Asymptotic limit

In addition, the intercept obtained by extrapolating this asymptote back to sin (0/2) = 0 equals (2M )". Note that both Mand are number averages when this asymptotic limit is used. This is illustrated schematically in Fig. 10.15 and indicates that even more information pertaining to polymer characterization can be extracted from an analysis of the curvature in Zimm plots. 

FIG. 7 Proof of the scaling result, (L) oc in the asymptotic limit of sufficiently long chains [65]. 

In this case we have to consider the transformation repeated n times (where n - oo) applied separately to the radius r and to the phase identical transformation to the first approximation. 

The precise formulation of the asymptotic condition can now be given as follows the operator [Pg.616]

A second difficulty which has only been alluded to—or rather neglected—in the present outline is the validity of the assumption that in and out fields exist. The existence of an asymptotic limit might well be rigorously provable for matrix elements of the form <0 r(o ) l particle), but not for more general situations without explicitly removing the Coulomb interaction between the particles. 

The bond energy per atom increases with the size of the cluster and reaches the asymptotic limit for a relatively small n value. 

These treatments of periodic parts of the dipole moment operator are supported by several studies which show that, for large oligomeric chains, the perturbed electronic density exhibits a periodic potential in the middle of the chain whereas the chain end effects are related to the charge transfer through the chain [20-21]. Obviously, approaches based on truncated dipole moment operators still need to demonstrate that the global polarization effects are accounted for. In other words, one has to ensure that the polymeric value corresponds to the asymptotic limit of the oligomeric results obtained with the full operator. 

In general, it is difficult to map contributions from different reaction paths onto the DCS. However, Eq. (6) teUs us that, in a reaction with a Cl, one can easily map the contributions from the e and o (Feynman) paths onto the DCS. Since Eq. (6) applies to the entire wave function, we can apply it to the asymptotic limit of the wave function in Eq. (14), and thus to and to obtain... 

This chapter has focused on reactive systems, in which the nuclear wave function satisfies scattering boundary conditions, applied at the asymptotic limits of reagent and product channels. It turns out that these boundary conditions are what make it possible to unwind the nuclear wave function from around the Cl, and that it is impossible to unwind a bound-state wave function. 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

II provides a transition between the two asymptotic limits. Viscous stresses now scale by the local thickness of the film, h, and the bubble shape varies from the constant thickness film to the spherical segment. Here the surfactant distribution along the interface may be important. Fortunately, for small capillary numbers, dh/dx < 1 and the lubrication approximation may be used throughout. Region II is quantified below. 

Statistical rate theories have been used to calculate rate constants for gas-phase Sn2 reactions.1,7 For a SN2 reaction like Cl" + CH3Clb, which has a central barrier higher than the reactant asymptotic limit (see Figure 1), transition state theory (TST) assumes that the crossing of the central barrier is rate-limiting. Thus, the TST expression for the SN2 rate constant is simply,... 

Case II. Instead, the rate will increase with increasing pressure of one reactant, eventually approaching an asymptotic limit corresponding to saturation of the type of site in question. 

In this case the initial rate increases as the square of the pressure at very low pressures and approaches a constant asymptotic limit at high pressures. 

Essentially, expressions (Eq. 34) have a physical sense only for values of quantities ev which are appreciably less than unity when the eigenvalues of operators Qdiff and Q coincide. The number of such values of sv will be the larger, the longer is the macromolecule. Hence, in the asymptotic limit l oo, expression (Eq. 38) for the two-point chemical correlator is reduced to the following form... 

At low-conversion copolymerization in classical systems, the composition of macromolecules X whose value enters in expression (Eq. 69) does not depend on their length l, and thus the weight composition distribution / ( ) (Eq. 1) equals 5(f -X°) where X° = jt(x°). Hence, according to the theory, copolymers prepared in classical systems will be in asymptotic limit (/) -> oo monodisperse in composition. In the next approximation in small parameter 1/(1), where (/) denotes the average chemical size of macromolecules, the weight composition distribution will have a finite width. However, its dispersion specified by formula (Eq. 13) upon the replacement in it of l by (l) will be substantially less than the dispersion of distribution (Eq. 69)... 

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