scholarly journals REEXAMINATION OF ASTROPHYSICAL RESONANCE-REACTION-RATE EQUATIONS FOR AN ISOLATED, NARROW RESONANCE

2011 ◽  
Vol 20 (01) ◽  
pp. 165-172
Author(s):  
J. J. HE ◽  
J. HU ◽  
L. Y. ZHANG ◽  
L. LI ◽  
S. W. XU ◽  
...  
Keyword(s):  
Reaction Rate ◽  
Charged Particles ◽  
Rate Equations ◽  
Broad Resonance ◽  
Narrow Resonance ◽  
Resonant Reaction ◽  
Stellar Temperatures ◽  
Definition Of ◽  
Resonance Equation

The well-known astrophysical resonant-reaction-rate (RRR) equations for an isolated narrow resonance induced by the charged particles have been reexamined. The validity of those assumptions used in deriving the classical analytic equations has been checked, and found that these analytic equations only hold for certain circumstances. It shows the customary definition of "narrow" is inappropriate or ambiguous in some sense, and it awakes us not to use those analytic equations without caution. As a suggestion, it is better to use the broad-resonance equation to calculate the RRR numerically even for a narrow resonance of a few keV width. The present conclusion may influence some work in which the classical narrow-resonant equations were used for calculating the RRRs, especially at low stellar temperatures for those previously defined "narrow" resonances.

Reactions and Reactors Basic Concepts

2001 ◽  
Author(s):  
L. K. Doraiswamy
Keyword(s):  
Reaction Time ◽  
Reaction Rate ◽  
Batch Reactor ◽  
Reaction Rates ◽  
Heterogeneous Reactions ◽  
Rate Equations ◽  
Continuous Reactor ◽  
Rigorous Treatment ◽  
Reactor Types ◽  
Definition Of

Organic synthesis is replete with countless classes of reactions, including several that are named after their discoverers (the name reactions), but fortunately they can all be conducted in less than a half-dozen broad types of reactors. Choosing a reactor for a given reaction is based on several considerations and combines reaction analysis with reactor analysis. Thus in this chapter we consider the following aspects of reactions and reactors, much of which should serve as an introduction to chemists and a refresher to chemical engineers: reaction rates, stoichiometry, and rate equations; the basic reactor types, as a prelude to a more rigorous treatment of these in Parts III and IV; transport of mass (represented by reactant and product molecules) and heat across phase boundaries for heterogeneous reactions; and types of laboratory reactors used by chemists and chemical engineers for their specific objectives. The first step in any consideration of reaction rates is the definition of reaction time. This depends on the mode of reactor operation, batch or continuous. For the batch reactor, the reaction time is the elapsed time; whereas for the continuous reactor, it is given by the time the reactant spends in the reactor, called the residence time, that is measured by the ratio of reactor volume to flow rate (volume/volume per unit time with units of time). An equally important consideration is the concept of reaction space (which can have units of volume, surface, or weight), leading to different definitions of the reaction rate. We begin this section by considering different ways of defining the reaction rate based on different definitions of reaction time and space. The basis of all reactor design is an equation for the reaction rate.

Potentiodynamic examination of electrode kinetics for electroactive adsorbed species: applications to the reduction of noble metal surface oxides

1969 ◽  
Vol 310 (1503) ◽  
pp. 541-563 ◽  
Keyword(s):  
Noble Metals ◽  
Reaction Rate ◽  
Substrate Surface ◽  
Intrinsic Property ◽  
Surface Heterogeneity ◽  
Surface Oxide ◽  
Rate Equations ◽  
Quasi Equilibrium ◽  
Linear Potential ◽  
Reaction Velocity

The relation between reaction rate and potential (or time) for electrochemical surface processes occurring under potentiodynamic control (linear potential-time programme) has been investigated with particular reference to the behaviour of thin surface oxide films on noble metals. The kinetics of processes involving adsorbed electroactive species are treated for several model cases; the rate equations are developed for mechanisms involving various reaction orders or for processes involving adsorbed reactant interactions and surface heterogeneity effects. By examination of the dependence of the reaction rate (current) with time and the effect of potential scan rate, v , on the maximum reaction velocity and the potential at which it occurs, the models may be distinguished. In this manner, the inter­dependence of v and the reaction velocity constants k a and k c for the anodic oxidation and the cathodic reduction processes respectively, can be quantitatively established. The relation between quasi-equilibrium situations where the reverse reaction is significant and irreversible situations where it is not can be demonstrated. Heterogeneity terms introduced into the kinetic relations express deviations from Langmuir adsorption behaviour and may be an intrinsic property of the substrate surface or a property of the adsorbed reactant (induced heterogeneity). Applications of the treatment are made to reduction of surface oxide species at the noble metals and the significance of hysteresis and time effects in the processes of electrochemical formation and reduction of surface oxide at platinum, rhodium, iridium and palladium is investigated.

Kinetic Studies on the Synthesis of Hydroxyapatite Nanowires by Solvothermal Methods

2007 ◽  
Vol 60 (2) ◽  
pp. 99 ◽  
Author(s):  
Shiying Zhang ◽  
Chen Lai ◽  
Kun Wei ◽  
Yingjun Wang
Keyword(s):  
Activation Energy ◽  
Reaction Rate ◽  
Kinetic Studies ◽  
Axial Ratio ◽  
Reaction Rate Constant ◽  
Rate Equations ◽  
Solvothermal Reaction ◽  
Growth Order ◽  
Constant Diffusion ◽  
Solvothermal Methods

Hydroxyapatite nanowires with a high axial ratio have been synthesized in reverse micelle solutions that consist of cetyltrimethylammonium bromide (CTAB), n-pentanol, cyclohexane, and the reactant solution by solvothermal methods. This paper focusses on the kinetic studies of the solvothermal reaction and the linear growth of hydroxyapatite nanowires. When the reaction was carried out at low temperatures (65°C), the experimental results showed that the reaction rate was of zero order since the whole reaction was diffusion controlled with constant diffusion coefficients. In the middle to high temperature range (130–200°C), the kinetics were characterized by second order reaction kinetics. Since the controlling factor was activation energy and the apparent activation energy was large, the reaction rate was more sensitive to the temperature. Therefore, the exponent of the reaction rate constant increased by two when the temperature was increased from 130 to 200°C. By calculating the yields of products and the specific surface areas at different times, the linear and overall growth rate equations of the hydroxyapatite nanowires could be obtained. The experimental effective growth order of the crystals was 11. The larger growth order indicated that the crystal could grow more effectively in one direction because of the induction of the surfactant in the experiment system.

Modeling of Gas Turbine Combustors—A Convenient Reaction Rate Equation

1972 ◽  
Vol 94 (3) ◽  
pp. 173-180 ◽  
Author(s):  
D. Kretschmer ◽  
J. Odgers
Keyword(s):  
Rate Equation ◽  
Reaction Rate ◽  
Reaction Order ◽  
Bimolecular Reaction ◽  
Rate Equations ◽  
Combustion System ◽  
Wide Range ◽  
Simple Mass ◽  
Rich Mixtures ◽  
Reaction Rate Equation

In order to model a practical combustion system successfully, it is necessary to develop one or more reaction rate equations which will describe performance over a wide range of conditions. The equations should be kept as simple as possible and commensurate with the accuracy needed. In this paper a bimolecular reaction is assumed, based upon a simple mass balance. Temperatures derived from the latter are related to measured practical ones such that, if required, an evaluation of the partly burned product composition can be made. A convenient reaction rate equation is given which describes a wide range of blow-out data for spherical reactors at weak mixture conditions. NVP2φ={1.29×1010(m+1)[5(1−yε)]φ[φ−yε]φe−C/(Ti+εΔT)}/{0.082062φyε[5(m+1)+φ+yε]2φ[Ti+εΔT]2φ−0.5} Analysis of the components used in the above equation (especially the variation of activation energy) clearly shows its empirical nature but does not detract from its engineering value. Rich mixtures are considered also, but lack of data precludes a reliable analysis. One of the major results obtained is the variation of the reaction order (n) with equivalence ratio (φ): weak mixtures, n = 2φ; rich mixtures, n = 2/φ. Some support for this variation has been noticed in published literature of other workers.

Computer-based derivation of rate equations for enzyme-catalyzed reactions. II. Rate equations for isotopic exchange

1970 ◽  
Vol 48 (8) ◽  
pp. 922-934 ◽  
Author(s):  
Arthur R. Schulz ◽  
Donald D. Fisher
Keyword(s):  
Initial Rate ◽  
Maximum Velocity ◽  
Rate Equations ◽  
Exchange Constant ◽  
Exchange Constants ◽  
Computer Based ◽  
Inhibition Constants ◽  
Catalyzed Reactions ◽  
Definition Of ◽  
Enzyme Catalyzed Reactions

A computer-based method is employed for the reformulation of rate equations for enzyme-catalyzed reactions from the coefficient form to the kinetic form. This method is applied to equations for the initial rate of enzyme-catalyzed isotope exchange. In the reformulated equations, the coefficients of each rate equation term are expressed as maximum velocity of the initial rate of the net reaction, Michaelis constants, inhibition constants, and exchange constants. The definition of the exchange constant for a given reactant may be identical to one of the inhibition constants for that reactant.

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