scholarly journals Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem

Physics ◽  
2019 ◽  
Vol 1 (2) ◽  
pp. 229-253
Author(s):  
Sergey Adzhiev ◽  
Janina Batishcheva ◽  
Igor Melikhov ◽  
Victor Vedenyapin
Keyword(s):  
Activation Energy ◽  
Reaction Rate ◽  
Detailed Balance ◽  
Kinetic Equations ◽  
Rate Coefficients ◽  
Arrhenius Law ◽  
Interacting Particles ◽  
H Theorem ◽  
Fragmentation Type ◽  
Frequency Functions

The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance is shown to be necessary to monitor, since it is proved that for accepted frequency functions, the condition of detailed balance is fulfilled and the H-theorem is valid. It is shown that for particular and very important cases, the H-theorem holds: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are found to be essential. In addition, based on the connection of the principle of detailed balance with the Boltzmann equation for the probability of state, the expressions for the reaction rate coefficients are obtained.

Lithium tracer diffusion in LiNi0.33Mn0.33Co0.33O2 cathode material for lithium-ion batteries

2021 ◽  
Vol 23 (10) ◽  
pp. 5992-5998
Author(s):  
Daniel Uxa ◽  
Helen J. Holmes ◽  
Kevin Meyer ◽  
Lars Dörrer ◽  
Harald Schmidt
Keyword(s):  
Activation Energy ◽  
Cathode Material ◽  
Lithium Ion Batteries ◽  
Lithium Ion ◽  
Tracer Diffusion ◽  
Arrhenius Law

Lithium tracer diffusivities in LiNi0.33Mn0.33Co0.33O2 cathode material for lithium-ion batteries follows the Arrhenius law with an activation energy of 0.85 eV.

scholarly journals Fluoride Leaching of Titanium from Ti-Bearing Electric Furnace Slag in [NH4+]-[F−] Solution

Metals ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 1176
Author(s):  
Fuqiang Zheng ◽  
Yufeng Guo ◽  
Feng Chen ◽  
Shuai Wang ◽  
Jinlai Zhang ◽  
...  
Keyword(s):  
Activation Energy ◽  
Electric Furnace ◽  
Kinetic Equations ◽  
Surface Chemical ◽  
Leaching Rate ◽  
Surface Chemical Reaction ◽  
Mechanism And Kinetics ◽  
Semi Empirical ◽  
Kinetics Of ◽  
Furnace Slag

The effects of F− concentration, leaching temperature, and time on the Ti leaching from Ti-bearing electric furnace slag (TEFS) by [NH4+]-[F−] solution leaching process was investigated to reveal the leaching mechanism and kinetics of titanium. The results indicated that the Ti leaching rate obviously increased with the increase of leaching temperature and F− concentration. The kinetic equation of Ti leaching was obtained, and the activation energy was 52.30 kJ/mol. The fitting results of kinetic equations and calculated values of activation energy both indicated that the leaching rate of TEFS was controlled by surface chemical reaction. The semi-empirical kinetics equation was consistent with the real experimental results, with a correlation coefficient (R2) of 0.996. The Ti leaching rate reached 92.83% after leaching at 90 °C for 20 min with F− concentration of 14 mol/L and [NH4+]/[F−] ratio of 0.4. The leaching rates of Si, Fe, V, Mn, and Cr were 94.03%, 7.24%, 5.36%, 4.54%, and 1.73%, respectively. The Ca, Mg, and Al elements were converted to (NH4)3AlF6 and CaMg2Al2F12 in the residue, which can transform into stable oxides and fluorides after pyro-hydrolyzing and calcinating.

scholarly journals Kinetic determination of As(III) in solution

2003 ◽  
Vol 68 (10) ◽  
pp. 765-769
Author(s):  
Sofija Rancic ◽  
Rangel Igov ◽  
Todor Pecev
Keyword(s):  
Acid Solution ◽  
Relative Error ◽  
Reaction Rate ◽  
Kinetic Method ◽  
Kinetic Equations ◽  
Strong Acid ◽  
Kinetic Determination

A new reaction is suggested and a new kinetic method is elaborated for the As(HI) traces determination in solution, on the basis of their catalyzing effect on komplexon III (EDTA) oxidation by KMnO4 in a strong acid solution (H2SO4). Using a spectrophotometric technique, a sensitivity of 72 ng/cm3 As(IIl) was achieved. The relative error of method varies from 5.5 to 13.9 % for As(HT) concentration range from 83 to 140 ng/cm-1. Appropriate kinetic equations are formulated and the influence of some other ions, including the As(V), upon the reaction rate is tested.

scholarly journals Equilibrium Spin Distribution From Detailed Balance

2021 ◽  
Vol 81 (9) ◽  
Author(s):  
Ziyue Wang ◽  
Xingyu Guo ◽  
Pengfei Zhuang
Keyword(s):  
Spin Polarization ◽  
Transport Theory ◽  
Axial Vector ◽  
Detailed Balance ◽  
Kinetic Equations ◽  
Distribution Functions ◽  
Spin Distribution ◽  
Balance Principle ◽  
Quark Level ◽  
Vector Distribution

AbstractAs the core ingredient for spin polarization, the equilibrium spin distribution function that eliminates the collision terms is derived from the detailed balance principle. The kinetic theory for interacting fermionic systems is applied to the Nambu–Jona-Lasinio model at quark level. Under the semi-classical expansion with respect to $$\hbar $$ ħ , the kinetic equations for the vector and axial-vector distribution functions are obtained with collision terms. For an initially unpolarized system, spin polarization can be generated at the first order of $$\hbar $$ ħ from the coupling between the vector and axial-vector charges. Different from the classical transport theory, the collision terms in a quantum theory vanish only in global equilibrium with Killing condition.

scholarly journals Apparent Activation Energy in Electrochemical Multistep Reactions: A Description via Degrees of Rate Control

2020 ◽  
Author(s):  
Alfredo Calderón-Cárdenas ◽  
Enrique A. Paredes-Salazar ◽  
Hamilton Varela
Keyword(s):  
Activation Energy ◽  
Catalytic Activity ◽  
Apparent Activation Energy ◽  
Chemical Kinetics ◽  
Rate Control ◽  
Rate Coefficients ◽  
Intrinsic Activity ◽  
Empirical Parameter ◽  
Heterogenous Catalysis ◽  
Multistep Reactions

<div> <div> <div> <p>Activation energy is a well-known empirical parameter in chemical kinetics that characterises the dependence of the chemical rate coefficients on the temperature and provides information to compare the intrinsic activity of the catalysts. However, the determination and interpretation of the apparent activation energy in multistep reactions is not an easy task. For this purpose, the concept of degree of rate control is convenient, which comprises a mathematical approach for analyzing reaction mechanisms and chemical kinetics. Although this concept has been used in catalysis, it has not yet been applied in electrocatalytic systems, whose ability to control the potential across the solid/liquid interface is the main difference with heterogenous catalysis, and the electrical current is commonly used as a measure of the reaction rate. Herein we use the definition of ‘degree of rate control for elementary step’ to address some of the drawbacks that frequently arise with interpreting apparent activation energy as a measure of intrinsic electrocatalytic activity of electrode. For this, an electrokinetic model Langmuir-Hinshelwood-like is used for making numerical experiments and verifying the proposed ideas. The results show that to improve the catalytic activity of an electrode material, it must act upon the reaction steps with the highest normalised absolute values of degree of rate control. On the other hand, experiments at different applied voltages showed that if the electroactive surface poisoning process take place, changes in 𝐸𝑎𝑝𝑝 can not be used to compare the catalytic activity of the electrodes. Finally, the importance of making measurements at steady-state to avoid large errors in the calculations of apparent activation energy is also discussed. </p> </div> </div> </div>

Zur Theorie der Elektronenübergangsreaktionen in molekularen Komplexen / Theory of Electron Transfer Reactions in Molecular Complexes

1974 ◽  
Vol 29 (6) ◽  
pp. 880-887 ◽  
Author(s):  
P. P. Schmidt
Keyword(s):  
Activation Energy ◽  
Electron Transfer ◽  
Charge Transfer ◽  
Rate Constant ◽  
Reaction Rate ◽  
Charge Transfer Complex ◽  
Rate Theory ◽  
Transfer Step ◽  
Donor And Acceptor ◽  
Inner Sphere

This paper reports a theory of the inner sphere-type electron transfer reaction. Inner sphere reactions, as opposed to the outer sphere variety, require that the solvate or ligand shells surrounding the electron donor and acceptor species undergo considerable change in the course of the electron transfer. In this paper we assume that the electron transfer step takes place in a molecular complex which exists in equilibrium with the reactants. The electron transfer step occurs as a non-radiative charge transfer-type transition. In this manner we treat the charge transfer kinetics, in particular, the evaluation of the reaction rate constant, in the same manner as is usual for non-radiative problems. The analysis leading to the rate constant expression is based on Yamamoto’s general chemical reaction rate theory. The rate constant expressions obtained are quite general, they hold for any degree of strength of coupling between subsystems comprising the entire system. The activation energy, in the Arrhenius form for the rate constant, shows a dependence on the energy (work) of formation of the intermediate charge transfer complex, on vibrational shift energies associated with the molecular motions of the ligands, and on solvent repolarization energies. The activation energy also shows an important dependence on coupling terms which link the vibrations of the molecular inner shell with the polarization states of the (assumed) dielectric continuum which surrounds the charge transfer participants. The approach we take in developing this theory we believe points the way towards the development of a more complete theory capable of accounting for the dynamics of the molecular reorganization leading to the intermediate charge transfer complex as well as accounting for the electron transfer step itself.

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