Derivation Of Rate Equations Research Articles

Full-wave analysis of photonic-crystal structures with multi quantum well active region

In this paper, a semi-classical formulation for modeling photonic-crystal (PC) structures with a four-layer multi quantum well (MQW) active region is presented. The formulation treats the electromagnetic fields as a classical entity, while employing the quantum-mechanically derived rate equations to analyze the active medium. Using the Poynting theorem, we interrelate the two sets of equations, namely the MQW rate equations and Maxwell's equations. The formulation involves the stimulately emitted photon density in the former and a virtual volume current density representing the generated photons in the latter. With the help of this formulation, we then model a PC optical amplifier as well as a PC hexagonal-defect cavity laser and solve the obtained equations after replacing the involved spatial and temporal derivatives by pseudo-spectral and Runge–Kutta's formulations, respectively. The formulation also takes the effect related to gain saturation and frequency broadening into account.

KINETIC STUDY OF THE OXIDATWE DECOLORIZATION OF XYLENOL ORANGE BY HYDROGEN PEROXIDE IN MICELLAR MEDIUM

ABSTRACT The kinetics of the oxidation of xylenol orange tetrasodium salt by hydrogen peroxide was investigated in the presence of cationic surfactant (N- dodecylpyridinium chloride) & strongly basic media over the temperature range 20 – 35 °C. The kinetics studies were carried out as a function of different variables like concentration, pH and temperature. The described reaction rate is greatly catalyzed by the presence of trace amount of Co(II) ions & followed pseudo-second- order with respect to dye concentration. Possible mechanism for the catalyzed and uncatalyzed reactions is proposed and found to be coincident with the experimental findings. The derived rate equation is in good agreement with the experimental observations. Thermodynamic parameters were also evaluated. Keywords: Kinetic; Hydrogen peroxide; Xylenol orange; Surfactant; Mechanism e-mail: nasiruk@uok.edu.pk INTRODUCTION Hydrogen peroxide has interesting biological properties and environmentally it is a friendly oxidant. It has been used for the oxidation of various organic compounds. [1 -3] Number of works is found in the literature in which hydrogen peroxide has been used as an oxidant for different dyes.[4 – 7] It is one of the principal oxidizing agents used for the treatment of fabrics, paper, cosmetic products and industrial wastewater due to its low cost, simple handling, effectiveness in destroying the chromospheres and being a friendly alternative to chlorine. The oxidation of dyes has got much attraction since last few years. [8- 10] This has been stimulated by environmental, commercial & social pressures, particularly environmental concerns over residual dyes present in wastewater streams and causing aesthetic pollution. Although, previous studies represent significant steps forward in generating empirical rules governing oxidation, further progress is required in developing the underlying science base.Xylenol orange [3, 3’ bis [N, N’ – Di(carboxymethyl)- aminomethyl- O - cresolsulphone – phthalein tetrasodium salt] (XO), a heterocyclic species which belongs to the acidic class of dyes. This anionic dye, which is stable and water soluble with an absorption peak at 577 nm, was chosen as a representative species for this study.A number of papers have been published on the oxidation of xylenol orange (reductive chromogenic agent) by various oxidizing agent. [11 – 13] In previous work , oxidation of xylenol orange was carried out but detailed kinetics study like reaction mechanism, effect of surfactants and thermodynamic parameters were not studied. In proposed work Co(II) has been used as a catalyst but in previous work other ions were used as catalyst except Co(II) in the oxidation of xylenol orange by bromate.The oxidation of xylenol orange is a very important redox reaction, as it has an environmental concern. It is a textile dye for industrial uses. Current environmental hazards with these dyes and their derivatives revolve around the carcinogenic potential. Different metal ions like copper (II), iron (II) and (III) etc. have been proved to enhance the decolorization of various dyes

Probabilistic and fractal approaches to deriving rate equations for heterogeneous oxide dissolution processes

Generalized equations of heterogeneous kinetics are derived which can be used to calculate kinetic parameters of oxide dissolution. Main rate equations are obtained, and their applicability to iron(III) oxide dissolution in sulfuric acid solutions is examined. The dependences of the specific dissolution rate on acid concentration, pH, and temperature are found. Kinetic curves represented as plots of α against t/t0.5 (where α is the fraction of oxide dissolved at time t and t/t0.5 is the time needed to dissolve 50% of the sample) are shown to be invariant to the kinetic parameters and the model of dissolution.

Autocatalytic Kinetics and Mechanism of Biogas Generation

The mechanism of anaerobic fermentation of biomass into biogas is explained in terms of autocatalytic reaction. Rate equation is derived for the first order kinetics and verified using the data obtained from a digester holding 20 kg of 10% cowdung slurry at 25°C. The derived rate equation based on the proposed autocatalytic mechanism is found to explain the experimental rate satisfactorily.

Kinetic and mechanism study of the oxidative decolorization of neutral red by bromate in micellar medium

The kinetics of the oxidation of neutral red by bromate was investigated in the presence of cationic surfactant (N-dodecylpyridinium chloride) and strongly acidic media over the temperature range 22-35 °C. The kinetic studies were carried out as a function of different variables like concentration and temperature. The described reaction rate is greatly catalyzed by the presence of trace amount of Fe(II) ions and followed by pseudo-third-order kinetics with respect to dye concentration. A possible mechanism for the catalyzed reaction is offered which is found to be coincidentally congruent with experimental findings. The derived rate equation is in good agreement with the experimental observations. Thermodynamic parameters were also evaluated.

Determination of rapid-equilibrium kinetic parameters of ordered and random enzyme-catalyzed reaction A+B=P+Q.

This article deals with the rapid-equilibrium kinetics of the forward and reverse reactions together for the ordered and random enzyme-catalyzed A + B = P + Q and emphasizes the importance of reporting the values of the full set of equilibrium constants. Equilibrium constants that are not in the rate equation can be calculated for random mechanisms using thermodynamic cycles. This treatment is based on the use of a computer to derive rate equations for three mechanisms and to estimate the kinetic parameters with the minimum number of velocity measurements. The most general of these three programs is the one to use first when the mechanism for A + B = P + Q is studied for the first time. This article shows the effects of experimental errors in velocity measurements on the values of the kinetic parameters and on the apparent equilibrium constant calculated using the Haldane relation.

Modeling Study of the Sorption-Enhanced Reaction Process for CO2 Capture. II. Application to Steam-Methane Reforming

In this paper, the reactor model introduced in part I will be verified using the results of an analytical solution for the increase of CH4 conversion over the bed and validated using the results of sorption-enhanced steam-methane reforming laboratory-scale experiments. An experimentally derived rate equation for the steam-methane reforming reaction is used, a literature rate equation for the water−gas shift reaction. An overview of modeling work on the sorption-enhanced reaction process for steam-methane reforming performed by other groups is presented. The CH4 and CO2 profiles obtained from laboratory-scale experiments are quite satisfactorily described using a Freundlich isotherm. A sensitivity analysis shows that both the CH4 and CO2 profiles are sensitive to the adopted isotherm model and its parameters. In addition to that, the CH4 and CO2 profiles are sensitive to the diffusion coefficient. Neither profile is sensitive to the particle size or the heat of adsorption.

Numerical Study of Doppler Dynamics in Self-Mixing Semiconductor Lasers

We present a new approach for performing numerical simulations of optical feedback semiconductor lasers for moving targets. The derived rate equation for the complex electric field is suitable for the numerical analysis of the dynamics in a self-mixing semiconductor laser with a Doppler frequency shift. We present numerical examples calculated using the rate equations.

An "Aufbau" Approach To Understanding How the King–Altman Method of Deriving Rate Equations for Enzyme-Catalyzed Reactions Works

The King–Altman method of deriving rate equations for enzymatic reactions is applied to the derivation of the Michaelis–Menten equation, along with an explanation for how (or why) the King–Altman method works in this case. The slightly more complicated cases of competitive inhibition and a two-substrate enzyme-catalyzed reaction are then treated in the same manner. It is the intent of the author that through this process of seeing how the King–Altman method works in progressively more complicated cases, a student is in a better position to appreciate the generality of the method and, perhaps, the general proof of the method.

Application of a Perrin-like Kinetics Model to the Photochemical Degradation of Polymers

The photochemical degradation rates of two solid-state polymer systems, a cross-linked poly(vinyl chloride) and a segmented polyurethane, that contain metal−metal bonds in the backbone were found to exhibit biphasic behavior. The polymers initially degraded quickly but then degraded slower at longer irradiation times. To investigate the origin of this biphasic rate phenomenon, the data were fit to four kinetic models: first- and second-order rate equations, a simple-diffusion model, and a diffusion-controlled model. None of the models adequately described the data. A new kinetic model was developed to describe a biphasic photodegradation rate that was inspired by Perrin’s model of fluorescence decay rates. Using this new Perrin-like model, the derived rate equation fit the data very accurately. The resulting fitting parameters obtained as a function of reaction temperature were also consistent with the model. To investigate the versatility of this Perrin-like model, literature data for the irradiation of polyoxymethylene were also evaluated. It was found that this system also exhibited Perrin-like kinetics.

Description of Desorption Kinetics at the Solid/Solution Interface Based on the Statistical Rate Theory

Desorption is one of the popular methods for the design and regeneration of catalysts. For better understanding and modeling of this process, it is important to have models with theoretical basis. In the present work, the statistical rate theory (SRT) approach was used for the description of desorption kinetics at the solid/solution interface. Based on the SRT approach, two rate equations at initial times of desorption have been derived. A comparison between these two rate equations was done based on numerically generated kinetic data points ( t, q) by the SRT equation. On the basis of experimental data, it has been shown that the kinetics of desorption can be analyzed by the SRT rate equation. Also, the experimental data approve the accuracy of derived rate equations at initial times of desorption.

Statistical Rate Theory Approach to Kinetics of Competitive Adsorption at the Solid/Solution Interface

Removal of pollutants from aqueous solution is an important process, and adsorption is one of the most popular methods for this. Wastewater usually contains several solutes, and therefore the adsorption process in some systems is competitive. The lack of an adequate model for kinetics of competitive adsorption and also difficulties in modeling led only to the reporting of experimental data in most publications. Here we propose the first description of the competitive adsorption kinetics at the solid/solution interface based on the statistical rate theory (SRT) approach. For derivation of rate equations based on the SRT approach, at first we derived the chemical potential of adspecies for a competitive adsorption based on statistical thermodynamics from the partition function of a canonical ensemble. The derived kinetic equations for competitive adsorption are able to describe quantitatively the kinetics of some experimental data. Since the analytical solution of the derived rate equations led to complex expressions, we used stochastic numerical simulation for modeling of experimental data by the derived SRT equations. The present study shows that the stochastic numerical simulation is a powerful technique for modeling adsorption kinetics based on statistical rate theory.

Theory of ultrafast nonequilibrium dynamics ind-wave superconductors

We use density-matrix theory to calculate the ultrafast dynamics of unconventional superconductors from a microscopic viewpoint. We calculate the time evolution of the optical conductivity as well as pump-probe spectra for a $d$-wave order parameter. Three regimes can be distinguished in the spectra. The Drude response at low photon energies is the only one of those which has been experimentally measured so far. At higher energies, we predict two more regimes: the pair-breaking peak, which is reduced as Cooper pairs are broken up by the exciting pulse, and a suppression above the pair-breaking peak due to nonequilibrium quasiparticles. Furthermore, we consider the influence of the electron-phonon coupling and derive rate equations that have been widely used so far.

Qubit Measurements with a Double-Dot Detector

We propose to monitor a qubit with a double-dot (DD) resonant-tunneling detector, which can operate at higher temperatures than a single-dot detector. In order to assess the effectiveness of this device, we derive rate equations for the density matrix of the entire system. We show that the signal-to-noise ratio can be greatly improved by a proper choice of the parameters and location of the detector. We demonstrate that quantum interference effects within the DD detector play an important role in the measurement. Surprisingly, these effects produce a systematic measurement error, even when the entire system is in a stationary state.

Diffusion Dynamics of Classical Systems Driven by an Oscillatory Force

We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small time-oscillating perturbation. Additionally, the equation involves an interaction operator which projects the distribution function onto functions of the fixed Hamiltonian. The paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. Here, the homogenization procedure leads to a diffusion equation in the energy variable. The presence of the interaction operator regularizes the limit process and leads to finite diffusion coefficients.

Phonon-mediated negative differential conductance in molecular quantum dots

Transport through a single-molecular conductor is considered, showing negative differential conductance behavior associated with phonon-mediated electron tunneling processes. This theoretical work is motivated by a recent experiment by Leroy et al. using a carbon nanotube contacted by a scanning tunneling microscope tip [Nature 432, 371 (2004)], where negative differential conductance of the breathing-mode phonon side peaks could be observed. A peculiarity of this system is that the tunneling couplings which inject electrons and those which collect them on the substrate are highly asymmetrical. A quantum dot model is used, coupling a single electronic level to a local phonon, forming polaron levels. A ``half-shuttle'' mechanism is also introduced. A quantum kinetic formulation allows us to derive rate equations. Assuming asymmetric tunneling rates and in the absence of the half-shuttle coupling, negative differential conductance (NDC) is obtained for a wide range of parameters. A detailed explanation of this phenomenon is provided, showing that NDC is maximal for intermediate electron-phonon coupling. In addition, in the absence of a gate, the ``floating'' level results in two distinct lengths for the current plateaus, related to the capacitive couplings at the two junctions. It is shown that the half-shuttle mechanism tends to reinforce the negative differential regions, but it cannot trigger this behavior on its own.

Kinetics of Contact Electrification between Metals and Polymers

Kinetics of charge transfer between metals and polymers was studied using an analytical rolling-sphere tool. The rates of charge transfer were related to the area of contact between contacting surfaces and the tunneling current between them. The derived rate equations accounted for the experimentally observed sigmoidal charging curves. Furthermore, for a model system of steel spheres rolling on modified polystyrene supports, it was shown that the magnitudes of separated charges can be varied by adjusting the polymer's surface properties and/or ambient conditions.

Theory for transport through a single magnetic molecule: EndohedralN@C60

We consider transport through a single N@C60 molecule, weakly coupled to metallic leads. Employing a density-matrix formalism we derive rate equations for the occupation probabilities of many-particle states of the molecule. We calculate the current-voltage characteristics and the differential conductance for N@C60 in a break junction. Our results reveal Coulomb-blockade behavior as well as a fine structure of the Coulomb-blockade peaks due to the exchange coupling of the C60 spin to the spin of the encapsulated nitrogen atom.

Quantum condensation of liquid 4He

We present experimental results on the condensation of 4He atoms on the surfaceof liquid 4He.We show that there is quantum condensation with the creation of phonons and R+rotons inone-to-one processes as atoms go into the Bose–Einstein condensate. These phonon and R+rotonsignals can be recognized by their time of flight, their angular distribution andtheir dependence on the ambient temperature. Condensing atoms also createripplons with a high probability. We derive rate equations for the growth of theripplon density and the ripplon decay by the creation of bulk phonons. During theatom pulse a dynamic equilibrium is established. The low-energy phonons that arecreated by ripplon–ripplon scattering can easily be detected and distinguishedfrom the phonons created by quantum condensation.From measured data,estimates are made of the probabilities of creating phonons, R+rotonsand ripplons in condensation processes as well as estimates of the quantumevaporation probabilities of phonons and rotons.We clearly detect R+rotonsdirectly with a bolometer in the liquid 4He and show why the signalof R+rotonsis obscured by the low-energy phonon signal when both the source and detectorare in the liquid helium. Values of the Kapitza conductance for the Zn bolometersare derived.

Kinetic determination of tight-binding impurities in enzyme inhibitors

A novel rate equation to characterize the dose-response behavior of a moderately potent (“classical”) enzyme inhibitor contaminated with a very potent (“tight-binding”) impurity is derived. Mathematical properties of the new rate equation show that, for such contaminated materials, experimentally observed I 50 values are ambiguous. The four-parameter logistic equation, conventionally used to determine I 50 values, cannot be used to detect the presence of tight-binding impurities in inhibitor samples. In contrast, fitting the newly derived rate equation to inhibitor dose- response curves can, in favorable cases, reveal whether the unknown material is chemically homogeneous or whether it is contaminated with a tight-binding impurity. The limitations of our method with respect to the detectable range of inhibition constants (both classical and tight-binding) were examined by using Monte-Carlo simulations. To test the new analytical procedure experimentally, we added a small amount (0.02 mole%) of a tight-binding impurity ( K i =0.065 nM) to an otherwise weak inhibitor of human mast-cell tryptase ( K i=50.4 μ M). The resulting material was treated as “unknown.” Our kinetic equation predicts that such adulterated material should show I 50=0.40 μ M, which was identical to the experimentally observed value. The best-fit value of the apparent inhibition constants for the tight-binding inhibitor was K i =(0.107±0.035) nM, close to the true value of 0.065 nM.