Phenomenological expressions

According to our analytical results on the solid-state redox reaction of LiNi02 based on the phenomenological expression for solid-state redox potentials of insertion electrodes [23], the reaction consists of three redox systems characterized by potentials of 4.23, 3.93, and 3.63V with re-... 

Other interesting features of elastomeric networks can be revealed using the plots of the reduced stress, crred = /( — -2) against inverse extension ratio 1. This can be analyzed from the stress-strain behavior described by a phenomenological expression suggested by Mooney [78] and Rivlin and Saunders [79] ... 

AC susceptibility measurements can yield valuable information on the dynamical properties of a system. The ac susceptibility of H0B22C2N shows frequency dependence, but could not be analyzed satisfactorily by the dynamical scaling theory of a three dimensional spin glass (Mori and Mamiya, 2003). Therefore, a detailed investigation of the behavior of relaxation times by the Cole-Cole analysis (Cole and Cole, 1941) was performed. The complex susceptibility can be phenomenologically expressed as ... 

The equations given in Table 9.1 simply describe the inhibition behaviors of enzymes and thus can be called phenomenological expressions. However, it is important to describe basic mechanisms, in the way that Michaelis and Menten did for a single enzyme. The mechanisms account for the form of the inhibition equations. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

For simplicity we shall neglect the reverse surface reaction and shall employ the phenomenological expression... 

The phenomenological expressions of chemical kinetics for w are discussed in Appendix B and are quoted in Chapter 1. 

Besides some measures of separation efficiency such as the separation factor and extent of separation defined above, some quantity indicative of the throughput rate of a membrane system is needed to compliment the permselectivity of the membrane. It is quite common and practical in the membrane technology to use a phenomenological expression to relate the permeate flux (Ja in the unit of cm (STP)/s-cm7) of a given gas (A) through the membrane to the driving force, the transmembrane pressure difference (Ap) as follows ... 

Simple phenomenological expressions, some involving free volume and entropy, can... 

A phenomenological expression for the single-pulse response in NMR can be derived from (1.1.7) in terms of an attenuated precession of the transverse magnetization around the direction of the magnetic field in a given volume cell, or voxel, in the 3D case, or a picture cell, or pixel, in the 2D case. 

Gierke also considered that these clusters are interconnected by short, narrow channels in the fluorocarbon backbone network. The diameter of these channels is about 1 mm estimated from hydraulic permeability data. He further considered that the Bragg spacing ( 5nm from SAXS data) can represent the distance between clusters. The cluster-network model is a phenomenological description. Recently, Hsu and Gierke " have derived a semi-phenomenological expression to correlate the variation of cluster diameter with water content, equivalent weight, and cation form of the membrane. They have shown that the short channels are thermodynamically stable. 

A more detailed analysis of the thermodynamic influence on ion mixing by Johnson et al. (1985) resulted in the following phenomenological expression... 

It is clear that these equations are closely related to the phenomenological expression equation (7-28) except that the "molecular rotational relaxation time" ris now replaced by an effective relaxation time t where... 

In order to obtain the theoretical expressions for the g- and /-tensor elements, we have to compare the theoretical Hamiltonian given in Eq. (IV.59) with the phenomenological Hamiltonian in Eq. (1.4). For this comparison we have to keep in mind that in the phenomenological expression corresponds to Pyjk in Eq. (IV.59) and that in order to make the phenomenological Hamiltonian Hermitian, products such as cos yZ Jy> must be symmetrisized to (cos yZJy- +Jy cos yZ)j2, etc. 

Farin and Avnir (369, 370) have published a phenomenological expression related to the geometry of fractals, which they write as follows ... 

A nonlinear least-squares program was written to fit the phenomenological expressions eqs 1 and 2 to the spectra of normal incidence (a = 0°) in... 

Kinetic modeling used for process development and process optimization has a historical tradition. Quite often power law models are still used to describe kinetic data. Such phenomenological expressions, although useful for some applications, in general are not reliable, as they do not predict reaction rate, concentration and temperature dependence outside of the range of the studied experimental conditions. Thus, in catalysis, due to the complex nature of this phenomenon, adsorption and desorption of reactants as well as several steps for surface reactions should be taken into account. Models based on the knowledge of elementary processes provide reliable extrapolation outside of the studied interval and also make the process intellectually better understood. 

It seems useful to summarize all optical relaxation data that have been obtained so far. In the preceding sections it was shown that for the temperature dependent part of 7 the following phenomenological expression fitted all presently available data T (7 )= 7 (oo)exp(AE/kT ). For... 

If we now compute the free energy of an interface bent into a sphere and that of one bent into a cylinder, we can extract K, K, and to by comparing these free energies with the phenomenological expressions Eqs. (26) and (27). Such calculations have been carried out both for the bent brush part of the ionic surfactant monolayer and for the bent electrical double layer part of the adsorbed ionic surfactant monolayer. 

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