Derivation Of Rate Equations Research Articles

Catalytic kinetics in nanoconfined space of acidic micro/mesoporous materials

Kinetics of heterogeneous acid catalyzed reactions is considered for micro-mesoporous materials taking into account nonuniformity of active sites, located in different pores. The methodology for deriving rate equations is developed accounting for reactions in single site nanopores, in macropores/external surfaces of crystals, in narrow nanopores with Gemini sites, as well as in more wider nanoconfined pores with limited pore occupancy. Contribution of such nanoconfined pores was considered using the acid site density as a kinetic descriptor for activity and selectivity. The rate equation wasderived for a mechanism with two kinetically significant steps,assuming thatthethe intermediate sizepores accommodateonly two molecules.The kinetic treatment accounts for different reactivity of acid sites and lateral interactions.Modification of zeolites to form mesopores was discussed from the kinetic viewpoint highlighting how such treatment influences activity and selectivity.Finally, the reactant and product shape selectivity is discussed in a quantitative way.

Pre‐equilibrium Approximation for Analysis of Chemical Kinetics in Chromatography: Application to Reversed‐Phase On‐column Kinetics of Decomposition of Murexide

AbstractA kinetic process can sometimes be observed during a chromatographic separation. The chemical reaction is coupled with the rapid distribution of the reacting compounds between the mobile and stationary phases. The use of the pre‐equilibrium approximation enables the derivation of the rate law for the chemical reaction(s) taking place in the mobile and/or stationary phase. A simple procedure is proposed for writing down this rate equation. It can be applied to any on‐column reaction, whether reversible or irreversible. For first‐ or pseudo‐first‐order kinetics, integration of the rate equation gives an exponential function of concentration with respect to time. To demonstrate the application of the proposed method, the kinetics of the decomposition of murexide in a reversed‐phase C8 column was studied. Murexide decomposes through a fast reversible protonation process to give purpuric acid, which then decomposes irreversibly. The stopped‐flow HPLC technique was used to obtain the entire kinetic profile for the on‐column reaction. Additionally, the kinetics was measured off‐column using the same mobile phase. The values of the kinetic parameters obtained from the batch experiments enabled the determination of all kinetic parameters of the on‐column reaction using non‐linear least squares fitting of the derived rate equation.

Reaction mechanism and intrinsic kinetics of sugar hydrogenation to sugar alcohols on solid foam Ru/C catalysts – From arabinose and galactose to arabitol and galactitol

Solid foam catalysts have two great benefits: thin catalyst layers guaranteeing a high effectiveness factor and a low pressure drop in continuous operation. The ability of ruthenium-based solid foam catalysts in the hydrogenation of monomeric sugars was illustrated with extensive experiments of L-arabinose and D-galactose hydrogenation at 90-120˚C and 20 bar hydrogen pressure. Kinetic experiments were carried out with individual sugars and binary sugar mixtures at different D-galactose-to-L-arabinose molar ratios to reveal the molecular interactions in the presence of the solid foam catalyst. High conversion of sugars and high selectivity to sugar alcohols were achieved in the isothermal and isobaric laboratory-scale reactor which operated under intrinsic kinetic control. The sugar hydrogenation process was considered from a viewpoint of elementary steps on the catalyst surface. By assuming plausible surface reaction mechanisms, it was possible to derive rate equations for the formation of sugar alcohols, both in case of individual sugars and binary sugar mixtures. The kinetic model based on the non-competitive adsorption of sugars and hydrogen on the ruthenium surface gave a very good description of the hydrogenation kinetics and product distribution on the solid foam catalysts. The work opens a perspective to the selective and very effective hydrogenation of several sugars to valuable sugar alcohols in the presence of open foam Ru/C catalysts, both in batch and continuous operation modes.

Dynamic signatures of mode competition in optically injected high-β lasers

In this paper we theoretically investigate the statistical light-emission properties of an optically injected bimodal quantum-dot micropillar laser with high spontaneous emission rates. The nanostructured device is described in terms of a stochastic, semiclassically derived rate equation model. We focus on the stochastic switching dynamics between the two fundamental modes and correlate the results with an in-depth bifurcation analysis of the underlying deterministic dynamics. By analyzing different statistical measures, e.g. average intensity, auto- and cross-correlation functions, as well as dwell-time distributions, we give a road map on how to unravel the different dynamic regimes in the presence of large noise from experimentally accessible quantities.

Emission spectrum of broadband quantum dot superluminescent diodes

We present a microscopic theory of the amplified spontaneous emission of a spectrally broadband quantum dot superluminescent diode with tilted end facets within the quantum white noise limit. From this multimode quantum theory, we have the ability to obtain all orders of temporal correlation functions. In particular, we derive rate equations for the optical power densities, the level occupation of inhomogeneous ensemble of quantum dots within the diode, as well as the emitted optical spectra. As the main result, we find the external power spectrum as a convolution of the intra-diode photon spectrum with a Lorentzian response. Assuming a Gaussian light-matter coupling results in a similar shaped Gaussian output spectrum, which agrees very well with available experimental data.

Method of Moments Applied to Most-Likely High-Temperature Free-Radical Polymerization Reactions

Many widely-used polymers are made via free-radical polymerization. Mathematical models of polymerization reactors have many applications such as reactor design, operation, and intensification. The method of moments has been utilized extensively for many decades to derive rate equations needed to predict polymer bulk properties. In this article, for a comprehensive list consisting of more than 40 different reactions that are most likely to occur in high-temperature free-radical homopolymerization, moment rate equations are derived methodically. Three types of radicals—secondary radicals, tertiary radicals formed through backbiting reactions, and tertiary radicals produced by intermolecular chain transfer to polymer reactions—are accounted for. The former tertiary radicals generate short-chain branches, while the latter ones produce long-chain branches. In addition, two types of dead polymer chains, saturated and unsaturated, are considered. Using a step-by-step approach based on the method of moments, this article guides the reader to determine the contributions of each reaction to the production or consumption of each species as well as to the zeroth, first and second moments of chain-length distributions of live and dead polymer chains, in order to derive the overall rate equation for each species, and to derive the rate equations for the leading moments of different chain-length distributions. The closure problems that arise are addressed by assuming chain-length distribution models. As a case study, β-scission and backbiting rate coefficients of methyl acrylate are estimated using the model, and the model is then applied to batch spontaneous thermal polymerization to predict polymer average molecular weights and monomer conversion. These predictions are compared with experimental measurements.

The photokinetics of phenothiazine dyes with titanium trichloride in different solvents

AbstractThe kinetic parameters of photoinduced electron transfer reaction of two phenothiazine dyes, methylene blue and methylene green with titanium trichloride, were determined in water and different aqueous‐alcoholic solvents at different acidities by using a specially designed optical system. The rate of photoinduced electron transfer reaction was measured by determining the quantum yield of the reaction. The methylene green had a higher reactivity as compared to methylene blue with titanium trichloride. The graphical analysis showed that the reaction of dye with titanium trichloride follows pseudo–first‐order kinetics. A reaction mechanism was proposed by considering the different excited states of dye and their possible interaction with the solvent and titanium trichloride. The different steps in the reaction mechanism were taken into consideration for deriving rate equations, which were used to determine the different rate constants in the reaction mechanism in different solvents.

Generalization of core percolation on complex networks.

We introduce a k-leaf removal algorithm as a generalization of the so-called leaf removal algorithm. In this pruning algorithm, vertices of degree smaller than k, together with their first nearest neighbors and all incident edges, are progressively removed from a random network. As the result of this pruning the network is reduced to a subgraph which we call the Generalized k-core (Gk-core). Performing this pruning for the sequence of natural numbers k, we decompose the network into a hierarchy of progressively nested Gk-cores. We present an analytical framework for description of Gk-core percolation for undirected uncorrelated networks with arbitrary degree distributions (configuration model). To confirm our results, we also derive rate equations for the k-leaf removal algorithm which enable us to obtain the structural characteristics of the Gk-cores in another way. Also we apply our algorithm to a number of real-world networks and perform the Gk-core decomposition for them.

Externally driven molecular ratchets on a periodic potential surface: a rate equations approach.

The long time dynamics of molecular ratchets on a 1D periodic potential energy surface (PES) subjected to an external stimulus is studied using the rate equation method. The PES consisting of repeated waveforms made of two peaks is considered as an example of a spatially symmetric or asymmetric PES. This PES may, for example, correspond to diffusion of a bipedal molecule that moves along an atomic track via an inchworm walk mechanism [Raval et al., Angew. Chem., Int. Ed., 2015, 54, 7101]. Generalisation to a PES consisting of an arbitrary number of peaks of various heights is straightforward. Assuming the validity of the transition state theory (TST) for the calculation of the transition rates between neighbouring potential wells, the probability of occupying each type of potential well on the PES is obtained analytically, and then the net current for the molecules to move preferentially in a particular direction under application of external fields over a long time is derived. Note that different to methods based on solving numerically the corresponding Fokker-Plank equation, our method is entirely analytical in the limit of weak external fields. The results of the analytical calculations are compared with the exact numerical solution of the derived rate equations. The following external stimuli are considered: constant, sinusoidal and shifted sinusoidal fields due to either a spatially uniform thermal gradient or an electrostatic field. The possible applications of the method for extracting energy from the Brownian motion under load and separating molecules of different chiralities on the surface are also discussed.

Threshold decrease in intracavity optical parametric oscillator synchronously pumped by mode-locked laser

An intracavity optical parametric oscillator (IOPO) pumped by a continuous-wave (CW) mode-locked laser is experimentally realized. The fundamental cavity and optical parametric oscillator cavity are designed to satisfy synchronous pumping. The output characteristics of signal and idle light are measured. Because of higher fundamental photon intensity in IOPO, the threshold of IOPO is lower than that of an extra-cavity optical parametric oscillator. The spectroscopy of signal light is obtained and the wavelength of idle light can be estimated to be 3.298 μm from noncritical phase matching. Based on the intensity fluctuation mechanism of fundamental locking, the dynamical model for IOPO pumped by a CW mode-locked laser is developed. The simulated results for the temporal shape of three lights are calculated from the derived rate equations. Because of dispersion, the signal pulse width of the theory is smaller than that of the experiment with the same pump energy.

Edge-based epidemic spreading in degree-correlated complex networks

Networks that grow through the addition of new nodes or edges may acquire degree-degree correlations. When one considers a short epidemic on a slowly growing network, such as the spread of a strain of influenza in a population for one season, it is reasonable to assume that the degree-correlated network is static during the course of an epidemic. In this case using only information about the network degree distribution is not enough to capture the exponential growth phase, the epidemic peak or the final epidemic size. Hence, in this paper we formulate an edge-based SIR epidemic model on degree-correlated networks, which includes the Miller model on configuration networks as a special case. The model is relatively low-dimensional; in particular, considering the fact that it captures degree correlations. Moreover, we derive rate equations to compute two node degree correlations in a growing network. Predictions of our model agree well with the corresponding stochastic SIR process on degree-correlated networks, such as the exponential growth phase, the epidemic peak and the final epidemic size. The basic reproduction number R0 and the final epidemic size are theoretically derived, which are equivalent to those based on the percolation theory. However, our model has the advantage that it can trace the dynamic spread of an epidemic on degree-correlated networks. This provides us with more accurate information to predict and control the spread of diseases in growing populations with biased-mixing. Finally, our model is tested on degree-correlated networks with clustering, and it is shown that our model is robust to degree-correlated networks with small clustering.

Validation of a Bitumen Oxidation Rate Model

This paper reports on experiments aimed at validating a model for predicting the rate of oxidation of bitumen, in asphalt mix and chip seal road surfacings, over time periods of practical significance. Experiments were conducted in which bitumen films of varying thickness were oxidized in the field for 20 years and in the laboratory for up to 5.5 years. Bitumen oxidation in the laboratory was carried out at 50°C in air at atmospheric pressure and resulted in an extent of oxidation approximately equivalent to 11.5 years in the field. Oxidation of bitumen in the films was followed for changes in carbonyl, sulfoxide, and hydroxyl group concentrations measured by infrared spectroscopy, weight change, and viscosity. The data were fitted to an earlier derived rate equation based on a model in which the overall reaction was approximated as the sum of a fast and slow reaction. The model had been found to describe oxidation product formation very well over short time frames. The current work has shown that the model also applies over time frames (extents of oxidation) relevant to actual surfacing lives. A key assumption in the model that depletion of bitumen reactants in the slow reaction phase was negligible was also supported over long time frames. Both fast and slow phases of the reaction were found to be dependent on oxygen concentration, as assumed in the model. The oxygen dependence of the rate is discussed as a two-stage reaction mechanism previously described in the literature.

Kinetic simulation of malate-aspartate and citrate-pyruvate shuttles in association with Krebs cycle

In the present work, we have kinetically simulated two mitochondrial shuttles, malate–aspartate shuttle (used for transferring reducing equivalents) and citrate–pyruvate shuttle (used for transferring carbon skeletons). However, the functions of these shuttles are not limited to the points mentioned above, and they can be used in different arrangements to meet different cellular requirements. Both the shuttles are intricately associated with Krebs cycle through the metabolites involved. The study of this system of shuttles and Krebs cycle explores the response of the system in different metabolic environments. Here, we have simulated these subsets individually and then combined them to study the interactions among them and to bring out the dynamics of these pathways in focus. Four antiports and a pyruvate pump were modelled along with the metabolic reactions on both sides of the inner mitochondrial membrane. Michaelis–Menten approach was extended for deriving rate equations of every component of the system. Kinetic simulation was carried out using ordinary differential equation solver in GNU Octave. It was observed that all the components attained steady state, sooner or later, depending on the system conditions. Progress curves and phase plots were plotted to understand the steady state behaviour of the metabolites involved. A comparative analysis between experimental and simulated data show fair agreement thus validating the usefulness and applicability of the model.

Exchange Current Density of SOFC Electrodes: Theoretical Relations and Partial Pressure Dependencies Rate-Determined by Electrochemical Reactions

As a theoretical consideration on electrode defect chemistry, general relations of exchange current density quantitatively representing Solid Oxide Fuel Cell (SOFC) electrode performance are systematically derived as a function of gas partial pressures, equilibrium constants of adsorption and dissociation reactions on electrode surfaces, and electrochemical reaction rate constants for possible elemental reactions at the cathode and the anode, in the case that an electrochemical reaction is the rate-determining electrode reaction. Simplified expressions are also derived, under the condition that one kind of neutral or charged adsorbed species is predominant at the electrode, to derive gas partial pressure dependence of exchange current density for given rate-determining electrochemical reactions. Importance of considering elementary steps is highlighted to derive rate equations and to clarify various dependencies. Partial pressure dependencies of the exchange current density are compiled and discussed by simulating normalized exchange current density values for given partial pressures. The applicability and limitation of the Butler-Volmer type expressions of exchange current density for SOFC electrodes are carefully discussed.

Competitive inhibition reaction mechanisms for the two-step model of protein aggregation

We propose three new reaction mechanisms for competitive inhibition of protein aggregation for the two-step model of protein aggregation. The first mechanism is characterized by the inhibition of native protein, the second is characterized by the inhibition of aggregation-prone protein and the third mechanism is characterized by the mixed inhibition of native and aggregation-prone proteins. Rate equations are derived for these mechanisms, and a method is described for plotting kinetic results to distinguish these three types of inhibitors. The derived rate equations provide a simple way of estimating the inhibition constant of native or aggregation-prone protein inhibitors in protein aggregation. The new approach is used to estimate the inhibition constants of different peptide inhibitors of insulin aggregation.

Systematic approach to thermal leptogenesis

In this work we study thermal leptogenesis using non-equilibrium quantum field theory. Starting from fundamental equations for correlators of the quantum fields we describe the steps necessary to obtain quantum kinetic equations for quasiparticles. These can easily be compared to conventional results and overcome conceptional problems inherent in the canonical approach. Beyond CP-violating decays we include also those scattering processes which are tightly related to the decays in a consistent approximation of fourth order in the Yukawa couplings. It is demonstrated explicitly how the S-matrix elements for the scattering processes in the conventional approach are related to two- and three-loop contributions to the effective action. We derive effective decay and scattering amplitudes taking medium corrections and thermal masses into account. In this context we also investigate CP-violating Higgs decay within the same formalism. From the kinetic equations we derive rate equations for the lepton asymmetry improved in that they include quantum-statistical effects and medium corrections to the quasiparticle properties.

Dehydrogenation of ethylbenzene over Fe–Ce–Rb and Fe–Ce–Cs mixed oxide catalysts

Fe–Ce mixed oxide catalysts promoted by Rb and Cs have been investigated for the dehydrogenation of ethylbenzene. Fe–Ce–Rb and Fe–Ce–Cs were found to be highly active. X-ray diffraction (XRD) analysis confirmed the formation of RbFeO2 and CsFeO2 in the catalyst. X-ray photoelectron spectroscopy (XPS) analysis suggests a surface composition of the catalysts with an atomic ratio of Rb/Fe = 1, Cs/Fe = 1 and O/Fe = 2, meaning the surface is composed of the binary oxides, RbFeO2 and CsFeO2. We conclude that the active phases of these catalysts are RbFeO2 and CsFeO2, similar to KFeO2 in the case of K–Fe and Fe–Ce–K mixed oxide catalysts. The rate equation of the reaction was determined supposing a Langmuir–Hinshelwood mechanism, in which the competitive adsorption of ethylbenzene and styrene dominate the reaction rate. However, the relative adsorption constant of styrene, Z ST, did not change significantly with the addition of Rb or Cs. The rate constant k increased with the addition of Rb or Cs. The activation energy increased with the addition of Rb or Cs compared with Fe–Ce, and the frequency factor also increased remarkably. Because the strong base points are most likely the active sites, addition of Rb and Cs increased the number of basic sites on the catalyst surface, resulting in higher activity. The kinetic factors, k and Z ST, were applied for the rate equation. The experimental results were in very good agreement with the derived rate equation.

Criticality in Transport through the Quantum Ising Chain

We consider thermal transport between two reservoirs coupled by a quantum Ising chain as a model for nonequilibrium physics induced in quantum-critical many-body systems. By deriving rate equations based on exact expressions for the quasiparticle pairs generated during the transport, we observe signatures of the underlying quantum phase transition in the steady-state energy current already at finite and different reservoir temperatures.

Modeling Inhomogeneous DNA Replication Kinetics

In eukaryotic organisms, DNA replication is initiated at a series of chromosomal locations called origins, where replication forks are assembled proceeding bidirectionally to replicate the genome. The distribution and firing rate of these origins, in conjunction with the velocity at which forks progress, dictate the program of the replication process. Previous attempts at modeling DNA replication in eukaryotes have focused on cases where the firing rate and the velocity of replication forks are homogeneous, or uniform, across the genome. However, it is now known that there are large variations in origin activity along the genome and variations in fork velocities can also take place. Here, we generalize previous approaches to modeling replication, to allow for arbitrary spatial variation of initiation rates and fork velocities. We derive rate equations for left- and right-moving forks and for replication probability over time that can be solved numerically to obtain the mean-field replication program. This method accurately reproduces the results of DNA replication simulation. We also successfully adapted our approach to the inverse problem of fitting measurements of DNA replication performed on single DNA molecules. Since such measurements are performed on specified portion of the genome, the examined DNA molecules may be replicated by forks that originate either within the studied molecule or outside of it. This problem was solved by using an effective flux of incoming replication forks at the model boundaries to represent the origin activity outside the studied region. Using this approach, we show that reliable inferences can be made about the replication of specific portions of the genome even if the amount of data that can be obtained from single-molecule experiments is generally limited.

The dissolution of chalcopyrite in chloride solutions. IV. The kinetics of the auto-oxidation of copper(I)

The use of chloride in the heap leaching of sulfide minerals requires the regeneration of the oxidants copper(II) and iron(III). While this is possible in a sulfate system by the use of bacterially catalysed oxidation of iron(II), this can only economically be achieved by chemical reaction with dissolved oxygen in chloride systems due to the sensitivity of bacteria to high concentrations of chloride ions. The kinetics of the reduction of dissolved oxygen by copper(I) in acidic chloride solutions has been studied under possible heap leach conditions. The results confirm some published data that the rate is second-order in copper(I). The rate increases with increasing acid concentrations but decreases with increasing chloride and copper(II) concentrations. Peroxide has been shown to be a detectable intermediate and a modified mechanism has been proposed which is consistent with the kinetic data. The use of the derived rate equation in a modified rate expression for the copper-catalysed auto-oxidation of iron(II) has been derived.