Gaussian behavior

Method of Moments The first step in the analysis of chromatographic systems is often a characterization of the column response to sm l pulse injections of a solute under trace conditions in the Henry s law limit. For such conditions, the statistical moments of the response peak are used to characterize the chromatographic behavior. Such an approach is generally preferable to other descriptions of peak properties which are specific to Gaussian behavior, since the statisfical moments are directly correlated to eqmlibrium and dispersion parameters. Useful references are Schneider and Smith [AJChP J., 14, 762 (1968)], Suzuki and Smith [Chem. Eng. ScL, 26, 221 (1971)], and Carbonell et al. [Chem. Eng. Sci., 9, 115 (1975) 16, 221 (1978)]. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

Now, we consider this phenomenon from the viewpoint of the non-Gaussian behavior of the network chain. As is well known, when we assume the idealized molecular network consisting... 

Mgauss is the molecular weight above which chains show Gaussian behavior. 

Equations (18) and (16) define a temperature where Gaussian behavior is observed (the phase separation temperature) where % — 1/2 and thermal energy is just sufficient to break apart PP and SS interactions to form PS interactions. Equation (12) using (17) for Vc is called the Flory-Krigbaum equation. This expression indicates that only three states are possible for a polymer coil at thermal equilibrium ... 

P5.06.12. REACTORS WITH CSTR. ERLANG AND GAUSSIAN BEHAVIOR... 

Even the simplest approximation to this system - that of totally ignoring the friction 6f - gives a Gaussian behavior for the solute friction at short times [1,2],... 

Fig. 24. a The three-functional regularly branched chain model with Gaussian behavior of the subchains, called the soft sphere model s). b The Berry-plot of the reciprocal particlescattering factor of the soft sphere model. Compare also Figs. 19, 25 and 27... 

In Chap. B II.4 we have shown that the angular dependence of the first cumulant of the electric field correlation function can be obtained by integration over the particle-scattering factor. This rule remains valid also for copolymers but is restricted to Gaussian behavior of the subchains. Although the whole q-region can be covered by this integration, which in most cases has to be carried out numerically, it is useful to discuss the... 

Equation (E.35) is very convenient for computational work but has two disadvantages. First, the equation is restricted to Gaussian behavior of the subchains, and second, the block-like structure is not immediately recognizable. The block character of the copolymer can, however, be made evident by a different factorization. We first realize that Eq. (E.34) can be written as... 

We shall now discuss the non-Gaussian behavior of our self-correlation functions. Rahman32 pointed out that it is convenient to do this by introducing the coefficients a N(t) which for Gs(r, t) are defined as... 

These coefficients are strongly dependent on the number of molecules used in the simulations. For example, Figures 37 and 38 present the coefficients from the Stockmayer simulation using 216 and 512 molecules, respectively. The corresponding coefficients from the 216 and 512 molecule systems differ substantially from each other. Therefore, we feel that these coefficients from our simulations are only qualitative indications of the non-Gaussian behavior of our self-correlation functions. Figure 41 presents the coefficients from the modified Stockmayer simulation. Comparing the results for the two simulations we see ... 

This appendix gives some of the properties of the Hermite polynomials, HeN(jc). These polynomials form a basis set for Rahman s32 expansion of C s(v)(r, t) and play a fundamental role in the discussion of the non-Gaussian behavior of this latter function. Brief sketches of this expansion and of the... 

Figure 12. Theoretically obtained plots of In (Q(f) 2(0)) versus t (where t is scaled by t 2, isc = [mcH cHii/kBT]1 2 1.1 ps ) for the first three quantum levels (n = 1,2,3) of the CH3-I mode in CH3I from the friction estimates (shown in Figs. 10 and 11) and the vibration-rotation contribution. The equilibrium CH3-I bond length was set to re = 2.14 A. The results show an increasing Gaussian behavior in the short-time scale with increasing quantum number n. This figure has been taken from Ref. 133.

The subquadratic n dependence clearly arises from the nonexponential component of (Q(t)Q(0)) (shown in Fig. 12) in the initial time scale which increases with increase in the quantum number n, which strongly reflects the presence of the Gaussian components of binary friction. This dominant Gaussian behavior is responsible for the nearly linear n dependence in the higher quantum levels. 

In ideal situations, all peak shapes are Gaussian. However, in certain cases, many eluted components do not show true Gaussian behavior. A measure of this nonideal behavior is the asymmetry factor, (b/a). This factor can be determined by drawing a line from the peak maxima perpendicular to the baseline and then measuring the widths (b and a) at one-tenth peak height (see Fig. 9.2.5). Values of —1.0 for b/a are recommended higher values indicate considerable skewing of the eluted component. 

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