Hyperbolic law

Later, the kinetics of the ITL of /i-irradiated vitreous solutions of Ph2 in methylcyclohexane was studied [55] over a much wider time interval (10 6 -103 s). Within this whole time interval the kinetics of ITL was found to obey one and the same hyperbolic law, i.e. eqn. (7) with m 1. These results are difficult to interpret in terms of conventional kinetic models, but are easy to account for in terms of the electron tunneling model. Indeed, as shown in Chap. 4, the drop in the intensity of recombinational luminescence in the case of the tunneling mechanism of recombination obeys the equation... 

Let there be the ensemble of particles reacting between themselves under the first-order kinetic law but with differing activation energies of transformation. What will be the shape of the activation energy distribution function of particles if it is known that the distribution function depends on the transformation rate constant according to a hyperbolic law n(k) l/k (kmm< k < Armax) ... 

In the case of constant viscosity, the thickness of a layer at a given point varies with time according to the hyperbolic law... 

From a statistical viewpoint, there is often little to choose between power law and hyperbohc equations as representations of data over an experimental range. The fact, however, that a particular hyperbolic equation is based on some land of possible mechanism may lead to a belief that such an equation may be extrapolated more safely outside the experimental range, although there may be no guarantee that the controlling mechanism will remain the same in the extrapolated region. 

J.B. Bell, P. Colella, and J.A. Trangenstein, Higher Order Godunov Methods for General Systems of Hyperbolic Conservation Laws, J. Comput. Phys. 82 (1989). 

I.L. Chern and P. Colella, A Conservative Front Tracking Method for Hyperbolic Conservation Laws, UCRL-97200, Lawrence Livermore National Laboratory, Livermore, CA, 1987. 

P. Colella, Multidimensional Upwind Methods for Hyperbolic Conservation Laws, LBL-17023, Lawrence Berkley Laboratory, Berkley, CA, 1984. 

Rules. Eliminate temperature terms in the denominator. (Terms with negative exponents in the power law model are considered to belong to the denominator, in the hyperbolic model. Author.)... 

Table (a) shows experimental data [24] for the initial charge density exiting a fuel filter Qq plus the charge density Q remaining 30 s downstream. At low conductivity the charge decays much faster than predicted by an exponential relaxation law [Eq. (2-3.7)] and instead follows a hyperbolic relaxation law [24] given by... 

Hyperbolic expansion The expansion of a fluid according to the law pV = Hypothalamus The temperature control center at the base of the brain, which regulates body temperature. Hypothermia The physiological state resulting when the deep core body temperature drops below 35 C. It results in vasoconstriction and shivering in an attempt to conserve body heat. 

It has been demonstrated that lin-log equations capture hyperbolic kinetics slightly better than either a linear approximation or the power-law approach [318, 320 322]. One reason for the improved performance is that for lin-log kinetics the elasticities (and kinetic orders) are not constant, but change with changing metabolite concentrations. For a monosubstrate reaction v(Sj, and omitting the dependence on the enzyme concentration, we obtain... 

The limiting cases are limvo 0 a = 1 and limy. x a = 0. To evaluate the saturation matrix we restrict each element to a well-defined interval, specified in the following way As for most biochemical rate laws na nt 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient = = ,> 1, the saturation parameter is restricted to the interval [0, n] and, analogously, to the interval [0, n] for inhibitory interaction with na = 0 and n = , > 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. 

Claims of perpetual motion create moments of mirth and consternation for those knowledgeable in the laws of thermodynamics. Yet, is it only hyperbole when a responsible journal such as the European Plastics News [1] proclaims that depolymerization of polyethylene terephthalate (PET) can be repeated indefinitely The second law of thermodynamics brings us back to reality. The depolymerization of PET does not operate at 100% yields, but does offer the opportunity for near-stoichiometric recovery of the monomers used to make the polyester. With high yields of potentially valuable monomers, the commercial potential for polyester depolymerization to regain feedstocks must be considered. 

The relationship between substrate concentration ([S]) and reaction velocity (v, equivalent to the degree of binding of substrate to the active site) is, in the absence of cooperativity, usually hyperbolic in nature, with binding behavior complying with the law of mass action. However, the equation describing the hyperbolic relationship between v and [S] can be simple or complex, depending on the enzyme, the identity of the substrate, and the reaction conditions. Quantitative analyses of these v versus [S] relationships are referred to as enzyme kinetics. 

The law of mass action describes the hyperbolic relationship between binding (B) and ligand concentration (c). This relationship is characterized by the drug s affinity (1/Itotal number of binding sites per unit of weight of membrane homogenate. 

If the recombination is delayed, e.g., by migration of excited electrons, luminescence takes place by a second-order bimolecular reaction. The probability of a luminescent recombination of the excited electron with the holes is then proportional to the product of the concentration of electrons and the concentration of holes. The lower the initial intensity is, and the further the decay has progressed, the slower the decay to the half value is. This hyperbolic decay law is only of limited validity. If the excited electron is momentarily trapped before recombination, very complex interactions can arise. 

Edmond Becquerel (1820-1891) was the nineteenth-century scientist who studied the phosphorescence phenomenon most intensely. Continuing Stokes s research, he determined the excitation and emission spectra of diverse phosphors, determined the influence of temperature and other parameters, and measured the time between excitation and emission of phosphorescence and the duration time of this same phenomenon. For this purpose he constructed in 1858 the first phosphoroscope, with which he was capable of measuring lifetimes as short as 10-4 s. It was known that lifetimes considerably varied from one compound to the other, and he demonstrated in this sense that the phosphorescence of Iceland spar stayed visible for some seconds after irradiation, while that of the potassium platinum cyanide ended after 3.10 4 s. In 1861 Becquerel established an exponential law for the decay of phosphorescence, and postulated two different types of decay kinetics, i.e., exponential and hyperbolic, attributing them to monomolecular or bimolecular decay mechanisms. Becquerel criticized the use of the term fluorescence, a term introduced by Stokes, instead of employing the term phosphorescence, already assigned for this use [17, 19, 20], His son, Henri Becquerel (1852-1908), is assigned a special position in history because of his accidental discovery of radioactivity in 1896, when studying the luminescence of some uranium salts [17]. 

It has been demonstrated that the oxidation of alcohols with hexacyanoferrate(III) (HCF) shows a hyperbolic variation with HCF concentration, and the reaction order varies from one to zero on increasing the HCF concentration. This rate law is obeyed during the initial moments of the reaction and at any subsequent time. These results rule out the possibility that any substance produced during the course of the reaction acts as an activator or inhibitor of the reaction rate. The mixed order has been attributed to the comparable rates of complex decomposition and catalyst regeneration steps.86 HCF acts as a selective oxidizing agent for the oxidation of catechols even in the presence of 2-mercaptobenzoxazole, as an easily oxidizable thiol, to produce related catechol thio ethers.87 Hexacyanoferrate(II) has a retarding effect on the oxidation of vanillin with HCF in alkaline solutions. A mechanism based on the observed kinetics has been proposed 88... 

Tools To determine the accuracy of a derived calibration law, computed masses of ions are compared to known values by using the calibration shown in equation (2) for the cubic cell or that shown in equation (3) for the hyperbolic. 

The structure of the paper itself affects the behavior with respect to Beer s law which establishes a relation between extinction E and concentration C of transparent solutions, according to the expression E = kC. When extinction measured on paper is plotted against concentration, not a linear but a hyperbolic curve is obtained (CIO, Cll). The cause of the error is the sievelike structure of the paper, since only the threads are covered with stained proteins while the meshes remain completely permeable to light (Fig. 34). This light falling directly on the photosensitive layer of the cell gives, on the microscopic scale, the same error as found for uneven distribution of stained spots on a clear... 

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