Weight, molecular number-average

The number average molecular weight,, is defined as the total mass of the sample divided by the total number of molecules (or moles of molecules) in the sample. 

This molecular weight average is called the number average molecular weight because it is based on the total number of molecules in the sample under consideration. 

K we substitute the proportion of species/, which have molecular weight M, by the corresponding mole fraction rii in Eq. (4.2), we obtain the de nition of number-average molecular weight, M , representing the number distribution  

The mole fraction n, is also the differential number function, and a plot of n, versus Mi represents a differential number distribution curve, as shown in Fig. 4.1(a). The distribution being normalized, the scale of the ordinate in this gure goes from 0 to 1, and the area imder the curve is unity. 

While Eq. (4.3) gives a simple de nition of the number average molecular weight M , we can derive other equivalent de nitions following a simple arithmetic. For this let us de ne the following terms, some of which have aheady been used above. 

Hi mole fraction of species i (that is, molecules of same size with molecular weight M ) in a sample Ni moles of species i N total of aUAj s Wi weight fraction of species i Wi weight of species i W sumofallW/ s 

The number-average degree of polymerization, DP , is defined as Number-average molecular weight 

The lignin solution is measured at various concentrations under the same experimental conditions as the calibration solution. The concentrations, cw, expressed in gkg-1, are in the range of 0.5 and lOgkg-1. Mn is calculated as described in Section 8.3.2. 

The effect of several operating parameters must be emphasized. They are the size of the drops on the thermistor beads, the response time, and the purity of both solvents and lignin samples. Furthermore, it must be noted that conflicting results have been reported on the constancy of the calibration factor (Bersted 1973, Brzezinski et al. 1973, Kamide et al. 1976, Burge 1979, Marx-Figini and Figini 1980, Kim 1985, Froment and Pla, unpubl. results, 1988). 

In this section, all these factors are briefly examined together with the influence of other experimental parameters. 

They also carried out end group analysis by means of a Chicago Nuclear liquid scintillation counter to count C14-labeled end groups. 

Higher number average molecular weights have been determined by osmometry, most frequently in benzene or toluene (25, 68, 74). 

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This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

The phenomena we discuss, phase separation and osmotic pressure, are developed with particular attention to their applications in polymer characterization. Phase separation can be used to fractionate poly disperse polymer specimens into samples in which the molecular weight distribution is more narrow. Osmostic pressure experiments can be used to provide absolute values for the number average molecular weight of a polymer. Alternative methods for both fractionation and molecular weight determination exist, but the methods discussed in this chapter occupy a place of prominence among the alternatives, both historically and in contemporary practice. 

In the next section we shall describe the use of Eq. (8.83) to determine the number average molecular weight of a polymer, and in subsequent sections we shall examine models which offer interpretations of the second virial coefficient. 

The number-average molecular weight of most commercially available acetal resins is between 20,000 and 90,000. Weight-average molecular weight may be estimated from solution viscosities. 

T is the glass-transition temperature at infinite molecular weight and is the number average molecular weight. The value of k for poly(methyl methacrylate) is about 2 x 10 the value for acrylate polymers is approximately the same (9). A detailed discussion on the effect of molecular weight on the properties of a polymer may be found in Reference 17. 

Molecular weights of polymers are determined by the weight—average molecular weight, and the number—average molecular weight, M. The... 

The width of molecular weight distribution (MWD) is usually represented by the ratio of the weight—average and the number—average molecular weights, MJM. In iadustry, MWD is often represented by the value of the melt flow ratio (MER), which is calculated as a ratio of two melt indexes measured at two melt pressures that differ by a factor of 10. Most commodity-grade LLDPE resias have a narrow MWD, with the MJM ratios of 2.5—4.5 and MER values in the 20—35 range. However, LLDPE resias produced with chromium oxide-based catalysts have a broad MWD, with M.Jof 10—35 and MER of 80-200. 

The dynamic shear behavior of the polymer melt can be used to determine the ratio of weight average, to number average, molecular weight (33). 

The number-average molecular weight is adjusted in the 12,000—15,000 range for apparel fibers, >20, 000 for high strength yams for tires and industrial end uses. 

The addition of small, but specific, amounts of a monofunctional acid to the polymerization is often used to control molecular weights and catalyze reactions. The polymerization is controlled to produce a number-average molecular weight of 18,000—30,000, depending on the end use. 

Na" > > Cs". The amount of unsaturation also iacreases with number-average molecular weight (M ) suggesting that the rate of polymerisation... 

For higher molecular weight polydimethyl siloxanes (Af > 2500), the number-average molecular weight is related to the bulk viscosity by the foUowiag formula, where the viscosity units are mm /s(=cSt). 

The number-average molecular weight of dimethylsiloxane can also be determined from the intrinsic viscosity [Tj, dL/g] (extrapolated to zero viscosity) ia toluene or methyl ethyl ketone according to the foUowiag equatioa (339,340) ... 

Polymetallocarbosilanes. PolymetaHocarbosilanes having a number-average molecular weight of 700—100,000 can be prepared by reaction of polycarbosilane, /2 2 fx where R is H, or lower alkyl, with a tetraalkyl titanate, to give a mono-, di-, tri-, or tetrafunctional polymer... 

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