Limiting cases

The general analysis, while not difficult, is complicated however, the limiting case of the very elongated, essentially cylindrical drop is not hard to treat. Consider a section of the elongated cylinder of volume V (Fig. II-18h). The centrifugal force on a volume element is u rAp, where w is the speed of revolution and Ap the difference in density. The potential energy at distance r from the axis of revolution is then w r Apfl, and the total potential energy for the... 

Due to such large numbers, it is usefiil to consider the limiting case of the thermodynamic limit, which is defined as... 

We have seen that the DFI theory in the limiting case neglects excluded volume effects in fact the excluded volume of the centra ion can be introduced into the theory as explained after A2.4.48. If the radius of the ions is taken as a for all ions, we have, in first order. 

While the time dependent populations Pj(t) may generally show a complicated behaviour, certain simple limiting cases can be distmguished and characterized by appropriate parameters ... 

Detemiining compositions is possible if the distribution of elements over the outer layers of the sample and the surface morphology is known. Two limiting cases are considered, namely a homogeneous composition tliroughout the outer layers and an arrangement in which one element covers the other. 

There are several practical limitations to the use of equation (B2.3.16) for the detennination of CM angidar distributions. The optimum kinematics for the use of this equation is the case where the speed c of the centre of mass is approximately equal to the product CM speed u hi the limiting case where the latter is small, the... 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

Therefore, in tire limiting case—tire surface concentration of tire reacting species is zero as all tire arriving ions immediately react—tire current density becomes voltage independent and depends only on diffusion, specifically, on tire widtli of tire Nerstian diffusion layer S, and of course tire diffusion coefficient and tire bulk concentration of anions (c). The limiting current density (/ ) is tlien given by... 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

It is also interesting to examine the relative importance of thermal transpiration and thermal diffusion in the two limiting cases. From equations (A. 1.12) and (A. 1.13)... 

In order to interpret the data in Table 5.1 in a quantitative fashion, we analysed the kinetics in terms of the pseudophase model (Figure 5.2). For the limiting cases of essentially complete binding of the dienophile to the micelle (5.If in SDS and 5.1g in CTAB solution) the following expression can be derived (see Appendix 5.2) ... 

Whereas the differentiation of trivalent carbenium and pentacoor-dinated carbonium ions serves a useful purpose in defining them as limiting cases, it should be clear that in carbocationic systems there always exist varying degrees of delocalization. This can involve participation by neighboring -donor atoms, 7r-donor gronps, or [Pg.148]

This method provides a reasonable estimate of the piQ, provided that the weak acid is neither too strong nor too weak. These limitations are easily appreciated by considering two limiting cases. For the first case let s assume that the acid is strong enough that it is more than 50% dissociated before the titration begins. As a result the concentration of HA before the equivalence point is always less than the concentration of A , and there is no point along the titration curve where [HA] = [A ]. At the other extreme, if the acid is too weak, the equilibrium constant for the titration reaction... 

The assumption that k values are constant over the entire duration of the reaction breaks down for termination reactions in bulk polymerizations. Here, as in Sec. 5.2, we can consider the termination process—whether by combination or disproportionation to depend on the rates at which polymer molecules can diffuse into (characterized by kj) or out of (characterized by k ) the same solvent cage and the rate at which chemical reaction between them (characterized by kj.) occurs in that cage. In Chap. 5 we saw that two limiting cases of Eq. (5.8) could be readily identified ... 

Based on considerations we have encountered earlier in this chapter, we can anticipate two limiting cases of this function P(0) approaches unity both in the limit of small particles and in the limit of small angles of observation. Interference is absent in both of these cases. 

On each of the lines that has been drawn, a mark is made at the value of the abscissa corresponding to one of the limiting cases above. Specifically, along the line where C2 = c, a mark is placed where the abscissa has the value kc. Since the abscissa is sin (0/2) + kc2, this mark corresponds to 0 = 0° for this concentration. Similarly, along a line for which 0 = 0, a mark is placed where the abscissa equals sin (6 12). This corresponds to the C2 = 0 limit at this angle. 

Consider the limit case corresponding to = 0 in (2.130). The restriction obtained in such a way describes approximately a mutual nonpenetration of the crack faces. Note that in reality a complete account of the thickness implies the dependence of the energy functional on . This dependence is as follows (Vol mir, 1972) ... 

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