Logarithmic comparison plot

FIGURE 5.11 Logarithmic comparison plot of the reduced dynamic loss modulus, G p (in dyne/cm = 0.1 Pa), against the logarithm of the reduced frequency, coaj-is ). The reduced reference temperatures give correspondence in the softening dispersion and match the positions of the loss tangent primary maxima. 

Fig. 4.26. Comparison plots for 10 preselected levels, graphed as the logarithm of heating rates versus the corresponding reciprocal temperatures at constant conversion.

The measured Jp t) curves of the other epoxy resins studied, Epon 1004, 1002, 1001, and 828/DDS, were successfully reduced at chosen reference temperatures and altogether the results are shown in Figure 5.4 as functions of the reduced time t/ar in a double logarithmic plot. This comparison plot was constructed by requiring all the reduced curves to cross at a compliance level of log 7p(f) = -8.5. The choice of Ibis 100.7,110.7,118.6,130,and205 for the DDS crosslinked 1007,1004, 1002, 1001, and 828 Epons, respectively. 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

Finally, in many cases the acidity equilibria cannot be measured but the rate of proton transfer or transmetallation can be measured to give an ionic or ion pair kinetic acidity. Studies using the rates of proton transfer have included the use of isotopes such as tritium and deuterium5,6. The rate is then used to calculate the Brpnsted slope, a, by plotting the logarithm of the proton transfer rate against the pK, as determined by the equilibrium acidity, for a series of compounds. From this plot, the approximate pKa of an unknown compound can be determined by comparison of the same type of compounds. 

Volume-normalized extinction is plotted in Fig. 11.2 as a function of photon energy for several polydispersions of MgO spheres both scales are logarithmic. For comparison of bulk and small-particle properties the bulk absorption coefficient a = Airk/X is included. Some single-particle features, such as ripple structure, are effaced by the distribution of radii. The information contained in these curves is not assimilated at a glance they require careful study. 

Figure 3.9 shows the double logarithmic plot of FBiW vs. as obtained for the said polyethylene. From the position of the dotted line a p/w-valiie of 25, as given in Table 3.3, is derived. Unfortunately, no exact data are known other than Mw, so that a comparison with those data cannot be given. As to a comparison with data obtained on the melt, reference is made to the next chapter. 

An advantageous nice feature of the er-moment approach is its ability to plot the dependence of the fitted logarithmic partition property as a function of the polarity, a. In Fig. 9.4, we show the respective curves for the log Roc model in comparison with the logRow model. It is apparent that log Roc exhibits a striking similarity to log Row, but it is much more sensitive to strong polarities at both ends of the partition coefficients increase with the amount of hydrogen-bond-donor surface. 

Fig. 2 Typical heterodyne diffraction signals measured for a critical PDMS/PEMS mixture for different distances T — Tc to the critical point. All curves have been normalized according to (15). The dashed line indicates the nearly constant initial slope of the concentration signal (an exponential function in the logarithmic plot) caused by the almost constant value of Dj- The inset shows, for comparison, a considerably smaller signal for an off critical PDMS/PEMS mixture (860 and 980g mol-1)...

This type of test is called parallel-line assay and is based on the comparison of a sample response with that of a reference standard (Finney, 1978). In general, it determines the response - at least by duplicates - of a series of dilutions of each preparation (sample and standard) while plotting the means of their corresponding doses on a logarithmic scale. As this test requires analysis of a linear portion of the curves, at least three points of each curve belonging to such a portion should be selected. The more selected points, the better the comparison. 

Figure 4.53 shows the dispersion spectra of hydrated Ca-HC, K-HC, and Na-HC at 300K [120], They are plotted in the x-axis, and the natural logarithm of the frequency (Hz) versus the natural logarithm of the real part of the relative permittivity in the y-axis. This experiment shows once more the higher mobility of monovalent cations in comparison with divalent cations and the higher mobility of Na+ with respect to K+. The cause of this effect is due to the inferior cationic radius of Na+ in comparison with that of K+. 

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