First-order Reaction Systems Research Articles

Macroscopic limit for stochastic chemical reactions involving diffusion and spatial heterogeneity

To model bio-chemical reaction systems with diffusion one can either use stochastic, microscopic reaction–diffusion master equations or deterministic, macroscopic reaction–diffusion system. The connection between these two models is not only theoretically important but also plays an essential role in applications. This paper considers the macroscopic limits of the chemical reaction–diffusion master equation for first-order chemical reaction systems in highly heterogeneous environments. More precisely, the diffusion coefficients as well as the reaction rates are spatially inhomogeneous and the reaction rates may also be discontinuous. By carefully discretizing these heterogeneities within a reaction–diffusion master equation model, we show that in the limit we recover the macroscopic reaction–diffusion system with inhomogeneous diffusion and reaction rates.

SAWI TRANSFORMATION FOR SOLVING A SYSTEM OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS

There are many problems in nature whose solutions are obtained through mathematical concepts. One of the most common mathematical concepts is a mathematical concept that is classified under initial value problems, such as a system of linear ordinary differential equations equipped with initial values. One tool that can solve the initial value problem is the Sawi transformation. This article describes the study of the initial value problem as a system of linear ordinary differential equations and its solution using the Sawi transformation. In addition, as part of applying the resulting theory, 2 (two) case examples are given (a first-order chemical reaction system with three certain chemicals and a mass-spring system with forced motion) to be solved using the Sawi transformation. So that problem solving can be interpreted and easily understood, in the 2 (two) case studies discussed and focused on the concentration of the chemical reactants, simulations were carried out for several different initial values and reaction rate constants. Compared to other methods (Laplace transform), the results obtained from using the Sawi transformation for the cases discussed show that the analytical solutions for the selected initial values have similar solutions.

Study on macroscopic properties and microstructure of thermosetting polyurethane asphalt binder (PAB) based on curing kinetics

Recently, novel thermosetting polyurethane asphalt binder (PAB) has been studied by more researchers. This paper investigated the curing mechanism, microstructure and macroscopic performance of PAB. The mechanical properties, rheological properties, and elastic recovery performance of PAB were evaluated. Fourier transform infrared spectroscopy (FTIR), fluorescence microscopy (FM) and gel permeation chromatography (GPC) were adopted to confirm the chemical reaction and microstructure, curing process was studied combined with kinetics. Results showed PAB was the first-order curing reaction system and slow curing progress was conducive to mixing application, polyurethane particles changed from granular to the reticular structure during curing. The ratio of large molecular weight showed an increasing trend as temperature/time up. Furthermore, multi-stress creep recovery demonstrated that elastic recovery performance of PAB-50 (0.1 kPa) can be improved to 2.2 times, 8.1 times approximately compared with styrene–butadienestyrene block copolymer asphalt binder (SBSMA) and unmodified asphalt binder (MA). The tensile strength of PAB-30 (cured at 4d-130 °C) was 3.1 times that of 2 h (130 °C). Viscosity temperature index (VTS) value gradually increased implied that temperature sensitivity was habited by curing time. Pearson correlation coefficient illustrated that the mechanical property of PAB was positively related to the large molecular weight and conversion rate of the isocyanate group. The non-recoverable creep compliance reduction positively relied on polyurethane content and microstructure.

Steady-state joint distribution for first-order stochastic reaction kinetics.

While the analytical solution for the marginal distribution of a stochastic chemical reaction network has been extensively studied, its joint distribution, i.e., the solution of a high-dimensional chemical master equation, has received much less attention. Here we develop an alternative method of computing the exact joint distributions of a wide class of first-order stochastic reaction systems in steady-state conditions. The effectiveness of our method is validated by applying it to four gene expression models of biological significance, including models with 2A peptides, nascent mRNA, gene regulation, translational bursting, and alternative splicing.

Machine-learning model selection and parameter estimation from kinetic data of complex first-order reaction systems.

Dealing with a system of first-order reactions is a recurrent issue in chemometrics, especially in the analysis of data obtained by spectroscopic methods applied on complex biological systems. We argue that global multiexponential fitting, the still common way to solve such problems, has serious weaknesses compared to contemporary methods of sparse modeling. Combining the advantages of group lasso and elastic net-the statistical methods proven to be very powerful in other areas-we created an optimization problem tunable from very sparse to very dense distribution over a large pre-defined grid of time constants, fitting both simulated and experimental multiwavelength spectroscopic data with high computational efficiency. We found that the optimal values of the tuning hyperparameters can be selected by a machine-learning algorithm based on a Bayesian optimization procedure, utilizing widely used or novel versions of cross-validation. The derived algorithm accurately recovered the true sparse kinetic parameters of an extremely complex simulated model of the bacteriorhodopsin photocycle, as well as the wide peak of hypothetical distributed kinetics in the presence of different noise levels. It also performed well in the analysis of the ultrafast experimental fluorescence kinetics data detected on the coenzyme FAD in a very wide logarithmic time window. We conclude that the primary application of the presented algorithms-implemented in available software-covers a wide area of studies on light-induced physical, chemical, and biological processes carried out with different spectroscopic methods. The demand for this kind of analysis is expected to soar due to the emerging ultrafast multidimensional infrared and electronic spectroscopic techniques that provide very large and complex datasets. In addition, simulations based on our methods could help in designing the technical parameters of future experiments for the verification of particular hypothetical models.

Multilayer network structure enhances the coexistence of competitive species.

The concept of a multiplex network can be used to characterize the dispersal paths and states of different species in a patch habitat system. The multiplex network is one of three types of multilayer networks. In this study, the effect of a multiplex network on the long-term stable coexistence of species is investigated using the concept of metapopulation. Based on the mean field approximation, the stability analysis of a two-species system shows that, compared to the single layer network, the multiplex network is more conducive to the stable coexistence of species when one species has a stronger colonization ability. That is, in such a patch habitat system, if the dispersal paths of the stronger species are different than those of the weaker species, then the larger the heterogeneity of the dispersal network of the stronger species is, the more likely the long-term stable coexistence of species. This result provides a different perspective for understanding the biodiversity in heterogeneous habitats.

Self-organizing dynamical networks able to learn autonomously

We present a model for the time evolution of network architectures based on dynamical systems. We show that the evolution of the existence of a connection in a network can be described as a stochastic non-Markovian telegraphic signal (NMTS). Such signal is formulated in two ways: as an algorithm and as the result of a system of differential equations. The autonomous learning conjecture (Kaluza P. and Mikhailov A. S., Phys. Rev. E, 90 (2014) 030901(R)) is implemented in the proposed dynamics. As a result, we construct self-organizing dynamical systems (networks) able to modify their structures in order to learn prescribed target functionalities. This theory is applied to two systems: the flow processing networks with time-programmed responses, and a system of first-order chemical reactions. In both cases, we show examples of the evolution and a statistical analysis of the obtained functional networks with respect to the model parameters.

A family of kinetic distributions for interpretation of experimental fluctuations in kinetic problems

The computation of confidence intervals frequently leads to arguable results due to lack of rigor when experimental errors are analyzed in kinetic experiments. Particularly, the usual Gaussian approach may not be adequate when the variable of interest is the reactant conversion, as this variable is constrained between very hard limits: 0 and 1. For this reason, the present work focuses on the development of analytical and numerical procedures for more accurate description of experimental errors in first-order reaction systems, which can be eventually extended to more complex reaction processes. Based on the proposed analytical and numerical schemes, new statistical distributions (named here as the kinetic distributions) can be derived to allow for more appropriate representation of conversion fluctuations and the respective statistical quantities, including the confidence intervals, which can be used more advantageously for analyses of kinetic data. In particular, it is shown that conversion errors are heteroscedastic, going through a point of maximum when conversion is allowed to increase from 0 to 1, and that confidence intervals are not symmetrical in respect to the averages, as assumed by Gaussian analyses.

On the ambiguity of the reaction rate constants in multivariate curve resolution for reversible first-order reaction systems

If for a chemical reaction with a known reaction mechanism the concentration profiles are accessible only for certain species, e.g. only for the main product, then often the reaction rate constants cannot uniquely be determined from the concentration data. This is a well-known fact which includes the so-called slow-fast ambiguity.This work combines the question of unique or non-unique reaction rate constants with factor analytic methods of chemometrics. The idea is to reduce the rotational ambiguity of pure component factorizations by considering only those concentration factors which are possible solutions of the kinetic equations for a properly adapted set of reaction rate constants. The resulting set of reaction rate constants corresponds to those solutions of the rate equations which appear as feasible factors in a pure component factorization.The new analysis of the ambiguity of reaction rate constants extends recent research activities on the Area of Feasible Solutions (AFS). The consistency with a given chemical reaction scheme is shown to be a valuable tool in order to reduce the AFS. The new methods are applied to model and experimental data.

Stochastic modeling analysis of sequential first-order degradation reactions and non-Fickian transport in steady state plumes

Stochastic analyses were performed to examine sequential first-order monomolecular reactions at the microscopic scale and both Fickian and non-Fickian plume reactive transport at the macroscopic scale. An analytical solution was derived for the chemical master equation (CME) for a closed system of irreversible first-order monomolecular reactions. Taking a Lagrangian reference frame of particles migrating from a source, analyses show that the relative concentration of each species in the deterministic analytical solution for 1-D steady state plug flow with first-order sequential degradation is mathematically equivalent to the mean of a multinomial distribution of plume particles moving at constant velocity with sequential transformations described by transition probabilities of a discrete state, continuous-time Markov chain. In order to examine the coupling of reaction and transport terms in subdiffusive-reactive transport equations, a closed-form multispecies analytical solution also was derived for steady state advection, dispersion, and sequential first-order reaction. Using a 1-D continuous-time random walk (CTRW) embedded in Markov chains, computationally efficient Monte Carlo simulations of particle movement were performed to more fully examine effects of subdiffusive-reactive transport with an application to steady state, sequentially degrading multispecies plumes at a site in Palm, Bay, FL. The simulation results indicated that non-Fickian steady state plumes can resemble Fickian plumes because linear reactions truncate the waiting time between particle jumps, which removes lower velocity particles from the broad spectrum of velocities in highly heterogeneous media. Results show that fitting of Fickian models to plume concentration data can lead to inaccurate estimates of rate constants because of the wide distribution of travel times in highly heterogeneous media.

A Numerical Model of the Soil Flushing Remediation in Heavy Metal Contaminated Soil

angela.antonucci@uniroma1.it This paper presents a 1D numerical model developed to simulate the EDTA chelation process of metal (lead) applied to a soil flushing remediation action. The model considers the non-stationary conditions, typical of unsaturated soil. Flow and transport equations are solved simultaneously. Metal mobilization is evaluated considering both the chemical aspects of chelating agents and the characteristics of the different soil fraction at which the metal can be bound. A system of first order reactions is implemented and, for each fraction, two different mobilization kinetics (slow and fast) are considered. The model was calibrated and validated using laboratory experimental data. Results confirm that the model can represent an evaluable tool to assess the feasibility of soil flushing application for heavy metals contaminated soil, especially in the case of surface layer contamination. It is useful to optimize the operating parameters (chelating dosage, application mode and treatment thickness) in order to achieve the maximum treatment efficiency while minimizing potential environmental impacts. Contamination of soils by heavy metals is a common problem throughout the world due to intensive use of the land, industrial activities and improper hazardous waste disposal. The remediation of soils is complex since many heavy metals show low mobility as in the case of Lead. Lead pollution may be caused, among others, by the emission of motor vehicles (before the use of unleaded gasoline), recovery operations of batteries or paint manufacturing. The mostly used remediation technologies are soil washing and soil flushing. Less invasive technologies such as phytoextraction (Mancini and Bruno, 2010) have been also investigated. The application of these technologies requires the use of fluids that mobilize the Lead bound to the soil. Solutions with chelating agents such as ethylenediamine tetraacetic acid (EDTA), nitriloacetic acid (NTA), diethylenetriamine pent acetic acid (DTPA) and S,S-ethylene-diaminedisuccinic acid (EDDS) are usefully applied. Some chelators (e.g., EDTA) can however be dangerous, because of their non- biodegradability and persistence in the environment, and could make the soil not reusable after the remediation. Therefore it can be critical to simulate the soil flushing process before proceeding the application of the chelator in order to determine the best and safer conditions and to prevent any negative effects. This paper presents a mathematical model describing the mechanism through which the Lead binds to a synthetic chelating agent (EDTA) as well as the transport of the complex into both saturated and unsaturated soil. This model represents an extension of a previous model (Luciano et al., 2013) implemented for taking into account different kinetics for each fraction of the soil at which the metal can be bound: 1) fraction retained by exchange sites of clay and amorphous materials, 2) organic fraction associated with carbonates, 3) fraction associated with Fe oxides, Al, Mn, 4) fraction associated with organic matter and 5) residual fraction (Zeien and Bruemmer, 1989). The model allows for evaluating the residual concentrations of the chelating agent, chelate-metal complex and metal still adsorbed to the soil. It assumes that there are different independent first-order reactions concurrently taking place. Each reaction is associated with a fraction of the lead adsorbed onto the soil. For each fraction slow and fast kinetics are considered. The model is able to simulate both saturated and unsaturated soil conditions. Few models are

Equilibrium relationships for non-equilibrium chemical dependencies

It is shown that equilibrium constants determine the time-dependent behavior of particular ratios of concentrations for any system of reversible first-order reactions. Indeed, some special ratios actually coincide with the equilibrium constant at any moment in time. This is established for batch reactors, and similar relations hold for steady-state plug-flow reactors, replacing astronomic time by residence time. Such relationships can be termed time invariants of chemical kinetics.

First-order parallel and consecutive reaction mechanisms — Isosbestic points criterium

A criterium for the selection of reaction mechanism derived from a condition for isosbestic points occurrence is presented. Analytical relationships involving the molar absorption coefficients of the species, which participate in a mechanism of parallel first-order reactions and the corresponding rate coefficients, are also reported. A model system of four species that present overlapping absorption spectra may correspond to the reactant and products of a system of parallel or consecutive first-order reactions. In the first case, under experimental conditions in which the absorbances are additive, the presence of an isosbestic point in the spectrum of the reaction mixture at a given wavelength leads to a time-independent ratio of the degree of advancement of reaction variables. From this, relevant kinetic information may be extracted, namely, the ratio of the reaction rate coefficients. Moreover, the occurrence of isosbestic points allows discarding the second mechanism. This conclusion is independent of the number of absorbing species. Model calculated examples show the application of the equations here derived. The resolution for the general case of mechanisms of N first-order reactions is provided.Key words: chemical kinetics, time-resolved absorption spectra, reaction mechanism.

Is a Proposed Reaction Mechanism Free from Unnecessary Assumptions? Occam's Razor Applied in a Mathematical Way To Complex First-Order Reaction Systems

Following Occam's principle, a proposed reaction mechanism should not contain assumptions about the existence of reactive intermediates and reaction paths that are unnecessary for a full description and interpretation of the available facts. A mechanism refers, in this paper, to a proposed reaction scheme or network that represents the reactions supposed to be going on in a complex reaction system with observable species as well as unobservable reactive intermediates. The scope is limited here to (pseudo) first-order reactions and the steady-state approximation is invoked in order to relate unknown mechanistic rate constants to experimentally determined ones, and, when available, theoretically calculated quantities. When the resulting, nonlinear system of equations admits a unique solution within a physically reasonable domain, it is concluded that the reaction mechanism fulfills Occam's principle. Otherwise, there are many or no solutions. No subjective or qualitative arguments enter the procedure and the outcome is not negotiable.

Modelling of Consecutive Reactions with a Semibatch Liquid Phase: Enhanced Kinetic Information by a New Experimental Concept

A semibatch reactor concept was proposed for the determination of the kinetics in complex catalytic liquid and gas−liquid systems with reactions that have highly varying rates. The method is based on continuous removal of the liquid phase from the reactor, while the catalyst remains inside the reactor. In this way, the catalyst bulk density (mass of catalyst, relative to the liquid volume) continuously increases, which enhances the secondary and tertiary reactions in the system. It becomes possible to determine all of the kinetic parameters from a single experiment. Mathematical models were presented and then solved analytically (first-order reaction systems) and numerically (general kinetics). The concept works in practice, which was demonstrated by an experimental study: the catalytic hydrogenation of citral on a nickel catalyst. The primary product (citronellal) is formed very rapidly, whereas the secondary (citronellol) and tertiary (3,7-dimethyloctanol) products appear much more slowly. A standard isothermal, constant-volume experiment in a slurry reactor cannot provide data from which all of the rate constants could be determined in a reliable way. With the proposed semibatch concept, the formation of the ultimate products was accelerated considerably, and all of the rate parameters were successfully estimated by nonlinear regression analysis. The proposed approach is not limited to semibatch slurry reactors; however, it can be extended to fixed beds with recycling, as demonstrated by computer simulations.

A general method for the computation of probabilities in systems of first order chemical reactions

We present a general method for the computation of molecular population distributions in a system of first-order chemical reactions. The method is based on the observation that the molecules in first-order reactions do not interact with each other. Therefore, the population distributions are a convolution of densities for one molecule. With this method one can study reactions involving any number of molecules. Such analysis is demonstrated on several examples, including an enzyme catalyst reaction and a first-order reaction chain.

A stochastic analysis of first-order reaction networks

A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other aspects, and the evolution equations for the first two moments are derived. We find the surprising, and apparently unknown, result that the time evolution of the second moments can be represented explicitly in terms of the eigenvalues and projections of the matrix that governs the evolution of the means. The model is used to analyze the effects of network topology and the reaction type on the moments of the probability distribution. In particular, it is shown that for an open system of first-order conversion reactions, the distribution of all the system components is a Poisson distribution at steady state. Two different measures of the noise have been used previously, and it is shown that different qualitative and quantitative conclusions can result, depending on which measure is used. The effect of catalytic reactions on the variance of the system components is also analyzed, and the master equation for a coupled system of first-order reactions and diffusion is derived.

An information-theoretic methodology for the resolution of pure component spectra without prior information using spectroscopic measurements

The resolution of pure component spectra based on spectroscopic measurements from a reaction system is a challenging task for chemometric systems in the absence of a priori knowledge about the reaction components involved. A popular approach in the literature is based on constrained entropy minimization of the second-order derivative of the resolved pure component spectra. Using an analytical information theoretic framework, it can however be shown that minimization of this cost function is not sufficient to completely separate the underlying components from a set of mixture spectra. Instead, an augmented objective function derived from this analysis is proposed for complete minimization of the mutual information between separated components. The final optimization approach is further shown to be analog to independent component analysis (ICA), a signal processing technique successfully applied to biomedical and speech data to separate linear source mixtures in the absence of a priori information. The developed theoretical insights and proposed methodologies in this paper are illustrated in a simulation study on the separation of three component spectra based on absorbance data acquired from a first-order kinetic reaction system.

Runaway behaviour and parametric sensitivity of a batch reactor — an experimental study

The present paper reports on the behaviour of a pseudo first-order reaction system in a batchwise operated stirred tank reactor. The experimental results refer to runaways, sensitive and stable operations and are compared with two different runaway criteria, i.e. the Semenov criterion and the Adler and Enig criterion. These criteria present the minimum heat-transfer capacity β c required for a safe design of a reacting system for a given adiabatic temperature rise α and a given ratio ϵ of the Arrhenius temperature and the modified coolant temperature. It is shown that: — the Semenov criterion is conservative but safe, — the Adler and Enig criterion holds at low values of α, while at high α values runaways already occur before β c is reached. For constant values of α and ϵ, the measured sensitivities of operating curves at β values in the vicinity of β c are compared with normalized objective sensitivities S β of Morbidelli and Varma.

Decomposition of diphenylamine in nitrocellulose based propellants—I. Optimization of a numerical model to concentration-time data for diphenylamine and its primary degradation products determined by liquid chromatography with dual-amperometric detection

The consumption of diphenylamine (DPA) in two nitrocellulose (NC) based propellants subjected to a heat storage test at 85° has been studied. A previously developed method based on reversed-phase liquid chromatography with dual-amperometric detection was used to monitor the concentrations of DPA, 2-nitro-DPA and 4-nitro-DPA during the test. A numerical model based on first order rate equations was fitted to the obtained analytical data with the use of a specially written curve fitting program. The model implemented in the program describes the initial nitrosation and nitration steps of DPA in aging NC propellants. The use of matrices in the calculation of concentration-time (CT) curves enables the introduction of a general algorithm which can be readily changed in order to simulate any system of first order reactions. The program can therefore be used in other applications such as mechanistic studies in organic synthesis. The general simulation algorithm allows inclusion of unknown (not analysed) components in the reaction mechanism. In this application, it was possible to simulate the course of N-nitroso-DPA which is not detectable by the amperometric principle.