Molecular weight distribution standard deviation

Thus, the molecular inhomogeneity U and, consequently, the value of Q depend also on the number-average molecular weight. The standard deviation remains as a more sensitive measure of the distribution width than either Q or U, but with one exception, it is not an absolute measure of the width of the distribution (see also Section 8.3.2.1). To be an absolute measure of the width of the molecular-weight distribution, the standard deviation must encompass a fraction of the original material that is independent of the width of the distribution, and this only holds for a Gaussian distribution. 

Determinarion of MW and MWD by SEC using commercial narrow molecular weight distribution polystyrene as calibration standards is an ASTM-D5296 standard method for polystyrene (11). However, no data on precision are included in the 1997 edition of the ASTM method. In the ASTM-D3536 method for gel-permeation chromatography from seven replicates, the M of a polystyrene is 263,000 30,000 (11.4%) for a single determination within the 95% confidence level (12). A relative standard deviation of 3.9% was reported for a cooperative determination of of polystyrene by SEC (7). In another cooperative study, a 11.3% relative standard deviation in M, of polystyrene by GPC was reported (13). 

Here, too, the deviation of this ratio Jromunity may be taken as a measure of polydisper-sity. The relationship between the ratio MJMn and the standard deviation of the molecular weight distribution is easily seen as follows. From Equation (16), it is clear that... 

From the general procedure for defining the mean, the left-hand side of Equation (18) may also be written as M2. Substituting this result into Equation (C.5) of Appendix C (with Min place of ), we can write the standard deviation a of the molecular weight distribution as... 

Fig. 15. Temperature dependence of the standard deviation o (in cm3) of the PDC elution curves corresponding to Fig. 14. The curves are parametrized by the molecular weight distribution w(P) as indicated and explained in section 5 the asymptote o(O) = 2.65 cm3 corresponds to a section P = 1 through the corresponding spreading surface (Kernel, cf. Sect. 4)...

The standard deviation of a weight-average molecular weight distribution can be written as (see Problem 1.2) ... 

Find the relationships between the average molecular weights and the standard deviations of the number- and weight-average molecular weight distributions and CT , respectively. 

It is impossible to manufacture a truly monodisperse polymer in which every molecule has the same value of M. Either the molecular weight distribution (MWD), or statistical averages of the MWD, are measured to characterise polymers. The mean and standard deviation are familiar statistical measures. An equivalent of the mean is used to characterise polymers, but the standard deviation is not used because the distribution shapes are skew rather than normal . 

The standard deviation is an absolute measure of the width for the Gaussian fimction only. Widths of molecular weight distribution for other functions have to be calculated for each case from the distribution function itself... 

A key term in the standard deviation is the ratio of Mw to Mn. This term is known as the polydispersivity index and it is used as a measure of the breadness of the molecular weight distribution. The polydispersity index or dispersity D is commonly used to measure the distribution of molecular mass in a given polymer sample ... 

In Chaps. 5 and 6 we shall examine the distribution of molecular weights for condensation and addition polymerizations in some detail. For the present, our only concern is how such a distribution of molecular weights is described. The standard parameters used for this purpose are the mean and standard deviation of the distribution. Although these are well-known quantities, many students are familiar with them only as results provided by a calculator. Since statistical considerations play an important role in several aspects of polymer chemistry, it is appropriate to digress into a brief examination of the statistical way of describing a distribution. 

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. 

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