A modified arrhenius equation

Pd-Catalyzed Decarbonylative sp2 C-H Arylation: Construction of Five- and Six-Membered (Hetero)Cyclic Compounds

The cyclic compounds have wide applications in the design and synthesis of drugs and materials; thus, their efficient construction attracts much attention from the synthetic community. In this letter, we report an efficient method for preparing cyclic compounds starting from the readily available carboxylic acids. This reaction takes place through intramolecular decarbonylative sp2 C-H arylation, enabling efficient synthesis of a wide range of five- and six-membered cyclic compounds. Both carbo- and heterocycles can be produced under the reaction conditions. Moreover, this reaction features a wide substrate scope with high functional group tolerance. The scale-up experiments also show its practicality in organic synthesis. Those experimental results indicate that this reaction would find wide applications in the synthetic community.

New prebiotic molecules in the interstellar medium from the reaction between vinyl alcohol and CN radicals: unsupervised reaction mechanism discovery, accurate electronic structure calculations and kinetic simulations

Vinyl alcohol (VyA) and cyanide (CN) radicals are relatively abundant in the interstellar medium (ISM). VyA is the enolic tautomer of acetaldehyde and has two low-lying conformers, characterized by the syn or anti placement of hydroxyl hydrogen with respect to the double bond. In this paper, we present a gas-phase model of the barrierless reactions of both VyA's conformers with CN employing accurate quantum chemical computations in the framework of a master equation approach based on the transition state theory. Our results indicate that both VyA conformers feature a similar reactivity with CN, starting with a barrierless addition to the double bond and followed by different isomerization, dissociation, and/or hydrogen elimination steps. The rate constants computed for temperatures up to 600 K show that several reaction channels are open even under the harsh conditions of the ISM, with the favoured one providing the first feasible formation route of a prebiotic molecule not yet detected in the ISM, namely cyanoacetaldehyde. This finding suggests looking for cyanoacetaldehyde in regions where both VyA and CN have already been detected, like, e.g., Sagittarius B2N or G+0.693-0.027.

A Review on a Class of Second Order Nonlinear Singular BVPs

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most cases. Since the exact solution does not exist, it is natural to look for the existence of the analytical solution and numerical solution. In this survey, we focus on both aspects of nonlinear singular boundary value problems (SBVPs) and cover different analytical and numerical techniques which are developed to deal with a class of nonlinear singular differential equations − ( p ( x ) y ′ ( x ) ) ′ = q ( x ) f ( x , y , p y ′ ) for x ∈ ( 0 , b ) , subject to suitable initial and boundary conditions. The monotone iterative technique has also been briefed as it gained a lot of attention during the last two decades and it has been merged with most of the other existing techniques. A list of SBVPs is also provided which will be of great help to researchers working in this area.

From the Kinetic Theory of Gases to the Kinetics of Rate Processes: On the Verge of the Thermodynamic and Kinetic Limits

A variety of current experiments and molecular dynamics computations are expanding our understanding of rate processes occurring in extreme environments, especially at low temperatures, where deviations from linearity of Arrhenius plots are revealed. The thermodynamic behavior of molecular systems is determined at a specific temperature within conditions on large volume and number of particles at a given density (the thermodynamic limit): on the other side, kinetic features are intuitively perceived as defined in a range between the extreme temperatures, which limit the existence of each specific phase. In this paper, extending the statistical mechanics approach due to Fowler and collaborators, ensembles and partition functions are defined to evaluate initial state averages and activation energies involved in the kinetics of rate processes. A key step is delayed access to the thermodynamic limit when conditions on a large volume and number of particles are not fulfilled: the involved mathematical analysis requires consideration of the role of the succession for the exponential function due to Euler, precursor to the Poisson and Boltzmann classical distributions, recently discussed. Arguments are presented to demonstrate that a universal feature emerges: Convex Arrhenius plots (super-Arrhenius behavior) as temperature decreases are amply documented in progressively wider contexts, such as viscosity and glass transitions, biological processes, enzymatic catalysis, plasma catalysis, geochemical fluidity, and chemical reactions involving collective phenomena. The treatment expands the classical Tolman’s theorem formulated quantally by Fowler and Guggenheim: the activation energy of processes is related to the averages of microscopic energies. We previously introduced the concept of “transitivity”, a function that compactly accounts for the development of heuristic formulas and suggests the search for universal behavior. The velocity distribution function far from the thermodynamic limit is illustrated; the fraction of molecules with energy in excess of a certain threshold for the description of the kinetics of low-temperature transitions and of non-equilibrium reaction rates is derived. Uniform extension beyond the classical case to include quantum tunneling (leading to the concavity of plots, sub-Arrhenius behavior) and to Fermi and Bose statistics has been considered elsewhere. A companion paper presents a computational code permitting applications to a variety of phenomena and provides further examples.

"Transitivity": A Code for Computing Kinetic and Related Parameters in Chemical Transformations and Transport Phenomena

The Transitivity function, defined in terms of the reciprocal of the apparent activation energy, measures the propensity for a reaction to proceed and can provide a tool for implementing phenomenological kinetic models. Applications to systems which deviate from the Arrhenius law at low temperature encouraged the development of a user-friendly graphical interface for estimating the kinetic and thermodynamic parameters of physical and chemical processes. Here, we document the Transitivity code, written in Python, a free open-source code compatible with Windows, Linux and macOS platforms. Procedures are made available to evaluate the phenomenology of the temperature dependence of rate constants for processes from the Arrhenius and Transitivity plots. Reaction rate constants can be calculated by the traditional Transition-State Theory using a set of one-dimensional tunneling corrections (Bell (1935), Bell (1958), Skodje and Truhlar and, in particular, the deformed ( d -TST) approach). To account for the solvent effect on reaction rate constant, implementation is given of the Kramers and of Collins-Kimball formulations. An input file generator is provided to run various molecular dynamics approaches in CPMD code. Examples are worked out and made available for testing. The novelty of this code is its general scope and particular exploit of d -formulations to cope with non-Arrhenius behavior at low temperatures, a topic which is the focus of recent intense investigations. We expect that this code serves as a quick and practical tool for data documentation from electronic structure calculations: It presents a very intuitive graphical interface which we believe to provide an excellent working tool for researchers and as courseware to teach statistical thermodynamics, thermochemistry, kinetics, and related areas.

Temperature Dependence of Rate Processes Beyond Arrhenius and Eyring: Activation and Transitivity

Advances in the understanding of the dependence of reaction rates from temperature, as motivated from progress in experiments and theoretical tools (e. g., molecular dynamics), are needed for the modeling of extreme environmental conditions (e.g., in astrochemistry and in the chemistry of plasmas). While investigating statistical mechanics perspectives (Aquilanti et al., 2017b, 2018), the concept of transitivity was introduced as a measure for the propensity for a reaction to occur. The Transitivity plot is here defined as the reciprocal of the apparent activation energy vs. reciprocal absolute temperature. Since the transitivity function regulates transit in physicochemical transformations, not necessarily involving reference to transition-state hypothesis of Eyring, an extended version is here proposed to cope with general types of transformations. The transitivity plot permits a representation where deviations from Arrhenius behavior are given a geometrical meaning and make explicit a positive or negative linear dependence of transitivity for sub- and super-Arrhenius cases, respectively. To first-order in reciprocal temperature, the transitivity function models deviations from linearity in Arrhenius plots as originally proposed by Aquilanti and Mundim: when deviations are increasingly larger, other phenomenological formulas, such as Vogel-Fulcher-Tammann, Nakamura-Takayanagi-Sato, and Aquilanti-Sanches-Coutinho-Carvalho are here rediscussed from the transitivity concept perspective and with in a general context. Emphasized is the interest of introducing into this context modifications to a very successful tool of theoretical kinetics, Eyring's Transition-State Theory: considering the behavior of the transitivity function at low temperatures, in order to describe deviation from Arrhenius behavior under the quantum tunneling regime, a "d-TST" formulation was previously introduced (Carvalho-Silva et al., 2017). In this paper, a special attention is dedicated to a derivation of the temperature dependence of viscosity, making explicit reference to feature of the transitivity function, which in this case generally exhibits a super-Arrhenius behavior. This is of relevance also for advantages of using the transitivity function for diffusion-controlled phenomena.

Recent Developments in Dispersive Kinetics

In general, chemical reactions proceeding on time scales comparable to, or shorter than, those of internal rearrangements in a reaction system renewing the environment of the reactants (mixing), are dispersive. For dispersive kinetics, as for dispersive transport and dispersive relaxation, many time scales coexist. The rate coefficients for dispersive processes depend on time. For a time-dependent specific reaction rate, using the concept of energy profile along the reaction path, one finds the potential energy barrier separating reactants from products to evolve in time during the course of reaction. The evolution of the energy barrier during the course of reaction is described in terms of energy distribution functions related directly to the distribution function of logarithms of lifetimes calculable from kinetic equations with a time-dependent specific reaction rate. This phenomenological approach is compared with that in which the kinetic equations with time-dependent specific reaction rates are interpreted in terms of the superposition of classical reaction patterns. Special attention is paid to renor-malization of rate coefficients following from the stochastic theory of renewals (structural relaxation) in the reaction system. This phenomenological approach to kinetics is taken as a convenient basis to present a number of comprehensive models of dispersive kinetics developed in the 1990s and to discuss some recently published experimental data to show what one derives directly from experimental data and what the detailed mechanistic models have to account for to be acceptable.