Molecular weight standard deviation

Besides the calculation of average molecular weights, several other means of characterizing the distribution were produced. These include tables of percentile fractions vs. molecular weights, standard deviation, skewness, and kurtosis. The data for the tables were obtained on punched cards as well as printed output. The punched cards were used as input to a CAL COMP plotter to obtain a curve as shown in Figure 3. This plot is normalized with respect to area. No corrections were made for axial dispersion. 

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. 

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). 

The data are dominated by a few low molecular weight components. Figure 15.11 presents an image that has been amplified by a factor of 30 many more components are visible. Figure 15.12 presents the same data amplified by another factor of 30 for a total amplification of 1000, and a sea of peaks is visible. An unsupervised routine was used to isolate all local maxima in the data 190 components were resolved with amplitude greater than 10 times the standard deviation of the background signal. 

MW = molecular weight TPSA = topological polar surface area S.D. = standard deviation Min. = minimum Max. = maximum. 

First, we estimated the parameters a to be 0.01 and 8 to be 0.05. For the estimation of the standard deviation of the DP values, twenty chromatograms of NBS 706 polystyrene were measured and elution volumes at the distinguished per cent points of the integral curve of each chromatogram were calculated. Then, the value a was obtained to be 0.042 mL. The value 6 was estimated to be 0.1 mL, which corresponds to 0.3 % of the elution volume at the center of the calibration curve of this SEC system and 5 % difference of molecular weight. 

Calibration of Gel Permeation Chromatograph Polystyrene Calibration. A plot of molecular size in (S) versus elution volume for polysty-rene standards in dichloromethane showed deviation from linearity at about 2,200 which may be attributed to Imperfect column resolution, peak broadening, axial dispersion and skewing. The extensive tailing of the chromatograms of high molecular weight polystyrene standards observed in dichloromethane has also been reported in the literature (23-26). 

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... 

Therefore the square root of the amount by which the molecular weight ratio exceeds unity measures the standard deviation of the distribution relative to the number average molecular weight. 

Figure 5. Architecture of the [Cr2(CO)io(M2-H)] monoanion of crystallographic Q-I symmetry showing the approximate D4/, geometry of the metal carbonyl framework and the two centrosym-metrically related (half-weighted) sites of the bridging hydrogen atom in the bent Cr-H-Cr molecular fragment. Internuclear distances and bond angles are given with their estimated standard deviations.

Because sample homogeneity is particularly important for a Standard Reference Material, it was carefully assessed for both polystyrenes using solution viscosity, a measure of molecular weight, as an index. There is essentially no variation with location within the lot, or from pellet to pellet, within the limits of error of the viscosity measurements. Viscosity measurements may be made with a standard deviation of a single determination of about 0.3%. 

The same precision as discussed above can be extended about 50 mass units by using N2 (molecular weight 28) and perfluoropropane (molecular weight 188) compared with C02 and SF6. For example, with a standard deviation in K of 0.5, a mass error standard deviation of 1 mass unit would be 300 instead of 250. Since the measurement of detector response is a function of the recorder (peak heights), integrator system (for areas), columns (absorption sites), electronics, temperature, etc., the overall precision of molecular weight measurement should be further improved in the future. 

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