Number-Average Molecular Weight Methods

Because the relative concentration of end groups is smaller for higher molecular-weight polymers, this method becomes less sensitive as the molecular weight 

For a solution of 1 g of polyethylene with M2 = 500 in 100 g of camphor Tf = 178.4°C, A//f = 2.58 kcal/mol, M = 152), Tj - r is 0.48°C. As the molecular weight of the solute increases, measurement of the temperature lowering demands more sensitive equipment and the method becomes impractical. In order to measure the number of molecules by colligative methods, it is necessary to use dilute solutions. In practice, we often extrapolate results to infinite dilution by some suitable linearizing techniques. 

FIGURE 6.14 Osmotic pressure cell containing polymer solution A and solvent B separated by a semipermeable membrane C. The osmotic pressure is it. 

p are the chemical potential of the solvent in the solution and the pure solvent, respectively 

By analogy to the pressure of a gas, the osmotic pressure of a solution of concentration c is expressed in terms of a virial expansion  

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Limits and Limitations for Number-Average Molecular Weight Methods"... 

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. 

We, therefore, propose an indirect method for obtaining the variation of the intrinsic viscosity and number average molecular weight across the chromatogram. First the intrinsic viscosity-molecular weight relationship for a polymer with long chain branching (LCB) is assumed to be expressable in a form similar to that used by Ram and Milts (6),... 

Having determined b by the above method, the true number average molecular weight and higher moments for the whole polymer can be calculated as follows ... 

Note that the weight and Z average molecular weights so obtained are only approximate and less than the true values. This is a consequence of the universal calibration method giving the instantaneous number average molecular weight across the chromatogram. 

Since our indirect method produces both the linear (b=0) and branched intrinsic viscosities across the chromatogram, it is possible to estimate several LCB parameters as a function of elution volume or number average molecular weight. The branching factor G(V) can be written as... 

After prolonged degassing of a large batch of PPG, analyses showed that its hydroxyl, unsaturation, and water contents were 0.97 meq/g, 0.033 meq/g, and 0.0035%, respectively. The hydroxyl content was determined by an acetylation method, carried out with acetic anhydride (10). The amounts of unsaturation and water were determined by the mecuric acetate and Karl Fischer methods (10), respectively. The obtained analytical results indicate that the number-average molecular weight of the dihydroxy material is 2062, provided its molecular weight is arbitrarily assumed to be twice that of the monohydroxy material, and that the mole fraction of the monohydroxy poly (propylene oxide) is 0.066. This value corresponds to a number-average functionality of 1.93 for the PPG. 

Determination of Molecular Weight. The number average molecular weight (TEi) was measured by the vapor pressure equilibrium method with a Hitachi Molecular Weight apparatus (Model 117). 

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