Determination of the Number-Average Molecular Weight

The first group of methods involves end-group analyses. Many types of syntheses leave a special group on one or both ends of the molecule, such as hydroxyl and carboxyl. These can be titrated or analyzed instrumentally by such methods as infrared. For molecular weights above about 25,000 g/mol, however, the method becomes insensitive because the end groups are present in too low a concentration. 

The second group of methods makes use of the colligative properties of solutions. Colligative properties depend on the number of molecules in a solution, and not their chemical constitution (22). The colligative properties include boihng point elevation, melting point depression, vapor pressure lowering, and osmotic pressure. The basic equations for the first two may be written (23) 

If the vapor pressure of the solute is small, and the solvent follows Raoult s vapor pressure law, 

The osmotic pressure 7t depends on the molecular weight as follows  

Typical values for the colligative properties for a polymer having a molecular weight of 20,000 g/mol are shown in Table 3.5 (23). Only osmotic pressure is large enough for fruitful studies at this molecular weight or higher. 

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Gel permeation chromatography is an attractive technique for determination of the number-average molecular weight (M ) distribution of petroleum fractions, especially the heavier constituents, and penoleum products (Altgelt, 1968 1970 ... 

As a consequence, the precise determination of the number-average molecular weights of PC samples by SEC is possible only if the nature of the oligomers present is taken into account. 

Vapor Pressure Osmometry The VPO became a popular method for the determination of the number-average molecular weight of nonvolatile solutes of less than about 20,000 g/mol and that tend to diffuse across the membrane in MO experiments [91], This method operates on the principle that the vapor pressure of a solution is lower than that of a pure solvent (P°) at constant pressure and temperature. This vapor pressure lowering (AP) is proportional to the molar mass of the solute (polymer) for dilute solutions. As it is known, the vapor pressure of a solvent in dilute solutions obeys the Raoult s law, Pj = Pfxj, where Pj is the partial vapor pressure of the solvent whose mole fraction in the solution is Xj. In terms of the mole fraction of the solute, Pj = P° (l - X2) or AP/pO = -X2. 

GRI-GG8, Standard Test Method for Determination of the Number Average Molecular Weight of Polyethylene Terephthalate (PET) Yams Based on a Relative Viscosity Rate. 

Direct determination of fn from the ratio of the number-average molecular weight, M , to the equivalent molecular weight, Meq, is applicable, with certain restrictions, to all the above three types [Eq. (2.1)]. 

Equations (10.33) and (10.35) can be used to determine the propagation rate constant. The equilibrium monomer concentration [M]e is obtained by direct analysis or as the intercept of a plot of polymerization rate versus initial monomer concentration (see Problem 10.4). The polymerization data are then plotted, in accordance with Eq. (10.36), as the left side of that equation versus time to yield a staright line where slope is A [M ]. Since [M ] for a living polymer can be obtained from measurements of the number-average molecular weight, one can determine the propagation rate constant. 

One final area for discussion is the use of NMR for the determination of polymer number average molecular weights. End group resonances were clearly visible in the 13c NMR spectrum of the ethylene-l-hexene copolymer in Figure 3. An opportunity to determine polymer degrees of polymerization or number average molecular weights should not be overlooked. 

The polymerization does not require an initiator of one of the types so far considered in this chapter but use of a slight excess of the ketonic reactant leads to the polymer having a fluorine atom at each chain-end. An accurate value of the fluorine content of the polymer would permit determination of its number-average molecular weight. 

The factor 2 has been introduced to account for differences in the definition of the rate of termination. In the analysis of the MWD results, termination was assumed to be exclusively by combination and, hence, 5 = 0 [49, 50]. Recall that equation 4.22 is a long chain approximation and assumes the absence of chain transfer events [51]. The rate of polymerization was determined from the gravimetric analyses of the conversion and the total time duration of the pulse experiment. The results of this number average molecular weight are shown in figure 4.7 for MA, EA and BA. To understand what can be concluded from this graph, let us first focus on the results for the monomer BA. It is clear from this figure that the data points for BA have been subdivided... 

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