Number average definition

The number average degree of polymerization for these mixtures is easily obtained by recalling the definition of the average from Sec. 1.8. It is given by the sum of all possible n values, with each multiplied by its appropriate weighting factor, provided by Eq. (5.24) ... 

Calculation of the N.A.S.L. (Number Average Sequence Length) can be performed, according to the definition given by Randall, given the model probabilities. Generation of the coefficients used in the equation required for this option, however, has not been automated. 

The obvious definition of the number average, Mn, of the distribution is the position on the M-axis that divides the area under the n (A/)- curve in equal parts (cf. Fig. 1.3). Because of the fact that n M) is normalized to 1, each of the subareas is equal to 0.5. As 50% of all the molecules are shorter than Mn, the other 50% are longer than Mn. Bearing in mind the normalization, the number average molecular mass is... 

Suppose we have a sample of polymer of weight w grams containing Nq molecules. By definition, the number average molecular weight is given by... 

This is a linear first-order differential equation that can be solved for [AM](t), and we can proceed to solve each of these successive Hnear differential equations for [AMj]. Without doing this, we can find the number average polymer length ii quite simply by noting the definition... 

In general a solution will contain a distribution of chain lengths, characterized by the reduced chain length distribution p(y) (Eq, (5.36)) and the number averaged chain length N. As pointed out in Sect. 11.6, p(y) is invariant under the renormalization group. The other physical variables characterizing the solution can be expressed in terms of N. From Eqs. (11.38), (11.83) we recall the definitions... 

The definition at Eq. (4) implies that the zeroth moment of the k-mer size distribution, Mo, is the reciprocal of the number average degree of polymerization in the system... 

For conversions lower than xgei the average molar mass of the polymer exhibits a continuous increase. The first two moments of the molar mass distribution are the number-average molar mass, Mn, and the mass-average molar mass, Mw, respectively. Mn is defined in terms of the number contribution of every species to the whole population. The weight factor used to define this average is the molar fraction. Mw is defined in terms of the mass contribution of every species to the whole mass, so that the mass fraction is the weight factor used in its definition. 

From a quantitative viewpoint, it is evident that the relative intensities of the resonances from carbons associated with branches and end groups can be compared to the intensity for the major methylene resonance, "n", at 30.00 ppm to determine branch concentrations and number average molecular weight or carbon number. The following definitions are useful in formulating the appropriate algebraic relationships ... 

With the previous definitions, the polymer carbon number and number average molecular weight are given by ... 

This moment is equal to the total number of floccules per unit volume, and Eq. 6.13a shows that it must always decline with increasing time if kmn > 0. The first moment M, is equal to the total number of primary particles "per unit volume, which should be constant according to the definition of the flocculation process and to Eq. 6.12 dM,/dt = 0.15 The number-average size is the ratio of M, to M0 16... 

Further, from the definition of the number average (eq C.2), one can write the following derivative, which combined with eq C.3, leads to... 

As mentioned above for copolymer based blends the quantities e and r in definitions (9) have to be replaced by number-averaged quantities . For a blend of a homopolymer A and a random copolymer B, P(CpD, p), comprising segments of type C and D with mole fractions p and (1 — p), respectively, the generalization of the parameters can be done along the same lines as sketched in Appendix III. It is useful to introduce the following notations ... 

The second definition we wish to introduce is the number average length of A (and B) runs. This is simply defined in Equation 6-37. 

Then recall the definition of the number average molecular weight and substitute to obtain an expression for c in terms of this quantity (Equations 12-6) ... 

The weight average expression follows directly from its definition, but obtaining the number average in terms of w. and M takes some manipulation, which we always leave as a homework problem. Also note that the intrinsic viscosity can be calculated from Equation 12-59 ... 

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