the initiator efficiency, f, and the propagation and termination rate constants, k p and k t , are diffusion-dependent from the onset of the reaction. We were able to calculate f, k p , and k t throughout the simulation using previously developed models. From a waiting time distribution, the time between two consecutive reactions was computed. A specific reaction pathway was chosen from knowledge of the reaction probability density function. Corresponding changes were made in the molecular distribution. With such an approach, we were able to obtain kinetic as well as structural information. The results presented here simulate the bulk polymerization of diethylene glycol diacrylate containing 1% 2,2-dimethoxy-2-phenylacetophenone as the photoinitiator. The simulation was able to predict experimentally observed reaction trends.

Molecular and phenomenological rate coefficients of fast reactions in solutions

Abstract: The paper is concerned with the problem of formulating chemical rate equations for reversible reactions in solution in terms of concentration‐independent, phenomenological rate coefficients These time‐dependent rate coefficients approach, after an initial transient, the rate constants that can be obtained in a relaxation experiment We start with the coupled evolution equations for the macroscopic concentrations, and for the two‐particle distribution functions describing association–dissociation (A+B=C), bimolecular isomerization (A+B=B+C), and double decomposition (A+B=C+D) The effects of interparticle forces and long‐ranged reactivity are included We derive general identities linking the reactants and products radial distribution functions For association–dissociation this leads to relations among the molecular rate coefficients which are valid for both contact and long ranged reactivities For the other two reaction types, we were able to derive analogous relations only for contact reactivities We demonstrate how the phenomenological rate coefficients can be defined via the solutions of the corresponding diffusional boundary‐value problems This approach is quite general, and valid for both contact and long‐ranged reactivities and if interaction forces are included

Kinetics and mechanisms for nitridation of floating aluminum powder

Abstract: Aluminum powder entrained by ammonia-containing nitrogen gas was reacted at various temperatures and time to form aluminum nitride powder. The kinetics of nitride formation were determined by a quantitative X-ray analysis and compared with those determined by a nitrogen analysis of the product. The conversion to aluminum nitride increased with the reaction time following a sigmoidal rate law. The reaction time for full conversion decreased as the temperature increased in the range 1,050--1,300. The reaction rate constant at a given temperature was evaluated using the Avrami equation. The activation energy for the reaction was 1,054 {+-} 91 kJ/mol in the temperature range of 1,050--1,200 C, and decreased to 322 {+-} 70 kJ/mol above 1,200 C. Comparative analysis of powders formed below and above 1,200 C suggested strongly that the rate-controlling step changed from chemical reaction to mass transport above 1,200 C.

Rate equation analysis of a multimode bipolarization Nd3+ doped fibre laser

Abstract: The authors derive the rate equations for a multimode bipolarization fibre laser They propose a simple model for describing the polarization properties of the laser

Measurement of Fokker-Planck diffusion with laser-induced fluorescence.

Abstract: Laser-induced fluorescence in plasmas is shown to be strongly influenced by velocity-space diffusion, under the proper conditions. The induced fluorescence is modeled by a set of coupled rate equations that include the Fokker-Planck operator for each state density, and the results are compared with data from a gas-discharge plasma in which the ArII $({3d}^{\ensuremath{'}}{)}^{2}{G}_{9/2}$ metastable state is optically pumped. The Fokker-Planck diffusion coefficient $D$ is determined and found to be in agreement with theory.

Detailed kinetics of titanium nitride synthesis

Abstract: A thermogravimetric analyzer is used to study the synthesis of TiN from Ti powder over a wide range of temperature, conversion and heating rate, and for two Ti precursor powders with different morphologies. Conversions to TiN up to 99% are obtained with negligible oxygen contamination. Nonisothermal initial rate and isothermal data are used in a nonlinear least-squares minimization to determine the most appropriate rate law. The logarithmic rate law offers an excellent agreement between the experimental and calculated conversions to TiN and can predict afterburning, which is an important experimentally observed phenomenon. Due to the form of the logarithmic rate law, the observed activation energy is a function of effective particle size, extent of conversion, and temperature even when the intrinsic activation energy remains constant. This aspect explains discrepancies among activation energies obtained in previous studies. The frequently used sedimentation particle size is a poor measure of the powder reactivity. The BET surface area indicates the powder reactivity much better.

FM response of quantum-well lasers taking into account carrier transport effects

Abstract: The FM response of a single-frequency semiconductor laser is very important for communication systems that use direct modulation of the laser injection current. A new rate equation for the optical phase in quantum-well lasers is proposed, and the corresponding FM response is derived. It is shown that the separate-confinement-heterostructure can have a significant effect on the laser performance. The theory is confirmed with experimental results. >

Kinetics of Carbothermal Reduction and Nitridation of Silicon Dioxide/Carbon Mixture

Abstract: The carbothermal reduction and nitridation kinetics of silicon dioxide/carbon pellet were studied under conditions of various reaction parameters including gas flow rate, pellet size, pellet-forming pressure, C/SiO 2 molar ratio, grain sizes of both carbon and silicon dioxide, and reaction temperature by using a thermal gravimetric analysis (TGA) technique. The rate equation and kinetic model expressions of the nitridation process in the chemical reaction-controlled region were presented. The activation energy investigated in the temperature range between 1623 and 1748 K was 448±10 kJ/mol. The reaction mechanism including the formation of a gaseous SiO intermediate was also discussed. The SiO formed from SiO 2 being a rate-determining step was confirmed by experimental results. These theoretical developments correlated sufficiently with observed experimental results

Apparent rate constant for diffusion-controlled three-molecule reactions.

Abstract: We present simple explicit estimates for the apparent reaction rate constant for three-molecule reactions. For diluted systems and dg1, it depends only on the diffusion coefficients and sizes of the reacting particles. For small concentrations and d\ensuremath{\le}1, it is also time dependent. For concentrated systems, it gains the dependence on concentrations.

Temperature dependencies of the reactions of CO−3(H2O)0,1 and O−3 with NO and NO2

Abstract: We have measured temperature dependencies of the rate constants for CO−3 and O−3 reacting with NO and NO2. In addition, the temperature dependence of the CO−3(H2O) reaction with NO was determined, and a 196 K rate constant was measured for the reaction of CO−3(H2O) with NO2. The reactions with NO all proceed by O− transfer to produce NO−2. The temperature dependencies of the rate constants for the reactions of CO−3 and O−3 with NO are represented as 1.5×10−7*T−1.64 and 4.4×10−7*T−2.15 cm3 s−1, respectively, and agree very well with previous measurements. The rate constant for the reaction of CO−3(H2O) with NO is 4.1×10−5*T−2.72 cm3 s−1. Previous measurements of the rate constants for CO−3, CO−3(H2O), and O−3 reacting with NO2 appear to be in error; our measured rate constants for the first two reactions are represented as 2.6×10−5*T−2.38 and 9.1×10−9*T−0.79 cm3 s−1, respectively. The rate constant for the reaction CO−3(H2O) with NO2 is 7.9×10−11 cm3 s−1 at 196 K. The reactions of CO−3 and CO−3(H2O) with N...