Double logarithm calculations

The value of n in the Eq. (5.49) is related with reaction mechanism, the rate of nucleation, and the size of crystal buds. The n values are given in Table 5.3. Equation 5.49 is often used to express a variety of local-chemical processes. Transformation constants k and n values can express all kinds of relationships. However, the weakness of this equation is obvious, because the equation during the application for the choice of the experimental data does not give a certain physical sense for conclusion. Graphic method can be used to test the applicabihty of Eq. (5.49). The graphical method involves double logarithm calculations, which further increase the uncertainty of the equation. However, the deviation of experimental from the theoretical relations can be minimized. 

Fig. 43 Double logarithmic graph of the ultimate strength of PpPTA fibre versus the degree of polymerisation for a monodisperse distribution for various values of the diameter and for /j=0.16. Calculation for aspect ratios fr=wa(2r) 1>10... 

Fig. 15.3 Calculated percentage of bound ligand (pb = [PL]/[L]0, where [PL] is the concentration of the protein-ligand complex, and [L]0 is the total ligand concentration) as a function of total ligand concentration. The graph is in double logarithmic scale. The dissociation constants, K0, are 10 pM (squares), 100 pM (diamonds), and 1 pM (triangles). Protein concentrations are 1 pM (solid curves) and 10 pM (dashed curves).

A careful examination of the Stockmayer and the percolation distributions reveals that both theories gives the same type of distribution [110]. In terms of the two exponents in Eqs. 52 and 53, the percolation calculation yields t=2.2 and 0 0.44, and the Stockmayer distribution yields r 2.5 and o 0.50. These differences in the exponents appear to be small in a double logarithmic plot, but they cause significant differences in the absolute values for w(x) when 3-4 decades in the degree of polymerization are covered. Another point is that the cut-off function could be calculated analytically in the FS-theory to be a single exponential function [110], while the percolation theory could only make a guess about its shape [7]. 

This calculation is repeated over a range of box sizes, and a double - logarithmic plot of the lacunarity versns the size of the shding box is then produced. FracLac then outputs a text file containing the valnes of r and A for each image. 

Figure 18.9 Molecular diffusion coefficients in air, Dla, at 25°C for different molecules plotted as a function of (a) their liquid molar volume, V(, (calculated as ratio of molar mass M, to liquid density, p,L), and (b) their molar mass, Mh Data from references reviewed by Fuller et al. (1966) plotted on double-logarithmic scale.

At this point it seems of interest to include a graph obtained on a quite different polymer, viz. cellulose tricarbanilate. Results from a series of ten sharp fractions of this polymer will be discussed in Chapter 5 in connection with the limits of validity of the present theory. In Fig. 3.5 a double logarithmic plot of FR vs. is given for a molecular weight of 720000. This figure refers to a 0.1 wt. per cent solution in benzophenone. It appears that the temperature reduction is perfect. Moreover, the JeR-value for fiN smaller than one is very close to the JeR value obtained from Figure 3.1 for anionic polystyrenes in bromo-benzene. As in the case of Fig. 3.1, pN is calculated from zero shear viscosity. The correspondence of Figs. 3.1 and 3.5 shows that also the molecules of cellulose tricarbanilate behave like flexible linear chain molecules. For more details on this subject reference is made to Chapter 5. 

These results could be complemented well with the curve slopes in the double logarithmic coordinates as plotted in Fig. 6.33(a) using idea of the intermediate critical exponent a(t), equation (4.1.68). In the traditional chemical kinetics its asymptotic limit ao = a(oo) = 1 is achieved already during the presented dimensionless time interval, t 104. For non-interacting particles and if one of two kinds is immobile, Da = 0, it was earlier calculated analytically [11] that the critical exponent is additionally reduced down to ao = 0-5. However, for a weak interaction (curve 1) it is observed that in the time interval t 104 amax 0.8 is achieved only for a given n(0) = 0.1, i.e., the... 

Fig. 4. Double logarithmic plots of Z1/2 vs Mw of poly(macromonomer) of 26 (m=4, n= 50) in 0.05 N NaCl H20 at 25 °C. Open symbols (o) are experimental results. The closed symbols (1) are calculated values by Eqs. (5) and (6). The thick solid line is theoretical, calculated from Eqs. (11) and (14-17) for the perturbed KP chain with q=17 nm, ML=1.03X104nm 1, and B=5.78 nm the dashed line is theoretical, calculated for the unperturbed KP chain (B=0)...

Examples of diagrams that lead to double logarithmic corrections are shown in Fig.2. The details of our calculation can be found in [3]. The final result, that agrees with the independent calculation in Refs. [17,18], reads ... 

When shear stress x was plotted against shear rate y on a double logarithmic scale, the intercept of the straight line on the x axis at y = 1 s l was taken as the value of the constant K. The apparent viscosity xapp at a given shear rate was then calculated from the equation... 

ZOpT can be used to study both self-similar and self-affine fractal objects. The data at low frequencies (u and v <10) is not to be included in the calculation of D j. Figure 17.25 from Tang and Marangoni (2006) illustrates how Df, and ZOpT are calculated from the double logarithmic plot ofX vs. Y for polarized light microscopy images of the fat crystal networks. 

Fig. 9. a Double-logarithmic power dependence of the population densities, Ni and N2, of levels 1 and 2 in Fig. 5 calculated from Eq. (10) using the same parameters as in Fig. 6 c, where power is taken as G. The dashed lines are the limiting slopes from Eqs. (14) and (15). b Double-logarithmic power dependence of the level 1 depopulation rate ratio between ETU (rate = 2 Wetu i) and linear downconversion (rate = kiNi). The horizontal dashed line indicates equal rates for the two. c Time-evolution of N2 following termination of a cw beam at f = 0 for the high, medium, and low powers indicated in (b), plotted on linear axes. The dashed lines are the limiting behaviors from Eqs. (22) and (24)... 

The Mark-Houwink plot for the pullulans is displayed in Figure 9 and indicates a smooth relationship with little scatter. A slope of 0.64 was obtained from the best fit. Figure 10 displays a double logarithmic plot of the radius of gyration versus the molecular weight for pullulans, and this plot has a slope of 0.37. The theoretical values of Rg were calculated by using the Ptitsyn-Eisner equation (as follows) and are shown in the same figure 11) ... 

Equation (9.72) provides a more precise method for determining the relaxation spectrum. The strategy to follow in the calculation of the relaxation spectrum involves the determination of provisional values of H x), at t = x, at a series of points equally spaced on the logarithmic scale using m = 0 in Eq. (9.72). Then from a double logarithmic plot of H x) against x, the slope —m is determined at each point. The reciprocal of r(w -f 1) multiplied by the provisional value of H gives the value of the relaxation spectra. 

In DNA separations the molecular mass calibration curve of a CGE system is usually described by double-logarithmic plotting of the reduced mobility i vs. molecular mass. The reduced mobility is calculated from the mobility i of the analyte in the sieving medium and that in free solution (p =plpo)-... 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

The slope of the regression lines was calculated from a double-logarithmic transformation of the data. 

The percent incidence in side effects was calculated from a double logarithmic transformation of the data using a robust linear regression model. dSide Effect ED25/Anxiolytic Efficacy ED25... 

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