Sphericity parameter

The application of (2.17)-(2.20) is shown in Fig. 2.18. The response depends on the sphericity parameter p = VD/rov7 [27]. Under the influence of increasing parameter p, the minimum of the forward component and the maximum of the backward component gradually vanish and both components acquire the form of a polarographic wave. At potentials much lower than the half-wave potential, both currents tend to the limiting value which is equal to —p. The net peak potential is equal to the reversible half-wave potential and independent of the sphericity parameter, but the dimensionless net peak current is a linear function of the parameter p. If n sw = 50 mV and uAE = -5 mV, this relationship is ... 

At an inlaid microdisk electrode, the dependence of the dimensionless net peak current on the sphericity parameter is given by the following equation [31] ... 

Figure 2.21 shows the dependence of dimensionless net peak currents of ferrocene and ferricyanide on the sphericity parameter (note that A0p = AT], andy = p)-The SWV experiments were performed at three different gold inlaid disk electrodes (ro = 30, 12.5 and 5 pm) and the freqnencies were changed over the range from 20 to 2000 Hz [26]. For ferrocene the relationship between AT], and p is linear A Fp = 0.88 + 0.74p. This indicates that the electrode reaction of ferrocene is elec-trochemically reversible regardless of the frequency and the electrode radius over the range examined. For ferricyanide the dependence of AT], on p appears in sequences. Each seqnence corresponds to a particular value of the parameter The results obtained with the same freqnency, but at different microelectrodes, are cormected with thin, broken lines. The difference in the responses of these... 

Fig. 4.7 Variation of the DDPV peak potential (solid line) and the half-wave potential (dotted line) with respect to the formal potential with the electrode sphericity parameter corresponding to the second pulse (Rq = rs/ D0Ti ) for different values of y indicated on the graph. AE = -50 mV, n = 5 s, t2 = 0.025 s...

One can speak about this convention in terms of the prolate spherical parameter system r. rutRl-... 

Bladon, P. (1989) A rapid method for comparing and matching the spherical parameter surfaces of molecules and other irregular objects. / Mol. Graph. 7, 130-137. 

It is convenient to describe the free fall of nonspherical isotropic particles by using the sphericity parameter... 

This equation expresses very clearly that the transition through linear and convergent diffusion mainly depends on the value of the sphericity parameter T, that is, on the electrode size, the time scale of the experiment and the diffusivity of the electroactive species. Indeed, in the limit of very fast scan rates, a, independently of the geometry of the electrode the voltammetry will show peaked response with a peak current given by the Randles-Sevcfk equation (see Figure 4.7) ... 

Ps = density of the solid particles = fluid density = particle diameter = acceleration of gravity = sphericity parameter of the solid particles mf = void fraction corresponding to minimum fluidization velocity /u = fluid viscosity. 

The theoretical basis for the molecular shape factors was derived in section 6.2. That analysis, which led to temperature-dependent shape factors, represents an idealized case where the non-spherical potential parameters may be incorporated with the spherical parameters through angle averaging. Although that approach is correct in certain circumstances, it is of limited practical use since the intermolecular potential function for real fluids is not known precisely. Hence, one is forced to use macroscopic thermodynamic measurements to determine the shape factors and then try to develop a generalized correlation for them which depend on known molecular parameters. We shall refer to the shape factors determined from experimental data as the apparent or exact shape factors and their generalized correlation as the correlated shape factors. 

P. Bladon, /, Mol. Graphics, 7, 130 (1989). A Rapid Method for Comparing and Matching the Spherical Parameter Surfaces of Molecules and Other Irregular Objects. 

T, and other magnetic parameters like and Pk- They vary fix)m one compound to another and linked to them are the level barycenters. Before introducing any cfp fitting, the barycenters have to be coimected as precisely as possible with the experimental values and therefore the spherical parameters have to be refined at first. Practically, the convergence of the procedure is rather quick, but the fitted parameters interact with each other, and there remains some uncertainty because of their mutual coupling. Judd and Crosswhite (1984) proposed an alternate set of parameters which corresponds to an orthogonalized set of operators for the f shell. The main result is that the conventional Hamiltonian... 

As to spherical parameters, at least the Slater integrals and the spin-orbit coupling constant, it should be possible to evaluate them with precision from the radial part of the f-electron wave function obtained by solving the Hartree or Hartree-Fock equations. Unfortunately, the F calculated this way are 30-50% greater than the fitted ones. It seems that the lanthanides are almost alone (even if there are fourteen elements ) in the Periodic Table in displaying such pronounced misbehaviour. Elements to help understand this may be found in a paper by Rajnak and Wyboume (1963) who showed that the effects of configuration interactions in configurations may be represented by... 

The relationship between the 5 and the spherical parameters may be found in Ng and Newman (1987a). 

Fig. 2.19 Dependence of dimensionless net peak current of reversible reaction (1.1) on the sphericity parameter n sw = 50 mV and nAE = —5 mV...

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