Journal articles: 'Reaction equations'

1

Slijepčević, Siniša. "Entropy of scalar reaction-diffusion equations." Mathematica Bohemica 139, no. 4 (2014): 597–605. http://dx.doi.org/10.21136/mb.2014.144137.

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Tykodi, R. J. "Annotating reaction equations." Journal of Chemical Education 64, no. 3 (1987): 243. http://dx.doi.org/10.1021/ed064p243.

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Gurevich, Pavel, and Sergey Tikhomirov. "Systems of reaction-diffusion equations with spatially distributed hysteresis." Mathematica Bohemica 139, no. 2 (2014): 239–57. http://dx.doi.org/10.21136/mb.2014.143852.

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4

Zhong, Wei, and Zhou Tian. "The Chemical Kinetic Numerical Computation and Kinetic Model Parameters Estimating of Parallel Reactions with Different Reaction Orders." Advanced Materials Research 560-561 (August 2012): 1126–32. http://dx.doi.org/10.4028/www.scientific.net/amr.560-561.1126.

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Abstract. Parallel reaction is a common reaction of chemical kinetics, and there are two types of parallel reactions according to the reaction orders equivalence: parallel reactions with same reaction orders and parallel reactions with different reaction orders. For the reason that the reaction orders are different, the chemical kinetic numerical computation and kinetic model parameters estimating of parallel reactions with different reaction orders is more complicated than parallel reactions with same reaction orders. In this paper, the 4th order Runge-Kutta method was employed to solve the numerical computation problems of complex ordinary differential equations, which was the chemical kinetic governing equations of parallel reactions with different reaction orders, and also, the Richardson extrapolation and Least Square Estimate were employed to estimate the kinetic model parameters of parallel reactions with different reaction orders. A C++ program has been processed to solve the problem and has been tested by an example of parallel reactions with different reaction orders.

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Sharma, Gitalee, Surashmi Bhattacharyya, and Niranjan Bora. "Matrix method for balancing chemical equations of few ‎significant inorganic reactions." International Journal of Basic and Applied Sciences 14, no. 2 (2025): 22–28. https://doi.org/10.14419/8g6xtb36.

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Balancing chemical equations provides a unified framework on understanding and quantifying chemical reactions, making it a fundamental ‎tool in chemistry. The prime objectives to balanced chemical equations are to make both sides of the reaction, the reactants as well as the ‎products, possess the same number of atoms per element. It is worth mentioning that understanding how and in what amounts certain mole-‎cules are created is made easier with the use of chemical reactions. It also indicates the quantity of reactants required to complete the reaction. ‎These two identities of a chemical reaction are specified by balancing the reaction, which also helps in understanding how to speed up or ‎stop the process. However, balancing long chemical reactions is a difficult and time-consuming task. Employing the principles of mathemat‎ical computation to the balancing of chemical equations may come as a remedy to this problem. Thus, in the current paper, we use a matrix-‎based method, Gauss Elimination mathematical model to balance the chemical equations of a few specific inorganic reactions. Computation ‎work was performed and validated with the help of Python software‎.

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Wei, G. W. "Generalized reaction–diffusion equations." Chemical Physics Letters 303, no. 5-6 (1999): 531–36. http://dx.doi.org/10.1016/s0009-2614(99)00270-5.

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7

Freidlin, Mark. "Coupled Reaction-Diffusion Equations." Annals of Probability 19, no. 1 (1991): 29–57. http://dx.doi.org/10.1214/aop/1176990535.

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Saxena, R. K., A. M. Mathai, and H. J. Haubold. "Fractional Reaction-Diffusion Equations." Astrophysics and Space Science 305, no. 3 (2006): 289–96. http://dx.doi.org/10.1007/s10509-006-9189-6.

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9

Manthey, Ralf, and Katrin Mittmann. "Stochastic reaction-diffusion equations with continuous reaction." Stochastics and Stochastic Reports 48, no. 1-2 (1994): 61–82. http://dx.doi.org/10.1080/17442509408833898.

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John, Zainab, Fadhel S. Fadhel, Samsul Ariffin Abdul Karim, and Teh Yuan Ying. "Stabilizability and Solvability of Fuzzy Reaction-Diffusion Equation using Modified Backstepping Control Method for Matrix Differential Equation." Malaysian Journal of Fundamental and Applied Sciences 21, no. 3 (2025): 2159–73. https://doi.org/10.11113/mjfas.v21n3.4249.

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In this article, an important type of fuzzy parabolic differential equations will be discussed, which is the one-dimensional fuzzy reaction-diffusion equation with fuzzy boundary conditions. This equation is one of the most widespread chemical fuzzy reaction-diffusion equations, as well as, studying the possibility of controlling and reducing the chemical pollution occurred in the chemical reactions. In order to reduce chemical contamination in the reaction medium, we observed that investigating this equation's stability is essential. In order to achieve stability, the fuzzy backstepping approach is proposed, which transforms the unstable system into a stable system after controlling the boundary conditions. Therefore, two different cases of Hukuhara derivatives must be considered, which are important in the study of fuzzy differential equations. Two cases are considered depending on the comparison between the lower and upper variable solution time derivative. Also, the proposed backstepping approach is applied based on the interval analysis of α-level sets. For this purpose, and in order to avoid the difficulty of separating the upper and lower solutions, the resulting non-fuzzy or crisp differential equations are converted into matrix differential equations, and then Consequently, we are able to remove the residual terms that are responsible for the instability of the open-loop. Moreover, this backstepping transformation is continuously invertible. Thus, the inverse transformation is used to obtain stabilizing state feedback for the original partial differential equation.

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11

Silaev, Michael M. "KINETIC EQUATIONS FOR RADICAL-CHAIN OXIDATION INVOLVING PROCESS-INHIBITING ALKYL (OR HYDRO)TETRAOXYL FREE RADICAL." American Journal of Applied Sciences 05, no. 06 (2023): 29–48. http://dx.doi.org/10.37547/tajas/volume05issue06-07.

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The derivation of kinetic equations for the oxidation processes by the free-radical nonbranched-chain mechanism is shown. This derivation is based on the proposed reaction scheme for the initiated addition of free radicals to the multiple bond of the molecular oxygen includes the addition reaction of the peroxyl free radical to the oxygen molecule to form the tetraoxyl free radical. This reaction competes with chain propagation reactions through a reactive free radical. The chain evolution stage in this scheme involves a few of free radicals, one of which – alkyl(or hydro)tetraoxyl – is relatively low-reactive and inhibits the chain process by shortening of the kinetic chain length. The rate equations (containing one to three parameters to be determined directly) are deduced using the quasi-steady-state treatment. These kinetic equations were used to describe the γ-induced nonbranched-chain processes of free-radical oxidation of liquid o-xylene at 373 K and hydrogen dissolved in water containing various amounts of oxygen at 296 K. The ratios of rate constants of competing reactions and rate constants of addition reactions to the molecular oxygen are defined. In these processes the oxygen with the increase of its concentration begins to act as an oxidation autoinhibitor (or an antioxidant), and the rate of peroxide formation as a function of the dissolved oxygen concentration has a maximum. It is shown that a maximum in these curves arises from the competition between hydrocarbon (or hydrogen) molecules and dioxygen for reacting with the emerging peroxyl 1:1 adduct radical. From the energetic standpoint possible nonchain pathways of the free-radical oxidation of hydrogen and the routes of ozone decay via the reaction with the hydroxyl free radical in the upper atmosphere (including the addition yielding the hydrotetraoxyl free radical, which can be an intermediate in the sequence of conversions of biologically hazardous UV radiation energy) were examined. The energetics of the key radical-molecule gas-phase reactions is considered.

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12

Johar, Dwindi Agryanti. "Application of the Concept of Linear Equation Systems in Balancing Chemical Reaction Equations." International Journal of Global Operations Research 1, no. 4 (2020): 130–35. http://dx.doi.org/10.47194/ijgor.v1i4.48.

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This study discusses the equalization of chemical reactions using a system of linear equations with the Gaussian and Gauss-Jordan elimination. The results show that there is a contradiction in the existing methods for balancing chemical reactions. This study also aims to criticize several studies that say that the equalization of the reaction coefficient can use a system of linear equations. In this paper, the chemical equations were balanced by representing the chemical equation into systems of linear equations. Particularly, the Gauss and Gauss-Jordan elimination methods were used to solve the mathematical problem with this method, it was possible to handle any chemical reaction with given reactants and products.

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Johar, Dwindi Agryanti. "Application of the Concept of Linear Equation Systems in Balancing Chemical Reaction Equations." International Journal of Global Operations Research 1, no. 4 (2020): 130–35. http://dx.doi.org/10.47194/ijgor.v1i4.48.

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This study discusses the equalization of chemical reactions using a system of linear equations with the Gaussian and Gauss-Jordan elimination. The results show that there is a contradiction in the existing methods for balancing chemical reactions. This study also aims to criticize several studies that say that the equalization of the reaction coefficient can use a system of linear equations. In this paper, the chemical equations were balanced by representing the chemical equation into systems of linear equations. Particularly, the Gauss and Gauss-Jordan elimination methods were used to solve the mathematical problem with this method, it was possible to handle any chemical reaction with given reactants and products.

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14

Serdyukov, Sergey. "Macroscopic Entropy of Non-Equilibrium Systems and Postulates of Extended Thermodynamics: Application to Transport Phenomena and Chemical Reactions in Nanoparticles." Entropy 20, no. 10 (2018): 802. http://dx.doi.org/10.3390/e20100802.

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In this work, we consider extended irreversible thermodynamics in assuming that the entropy density is a function of both common thermodynamic variables and their higher-order time derivatives. An expression for entropy production, and the linear phenomenological equations describing diffusion and chemical reactions, are found in the context of this approach. Solutions of the sets of linear equations with respect to fluxes and their higher-order time derivatives allow the coefficients of diffusion and reaction rate constants to be established as functions of size of the nanosystems in which these reactions occur. The Maxwell-Cattaneo and Jeffreys constitutive equations, as well as the higher-order constitutive equations, which describe the processes in reaction-diffusion systems, are obtained.

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15

Angstmann, Christopher N., and Bruce I. Henry. "Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations." Entropy 22, no. 9 (2020): 1035. http://dx.doi.org/10.3390/e22091035.

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A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

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16

HE, J. J., L. LI, J. HU, et al. "DEVELOPMENT OF A LORENTZIAN-FUNCTION APPROXIMATION UTILIZING IN THE CHARGED- PARTICLE-INDUCED NONRESONANT REACTION RATE." International Journal of Modern Physics E 20, no. 03 (2011): 747–52. http://dx.doi.org/10.1142/s0218301311018204.

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A development has been made for the charged-particle-induced nonresonant reaction-rate equations. The forms of reaction-rate equations for nonresonant and resonant reactions have been united in a frame of Lorentzian-Function Approximation (LFA) mathematically. In the frame of LFA, the nonresonant reaction taken place within the Gamow window can be considered, in form, as a "resonance" reaction with a full width at half maximum (FWHM, Γ nr ) equal to the 1/e width (Δ) in a well-known Gaussian-Function Approximation (GFA).

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17

HORSTHEMKE, WERNER. "EXTERNAL NOISE AND FRONT PROPAGATION IN REACTION-TRANSPORT SYSTEMS WITH INERTIA: THE MEAN SPEED OF FISHER WAVES." Fluctuation and Noise Letters 02, no. 04 (2002): R109—R124. http://dx.doi.org/10.1142/s0219477502000932.

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We review the effect of spatiotemporal noise, white in time and colored in space, on front propagation in systems of reacting and dispersing particles, where the particle motion displays inertia or persistence. We discuss the three main approaches that have been developed to describe transport with inertia, namely hyperbolic reaction-diffusion equations, reaction-Cattaneo systems or reaction-telegraph equations, and reaction random walks. We focus on the mean speed of Fisher waves in these systems and study in particular reaction random walks, which are the most natural generalization of reaction-diffusion equations. Hyperbolic reaction-diffusion equations account for inertia in the transport process in an ad hoc way, whereas the other reaction-transport systems have a proper macroscopic or microscopic foundation. For the former, external noise affects neither the mean wave speed nor the region in parameter space for which Fisher waves exist. For the latter, external noise increases the mean wave speed of Fisher waves and decreases the upper limit for the characteristic time of the transport process, below which propagating fronts exist.

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18

Jensen, William B. "Kinetic versus thermodynamic control: Some historical landmarks." Bulletin for the History of Chemistry 39, no. 2 (2014): 107–21. https://doi.org/10.70359/bhc2014v039p107.

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The study of chem. reactivity may be broadly divided into the subject areas of reaction stoichiometry, reaction kinetics, and reaction thermodn. The first deals with the classification of chem. reactions, their expression as properly balanced net chem. equations, and the various quant. calcns. that are based upon these balanced equations. The second deals with the detn. of rate laws and the deduction of reaction mechanisms, while the third deals with reaction efficiency and chem. equil. as a function of the relative stabilities of the various reactants and products, their concns., and the ambient temp. and pressure.

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Korakianitis, T., R. Dyer, and N. Subramanian. "Pre-integrated Nonequilibrium Combustion-Response Mapping for Gas Turbine Emissions." Journal of Engineering for Gas Turbines and Power 126, no. 2 (2004): 300–305. http://dx.doi.org/10.1115/1.1688769.

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In gas turbine combustion the gas dynamic and chemical energy release mechanisms have comparable time scales, so that equilibrium chemistry is inadequate for predicting species formation (emissions). In current practice either equilibrium chemical reactions are coupled with experimentally derived empirical equations, or time-consuming computations are used. Coupling nonequilibrium chemistry, fluid dynamic, and initial and boundary condition equations results in large sets of numerically stiff equations; and their time integration demands enormous computational resources. The response modeling approach has been used successfully for large reaction sets. This paper makes two new contributions. First it shows how pre-integration of the heat release maps eliminates the stiffness of the equations. This is a new modification to the response mapping approach, and it performs satisfactorily for non-diffusion systems. Second the theoretical framework is further extended to predict species formation in cases with diffusion, which is applicable to gas turbine combustion systems and others. The methodology to implement this approach to reacting systems, and to gas turbine combustion, is presented. The benefits over other reaction-mapping techniques are discussed.

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20

Polyanin, A. D., A. I. Zhurov, and A. V. Vyazmin. "Time-Delayed Reaction-Diffusion Equations." Vestnik Tambovskogo gosudarstvennogo tehnicheskogo universiteta 21, no. 1 (2015): 071–77. http://dx.doi.org/10.17277/vestnik.2015.01.pp.071-077.

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21

Bocharov, G. A., V. A. Volpert, and A. L. Tasevich. "Reaction–Diffusion Equations in Immunology." Computational Mathematics and Mathematical Physics 58, no. 12 (2018): 1967–76. http://dx.doi.org/10.1134/s0965542518120059.

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Tasevich, A., G. Bocharov, and V. Wolpert. "Reaction-diffusion equations in immunology." Журнал вычислительной математики и математической физики 58, no. 12 (2018): 2048–59. http://dx.doi.org/10.31857/s004446690003551-7.

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23

Shah, Jayant. "Reaction–Diffusion Equations and Learning." Journal of Visual Communication and Image Representation 13, no. 1-2 (2002): 82–93. http://dx.doi.org/10.1006/jvci.2001.0478.

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24

Field, Richard J. "Chaos in the Belousov–Zhabotinsky reaction." Modern Physics Letters B 29, no. 34 (2015): 1530015. http://dx.doi.org/10.1142/s021798491530015x.

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The dynamics of reacting chemical systems is governed by typically polynomial differential equations that may contain nonlinear terms and/or embedded feedback loops. Thus the dynamics of such systems may exhibit features associated with nonlinear dynamical systems, including (among others): temporal oscillations, excitability, multistability, reaction-diffusion-driven formation of spatial patterns, and deterministic chaos. These behaviors are exhibited in the concentrations of intermediate chemical species. Bifurcations occur between particular dynamic behaviors as system parameters are varied. The governing differential equations of reacting chemical systems have as variables the concentrations of all chemical species involved, as well as controllable parameters, including temperature, the initial concentrations of all chemical species, and fixed reaction-rate constants. A discussion is presented of the kinetics of chemical reactions as well as some thermodynamic considerations important to the appearance of temporal oscillations and other nonlinear dynamic behaviors, e.g., deterministic chaos. The behavior, chemical details, and mechanism of the oscillatory Belousov–Zhabotinsky Reaction (BZR) are described. Furthermore, experimental and mathematical evidence is presented that the BZR does indeed exhibit deterministic chaos when run in a flow reactor. The origin of this chaos seems to be in toroidal dynamics in which flow-driven oscillations in the control species bromomalonic acid couple with the BZR limit cycle.

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Li, Changhao, Jianfeng Li, and Yuliang Yang. "A Feynman Path Integral-like Method for Deriving Reaction–Diffusion Equations." Polymers 14, no. 23 (2022): 5156. http://dx.doi.org/10.3390/polym14235156.

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This work is devoted to deriving a more accurate reaction–diffusion equation for an A/B binary system by summing over microscopic trajectories. By noting that an originally simple physical trajectory might be much more complicated when the reactions are incorporated, we introduce diffusion–reaction–diffusion (DRD) diagrams, similar to the Feynman diagram, to derive the equation. It is found that when there is no intermolecular interaction between A and B, the newly derived equation is reduced to the classical reaction–diffusion equation. However, when there is intermolecular interaction, the newly derived equation shows that there are coupling terms between the diffusion and the reaction, which will be manifested on the mesoscopic scale. The DRD diagram method can be also applied to derive a more accurate dynamical equation for the description of chemical reactions occurred in polymeric systems, such as polymerizations, since the diffusion and the reaction may couple more deeply than that of small molecules.

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Kaleeswari. S. "Analytical Solution of Chemical Reaction Systems using Kinetic Models." Advances in Nonlinear Variational Inequalities 28, no. 1s (2024): 137–46. http://dx.doi.org/10.52783/anvi.v28.2234.

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It encompasses multiple model kinds, such as spatially distributed versus homogenous, discrete versus continuous, and stochastic versus determinist. Among all chemical reactions, oscillating reactions are the most exciting. Here, the reactant and auto catalyst are the two chemical species that make up the system. The analysis of chemical reaction processes by mathematical approaches has also a long history. We now shift our focus to mathematical frameworks for the deterministic kinetics of mass motion. On the positive orthant, these models are systems of related nonlinear differential equations. Reaction-diffusion equations with a nonlinear term resembling Michaelis-Menten's enzymatic reaction kinetics form the basis of the model. In this work, the New Homotopy perturbation method yields a preliminary analytical solution for all values of parameter to the nonlinear differential equations analysing the concentrations in mass-action kinetics and Michaelis - Menten enzyme kinetics, also known as the hill binding kinetics. A satisfactory level of agreement is found when these results are compared to the numerical result.

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27

Razani, Abdolrahman. "Chapman-Jouguet detonation profile for a qualitative model." Bulletin of the Australian Mathematical Society 66, no. 3 (2002): 393–403. http://dx.doi.org/10.1017/s0004972700040259.

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In this article, the existence of traveling wave fronts for a one step chemical reaction with a natural discontinuous reaction rate function is studied. This discontinuity occurs because of the cold boundary difficulty and implies a discontinuous system of ordinary differential equations. By some general topological arguments in ordinary differential equations, the Chapman-Jouguet detonation for exothermic reactions is shown to exist. In addition, the uniqueness of this wave is considered.

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Hanyka, Jiří, and Alena Fialová. "The course of consecutive reactions inside a nonisotropic catalyst particle, affected by internal difusion." Collection of Czechoslovak Chemical Communications 51, no. 1 (1986): 54–65. http://dx.doi.org/10.1135/cccc19860054.

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A system of differential equations for consecutive reactions inside a nonisotropic catalyst particle under conditions of internal diffusion is solved. The system of diffusion equations for the spherical geometry of the catalyst grain is numerically solved by using the collocation method. The solution is sought for various radial activity profiles across the catalyst particle and for various values of Thiele's modulus for the two consecutive reactions. The effect of the reaction orders with respect to the reactants on the degree of utilization of the internal catalyst surface and on the reaction selectivity is examined.

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E. Kloeden, Peter, Thomas Lorenz, and Meihua Yang. "Reaction-diffusion equations with a switched--off reaction zone." Communications on Pure & Applied Analysis 13, no. 5 (2014): 1907–33. http://dx.doi.org/10.3934/cpaa.2014.13.1907.

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30

Horno, José, and Carlos F. González-Fernández. "Analysis of chemical reaction systems by means of network thermodynamics." Collection of Czechoslovak Chemical Communications 54, no. 9 (1989): 2335–44. http://dx.doi.org/10.1135/cccc19892335.

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The simple network thermodynamics approach is applied to chemical reaction systems, whereby chemical reactions can be studied avoiding complex mathematical treatment. Steady state reaction rates are obtained for two chemical reaction systems, viz. the decomposition of ozone and the reaction of hydrogen with bromine. The rate equations so obtained agree with those derived from the chemical kinetics concept.

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Sulcius, Algirdas, and Sergey Teleshov. "WRITING CHEMICAL FORMULAS AND REACTION EQUATIONS: THE HISTORY AND PRACTICE OF BUILDING BLOCK METHOD." GAMTAMOKSLINIS UGDYMAS / NATURAL SCIENCE EDUCATION 16, no. 1 (2019): 54–62. http://dx.doi.org/10.48127/gu-nse/19.16.54.

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One of the most difficult elements for students in chemistry is writing and balancing equations of chemical reactions. The aim of this research was to investigate the ability of the 8-9th grade students to write chemical formulas and equations of reactions. Verhovskij’s method, which involves using white and full-colour paper cards, was proposed as a chemistry teaching method for students. Some of the students suggested using non-paper-based cards, i.e. “domino sticks”. The obtained results proved that students who had participated in the building blocks activity showed significantly higher post-test scores than students who had not participated. The average grade of the students who used the cards for chemical formula formation and reaction equations increased respectively by 1.30 and 1.20. The average grade for writing and balancing of the reactions equations was lower because balancing requires strong knowledge in mathematics. The results have shown that the use of cards is a good method at an early stage of chemistry education. Keywords: general chemistry, hands-on learning, domino-stick, graphical formula, reaction equation.

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Harrington, David A. "Theory of electrochemical impedance of surface reactions: second-harmonic and large-amplitude response." Canadian Journal of Chemistry 75, no. 11 (1997): 1508–17. http://dx.doi.org/10.1139/v97-181.

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The theory for the electrochemical impedance of surface reactions involving a single adsorbed species is presented. A new methodology is used, in which many harmonics are considered, and the differential equations are reduced to algebraic matrix equations. The amplitude of the ac potential perturbation is not assumed to be small, and nonlinear effects are taken into account. The amplitude dependence of the impedance and the second-harmonic response are investigated. The quasi-reversible electrosorption reaction and the hydrogen evolution reaction are considered in detail, assuming that the adsorbed species obeys the Langmuir isotherm. Keywords: electrochemistry, impedance, adsorption, hydrogen evolution reaction, second harmonic.

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Hou, Qingzhi, Jiaru Liu, Jijian Lian, and Wenhuan Lu. "A Lagrangian Particle Algorithm (SPH) for an Autocatalytic Reaction Model with Multicomponent Reactants." Processes 7, no. 7 (2019): 421. http://dx.doi.org/10.3390/pr7070421.

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For the numerical simulation of convection-dominated reacting flow problems governed by convection-reaction equations, grids-based Eulerian methods may cause different degrees of either numerical dissipation or unphysical oscillations. In this paper, a Lagrangian particle algorithm based on the smoothed particle hydrodynamics (SPH) method is proposed for convection-reaction equations and is applied to an autocatalytic reaction model with multicomponent reactants. Four typical Eulerian methods are also presented for comparison, including the high-resolution technique with the Superbee flux limiter, which has been considered to be the most appropriate technique for solving convection-reaction equations. Numerical results demonstrated that when comparing with traditional first- and second-order schemes and the high-resolution technique, the present Lagrangian particle algorithm has better numerical accuracy. It can correctly track the moving steep fronts without suffering from numerical diffusion and spurious oscillations.

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Boeker, E. A. "Integrated rate equations for irreversible enzyme-catalysed first-order and second-order reactions." Biochemical Journal 226, no. 1 (1985): 29–35. http://dx.doi.org/10.1042/bj2260029.

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Integrated rate equations are presented that describe irreversible enzyme-catalysed first-order and second-order reactions. The equations are independent of the detailed mechanism of the reaction, requiring only that it be hyperbolic and unbranched. The results should be directly applicable in the laboratory.

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35

Ptashnyk, Mariya. "Nonlinear pseudoparabolic equations as singular limit of reaction–diffusion equations." Applicable Analysis 85, no. 10 (2006): 1285–99. http://dx.doi.org/10.1080/00036810600871909.

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36

Rios, William Q., Bruno Antunes, Alírio E. Rodrigues, Inês Portugal, and Carlos M. Silva. "Revisiting Isothermal Effectiveness Factor Equations for Reversible Reactions." Catalysts 13, no. 5 (2023): 889. http://dx.doi.org/10.3390/catal13050889.

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Ion exchange resins have many industrial applications, namely as sorbents and catalysts. In solid-catalyzed reactions, intraparticle reaction-diffusion competition is generally described by effectiveness factors calculated numerically or analytically in the case of isothermal particles and simple rate laws. Although robust, numerical calculations can be time-consuming, and convergence is not always guaranteed and lacks the flexibility of user-friendly equations. In this work, analytical equations for effectiveness factors of reversible reactions derived from the general scheme A+B⇌C+D are developed and numerically validated. These effectiveness factors are analytically expressed in terms of an irreversible nth order Thiele modulus (specifically written for the nth order forward reaction), the thermodynamic equilibrium constant, the ratios of effective diffusivities, and the ratios of surface concentrations. The application of such analytical equations is illustrated for two liquid phase reactions catalyzed by Amberlyst-15, specifically the synthesis of ethyl acetate and acetaldehyde dimethyl acetal. For both reactions, the prediction of the concentration profiles in isothermal batch reactors achieved errors between 1.13% and 3.38% for six distinct experimental conditions. Finally, the impact of non-ideal behavior upon the multicomponent effective diffusivities, subsequently conveyed to the effectiveness factors, is enlightened.

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Din, Qamar, and Umer Saeed. "Stability, Discretization, and Bifurcation Analysis for a Chemical Reaction System." Match Communications in Mathematical and in Computer Chemistry 90, no. 1 (2023): 151–74. http://dx.doi.org/10.46793/match.90-1.151d.

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Chemical reactions reveal all types of exotic behavior, that is, multistability, oscillation, chaos, or multistationarity. The mathematical framework of rate equations enables us to discuss steadystates, stability and oscillatory behavior of a chemical reaction. A planar cubic dynamical system governed by nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction is studied. It is investigated that system has unique positive steady state. Moreover, local dynamics of system is studied around its positive steady state. Existence and direction of Hopf bifurcation about positive equilibrium are carried out. In order to modify the bifurcating behavior, bifurcation control is investigated. Keeping in mind, a consistency preserving discretization for continuous chemical reaction system, a discrete counterpart is proposed, and its qualitative behavior is investigated. Numerical simulation along with bifurcation diagrams are provided to illustrate the mathematical investigations.

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Ahmed, Kamran, Tanvir Akbar, and Taseer Muhammad. "Physical Aspects of Homogeneous-Heterogeneous Reactions on MHD Williamson Fluid Flow across a Nonlinear Stretching Curved Surface Together with Convective Boundary Conditions." Mathematical Problems in Engineering 2021 (November 15, 2021): 1–13. http://dx.doi.org/10.1155/2021/7016961.

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This article is concerned with the fluid mechanics of MHD steady 2D flow of Williamson fluid over a nonlinear stretching curved surface in conjunction with homogeneous-heterogeneous reactions with convective boundary conditions. An effective similarity transformation is considered that switches the nonlinear partial differential equations riveted to ordinary differential equations. The governing nonlinear coupled differential equations are solved by using MATLAB bvp4c code. The physical features of nondimensional Williamson fluid parameter λ , power-law stretching index m , curvature parameter K , Schmidt number Sc , magnetic field parameter M , Prandtl number Pr , homogeneous reaction strength k 1 , heterogeneous reaction strength k 2 , and Biot number γ are presented through the graphs. The tabulated form of results is obtained for the skin friction coefficient. It is noted that both the homogeneous and heterogeneous reaction strengths reduced the concentration profile.

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MORAN, JEFFREY L., and JONATHAN D. POSNER. "Electrokinetic locomotion due to reaction-induced charge auto-electrophoresis." Journal of Fluid Mechanics 680 (June 13, 2011): 31–66. http://dx.doi.org/10.1017/jfm.2011.132.

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Mitchell originally proposed that an asymmetric ion flux across an organism's membrane could generate electric fields that drive locomotion. Although this locomotion mechanism was later rejected for some species of bacteria, engineered Janus particles have been realized that can swim due to ion fluxes generated by asymmetric electrochemical reactions. Here we present governing equations, scaling analyses and numerical simulations that describe the motion of bimetallic rod-shaped motors in hydrogen peroxide solutions due to reaction-induced charge auto-electrophoresis. The coupled Poisson–Nernst–Planck–Stokes equations are numerically solved using Frumkin-corrected Butler–Volmer equations to represent electrochemical reactions at the rod surface. Our simulations show strong agreement with the scaling analysis and experiments. The analysis shows that electrokinetic locomotion results from electro-osmotic fluid slip around the nanomotor surface. The electroviscous flow is driven by electrical body forces which are generated from a coupling of a reaction-induced dipolar charge density distribution and the electric field it creates. The magnitude of the electroviscous velocity increases quadratically with the surface reaction rate for an uncharged motor, and linearly when the motor supports a finite surface charge.

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Syam, Azhari M., Hamidah A. Hamid, Robiah Yunus, and Umer Rashid. "Dynamic Modeling of Reversible Methanolysis ofJatropha curcasOil to Biodiesel." Scientific World Journal 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/268385.

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Many kinetics studies on methanolysis assumed the reactions to be irreversible. The aim of the present work was to study the dynamic modeling of reversible methanolysis ofJatropha curcasoil (JCO) to biodiesel. The experimental data were collected under the optimal reaction conditions: molar ratio of methanol to JCO at 6 : 1, reaction temperature of 60°C, 60 min of reaction time, and 1% w/w of catalyst concentration. The dynamic modeling involved the derivation of differential equations for rates of three stepwise reactions. The simulation study was then performed on the resulting equations using MATLAB. The newly developed reversible models were fitted with various rate constants and compared with the experimental data for fitting purposes. In addition, analysis of variance was done statistically to evaluate the adequacy and quality of model parameters. The kinetics study revealed that the reverse reactions were significantly slower than forward reactions. The activation energies ranged from 6.5 to 44.4 KJ mol−1.

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Keyfitz, Barbara Lee, and Joel Smoller. "Shock Waves and Reaction-Diffusion Equations." American Mathematical Monthly 93, no. 4 (1986): 315. http://dx.doi.org/10.2307/2323701.

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Bland, J. A., and J. Smoller. "Shock Waves and Reaction-Diffusion Equations." Mathematical Gazette 69, no. 447 (1985): 70. http://dx.doi.org/10.2307/3616482.

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Da Prato, Giuseppe, and Piermarco Cannarsa. "Invariance for stochastic reaction-diffusion equations." Evolution Equations and Control Theory 1, no. 1 (2012): 43–56. http://dx.doi.org/10.3934/eect.2012.1.43.

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DELLNITZ, MICHAEL, MARTIN GOLUBITSKY, ANDREAS HOHMANN, and IAN STEWART. "SPIRALS IN SCALAR REACTION–DIFFUSION EQUATIONS." International Journal of Bifurcation and Chaos 05, no. 06 (1995): 1487–501. http://dx.doi.org/10.1142/s0218127495001149.

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Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction–diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction–diffusion equation [ in u(x, t)] on a disk, if one assumes "spiral" boundary conditions (ur = muθ). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archimedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the "red" and "blue" arms of the spiral.

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Dunyak, James P. "Reaction - diffusion equations in perforated media." Nonlinearity 10, no. 2 (1997): 377–88. http://dx.doi.org/10.1088/0951-7715/10/2/004.

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Durrett, R., and C. Neuhauser. "Particle Systems and Reaction-Diffusion Equations." Annals of Probability 22, no. 1 (1994): 289–333. http://dx.doi.org/10.1214/aop/1176988861.

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Wilhelmsson, H. "Explosive instabilities of reaction-diffusion equations." Physical Review A 36, no. 2 (1987): 965–66. http://dx.doi.org/10.1103/physreva.36.965.

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Le Vent, S. "Rate of Reaction and Rate Equations." Journal of Chemical Education 80, no. 1 (2003): 89. http://dx.doi.org/10.1021/ed080p89.

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Hadeler, K. P. "Reaction transport equations in biological modeling." Mathematical and Computer Modelling 31, no. 4-5 (2000): 75–81. http://dx.doi.org/10.1016/s0895-7177(00)00024-8.

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Lang, Jens. "Adaptive FEM for reaction—diffusion equations." Applied Numerical Mathematics 26, no. 1-2 (1998): 105–16. http://dx.doi.org/10.1016/s0168-9274(97)00088-3.

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Journal articles on the topic 'Reaction equations'

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