Distribution curves worked example

This paper outlines the basic principles and theory of sedimentation field-flow fractionation (FFF) and shows how the method is used for various particle size measurements. For context, we compare sedimentation FFF with other fractionation methods using four criteria to judge effective particle characterization. The application of sedimentation FFF to monodisperse particle samples is then described, followed by a discussion of polydisperse populations and techniques for obtaining particle size distribution curves and particle densities. We then report on preliminary work with complex colloids which have particles of different chemical composition and density. It is shown, with the help of an example, that sedimentation FFF is sufficiently versatile to unscramble complex colloids, which should eventually provide not only particle size distributions, but simultaneous particle density distributions. 

Plotting the concentrations on logarithmic paper would therefore yield a line whose slope indicates the number of simple molecules in the polymer [see, for example, the work of Hendrixson (23)]. Campbell (11) proposed the general use of logarithmic coordinates in this manner for various distributions, but, as has been shown (30), the method cannot be relied upon to describe data near the plait point. The simple distribution curve itself is useful for interpolating data when a relatively large number of tie lines are available but should not be used for extrapolation. 

Equations 3.22, 3.23 and 3.24 require plotting the composite functions of two solid size distributions and the total efficiency and finding the maximum (see the worked example elsewhere ), which requires no differentiation and can be performed easily in industrial production situations. Naturally, the test information necessary for this calculation is the same as for the grade efficiency curve itself two particle size distributions and the total efficiency Ej. 

For the foregoing example that was worked to illustrate the use of the standard normal distribution curve, the probability of a measurement falling between 3.531 (1 <7) and 3.583 (3cr) is found to be 0.159 from Table 14.3. Similarly, the probability of a measurement falling above 3.583 would be 0.0001 (essentially zero). By symmetry, the probability of a point falling between 3.479 (-1 <7) and 3.427 (-3 <7) would be 0.159 while the probability of a point falling below 3.427 would be essentially zero (0.001). 

Whereas at the lower end of its range mercury porosimetry overlaps with the gas adsorption method, at its upper end it overlaps with photomicrography. An instructive example is provided by the work of Dullien and his associates on samples of sandstone. By stereological measurements they were able to arrive at a curve of pore size distribution, which was extremely broad and extended to very coarse macropores the size distribution from mercury porosimetry on the other hand was quite narrow and showed a sharp peak at a much lower figure, 10nm (Fig. 3.31). The apparent contradiction is readily explained in terms of wide cavities which are revealed by photomicrography, and are entered through narrower constrictions which are shown up by mercury porosimetry. 

In a preliminary screening, the alkylation of 2-methylnaphthalene was studied using a variety of acid zeolites with different pore widths. In principal agreement with the earlier work of Fraenkel et al. [22-25] it was found that the best selectivities for the slim alkylation products, i. e., 2,3-, 2,6- and 2,7-dimethylnaphthalene, are obtained on HZSM-5 and HZSM-11. On these catalysts it was observed that the alkylation is always accompanied by the undesired isomerization into 1-methylnaphthalene. Moreover, a peculiar deactivation behavior was encountered With time on stream, the yield of 1-methylnaphthalene always dropped while the yield of alkylation products remained practically constant or even slightly increased. An example for the conversion and yield curves is given in Fig. 4. The distribution of the dimethylnaphthalene isomers is shown for the same experiment in Fig. 5. Bearing in mind that in equilibrium one would expect roughly 12 mole-% of each of the slim isomers, the... 

The influence of the density of the matrix is depicted in Figure 3c for the X orientation alone These curves demonstrate the variability that may possibly be achieved by altering the matrix between yarns. More work is needed to further clarify the mechanisms for such variations. For example, the fibers may vary slightly because the high-density sample received four graphitization cycles rather than three. Other factors could contribute, such as variations of yarn distributions between samples. 

Then work out how many standard deviations corresponding to the area under the normal curve calculated in step 3, using normal distribution tables or standard functions in most data analysis packages. For example, a probability of 0.9286 (coefficient b2) falls at 1.465 standard deviations. See Table A.l in which a 1.46 standard deviations correspond to a probability of 0.927 85 or use the NORMINV function in Excel. 

Initiated by the work of Bunker [323,324], extensive trajectory simulations have been performed to determine whether molecular Hamiltonians exhibit intrinsic RRKM or non-RRKM behavior. Both types have been observed and in Fig. 43 we depict two examples, i.e., classical lifetime distributions for NO2 [271] and O3. While Pd t) for NO2 is well described by a single-exponential function — in contrast to the experimental and quantum mechanical decay curves in Fig. 31 —, the distribution for ozone shows clear deviations from an exponential decay. The classical dynamics of NO2 is chaotic, whereas for O3 the phase space is not completely mixed. This is in accord with the observation that the quantum mechanical wave... 

In most statistieal work, data that closely approximate a particular symmetrical curve, called the normal curve, are required. Both curves A and B in Figure 132 are examples of normal curves. In dealing with skewed curves, such as C in the same figure, transforming the data in some way so that a symmetrical curve resembling the normal curve results is desirable. Referring to the fi equeney table of the data used earlier in Problem STT.2, it can be seen that for this set of data the distribution is skewed in the opposite direction as depicted by curve C in Figure 132. 

In many cases additional work may be required to reparameterize models into the form required for the current analysis. This may involve, for example, a reparameterization between rate constants and clearance and volume terms or between derived parameters, such as volume of distribution by area (E ) and volume of distribution at steady state (Ess), or even extraction of parameter values from data summary variables (such as peak concentration, Cmaxi time to peak concentration, and area under the concentration curve, Af/C). The latter process is sometimes not straightforward and ultimately some data summaries may provide little useful information. See Dansirikul et al. (20) for methods of conversion of data summary variables into model-based parameters. 

---