Showing papers on "Rate equation published in 2010"

Kinetics of the Hydrogen Electrode Reaction

Abstract: It is well recognized that the standard Butler–Volmer equation is lacking in an adequate description of the kinetics of the hydrogen electrode reaction over the complete range of potentials for the alkaline as well as the acid electrolytes. Further, it is unable to explain the asymmetry in current vs potential observed in the hydrogen evolution reaction HER vs the hydrogen oxidation reaction HOR. In fact, even kinetic descriptions via two-step mechanisms Volmer–Heyrovsky, Volmer–Tafel, or Heyrovsky– Tafel are individually applicable only in limited potential ranges. We present an approach that provides explicit rate expressions involving kinetics of all the three steps Tafel–Volmer–Heyrovsky simultaneously, as well as more limiting rate expressions based on two-step pathways. The analysis is based on our recently developed graph–theoretic approach that provides accurate rate laws by exploiting the electrical analogy of the reaction network. The accuracy of the resulting rate expressions, as well as their asymmetric potential dependence, for both HOR and HER is illustrated here based on step kinetics provided in the literature for Pt catalyst in 0.5 M NaOH solution.

An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions

Abstract: Chemical master equations provide a mathematical description of stochastic reaction kinetics in well-mixed conditions. They are a valid description over length scales that are larger than the reactive mean free path and thus describe kinetics in compartments of mesoscopic and macroscopic dimensions. The trajectories of the stochastic chemical processes described by the master equation can be ensemble-averaged to obtain the average number density of chemical species, i.e., the true concentration, at any spatial scale of interest. For macroscopic volumes, the true concentration is very well approximated by the solution of the corresponding deterministic and macroscopic rate equations, i.e., the macroscopic concentration. However, this equivalence breaks down for mesoscopic volumes. These deviations are particularly significant for open systems and cannot be calculated via the Fokker-Planck or linear-noise approximations of the master equation. We utilize the system-size expansion including terms of the order of Omega(-1/2) to derive a set of differential equations whose solution approximates the true concentration as given by the master equation. These equations are valid in any open or closed chemical reaction network and at both the mesoscopic and macroscopic scales. In the limit of large volumes, the effective mesoscopic rate equations become precisely equal to the conventional macroscopic rate equations. We compare the three formalisms of effective mesoscopic rate equations, conventional rate equations, and chemical master equations by applying them to several biochemical reaction systems (homodimeric and heterodimeric protein-protein interactions, series of sequential enzyme reactions, and positive feedback loops) in nonequilibrium steady-state conditions. In all cases, we find that the effective mesoscopic rate equations can predict very well the true concentration of a chemical species. This provides a useful method by which one can quickly determine the regions of parameter space in which there are maximum differences between the solutions of the master equation and the corresponding rate equations. We show that these differences depend sensitively on the Fano factors and on the inherent structure and topology of the chemical network. The theory of effective mesoscopic rate equations generalizes the conventional rate equations of physical chemistry to describe kinetics in systems of mesoscopic size such as biological cells.

Modular rate laws for enzymatic reactions

Abstract: Motivation: Standard rate laws are a key requisite for systematically turning metabolic networks into kinetic models. They should provide simple, general and biochemically plausible formulae for reaction velocities and reaction elasticities. At the same time, they need to respect thermodynamic relations between the kinetic constants and the metabolic fluxes and concentrations. Results: We present a family of reversible rate laws for reactions with arbitrary stoichiometries and various types of regulation, including mass–action, Michaelis–Menten and uni–uni reversible Hill kinetics as special cases. With a thermodynamically safe parameterization of these rate laws, parameter sets obtained by model fitting, sampling or optimization are guaranteed to lead to consistent chemical equilibrium states. A reformulation using saturation values yields simple formulae for rates and elasticities, which can be easily adjusted to the given stationary flux distributions. Furthermore, this formulation highlights the role of chemical potential differences as thermodynamic driving forces. We compare the modular rate laws to the thermodynamic–kinetic modelling formalism and discuss a simplified rate law in which the reaction rate directly depends on the reaction affinity. For automatic handling of modular rate laws, we propose a standard syntax and semantic annotations for the Systems Biology Markup Language. Availability: An online tool for inserting the rate laws into SBML models is freely available at www.semanticsbml.org Contact: wolfram.liebermeister@biologie.hu-berlin.de Supplementary Information:Supplementary data are available at Bioinformatics online.

Kinetics of the boron-oxygen related defect in theory and experiment

Abstract: The formation of boron-oxygen complexes in boron-doped crystalline silicon can lead to a severe reduction in the minority charge carrier lifetime. This strongly influences, e.g., solar cell efficiencies if the material is used for photovoltaic application. Recent investigations have shown that a recovery of the carrier lifetime can be achieved by a subsequent thermally enhanced reaction induced by charge carriers. A model of the reaction dynamics of the boron-oxygen complex by means of rate equations is presented in this paper. Following a mathematical description of the reactions involved, the consequences based on the calculations are presented and allow a prediction of the observable electrical parameters. The fundamental agreement with measured data is proven experimentally for different phenomena.

Steepest descent reaction path integration using a first-order predictor-corrector method.

Abstract: The theoretical treatment of chemical reactions inevitably includes the integration of reaction pathways. After reactant, transition structure, and product stationary points on the potential energy surface are located, steepest descent reaction path following provides a means for verifying reaction mechanisms. Accurately integrated paths are also needed when evaluating reaction rates using variational transition state theory or reaction path Hamiltonian models. In this work an Euler-based predictor–corrector integrator is presented and tested using one analytic model surface and five chemical reactions. The use of Hessian updating, as a means for reducing the overall computational cost of the reaction path calculation, is also discussed.

Interaction of dielectrics with femtosecond laser pulses: application of kinetic approach and multiple rate equation

Abstract: We calculate the transient free-electron density in laser-irradiated dielectrics with two different approaches, both considering the energy distribution of excited electrons. The kinetic approach solves a system of complete Boltzmann collision integrals describing different excitation and relaxation processes in detail. The multiple rate equation (MRE) is an approximative way to keep track of the energy distribution of excited electrons with reduced numerical effort. Both methods are applied to trace dielectric breakdown, considering the changing optical parameters during irradiation with a high-intensity laser pulse. In the MRE approach we include also fast recombination, leading to a delay of the increase of the electronic density and to a decrease of the maximum number of free electrons.

Anomalous mixing and reaction induced by superdiffusive nonlocal transport.

Abstract: Spatially nonlocal transport describes the evolution of solute concentration due to mass transfer over long ranges. Such long-range mass transfer, present in many flow situations, changes the character of mixing and consequent chemical reactions. We study mixing in terms of the scalar dissipation and reaction rates for mixing-limited equilibrium reactions, using the space-fractional advection-dispersion equation (fADE) to model long range mass transfer. The scalar dissipation and global reaction rates decay as power-laws at late time. As opposed to the Fickian (local) transport model, local reaction rates are not zero where the concentration has zero gradient. As α , the fractional derivative exponent, decreases from two in the fADE, the reaction rate grows larger at the position of zero gradient, due to long-range transfer of reactants from distances larger than Fick's law allows. The reaction rates are also greater far from the reactant source for non-Fickian transport; however, the globally integrated reaction rate decreases with smaller α . This behavior may provide a method to investigate spatial nonlocality as a proper model of upscaling: the reaction products would be found in places precluded by Fickian dispersion, and overall reaction rates are suppressed.

Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation.

Abstract: The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions t...

A novel approach for negative ion analysis using 160?GHz microwave interferometry and laser photodetachment in oxygen cc-rf plasmas

Abstract: Microwave interferometry at 160.28 GHz with Gaussian beam propagation (beam waist: 5 mm) and laser photodetachment were combined for the analysis of negative atomic oxygen ions in the bulk plasma of an asymmetric capacitively coupled 13.56 MHz discharge (cc-rf). The line-integrated negative oxygen ion density amounts to between 2.5 × 1014 and 1015 m−2 depending on the oxygen pressure and rf power. Furthermore, the measured decay of the detachment signal reveals two modes of rf oxygen plasma characterized by different electronegativities. High electronegativity, α > 2, is associated with a low decay time constant of only a few microseconds, whereas in oxygen plasmas with low electronegativity, α < 1, the relaxation of electron density needs much longer with typical decay time constants of up to about 100 µs. The transition between the two modes shows a step-like characteristic and was observed at a specific rf power depending on the oxygen pressure. In the case of high electronegativity the electron density relaxation can be described by a simple 0D-attachment–detachment model, taking into consideration a constant density for positive ions and neutral oxygen species. Using the appropriate rate coefficients from the literature and the experimentally determined effective rate coefficients of first order kinetics, the evaluation of the attachment and detachment rates indicates the significant role of O2(a 1Δg) in the formation and loss of negative atomic oxygen ions.

Mathematical modeling of CO2 removal using carbonation with CaO: The grain model

Abstract: CaO carbonation with CO2 is potentially a very important reaction for CO2 removal from exhaust gas produced in power plants and other metallurgical plants and for hydrogen production by promoting water gas shift reaction in fossil fuel gasification. A mathematical model based on the grain model was applied for modeling of this reaction. Diffusion of gaseous phase through the product layer and structural change of the grains were considered in the model. The modeling results show that ignoring the reaction kinetics controlling regime in the early stage of the reaction and replacing it with a regime considering both the reaction kinetics and diffusion can generate good simulation results. The frequency factor of the reaction rate equation and the diffusivity of CO2 through the CaCO3 layer were justified to get the best fit at different temperature range from 400 to 750 °C with respect to experimental data in the literature. The mathematical model switches to a pure diffusion controlling regime at final stage of reaction.

Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models

Abstract: Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker–Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis–Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.

Determination of the Low-Temperature Water-Gas Shift Reaction Kinetics Using a Cu-Based Catalyst

Abstract: An integral packed-bed reactor was used to determine the kinetics of the water−gas shift (WGS) reaction over a CuO/ZnO/Al2O3 catalyst, under operating conditions such that there was no film or intraparticle resistance. Experiments were carried out over a wide range of temperatures and space times using a typical reformate gas mixture (4.70% CO, 34.78% H2O, 28.70% H2, 10.16% CO2, balance N2). In the first part of the work, three different mechanistic-rate equations and two empirical kinetic models are proposed to describe the WGS kinetic data throughout the entire range of temperatures. To improve the independence of the parameters in using the Arrhenius and van’t Hoff equations, the temperature was centered. Good agreement was obtained between the Langmuir−Hinshelwood (LH) rate equations and the experimental results. Further, analysis using two different temperature ranges for parameter estimation revealed distinct rate-controlling mechanisms for each range. For temperatures of 180−200 °C, the associative...

The Effect of Temperature on the Enzyme-Catalyzed Reaction: Insights from Thermodynamics

Abstract: When teaching the effect of temperature on biochemical reactions, the problem is usually oversimplified by confining the thermal effect to the catalytic constant, which is identified with the rate constant of the elementary limiting step. Therefore, only positive values for activation energies and values greater than 1 for temperature coefficients (Q10, defined as the rate change that parallels a 10 °C temperature shift) can be expected. We show why this approach, which is appropriate under saturation conditions, is unsuitable for those enzymes working at subsaturating concentrations of their substrates. To overcome this limitation, we present and discuss a simple thermokinetic treatment of the Michaelis−Menten model for the enzyme-catalyzed reaction.

Kinetic and thermodynamic aspects of enzyme control and regulation.

Abstract: This paper develops concepts for assessing and quantifying the regulation of the rate of an enzyme-catalyzed reaction. We show how generic reversible rate equations can be recast in two ways, one making the distance from equilibrium explicit, thereby allowing the distinction between kinetic and thermodynamic control of reaction rate, as well as near-equilibrium and far-from-equilibrium reactions. Recasting in the second form separates mass action from rate capacity and quantifies the degree to which intrinsic mass action contributes to reaction rate and how regulation of an enzyme-catalyzed reaction either enhances or counteracts this mass-action behavior. The contribution of enzyme binding to regulation is analyzed in detail for a number of enzyme-kinetic rate laws, including cooperative reactions.

Hydroformylation of 1-Octene Using [Bmim][PF6]−Decane Biphasic Media and Rhodium Complex Catalyst: Thermodynamic Properties and Kinetic Study

Abstract: A chemical reaction engineering approach is reported to investigate the biphasic hydroformylation of 1-octene using [bmim][PF6] ionic liquid It is based both on a process parameter investigation (temperature, concentrations, and pressures) and a thermodynamic study of the reaction medium (gas−liquid and liquid−liquid equilibria) Initial rate data show complex behavior with respect to operating parameters and are best described by a rate equation based on a mechanistic model Complete reaction scheme including isomerization is then modeled accounting from the time dependent concentration of the organic substrates measured in organic phase and recalculated in ionic liquid phase from liquid−liquid equilibria

Calculations of Internal Oxidation Rate Equations and Boundary Conditions between Internal and External Oxidation in Silicon Containing Steels

Abstract: mechanism of fayalite scale, which can form as a ‘‘sub-scale’’ in Si containing steels. The diffusion coefficient of oxygen in the oxide layer, DO, and the oxygen concentration at specimen surface, NO (s) , which are constituents of the internal oxidation rate constant, (2DONO (s) =NB (O) n), were calculated under various oxidation conditions, and the rate equation for the internal oxide layer was derived. Comparing the calculated and the measured values of (2DONO (s) =NB (O) n), we confirmed that the rate equation determined for the internal oxide layer was reasonable. The conditions at the boundary between internal to external oxidation of Si containing steels (Fe-Si alloys) at 850 � C were also calculated by substituting the calculated values of DO and NO (s) at the boundary into the rate equation. [doi:10.2320/matertrans.M2009256]

Rate Equation Analysis of High Speed Q-Modulated Semiconductor Laser

Abstract: Dynamic response of a Q-modulated semiconductor laser is simulated based on a rate equation model. Numerical results from both small-signal and large-signal analyses show that the Q-modulation has much higher bandwidth limit and smaller wavelength chirp than the direct modulation. It is shown that a high-extinction-ratio, low-chirp modulation of 40 GHz RZ signal can be achieved and the effects of various parameters on the Q-modulation are discussed. In addition to high-speed communications, the Q-modulated laser is particularly suitable for microwave carrier generation in radio-over-fiber systems.

Kinetic modeling of esterification of cardanol‐based epoxy resin in the presence of triphenylphosphine for producing vinyl ester resin: Mechanistic rate equation

Abstract: In this study, cardanol-based epoxidized novolac resins and methacrylic acid were used to produce cardanol-based epoxidised novolac vinyl ester resins. The reactions were conducted under nonstoichiometric condition using triphenylphosphine as catalyst in the temperature range of 80–100°C with an interval of 5°C. The first-order rate equation and mechanism based rate equation were examined. Parameters were evaluated by least square method. A comparison of mechnism based rate equation and experimental data showed an excellent agreement. Finally, Arrhenius equation and activation energy were presented. The specific rate constants, based on linear regression analysis, were found to obey Arrhenius equation. The values of activation energy, frequency factor, enthalpy, entropy, and free energy of the reaction revealed that the reaction was spontaneous and irreversible and produced a highly activated complex. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2010

A Microkinetic Vision on High-Throughput Catalyst Formulation and Optimization: Development of an Appropriate Software Tool

Abstract: An advanced (micro) kinetic modeling tool is presented. It can be used in the assessment of chemical kinetics going from power law models to full microkinetic models in terms of elementary steps. Reactants and products are considered to be present in a single, ideal aggregation state. Rate equations are automatically generated from the reaction network as specified by the user of the engine. Combined stochastic and deterministic algorithms are used for the optimization procedure. The flexibility of the code in its integration with different graphical user-friendly interfaces is illustrated. Thus, researchers with little programming skills can both implement advanced micro-kinetic models and perform their assessment. O-xylene hydrogenation data are used for illustration purposes.

Finite-Difference Time-Domain Analysis of Laser Action in Cholesteric Photonic Liquid Crystal

Abstract: We have numerically investigated lasing dynamics in cholesteric liquid crystal (CLC) with gain by an auxiliary differential equation finite-difference time-domain (ADE-FDTD) method in which the FDTD method is coupled with a rate equation in a four-level energy structure. Circularly polarized lasing was achieved at the photonic band edge above threshold pumping. Our model opens a way for a computational design of the CLC laser on the basis of numerical simulation to realize a more efficient device architecture for a greatly reduced lasing threshold, which is still under extensive study.

On the applicability of a few level rate equation model to the determination of exciton versus biexciton kinetics in quasi-zero-dimensional structures

Abstract: Hereby, we present a few level rate equation model in a context of the interpretation of excitation power dependent exciton and biexciton emission intensity from single quantum-dot-like structures. We emphasize that it not only allows identifying the excitonic and biexcitonic emission from one quasi-zero-dimensional object, but gives also an insight into the kinetics of the carriers confined in the system (both the internal dynamics of the exciton within its fine structure and the relative exciton to biexciton lifetimes ratio), the regime of the confinement itself and the importance of the higher energy levels occupation. Eventually, there are presented and discussed examples of the rate equation model application for an analysis of the experimental data for several kinds of epitaxial nanostructures.

Performance Limits for a Class of Irreversible Internal Combustion Engines

Abstract: The performance of a class of irreversible internal combustion engines with finite rate heat exchange with the environment and nonzero entropy generation due to combustion chemical reactions in the cylinder is studied in this paper by using finite time thermodynamics. It is assumed that the heat transfer between the working fluid in the cylinder and the environment obey linear phenomenological law [q ∞ Δ(T 1 )] in the irreversible thermodynamics, and the combustion chemical reactions in the cylinder obey a general rate equation of reactions. The upper bounds ofpower output and efficiency of the internal combustion engines are derived by applying optimal control theory. For the special examples with one chemical reaction and some linearly independent chemical reactions, the average optimal control problems are transformed into nonlinear programming problems, and the Kuhn-Tucker conditions corresponding to the optimal solutions are found. Analytical solutions of minimum entropy generation for the two cases are provided. The results obtained herein are compared with those obtained with different rate equation of reactions and Newton's heat transfer law ([q ∝ Δ(T)]). The methods used and the results obtained in this paper can provide some theoretical guidelines for the optimal design and operation of practical internal combustion engines.

Spin dynamics of excited trion states in a single InAs quantum dot

Abstract: We measure photon cross-correlation between two positively charged biexcitonic emission lines and a positively charged excitonic line for a single InAs/GaAs quantum dot (QD). Marked difference in the correlation function is observed, which originates from multiple spin configurations of intermediate states, i.e., excited positively charged exciton (trion) states. With the aid of a rate equation simulation, we evaluate the transition rate with $p$-shell hole-spin flip to be 0.8--1.0 GHz, which is almost comparable to the radiative decay rate of the ground charged exciton, presumably due to the spin scattering between carriers in a QD and a wetting layer and/or mixing of the excited trion states. In contrast, the relaxation rate with conserving $p$-shell hole-spin projection is estimated to be about one order of magnitude higher than that with hole-spin flip.

Two Optimization Methods to Determine the Rate Constants of a Complex Chemical Reaction Using FORTRAN and MATLAB

Abstract: Problem statement: For chemical reactions, the determination of the rate constants is both very difficult and a time consuming process. The aim of this research was to develop computer programs for determining the rate constants for the general form of any complex reaction at a certain temperature. The development of such program can be very helpful in the control of industrial processes as well as in the study of the reaction mechanisms. Determination of the accurate values of the rate constants would help in establishing the optimum conditions of reactor design including pressure, temperature and other parameters of the chemical reaction. Approach: From the experimental concentration-time data, initial values of rate constants were calculated. Experimental data encountered several types of errors, including temperature variation, impurities in the reactants and human errors. Simulations of a second order consecutive irreversible chemical reaction of the saponification of diethyl ester were presented as an example of the complex reactions. The rate equations (system of simultaneous differential equations) of the reaction were solved to get the analytical concentration versus time profiles. The simulation results were compared with experimental results at each measured point. All deviations between experimental and calculated values were squared and summed up to form a new function. This function was fed into a minimizer routine that gave the optimal rate constants. Two optimization techniques were developed using FORTRAN and MATLAB for accurately determining the rate constants of the reaction at certain temperature from the experimental data. Results: Results showed that the two proposed programs were very efficient, fast and accurate tools to determine the true rate constants of the reaction with less 1% error. The use of the MATLAB embedded subroutines for simultaneously solving the differential equations and minimization of the error function was very fast in solving such problems, as compared to the FORTRAN program, which, although resulting in fast and accurate results, yet, requiring the use of a library of external subroutines. Conclusion: Any of the two proposed methodologies could be used to determine the rate constants of any complex reaction at a certain temperature. The proposed programs were independent of the nature of the reaction, only the rate equations and the initial conditions had to be modified for any new reaction.