Order Chemical Kinetics Research Articles

Solving Partial Differential Equations for Physical and Chemical Problems Using Physics-Informed Neural Networks

Numerous physical and chemical problems at a high school level can be described by ordinary differential equations (ODEs) and partial differential equations (PDEs). However, the underlying equations troubled high school students because they often lack advanced mathematical skills, such as discrete calculus. Our goal is not to elaborate on those skills, but to offer a shortcut to the solution. In this paper, we demonstrated the use of Physics-Informed Neural Networks (PINNs), a neural network which solves the PDEs by incorporating the PDEs into the loss functions. The heat transfer equation and second order chemical kinetics are the two chosen model problems for high school seniors. Using PINNs, we were able to solve these two problems without recurring to university math. Hence, we strongly recommend peers to employ this method for physical or chemical problems for high school students and beyond.

Theory of First Order Chemical Kinetics at the Critical Point of Solution.

Liquid mixtures, which have a phase diagram exhibiting a miscibility gap ending in a critical point of solution, have been used as solvents for chemical reactions. The reaction rate in the forward direction has often been observed to slow down as a function of temperature in the critical region. Theories based upon the Gibbs free energy of reaction as the driving force for chemical change have been invoked to explain this behavior. With the assumption that the reaction is proceeding under relaxation conditions, these theories expand the free energy in a Taylor series about the position of equilibrium. Since the free energy is zero at equilibrium, the leading term in the Taylor series is proportional to the first derivative of the free energy with respect to the extent of reaction. To analyze the critical behavior of this derivative, the theories exploit the principle of critical point isomorphism, which is thought to govern all critical phenomena. They find that the derivative goes to zero in the critical region, which accounts for the slowing down observed in the reaction rate. As has been pointed out, however, most experimental rate investigations have been carried out under irreversible conditions as opposed to relaxation conditions [Shen et al. J. Phys. Chem. A 2015, 119, 8784-8791]. Below, we consider a reaction governed by first order kinetics and invoke transition state theory to take into account the irreversible conditions. We express the apparent activation energy in terms of thermodynamic derivatives evaluated under standard conditions as well as the pseudoequilibrium conditions associated with the reactant and the activated complex. We show that these derivatives approach infinity in the critical region. The apparent activation energy follows this behavior, and its divergence accounts for the slowing down of the reaction rate.

Assessment of a POD method for the dynamical analysis of a catalyst pellet with simultaneous chemical reaction, adsorption and diffusion: Uniform temperature case

Abstract A model reduction method applied to the dynamic model of a single isothermal catalyst pellet with simultaneous chemical reaction, diffusion and adsorption is presented. The model of the pellet accounts for both internal and external resistances to mass transfer, and variable bulk gas concentration. The reduction of the model was performed by means of proper orthogonal decomposition (POD). Accuracy and computational efficiency of the reduced order model (ROM) were discussed for two kinetic models, i.e. a first order chemical kinetics and a Langmuir–Hinshelwood kinetics. To improve the computational efficiency of the model described by the Langmuir–Hinshelwood equation, the non-linear terms were approximated by a discrete empirical interpolation method (DEIM). High accuracy and efficiency of the proposed reduction approach was demonstrated.

Time-evolving collagen-like structural fibers in soft tissues: biaxial loading and spherical inflation

This work considers a previously developed constitutive theory for the time dependent mechanical response of fibrous soft tissue resulting from the time dependent remodeling of a collagen fiber network that is embedded in a ground substance matrix. The matrix is taken to be an incompressible nonlinear elastic solid. The remodeling process consists of the continual dissolution of existing fibers and the creation of new fibers. Motivated by experimental reports on the enzyme degradation of collagen fibers, the remodeling is governed by first order chemical kinetics such that the dissolution rate is dependent upon the fiber stretch. The resulting time dependent mechanical response is sensitive to the natural configuration of the fibers when they are created, and different assumptions on the nature of the fiber’s stress free state are considered here. The response under biaxial loading, a type of loading that has particular significance for the characterization of biological materials, is studied. The inflation of a spherical membrane is then analyzed in terms of the equal biaxial stretch that occurs in the membrane approximation. Different assumptions on the natural configuration of the fibers, combined with their time dependent dissolution and reforming, are shown to emulate alternative forms of creep and relaxation response. This formal similarity to viscoelastic phenomena occurs even though the underlying mechanisms are fundamentally different from the mechanism of macromolecular reconfiguration that one typically associates with viscoelastic response.

Probing Radiation Damage in Plutonium Alloys with Multiple Measurement Techniques

AbstractA material subjected to radiation damage will usually experience changes in its physical properties. Measuring these changes in the physical properties provides a basis to study radiation damage in a material which is important for a variety of real world applications from reactor materials to semiconducting devices. When investigating radiation damage, the relative sensitivity of any given property can vary considerably based on the concentration and type of damage present as well as external parameters such as the temperature and starting material composition. By measuring multiple physical properties, these differing sensitivities can be leveraged to provide greater insight into the different aspects of radiation damage accumulation, thereby providing a broader understanding of the mechanisms involved. In this report, self-damage from α-particle decay in Pu is investigated by measuring two different properties: magnetic susceptibility and resistivity. The results suggest that while the first annealing stage obeys second order chemical kinetics, the primary mechanism is not the recombination of vacancy-interstitial close pairs.

Fitting isoperibolic calorimeter data for reactions with pseudo-first order chemical kinetics

The non-linear least squares fitting of the chemical heat flow and the reactor temperature are compared for reactions with pseudo-first order chemical kinetics carried out in an isoperibolic calorimeter operating quasi-isothermally. Both methods give very similar results for the reaction rate constant and enthalpy of reaction but fitting the reactor temperature appears to have some advantages especially when there is an enthalpy of mixing of the reagents.

Electrical Dissipative Structures in Membrane‐Coupled Compartment Systems

AbstractReaction‐diffusion systems with charged particles are studied. Conditions for the arising of electrical dissipative structures in a compartment system consisting of two boxes separated by a membrane are derived. The appearance of a polar dissipative structure is proved for a simple capacitor model in combination with a simple second order chemical kinetics which leads to an analytically solvable problem. Electrical dissipative structures can in principle be considered as non equilibrium electrical batteries. The theoretical efficiency of such batteries is estimated.

34 Catastrophe theory and chemical reactors: Exact uniqueness criteria for the CSTR, catalyst particle, and packed bed reactor

The steady state manifolds for simple nth order chemical kinetics in lumped parameter reactor systems, such as the CSTR, the single catalyst particle, and the mixing cell model of the packed bed reactor, are all equivalent to the cusp catastrophe. Exact bounds between regions of unique and multiple solutions in parameter space issue from a topological approach based on this similarity.

Electrochemical reaction mechanisms with following first order chemical kinetics in controlled potential voltammetry (potential step)

The general solution and the theoretical curves of the limiting current against log (lkt) are given for the mechanism:O+ne⇌RlR→kqO+pWW+me⇌V with different ratios q/l, for the controlled potential voltammetry with periodical renewal of the diffusion layer.

Electrochemical reaction mechanisms with following first order chemical kinetics in controlled potential voltammetry (potential step)

The general solution and the theoretical curves of the limiting current against log ( lkt) are given for the mechanism: O + n e ⇌ R l R → k q O + p W W + m e ⇌ V with different ratios q/l, for the controlled potential voltammetry with periodical renewal of the diffusion layer.

Quantitation of fluorescence fading phenomena for identifying intracellular biopolymers.

The fading behavior of the 670 nm fluorescence emission band produced by unfixed rat mast cells stained with acridine orange (AO) has been found to be in excellent agreement with the behavior predicted by second order chemical kinetics. The reciprocal of fluorescence intensity plotted against time yields a straight line. When due account is taken of dye/cell ratio and the intensity of fluorescence-exciting radiation, Io (measured with the standard phosphor particle), the slope of this straight line is a constant, k'', which is independent of dye/cell ratio and Io. k'' differs from the second order photochemical rate constant by a constant factor. The fading of a given AO-biopolymer complex is described by a particular value of k''. Two values of k'' have been found for rat mast cell granules, indicating the presence of two different AO-biopolymer complexes. Fading of fluorescence may serve to identify particular intracellular biopolymers in individual cells even when present in a heterogeneous population.

A kinetic model for static bioassay of insecticides

Static bioassays, although widely used in evaluating the effects of toxins in aquatic environments, provide rather imprecise data, especially when used with relatively insoluble toxins. A mathematical model of such a system was developed, assuming only first order chemical kinetics. The model was tested using DDT as toxin and stoneflies as assay organisms. Monitoring of DDT concentrations in test and control aquaria and fitting the data to the model permitted the calculation of mean dosage per insect, thus yielding relatively precise and useful toxicological information.

Ignition in the laminar wake of a flat plate

The interaction of a cool, combustible mixture with its hot combustion products is studied from the point of view of determining where the first local maximum in the temperature profile occurs with respect to the initial contact point between the two streams. This work represents a modifieation of Marble and Adamson's study of a similar interaction in which the effect of preformed velocity boundary layers in the two flows was not considered. Marble and Adamson determined this “ignition length” for a system with first order chemical kinetics assuming the velocities of the two streams were uniform before mixing. They found that this length depended linearly on the velocity of the hot stream and exponentially on the hot stream temperature. The inclusion of non-uniform velocity boundary layers considerably changes the results. There is a foreshortening of the temperature and concentration fields both toward the origin and the x 1 -axis. The distance to the first bulge, for the same combustible, is proportional to the slope of the velocity profiles at the initial contact point between the two streams. Furthermore, the “ignition length” becomes more sensitive to velocity and temperature changes than in the uniform flow. Nondimensional scaling factors are discussed which show the dominance of free stream velocity and chemistry compared to geometry in determining the “ignition length”.