Showing papers on "Rate equation published in 1986"

Metamorphic reaction rate laws and development of isograds

Abstract: A n S T R A C T. New data on the kinetics of dehydration o f muscovite+quartz suggest the necessity for a careful treatment of both surface kinetics and diffusion processes in metamorphic reactions. A new model is proposed that illustrates the relative role of diffusion and surface reactions in the overall metamorphic process. The rate law for the reaction at mineral surfaces derived from the experimental data is shown to be probably non-linear and similar to rate laws derived from Monte Carlo calculations. The experimental rate data is then used in a heat flow calculation to model the evolution of the muscovite isograd in the field. The position of the isograd, the temperature oversteps above equilibrium, and the width of 'reaction zones' are then analysed as a function of intrusion size and kinetic parameters.

Electron and Chemical Kinetics in the Low-Pressure RF Discharge Etching of Silicon in SF6

Abstract: Monte Carlo simulation and Boltzmann equation solutions have been used to study the electron kinetics. All electronic excitation of SF6 is assumed to be dissociative in analogy with the known product channels in ionization and multiphoton dissociation. The electric-field-to-gas-density ratios are high (E/n ? 1000 Td, where 1 Td (Townsend) = 1 × 10-17 V . cm2) in low-pressure (p < 0.3 torr) radiofrequency (RF) discharges. At these high E/n values, the electron energy relaxation time is much shorter than the 74-ns period at 13.56 MHz. Furthermore, the time scale of the chemical kinetics is much longer than the period of the applied RF voltage. Therefore the electron energy distribution can "track" the time-varying electric field, and time- and space-averaged rate coefficients can be used in chemical kinetics models. A rate equation model has been used to study the chemical kinetic processes. Electron-impact dissociation and ionization are the dominant sources of chemically active species. An electron density of 1 × 108 cm-3 is estimated from the known average values of E/n and the discharge input power. Two limiting cases are studied for the positive and negative ion diffusion losses: a) trapped negative ions and positive ion loss at the ambipolar diffusion rate; and b) positive and negative ion losses at the free diffusion rates. Neutral particle diffusion losses are estimated by using an effective diffusion length which takes surface reflection into account and increases as the surface reflection probability increases.

Optical bistability in semiconductors induced by thermal effects

Abstract: We investigate thermally induced optical bistability (OB) in semiconductors using a cw-Argon-ion-laser beam pulsed by an electro-optic modulator. We obtain different types of OB due to absorptive and refractive changes with switching times in the order of ms. The different processes are modeled using a simple rate equation for the lattice temperature. Good agreement between theoretical predictions and the measurements are clearly to be seen.

Kinetics of Pd2Si layer growth measured by an x‐ray diffraction technique

Abstract: An x‐ray diffraction approach has been developed for determination of the kinetics of growth of Pd2Si layers. Epitaxial Pd2Si films were grown on Si(111) substrates over a temperature range of 160–222 °C by a solid‐state reaction between the substrates and the Pd overlayers. The parabolic rate equation was verified and rate constants showed Arrhenius behavior with an activation energy Ea=1.06 eV and prefactor k0=7×10−4 cm2/s. The low value of Ea suggests a short‐circuit diffusion mechanism. It is reasonable to expect that impurities and microstructure may play important roles in the growth process. Impurity levels in the specimens were evaluated by analytic techniques suited to thin‐film study: Rutherford backscattering spectrometry, secondary ion mass spectrometry, and Auger electron spectrometry. No impurities were present at concentrations approaching 1 at. %. Some O, C, and F were detected at the Pd2Si/Si interfaces. The annealing ambient was the major source of further contamination. Upon emergence o...

Relaxation behavior in a bistable chemical system near the critical point and hysteresis limits

Abstract: Relaxation half‐times have been measured in the bistable iodate‐arsenous acid reaction near a hysteresis limit and the critical point. A one‐variable empirical rate law model is used to examine the apparent divergence in the relaxation times at these points. Relaxation behavior very close to, and exactly at the critical point and hysteresis limits is considered.

Reaction of N2O5 with H2O on carbonaceous surfaces

Abstract: The heterogeneous reaction of N2O5 with commercially available ground charcoal in the absence of H2O revealed a physisorption process (gamma = 0.003), together with a redox reaction generating mostly NO. Slow HNO3 formation was the result of the interaction of N2O5 with H2O that was still adsorbed after prolonged pumping at 0.0001 torr. In the presence of H2O, the same processes with gamma = 0.005 are observed. The redox reaction dominates in the early stages of the reaction, whereas the hydrolysis gains importance later at the expense of the redox reaction. The rate law for HNO3 generation was found to be d(HNO3)/dt = k(bi)(H2O)(N2O5) with k(bi), the effective bimolecular rate constants, for 10 mg of carbon being (1.6 + or - 0.3) x 10 to the -13th cu cm/s.

New Integration Techniques for Chemical Kinetic Rate Equations. I. Efficiency Comparison

Abstract: A comparison of the efficiency of several recently developed numerical techniques for solving chemical kinetic rate equations is presented. The solution procedures examined include two general-purpose codes, EPISODE and LSODE, developed as multipurpose differential equation solvers, and three specialized codes, CHEMEQ, CREK1D, and GCKP84, developed specifically for chemical kinetics. The efficiency comparison is made by applying these codes to two practical combustion kinetics problems. Both problems describe adiabatic, constant-pressure, gas-phase chemical reactions and include all three combustion regimes: induction, heat release, and equilibration. The comparison shows that LSODE is the fastest routine currently available for solving chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for temperature can be significantly faster than evaluation of the temperature by integration of its time derivative. Significant in...

Kinetic analysis of complex reactions

Abstract: TG and DSC techniques have been extensively used to study complex solid state reactions. For complex reactions (constituting of all exothermic or all endothermic one step first order individual reactions), it has been shown that the results of TG and DSC instruments may not be identical. This is because the TG instrument is incapable of correctly recording the true effective reaction rate of complex reactions (if the reaction rate is not proportional to the total amount of reactants). This may happen when the reaction rates of the individual reactions in the composite reaction mixture are significantly different. In this communication it has also been suggested that the Friedman analysis of obtaining activation energy (E) may be inapplicable for complex or composite reaction due to the fact that, there may be no unique effective constant conversion at various heating rates.

Temperature and pressure dependence of the reaction of He+ ions with H2

Abstract: The reaction of He+ ions with H2 has been studied in the temperature range from 18 to 408 K using a selected‐ion drift‐tube apparatus. The reaction rate coefficients are very small and do not exceed 2×10−13 cm3 s−1 at low pressure in the whole temperature range. The rate coefficients are pressure dependent. This indicates that a two‐body and a three‐body collision process contributes to the observed reaction rate. The three‐body reaction rate coefficient is steadily increasing with decreasing temperature, approximately k3=1.6×10−30(100/T)1.27. To explain the three‐body reaction process it has to be assumed that an intermediate complex He+–H2 with a lifetime of just a few vibrational periods exists.

Local-field rate equations for coupled optical resonators

Abstract: We present a general formalism for obtaining a coupled set of first-order rate equations for the optical field at a finite number of points in an optical resonator. We compare these local-field equations to an expansion in modes of the individual resonators and point out a shortcoming of the latter technique that has not been recognized before. We consider the special case of a set of two coupled resonators and derive the coupled rate equations for the fields in each of the two cavities with coupling coefficients. The resulting formulas are both simpler than those derived from coupled-mode theory and more accurate in the sense that they give a steady-state response which agrees with that calculated from the composite cavity modes. We show how the use of a single local-field rate equation can carry the same information as a full set of multimode rate equations in some cases, in particular, that of partial reflection back into a laser from a distant source.

Rate Equation Approach to Aggregation Phenomena

Abstract: A general qualitative theory for the mean-field rate equation describing aggregation processes is described. Various exponents describing the cluster-size distribution and its time evolution are derived. It is found that this theory frequently gives a good account of systems with more realistic kinetics, such as cluster-cluster aggregation, but that considerable caution must be exercised to ensure being in the actual asymptotic regime, since very long crossovers are sometimes observed.

Two-dimensional device simulator for laser diodes: HILADIES

Abstract: A two-dimensional device simulator for designing laser diodes has been developed. The wave equation and the rate equation for photons, as well as Poisson's equation and the current-continuity equations for electrons and holes, are solved numerically. The letter can reveal precisely the operation mechanism of laser diodes.

Improved cw lasers in the 11-13-um wavelength region produced by optically pumping NH3.

Abstract: An improved technique for creating gain in the ν2 band of NH3 by optically pumping mixtures of NH3 in N2 is described. Conventional (<10-W power) cw CO2 lasers were used to pump both 14NH3 and 15NH3. Lasing was obtained on more than thirty NH3 transitions in the 11–13-μm region, with a maximum output power of 1.2 W and conversion efficiencies as high as 17%. A simple rate equation model is used to explain the inversion mechanism.

Topochemistry of the wood-isocyanate reaction an analysis of reaction profiles.

Abstract: The rate of reaction of Scots pine (Pinus sylvestris) with n-butyl isocyanate was followed by measuring weight percent gain (WPG) as a function of time (t). A model was developed to predict the form that an equation representing the WPG versus t ‘reaction profile’ might take. A curve of that form was fitted to the data using standard commercial software. The resulting ‘whole wood’ integrated rate equation was split into its component equations to yield information about the rate and extent of reaction in the lignin and holocellulose fractions of the wood. Assignment of equations, based on defined assumptions, indicate that in the initial stages of the reaction lignin reacts faster and to a higher degree of substitution than the holocellulose.

A kinetic study of the silver-mercury contact reaction

Abstract: The solid-liquid diffusion couple technique was employed to determine the interdiffusion coefficient of the gamma phase in the Ag-Hg contact reaction. Diffusion coefficients were calculated with the aid of an equation given by Wagner. The composition range of the gamma phase was determined to be between 55.3 and 57.5 at% by electron microprobe analysis, and values for the average interdiffusion coefficient of the gamma phase were found to beD av(cm2sec−1)=3.181×10−5exp (−32539 (J mol−1)/RT) in the temperature range 40 to 115° C. The amalgamation reaction between silver and liquid mercury proceeded with the formation of gamma phase and a solid solution of Ag-Hg. The growth of gamma phase followed a parabolic rate law. The penetration of liquid mercury into grain boundary of the gamma phase caused the gamma to be crumbled off. The possibility of short-circuit diffusion is discussed.

Correlation of the rate constants of propagation with the structures of monomers and active centers in chain-growth polymerization

Abstract: Structure-reactivity correlation is one of the most fundamental problems in chemistry. In organic chemistry the influence of the structure of substituents on the rate and equilibrium constants of chemical reactions was first quantitatively established and presented in the form of the well-known Hammett equation.

A rate equation analysis of actively mode-locked semiconductor lasers

Abstract: A rate equation has been used to analyze the actively mode-locked diode laser which is biased with a subthreshold dc current upon which a sinusoidal signal, whose frequency is equal to the mode spacing (or its multiple) of the external cavity, is superimposed. Steady-state operation of the laser enables us to find out many important parameters of the system. Based upon the experimental observations, we have established the time evolution of the injection carrier density which suggests the existence of a subpulse following the main pulse and provides the possibility of predicting the time gap between them. Meanwhile, the average output power of the laser, the threshold condition of the mode-locked oscillator, the arrival time of the main pulse at the gain medium, the temporal difference between the main pulse, and the peak of the small-signal gain have also been derived.

Kinetic studies of ligand substitution rates for the Ru(NH3)5(H2O)2+ ion in Nafion films

Abstract: Substitution rates have been measured for reaction of a number of pyridines with the Nafion-bound Ru(NH/sub 3/)/sub 5/(H/sub 2/O)/sup 2 +/ complex. Reaction activities have been determined by electrochemical techniques, which also allow for determination of site thermodynamics and heterogeneity during the course of the reaction. Diffusion-coefficient effects are investigated by variation in polymer film thickness, and partition coefficients have been determined under equilibrium conditions by optical absorbance techniques. The partition-coefficient corrected rate law is found to be first order in Nafion-bound (Ru/sup II/) and first order in ligand concentration in the polymer phase. The partition-coefficient corrected bimolecular rate constants for a variety of pyridine ligands are found to vary by a factor of 5, which contrasts with the relatively constant substitution rates observed in aqueous solution. Also, sterically hindered ligands, such as 2-propylpyridine, exhibit surprisingly high substitution rate constants on the Nafion-bound Ru/sup II/ ion. These rate data indicate that pronounced molecular reactivity changes can occur upon electrode modification and have implications with respect to the design of chemically modified electrodes for use in electrocatalysis.

A theoretical analysis of stimulated Raman conversion

Abstract: The rate equations for stimulated electronic Raman scattering are presented for a general case, including linear absorption, atom and pump photon depletion, and variable beam geometry. Nondimensional variables are introduced, resulting in a simple procedure for optimization of Raman cell design and performance. A numerical solution of the detailed model is shown for a specific case, and the effects of the various parameters on the conversion efficiency are discussed.

On the law of mass action in fluid mixtures with several temperatures

Abstract: A mixture is considered in which each constituent is an electrically charged fluid and has its own temperature field. On starting from the usual balance equations and by adopting suitable constitutive equations, a reduced dissipation inequality is derived; this inequality is investigated in the case of constituents with the same velocity. Thus the law of mass action is obtained in a form which generalizes that for constituents at a common temperature. As an application, a plasma is considered, the chemical reaction being an ionization reaction. As a consequence of the law of mass action a relation between the number densities (Saha’s equation) is determined for mixtures with several temperatures.

Fluctuations in diffusion reaction systems. I: Adiabatic elimination of transport modes from a mesoscopicN-body system and the Ω-expansion

Abstract: We develop a concise method to compute the corrections to the master equation for chemically reacting systems in particle number space that arise if the system is not a well-stirred tank reactor, but the transport occurs by diffusion. Starting from the master equation in theR N space of all reactant particle positions, we expand in inverse powers of the diffusion constant and eliminate all transport modes adiabatically. It is found that the overall effect of spatially nonuniform fluctuations cannot be treated as a mere renormalization of the reaction rate constants. From second order on there appear correction terms with a new structure that corresponds formally to additional virtual reaction paths. An intuitive interpretation along this line is impeded, however, by the formal occurrence of negative reaction rate constants in these terms, i.e., the reaction rate may depend on the concentrations of the final products of the virtual reaction rather than on the ingoing products. We also identify Avogadro's constant as the suitableΩ parameter and extend van Kampen'sΩ-expansion systematically, to spatially continuous systems. This secondary expansion then serves to interpret the corrections to the rate equation, and the average and autocorrelation of the density in the stationary state. It is seen that the limitsD→∞ andΩ→∞ do not commute. The relevant length and time scales are discussed.