Mass distribution curve

The equations giving the number distribution curve for a powdered material arc dn/dd = d for the size range 0-10 xm, and dn/dd = 100,000/cf4 for the size range 10-100 p,m where d is in p,m. Sketch the number, surface and mass distribution curves and calculate the surface mean diameter for the powder. Explain briefly how the data for the construction of these curves may be obtained experimentally. 

It follows that / (M) = 1 as well for M = 0 —> M = oo. Using the thus normalized integral mass distribution curves of the molecular weights - as can be determined by fractionated precipitation of a polymer - it is possible to calculate the averages of the molecular weights according to the above equations. 

The thus obtained integral mass distribution curves of the molecular weight can be transformed into the differential mass distributions w(M) of the molecular weight by differentiation with respect to M ... 

Figure I. Molecular mass distribution curves from phosphorus-containing poly-ethyleneterephlhalate sample with 0.74% phosphorus modified with sodium salt of diethyl phosphite. Key l, integral curve 2. differential curve and 3, integral curve of phosphorus distribution.

A comparison of the degrees of polymerization and of the molecular mass distribution curves of the products may help to reveal the nature of the reactions which occur. Another kind of degradative transfer was described by Scott and Senogles [12]. It involves intramolecular transfer with the formation of a non-propagating centre and occurs during the generation of chains with a tendency to form five- and six-membered cyclic structures as a transition stage... 

RATE OF LIVING POLYMERIZATIONS AND MOLECULAR MASS DISTRIBUTION CURVES OF PRODUCTS... 

When initiation is not sufficiently rapid, living centres will be generated even during macromolecular growth. The propagation times of various chains will be different, and the molecular mass distribution curve of the product will be broadened. 

From the start of the polymerization, the molecular mass distribution curve widens up to the total consumption of the transfer agent. The rate of the distribution changes depends on the intensity of the transfers and the starting conditions. The populations of various lengths in dead chains and in... 

K (curve number 3). The molar mass distribution curves of the samples exposed at higher temperatures in the absence of catalyst were identical to the curve of the original polymer sample. 

However all the samples heated in the presence of US-Y catalyst (polymer-to-catalyst mass ratio 2 1) showed a deviation from the original polymer molar mass distribution in the region of lower molar masses. In the first experiment, the polymer/US-Y-zeolite sample was exposed at a temperature of 378 K, which is below its melting point, for 120 min and then for 30 min at 418 K. No volatile products were initially observed, but traces of isobutane and isopentane were detected when the temperature was raised to 418 K. Although these conditions were much milder than in the equivalent experiment with pure polymer (curve number 2), the molar mass distribution, curve number 5 in Figure 7.5, was different from that of the original polymer. 

In Fig. 8.13 the yield of fission products obtained by thermal fission of is plotted as a function of the mass number A (mass distribution). The maxima of the yields are in the ranges of mass numbers 90-100 and 133-143. In these ranges the fission yields are about 6%, whereas symmetrical fission occurs with a yield of only about 0.01%. The peaks in the mass distribution curve A = 100 and at 4 = 134 are explained by the fact that formation of even-even nuclei is preferred in the fission of the even-even compound nucleus It should be taken into account that the sum of the fission yields is 200%, because each fission gives two fission products. 

The mass distribution curves in Figs. 8.13 to 8.15 give the total yields of the decay chains of mass numbers A. The independent yields of members of the decay chains, i.e. the yields due to direct formation by the fission process, are more diflicult to determine, because the nuclides must be rapidly separated from their precursors. Only a few so-called shielded nuclides (shielded from production via decay by a stable isobar one unit lower in Z) are unambiguously formed directly as primary... 

Using mass spectrometry techniques, the mass distribution of peptide mixtures can be experimentally determined and compared to the theoretical mass distribution this distribution has the shape of a beU curve, and can be calculated using various computer programs. These data provide important information on the composition of the analyzed peptide mixture. While it is not possible to verify the correct representation of each individual peptide within the nnixture, the mass distribution curve can be used to confirm the correct overall composition of the nnixture. It is also often possible to confirm the correct amino acid at a defined position of a peptide nnixture. For example, the average mass (i.e., the peak of the mass distribution bell curve) of the mixture AC-AXXXXX-NH2 (716) is sufficiently different from that of the mixture AC-YXXXXX-NH2 (808). Furthermore, an incorrect mass distribution of the peptide nnixture can indicate problems that occurred during the synthesis, such as incomplete side-chain deprotection of a particular annino acid. 

This results in values of du /d log M which are plotted against log M to yield the differential molecular weight/mass distribution curve (MWD). 

Particle-size and mass distribution curves, along with information on particle porosity, density, shape, and aggregation, can be obtained for submicrometer- and supramicrometer-size silica materials suspended in either aqueous or nonaqueous media by field-flow fractionation (FFF). Narrow fractions can readily be collected for confirmation or further characterization by microscopy and other means. Among the silicas examined were different types of colloidal microspheres, fumed silica, and various chromatographic supports. Size distribution curves for aqueous silica suspensions were obtained by both sedimentation FFF and flow FFF and for nonaqueous suspensions by thermal FFF. Populations of aggregates and oversized particles were isolated and identified in some samples. The capability of FFF to achieve the high-resolution fractionation of silica is confirmed by the collection of fractions and their examination by electron microscopy. 

In Figure 14.6 at 40 MeV only the second, evaporation, stage is observed as seen by the narrow mass distribution curve. The curves for 480 and 3000 MeV reflect the increased importance of the first, direct interaction stage which leads to a broad spectrum of product mass numbers. 

In order to compare the results of critical chromatography with results of an independent method, SEC with coupled density (D) and refractive index (RI) detection was used, which has been shown to be very useful for the characterization of copolymers with respect to their chemical composition [39,40]. The MMD curve for one of the block copolymers and the mass distribution curves of the components are shown in Fig. 18. From these the overall chemical composition may be calculated. An excellent agreement between the results of critical chromatography and the SEC experiments was obtained. 

Fig. 4 Typical differential molar mass distribution curve, example of polystyrene ----------------wide...

These relationships are shown in Figure 9.4 where it is seen that the mixed product shows a dominant size fraction for = 3, at the peak of the mass distribution curve. This is sometimes referred to as the modal or dominant size. The median size of the distribution, at 0.5 mass fraction on the cumulative curve, occurs at X = 3.67. 

The values of 2.84%, Lso%( = Lm) and Li % may be obtained from a cumulative mass distribution curve, as described in section 2.14.4. The higher the CV the broader the spread, CV = 0 denoting a monosized distribution. The CV for a Gaussian distribution is 52%, but the product from an MSMPR crystallizer, which generally conforms more to a gamma function distribution (equation 2.78), has a CFof 50%. 

Figure 2 Differential molar mass distribution curve of poly-disperse polystyrene. Averages by SEC-MALS Mn = 100000gmor yi4=334000gmor At=704000gmor Mj (by viscometry) = 307000gmol Zimm plot for this sample is shown in Figure 4.

Sometimes it is reflecting the existence of a second maximum on the molecular mass distribution curve. Incorporation of a chainOtransformation agent reduces the polymer molecular mass both in the light and in the dark polymerization regimes. 

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