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lost reactant. The amount of added reactant must be precisely calculated based on a chemical analysis of the r value for the reaction system as a function of conversion. Also the additional amount of the lost reactant must be added at the appropriate time. Premature addition or the addition of an incorrect amount results in performing the polymerization at other than the required stoichiometric ratio. There is a reaction condition in which high polymer molecular weights have been obtained independent of the stoichiometric ratio when one of the two monomers is only slightly soluble in the reaction mixture and is present in excess. An example is the reaction of bis(o-aminophenol) with terephthalic acid in polyphosphoric acid (Sec. 2-14h) [So, 2001]. Bis(o-aminophenol) is soluble but terephthalic acid is only slightly soluble in the reaction mixture. At any instant the polymer chain ends are those from the soluble monomer bis(oaminophenol), but these chain ends react quickly with the small amount of dissolved terephthalic acid, and then there is reaction again with bis(o-aminophenol). Polymerization proceeds to high molecular weight because there is a continuous dissolution of terephthalic acid to maintain its saturation concentration. On the other hand, high molecular weights are not achieved when the soluble monomer is present in excess. Similar results have been reported for the polymerization of terephthalic acid with ethylene glycol, p-dichlorobenzene with sodium sul de, and some aromatic diacid chlorides with aromatic diamines [Goodman, 1988; Liou and Hsiao, 2001a]. 2-6c Kinetics of Nonstoichiometric Polymerization
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The kinetics of polymerizations involving nonstoichiometric amounts of A and B functional groups can be handled in a straightforward manner. Consider the external acid-catalyzed A plus B polymerization with r < 1. The polymerization rate, de ned as the rate of A B disappearance of the functional groups present in de cient amount, is given by
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The following stoichiometry holds:
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where A 0 and B 0 are the initial concentrations of A and B groups. Combination of Eqs. 2-82 and 2-83 followed by integration [Moore and Pearson, 1981] yields
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which is combined with r A 0 = B 0 to give
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A plot of ln B = A versus t is linear with a positive slope of B 0 1 r k and an intercept of ln r. When r is close to unity the polymerization rate is adequately described by the expressions in Secs. 2-2a and 2-2b for the case of r 1. Only when r is considerably different from unity does it become necessary to use Eq. 2-85 or its equivalent. Most step polymerizations
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are carried out with close to stoichiometric amounts of the two reacting functional groups. The main exceptions to this generalization are some of the reaction systems containing polyfunctional reactants (Secs. 2-10 and 2-12). Web Forms barcode developmentin .net
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2-7 MOLECULAR WEIGHT DISTRIBUTION IN LINEAR POLYMERIZATION The product of a polymerization is a mixture of polymer molecules of different molecular weights. For theoretical and practical reasons it is of interest to discuss the distribution of molecular weights in a polymerization. The molecular weight distribution (MWD) has been derived by Flory by a statistical approach based on the concept of equal reactivity of functional groups [Flory, 1953; Howard, 1961; Peebles, 1971]. The derivation that follows is essentially that of Flory and applies equally to A and stoichiometric A plus B types B A B of step polymerizations. 2-7a Derivation of Size Distributions
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Consider the probability of nding a polymer molecule containing x structural units. This is synonymous with the probability of nding a molecule with x 1 A groups reacted and one A group unreacted. The probability that an A group has reacted at time t is de ned as the extent of reaction p. The probability that x 1 A groups have reacted is the product of x 1 separate probabilities or px 1. Since the probability of an A group being unreacted is 1 p , the probability Nx of nding the molecule in question, with x structural units, is given by
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Since N x is synonymous with the mole or number fraction of molecules in the polymer mixture that are x-mers (i.e., that contain x structural units), then
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where N is the total number of polymer molecules and Nx is the number that are x-mers. If the total number of structural units present initially is N0 , then N N0 1 p and Eq. 2-87 becomes
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Neglecting the weights of the end groups, the weight fraction wx of x-mers (i.e., the weight fraction of the molecules that contains x structural units) is given by wx xNx =N0 and Eq. 2-88 becomes
wx x 1 p 2 px 1 2-89
Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a