Equations number-average molecular weight

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 number-average Molecular weight, has been inserted to make the equation applicable to... 

Functionality. For reactive liquid polymers, the most important property is the functionality. Functionality is determined from number average molecular weight (Mn) and hydroxyl number by the following equation ... 

It is possible to uniquely determine both the scission efficiency Gis) and the crosslinking efficiency G(x) via Equations 9 and 10 providing both the number average molecular weight (M ) and the weight average molecular weight (Af ) can be measured. 

Next the coefficients of the effective relation of each sample with known vreight and number average molecular weight are evaluated by iteration. Combining Equation 3,U and 6, the average molecular weights and inhomogeneity index may be expressed as... 

Three of the four resins yield extracts of the functionality of Equation 4. The slope of the exponential decay allows for the evaluation of x. The resin, see Table II, initiated by benzyl dimethyl amine (BDMA) at the stated cure cycle, when subjected to leaching yields an extract of low solubility and a distribution of oligomeric molecules of low number average molecular weight. 

The relationship shown in Equations 4.91 and 4.92 shows that high values of p (0.99) are essential in producing high N and W values. The number-average molecular weight M calculated from Equations 4.89 and 4.90 is as follows ... 

In both equations, Dw is the solute diffusion coefficient in pure water, rs is the molecular radius of the solute, ls is its characteristic size, Vw is the water free volume, Mc is the molecular weight between crosslinks in the amorphous phase, Mn is the number average molecular weight of the polymer before crosslinking, M is the minimum value of Mc below which the solute cannot diffuse, 4>(V) is the free volume function mentioned earlier, and k3 is a constant. 

Now we compare the above osmotic pressure data with the scaled particle theory. The relevant equation is Eq. (27) for polydisperse polymers. In the isotropic state, it can be shown that Eq. (27) takes the same form as Eq. (20) for the monodisperse system though the parameters (B, C, v, and c ) have to be calculated from the number-average molecular weight M and the total polymer mass concentration c of a polydisperse system pSI in the parameters B and C is unity in the isotropic state. No information is needed for the molecular weight distribution of the sample. On the other hand, in the liquid crystal state2, Eq. (27) does not necessarily take the same form as Eq. (20), because p5I depends on the molecular weight distribution. 

Equation (44) shows the average molecular weight determined from osmometry to be the number average molecular weight as defined by Equation (1.14). 

This same conclusion may also be reached by the following argument. The product nJA, in Equation (43) equals the weight of component i in the solution the total weight of solute in the solution equals L,n,M,. The experimental osmotic pressure depends on and therefore measures the total number of moles of solute The ratio of the total weight to the total number of moles of solute defines the number average molecular weight. 

Viscometry The specific viscosity of each polymer from the bulk polymerization was measured in acetone at 30°C using an Ubbelohde dilution viscometer. Five concentrations in the range of 1.120 to 0.242 g/d poly(vinyl acetate) and polyvinyl trideuteroacetate) and 0.385 to 0.084 g/dl (poly(trideu-terovinyl acetate)) were run. Intrinsic viscosity was calculated by extrapolation of the Tlsp/c versus c plot to zero concentration. Number average molecular weights were calculated using the equation(20) [q] =1,0 x 10 1 [Mn] 0 72 which is in the mid range of the equations listed. 

Viscosity. The viscosity measurements of the PMMA graft chains obtained by hydrolysis of graft copolymer were carried out in benzene at 30°C using an Ubbelhode viscometer, and the number average molecular weight of the grafted chain was calculated using the equation(25) ... 

The number of repeating units between crosslinks, N, a parameter appearing in equation (1), can be simply defined as the jratio of the number average molecular weight between crosslinks, Mc, to the average molecular weight of the clusters, Mq. 

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