Search

The Gauss Jordan method is used in this study to equalize chemical reactions using a system of linear equations. One of the most common topics in Chemistry is balancing chemical reaction equations. Students typically struggle with balancing the chemical equation and find it difficult to understand; sometimes teachers struggle as well to balance chemical equations. The results of the equation balancing comply with the law on the conservation of matter and confirm that the existing methods for balance of chemical equations do not contradict each other. To solve the mathematical problem, the Gauss-Jordan method was used. Any chemical reaction can be handled using this method by certain reactants and products. we can add several other features to the python application which can check whether the chemical equation is already balanced or not, we can also categorise the type of the chemical reaction.

An unconditionally-positive finite difference (UPFD) and the standard explicit finite difference schemes are compared to the analytical solution of the advection–diffusion reaction equation which describes the exponential traveling wave in heat and mass transfer processes. It is found that although the unconditional positivity of the UPFD scheme, this scheme is less accurate than the standard explicit finite difference scheme. This is because the UPFD scheme contains additional truncation-error terms in the approximations of the first and second derivatives with respect to x, which are evaluated at different moments in time. While these terms tend to zero as the mesh is refined, the UPFD scheme nevertheless remains less accurate than its standard explicit finite difference counterpart. The presented results are important when modeling a heat and mass transfer processes using the investigated advection–diffusion reaction equation. Furthermore, current and future developers of coupled multi-species transport models may draw on the ideas of solutions methods employed in this study to further develop numerical models for various types of coupled multi-species transport problems.

Photosynthesis and cellular respiration are core concepts in high school biology, as they are at the core of all living things. However, perhaps because students can’t see them happening, they very often have difficulty understanding them. Further, students often come into biology with a limited understanding of the law of conservation of matter as it relates to chemical reactions. These problems compound to create a great deal of misconceptions about these two important processes. In these lessons, I implemented the use of modeling to overcome these misconceptions and give my students a firm understanding of chemical equations, photosynthesis, and cellular respiration.

The present paper addresses a diffusion-reaction equation describing the dynamics of dissolved oxygen in a polluted stream of a river. The diffusion-reaction equation is a mass-balanced partial differential equation which relates the concentration of dissolved oxygen with the effect of other natural processes, viz. diffusion, natural aeration and reaction with pollutants. The well-known method of lines is used to solve the one-dimensional non-steady state case with Dirichlet boundary conditions. The study is motivated by the miserable condition of most of the rivers in India. Water pollution has now become a global concern and this study furnishes a better apprehension of complex phenomenon of maintaining desired level of oxygen and will aid water resource management.

The experimental execution of chemical reactions is a context-dependent and time-consuming process, often solved using the experience collected over multiple decades of laboratory work or searching similar, already executed, experimental protocols. Although data-driven schemes, such as retrosynthetic models, are becoming established technologies in synthetic organic chemistry, the conversion of proposed synthetic routes to experimental procedures remains a burden on the shoulder of domain experts. In this work, we present data-driven models for predicting the entire sequence of synthesis steps starting from a textual representation of a chemical equation, for application in batch organic chemistry. We generated a data set of 693,517 chemical equations and associated action sequences by extracting and processing experimental procedure text from patents, using state-of-the-art natural language models. We used the attained data set to train three different models: a nearest-neighbor model based on recently-introduced reaction fingerprints, and two deep-learning sequence-to-sequence models based on the Transformer and BART architectures. An analysis by a trained chemist revealed that the predicted action sequences are adequate for execution without human intervention in more than 50% of the cases., In organic chemistry, synthetic routes for new molecules are often specified in terms of reacting molecules only. The current work reports an artificial intelligence model to predict the full sequence of experimental operations for an arbitrary chemical equation.

A technique is presented for calculating the main parameters along the flame tube length in the combustor with combustion of premixed air–fuel mixtures. The calculation in the combustion zone is carried out by numerical solution of the system of equations based on the well-known brutto reaction equation of material balance for a homogeneous reactor.

In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in R 1 . The analysis is performed in the weighted Sobolev spaces, H ( a , b ) s ( I ) . Three different characterizations of H ( a , b ) s ( I ) are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters α and r defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.

This study compares students’ explanations of the oxidation–reduction reaction between silver nitrate and copper metal after viewing a chemical demonstration and one of four different particulate-level computer animations. The animations differed in the way the ionic charges were depicted (shown or omitted) and the way the transferred electrons were depicted (as discrete “e−” particles or as fuzzy electron “halos” around the metal atoms). Students viewing animations explicitly showing the ion charges were better at determining the number of electrons gained by each silver ion and lost by each copper atom and wrote more correct balanced chemical equations for the reaction than students viewing animations with charges omitted. Compared to students who viewed animations depicting the transferred electrons as halos around the metal atoms, students viewing animations depicting transferred electrons as discrete particles provided better explanations for the number of electrons gained by the silver ions and lost by the copper atoms and the relative sizes of the silver/copper atoms and their respective cations. Based on comments from student interviews, it appears that several students did not know the proper charges of the silver, copper(ii), and nitrate ions and that animations showing ionic charges provided students with relevant visual information they could use to improve their pictorial models of the reaction, which led to improved student-generated balanced equations for this reaction. Student interview comments also suggested that animations depicting the transferred electrons as discrete particles provided more relevant visual images (the number of electrons being transferred between silver ions and copper atoms and the relative sizes of the metal atoms and their cations) that were not provided by animations depicting the transferred electrons as halos, and this led students to construct more accurate pictorial models of the oxidation–reduction reaction. The results of this study and previous research by our group suggest that, although showing or omitting water molecules in animations depicting simple oxidation–reduction reactions had little effect on students explanations of the oxidation–reduction process, showing ion charges and depicting transferred electrons as particles (instead of halos) led to significant improvement of students’ explanations of the oxidation–reduction process.

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.

Stoichiometry, the ideal gas law, and the concept of limiting reagent are challenging principles for students to conceptualize in introductory chemistry. These topics are fundamental to successfully mastering other chemistry content. Therefore, a handheld microfluidic device was designed as a new tool to visualize these principles. Using food-grade baking soda, vinegar, and carbonated water as reagents, the students were engaged in a hands-on learning experience. The baking soda and vinegar were the source of sodium bicarbonate and acetic acid which were combined in a 1:1 mol ratio to form carbon dioxide gas which was then quantified using the device, thereby illustrating the ideal gas law. Experiments were devised to support learning outcomes that involved using the ideal gas law to interconvert moles and volume, balancing chemical equations, quantifying the amount of product generated from a known amount of reactant, conveying measurement uncertainty, and postulating sources of experimental error. The microfluidic device is a fast and cost-effective tool to teach stoichiometry. Moreover, the use of food-grade reagents makes the activity accessible as well as safe enough to be conducted outside of a traditional laboratory setting.

This study presents a three-level explicit time-split MacCormack method to compute approximate solutions of two-dimensional time-dependent linear convection-diffusion-reaction equations with source...

The state of health of polyfluorinated sulfonic-acid ionomer membranes (e.g. Nafion®) in low-temperature proton exchange membrane fuel cells (LT-PEMFCs) is negatively influenced by degradation phenomena occurring during their operation. As a consequence, the performance and durability of the membrane are decreased. In this article, we focus on simulating and predicting chemical membrane degradation phenomena using a holistic zero-dimensional kinetic framework. The knowledge of chemical degradation mechanisms is widely spread. We have collected and evaluated an extensive set of chemical mechanisms to achieve a holistic approach. This yields a set of 23 coupled chemical equations, which provide the whole cause and effect chain of chemical degradation in LT-PEMFCs (based on the Fenton reaction between Fe2+ and H2O2via the attack of hydroxyl radicals on the membrane, loss of ionomer moieties and emission of fluoride). Our kinetic framework allows the reproduction of experimentally accessible data such as fluoride emission rates and concentrations of ionomer moieties (from both in situ and ex situ tests). We present an approach, which allows estimations of the membrane lifetime based on fluoride emission rates. In addition, we outline the demetallation of Fe-N-C catalysts as a source of additional harmful iron species, which accelerate chemical membrane degradation. To demonstrate the expandability and versatility of the kinetic framework, a set of five chemical equations describing the radical scavenging properties of cerium agents is coupled to the main framework and its influence on membrane degradation is analysed. An automated solving routine for the system of coupled chemical equations on the basis of the chemical kinetic simulation tool COPASI has been developed and is freely accessible online ().

In the never ending quest for clean energy, storing chemical energy in the form of hydrogen molecules in solid state materials is a premier choice. The release of H2 from the H-storage (absorbent) material is hindered by the on-set of reaction mechanism which depends on the energy barrier. In a search for efficient onset (re)hydrogenation temperature, stimuli like catalysts have drawn more attention in the hydrogen fuel economy. It is laborious to down-select the aspirant candidates by screening their material properties and the thermochemical energy which are required for bond breaking. An understanding on perceptible energies involved in breaking/creating bonds has been obtained using the novel “Interface Reaction tool”. To enlighten the underpinning catalytic reaction mechanism to the absorbent, the energetics of interactions between the catalyst and the solid metal hydrides have been derived. For this test study, a leading catalyst TiFx (x = 4,3, and 2) has been added to the well-studied high gravimetric metal hydrides such as MgH2, Mg(BH4)2, and Mg(AlH4)2. The reaction equation at ambient condition has been validated using the total energies from Kohn-Sham density-functional-theory, and the outcome emphasizes the importance of bonding analysis. Hence, we exemplified the bonding states of all the Ti–F derivatives namely TiF4, TiF3 and TiF2 in an effort to elucidate the role of fluorine for the onset of catalytic reaction. Our detailed analysis of reaction pathways indicate the vitality of TiF4 as an additive before and after the H2 release. Our work pioneers the study on often overlooked reaction possibilities at ambient condition. The calculated reaction energy of less than 40 kJ/mol and 4.64 wt % of H2 release for TiF4 and TiF3 underscores their catalytic activity and meet the system targets set by DoE for the year 2020. This work presents “significant disclosures” on the hydrogen decomposition mechanism and provides a platform for the feasibility study of a new material. This proposes an innovative methodology for predicting the suitability of a new material for efficient production of H2 fuel.

An open-source repository of basic building block models design files for writing chemical formulas, equations, and ionic states are provided. Writing chemical symbols, molecules, and ions in their...

By using improved F-expansion method (IFEM), we study density-dependent space–time-fractional diffusion–reaction equation with quadratic nonlinearity (DDFDRE), which arises in mathematical biology. The fractional derivative is described in the sense of the CF derivative. Exact traveling wave solutions for DDFDRE are derived and expressed in terms of hyperbolic functions and rational functions. The IFEM is brief, efficient and easy to apply. It also can be used to solve many other fractional nonlinear evolution equations.

A series of ReaxFF-MD simulations are employed to illuminate the oxidation mechanism of HFO-1336mzz(Z) as an environmentally friendly refrigerant in O2/H2O environment. The results showed that H2O molecule has a significant impact on the HFO-1336mzz(Z) oxidation reactions and promotes the rate of HFO-1336mzz(Z) oxidation decomposition. H2O molecule can react with H and O radicals to form OH radical and will provide more H atoms to combine with F radicals to generate HF molecule. The OH radical plays a crucial role in the HFO-1336mzz(Z) oxidation reactions because the reactivity of OH radical. CO2, COF2 and HF are dominant products and the other radicals COF, O, F, CF3 and CO2F have a significant impact on the HFO-1336mzz(Z) oxidation reactions. The chemical equations of HFO-1336mzz(Z) oxidation in different conditions are presented to expound the comprehensive oxidation mechanism.

A recently proposed reaction network induces the classical Gompertz equations. The network consist of two reactions, one reaction for a decaying species $S$ and another reaction for a sigmoidally growing species $X$ which is catalyzed by species S. The proposed reaction network provides for a separation of the dynamical evolutions of the two species inherently present in the Gompertz reaction network. More specifically, the reaction equation for the decaying species S is totally independent on (uncoupled from) the growing species X, while the species X uses species S simultaneously both as a catalyst and as a resource (food). Based on the idea of such a separation, in this work we propose a class of growth-decay models formulated in terms of reaction networks that includes the Bateman exponential decay chain for the evolution of the catalist/resource species. An important advantage of the new class of models is the presence of a prolonged lag time of the sigmoidal solutions of the growing species. In this note we show that the Gompertz-type reaction network can be generalized into a class of Gompertzian-type growth-decay models by replacing the first species from the Bateman exponential decay chain by the second species in the chain, that now can play the role of a catalyst/resource for the growing species. A further generalization would be to use the third species (or the k-th species) in the Bateman decay chain as a catalyst/resource for the growing species.

A kinetic model is proposed to characterize surface segregation in the case of evaporation. The kinetics is formulated using the difference between the diffusion rate of solute atom in the bulk and the evaporation rate of solute atom from the surface layer. The first-order reaction equation is applied to quantify evaporation flux. A balanced state is quantified when the vapor and ambient pressure is equal and thus a complete quantification of the kinetics forms. The kinetic curve is convex prior to critical time and concave after. The maximum surface concentration decreases as evaporation parameter decreases. The kinetics of Mg surface segregation of Al-0.8 wt%Mg alloy is fitted. An enrichment ratio of 16.484 and evaporation parameter of 0.57 are the values that fit the experimental results the best. The balanced surface concentration of Mg is calculated and implies further evaporation with prolonged time. The proposed model can adequately describe surface segregation in the case of evaporation, and the formulated kinetics can satisfactorily quantify this phenomenon.

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H 0 1 ⁢ ( Ω ) {H_{0}^{1}(\Omega)} , the Banach Sobolev space W 0 1 , q ⁢ ( Ω ) {W^{1,q}_{0}(\Omega)} , 1 < q < ∞ {1 , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W 0 1 , q ⁢ ( Ω ) {W^{1,q}_{0}(\Omega)} - W 0 1 , q ′ ⁢ ( Ω ) {W_{0}^{1,q^{\prime}}(\Omega)} functional setting, 1 q + 1 q ′ = 1 {\frac{1}{q}+\frac{1}{q^{\prime}}=1} . The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of W 1 , q ′ {W^{1,q^{\prime}}} -stability of the H 0 1 {H_{0}^{1}} -projector.

In this article, a characteristic radial basis function-generated finite differences method (RBF-FD) is presented to numerically solve the convection-diffusion-reaction equation on implicit surface...

This comment is devoted to the paper "Exact solutions of nonlinear diffusion-convection-reaction equation: A Lie symmetry approach" (CNSNS, vol. 67 (2019), 253-263) in which several results are not new because were derived much earlier. Moreover, some results in the paper are incorrect or incomplete., Commun Nonlinear Sci Numer Simulat

Implementation of finite element method (FEM) needs special cares, particularly for essential boundary conditions that have an important effect on symmetry and number of unknowns in the linear systems. Moreover, avoiding numerical integration and using (off-line) calculated element integrals decrease the computational cost significantly. In this chapter we briefly present theoretical topics of FEM. Instead we focus on what is important (and how) to carefully implement FEM for equations that can be the core of a numerical simulator for a diffusion–advection-reaction problem. We consider general 2D and 3D domains, high contrast and heterogeneous diffusion coefficients and generalize the method to nonlinear parabolic equations. Although we use Matlab codes to simplify the explanation of the proposed method, we have implemented it in C++ to reveal the efficiency and examples are presented to admit it.

This study aimed to investigate the degradation kinetics of the total organic carbon (TOC) in real pharmaceutical wastewater. Because of the complexity of real wastewaters and the lack of knowledge on all reactions involved in their chemical degradation, the use of a kinetic approach becomes difficult. To overcome this difficulty, we adopted a kinetic model based on chemical equations validated for mechanisms of free radical generation. Photo-Fenton reactions were performed in a 1-L tubular reactor. The developed model satisfactorily described the degradation kinetics of TOC in wastewater, with an R2 of 0.9997. The TOC degradation rate constant (kd) was estimated at 1 × 1011 L2 mol2 s−1, in agreement with literature values for the degradation kinetics of organic compounds. UV radiation had a positive effect on degradation, contributing to HO• formation. The adopted mathematical approach was highly sensitive to the formation rate constant of HO• from H2O2 (k4). By assuming the reaction to follow variable order kinetics (β = 1 and β = 2), it was possible to improve the goodness of fit of the model. Our results suggest, albeit in an incipient way, that the approach used in this study can be extended to the kinetic understanding of other complex wastewaters.

Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility of a sixth order compact finite difference scheme in space and low-storage third order total variation diminishing Runge-Kutta (TVD-RK3) scheme in time combined to find the numerical solution of Nagumo equation. The comparison of the computed numerical results with the exact solution revealed that the proposed method to solve Nagumo equation is capable of achieving higher accuracy than some existing compact finite difference methods.

From 1990 to 2000, Harald A. Oye supervised work to establish a model of the chemistry of the melt from the electrolyte that goes into the porosity of the carbon cathode blocks during start-up and operation of Hall–Heroult cells. The model was effectively complete by 1995. The work had two parts, where the first was laboratory-scale testing and the second was analysis of industrial cathode materials from cells. In Part I, the laboratory-scale electrolysis, a comprehensive test program was run by visiting scientist Philippe Brilloit with tests of a range of carbon materials and electrolyte compositions, and operation from less than 1 h to a few days. In Part II, industrial cells, Lorentz Petter Lossius collected cores from cathodes from cells stopped after 10 days to 2534 days. The understanding built with the laboratory-scale results was applied to interpret the chemical processes occurring through the lifetime of industrial cathodes. Briefly, in a new cell a sodium front (metallic Na) will move from the electrolyte side down into the cathode carbon. Electrolyte will follow, as melt in the pores in the carbon. Chemical changes will occur after a pattern of dynamic change downward and over time. This was described with a set of model chemical equations and thermodynamic calculations that supported the observed changes. The paper recounts key analysis experiences with focus on the development of the sample preparation, the XRD method, and supplemented with later experiences with newer analysis methods.

We present a deep-learning model for inferring missing molecules in reaction equations. Such an algorithm features multiple interesting behaviors. First, it can infer the necessary reagents and solvents in chemical transformations specified only in terms of main compounds, as often resulting from retrosynthetic analyses. The completion with necessary reagents ensures that reaction equations are compatible with deep-learning models relying on a complete reaction specification. Second, it can cure existing datasets by detecting missing compounds, such as reagents that are essential for given classes of reactions. Finally, this model is a generalization of models for forward reaction prediction and retrosynthetic analysis, as both can be formulated in terms of incomplete reaction equations. We illustrate that a single trained model, based on the transformer architecture and acting on reaction SMILES strings, can address all three points.Workshop paper at the Machine Learning for Molecules Workshop at NeurIPS 2020.

We derive some exact solutions of a nonlinear diffusion-convection-reaction equation which models biological, chemical and physical phenomena. The Lie symmetry classification approach is employed to specify the model parameters and then the symmetries of resulting submodels are utilized for construction of exact solutions.

In this paper, a formal and systematic method for balancing chemical reaction equations was presented. The results satisfy the law of conservation of matter, and confirm that there is no contradiction to the existing way(s) of balancing chemical equations. A chemical reaction which possesses atoms with fractional oxidation numbers that have unique coefficients was studied. In this paper, the chemical equations were balanced by representing the chemical equation into systems of linear equations. Particularly, the Gauss elimination method was used to solve the mathematical problem with this method, it was possible to handle any chemical reaction with given reactants and products.