Distributional plots

Instead of probability distributions it is more common to represent orbitals by then-boundary surfaces, as shown m Figure 1 2 for the Is and 2s orbitals The boundary sur face encloses the region where the probability of finding an electron is high—on the order of 90-95% Like the probability distribution plot from which it is derived a pic ture of a boundary surface is usually described as a drawing of an orbital... 

Analysis. Analyses of a number of lignitic coals are given in Table 3. Figure 1, a distribution plot of 300 U.S. coals according to ASTM classification by rank, indicates the broad range of fixed carbon values (18). According to the ASTM classification, fixed carbon for both lignite and subbituminous coals has an upper limit of 69%, but in practice this value rarely exceeds 61%. 

Fig. 3. Cumulative frequency distribution plotted by A, number B, surface area and C, volume, for the data in Table 1.

Coefficient of Variation One of the problems confronting any user or designer of crystallization equipment is the expected particle-size distribution of the solids leaving the system and how this distribution may be adequately described. Most crystalline-product distributions plotted on arithmetic-probability paper will exhibit a straight line for a considerable portion of the plotted distribution. In this type of plot the particle diameter should be plotted as the ordinate and the cumulative percent on the log-probability scale as the abscissa. 

A common form of societal risk measure is an F-N curve, which is normally presented as a cumulative distribution plot of frequency F... 

Table 4.2 Analysis of histogram data for SAE 1018 to obtain the Normal distribution plotting positions...

It has been shown that equivalent information can be obtained by analysis of logfnumbcr chain length distribution) plots (the log CLD method)."4" 7 For the case where termination is wholly by disproportionation or chain transfer, it is possible to show that (eq. 6) applies ... 

Fig. 1.21. Ratio of free-path distribution Xp to the scaled zero-density free-path distribution plotted as a function of reduced free-path length r/X for two-dimensional (a) and three-dimensional (b) liquids. Circles, inverted triangles and upright triangles refer to reduced volumes V/V0 of 1.6, 2, and 3, respectively (V0 is the volume of the system at close packing) [74].

Fig. 5 Sedimentation concentration distribution plots for guar gum using SEDFIT. a g (s) vs. s b c(s) vs. s. A Gaussian fit to the data lighter line) is also shown in (a). Rotor speed was 40000 rpm at 20.0 °C, concentration was 0.75 mg/ml in 0.02% NaNs. The guar had been heated at 160 °C for 10 min at a pressure of 3bar. From [49]...

The output of a Nd YLF laser is focussed by a series of lenses to a spot size of 0.5 pm upon a sample which may be positioned by an x-y-z stepping motor stage and scanned by a computer-controlled high frequency x-y-z piezo stage. Ions are accelerated and transmitted through the central bore of the objective into a time-of-flight (TOF) mass spectrometer. The laser scans an area of 100 x 100 pm with a minimum step size of 0.25 pm. TOF mass spectra of each pixel are evaluated with respect to several ion signals and transformed into two-dimensional ion distribution plots. 

Size exclusion chromatography (which is also known as gel permeation chromatography) is based on the premise that a polymer molecule in solution adopts a random coil configuration, which encompasses a volume (known as its hydrodynamic volume) that is proportional to its molecular weight. We fractionate polymers according to their hydrodynamic volumes to generate a molecular weight distribution plot. 

Firstly, a point dipole placed at the surface of a glass substrate is considered. The orientation of the dipole is considered to be random. The angular distribution plotted in spherical co-ordinates is shown in Figure 12. The radiation propagating in the positive (down) and negative (up) z-directions corresponds to that radiated into the glass substrate and the environment (air), respectively. 

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

The histograms of the resolved distributions, plotted on a log scale to show the number of particles in the high diameter cells, is shown in Figure 6. 

Fig. 4.1 Liquid-liquid distribution plots, (a) The distribution ratios D for three different substances A, B, and C, plotted against the variahle Z of the aqueous phase. Z may represent pH, concentration of extractant in organic phase ([HA]org), free ligand ion concentration in the aqueous phase ([A ]), aqueous salt concentration, etc. (b) Same systems showing percentage extraction %E as a function of Z. D and Z are usually plotted on logarithmic scale. 

Figure 8. Weight differential and integral molecular weight distribution plots.

Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a... 

Fig. 2-10 Weight fraction distribution plot for linear polymerization. Plot 1, p = 0.9600 plot 2, p = 0.9875 plot 3, p = 0.9950. After Howard [1961] (by permission of Iliffe Books, London and Elsevier, Oxford).

Fig. 2-14 Weight fraction distribution plot for multichain polymerization. Plot I. / 1 plot 2, / = 2 ...

L is obtained from the first point of inflection on the low-molecular-weight side of the major peak of the molecular weight distribution plot. In practice this is most easily determined from the maxima in a differential plot of the molecular weight distribution, chi (log M)/d log M versus log M, the dotted plot in Fig. 3-12. For this polymerization it is easy to detect not only L. but also Li, L3, and Lj. The detection of several Lt values acts as an internal check on the PLP-SEC method. For Fig. 3-12, the kp values calculated from L, La, and L are within 2% of each other the value from L4 is within 6-7% of the other values. In practice, closeness between kp values from L and L2 are considered sufficient to validate the PLP-SEC method. 

Most readers will be familiar with the bell-shaped normal distribution plotted in Fig. 9.12. When applied to the size distribution of particles, for example, such a distribution is fully characterized by the arithmetic mean D and the standard deviation a, where a is defined such that 68% of the particles have sizes in the range D a In the log-normal distribution, the logarithm of the diameter D is assumed to have a normal distribution. (Either logarithms to the base 10 or loga-... 

Other metals give initial distributions of products equivalent to a combination of the distributions described in 1 and 2 i.e., substantial amounts of both C2H5D and C2D6 are formed, and the distribution plotted against deuterium content is U-shaped. The values of M for the metals showing this behavior are zirconium 2.3, chromium 2.5, vanadium 2.6, and platinum 3.5. 

Fig. 11.5. Distribution plots of (a) molecular weight (MW), (b) number of rotatable bonds (NRot), (c) SlogP, (d) number of pharmacophore (ph4) triangles, and (e) number of rings for the whole library (VER ref), all hits (Class 1-3), Class 1 hits, and nonhits. With kind permission from Springer Science+Business Media Journal of Computer-Aided Molecular Design, 23, 2009,603, l-Jen Chen and Roderick E. Hubbard, Fig. 6.

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