wA U Mo wB Mo pB U wB0 Mo pB0 U 2-109a 2-109b 2-109c in .NET

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wA U Mo wB Mo pB U wB0 Mo pB0 U 2-109a 2-109b 2-109c
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U 2wiA 2Mo pB Mo pB0 Mo 1 p2 p2 0 =2 B B 2-110
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The weight-average molecular weight, obtained by a weight averaging of wA , wB , and wB0 , is given by
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Mw wA wB wB0 2 4 4 2-111
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Mo Mw 4 ( 2 pB pB0 2 1 p2 p2 0 =2 B B ) 2-112
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on substitution for wA, wB , and wB0 . The number-average molecular weight, obtained by dividing the total number of moles of reactants initially present divided by the total present at any time, yields
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It is useful to introduce the fractions of unreacted A, B, and B0 functional groups
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For the case where B and B0 groups have the same reactivity:
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Combination of Eqs. 2-108, 2-112 through 2-114, and 2-29 yields the polydispersity index as
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PDI 2 a 2 2 b2 g2 =2a 2-115
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Equation 2-115 yields PDI as a function of conversion in a straightforward manner without having to solve differential equations to obtain the number- and weight-average molecular weight distributions. One need only take a set of b values and then calculate the corresponding g, a, and PDI values from Eqs. 2-114 and 2-115. The limit of PDI at complete conversion pA 1; a 0 is 2 as for the Flory distribution. The Macosko-Miller method has also been applied to other polymerizations, including the A plus A0 0 system. In B B addition to being a simpler method for obtaining the average properties compared to the Flory and similar methods, it more readily allows an evaluation of the effect of various reaction parameters such as unequal group reactivity or nonstoichiometric amounts of reactants on M n , M w , and PDI (see Sec. 2-7e).
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Effect of Reaction Variables on MWD Unequal Reactivity of Functional Groups
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The molecular weight distribution and/or PDI has been described for several cases where the assumption of equal reactivity of functional groups is not valid. Unequal reactivity is easily B handled by the Macosko Miller method. For the A B B0 0 system described in A B the previous section, we simply rede ne the relationship between b and g by
g bs 2-47
(which was derived previously in Sec. 2-2d-2) where s is the ratio of rate constants for reaction of B0 and B groups [Ozizmir and Odian, 1980]. The polydispersity index increases (for conversions <100%) when the reactivities of B and B0 groups are different compared to the case where the reactivities are the same. The opposite trend is found for the A B 0 system [Gandhi and Babu, 1979, 1980]. There A B is a basic difference between the two- and the three-species reaction systems. In the threespecies system, when B0 groups have higher reactivity than B groups, the B0 0 monomer B continues polymerization with A monomer, independently of the B monomer, to form A B large-sized polymer chains. In the two-species system, long chains cannot be formed without participation of B groups in the polymerization process. The extents of the broadening and narrowing effects in the two systems increase as the difference in reactivities of B and B0 become larger. The effects are signi cant at low conversions; for instance, PDI is 2.10 at 50% conversion for the three-species system when B0 is 20-fold more reactive than B compared to a PDI of 1.50 for the equal reactivity case. However, the PDI at complete converson equals 2 in both the two- and three-species systems, which is the same limit as for the equalreactivity case.
Change in Reactivity on Reaction
An increase in the reactivity of one functional group in a bifunctional reactant on reaction of the other functional group results in an increase in PDI [Gandhi and Babu, 1979, 1980]. The PDI at complete conversion exceeds 2 when there is a greater than twofold increase in reactivity. PDI decreases when there is a decrease in reactivity of one functional group on reaction of the other group but the PDI at complete conversion is 2. The same trends are found but are more exaggerated when functional group reactivity varies continuously with molecular size [Gupta et al., 1979a,b; Nanda and Jain, 1968].