Number average degree of polymerization

The number-average degree of polymerization of the reaction mixture, X , is defined (Ghosh, 1990 Odian, 1991) as the total number (No) of monomer molecules initially present, divided by the total number (N) of molecules present at time t. 

It is easy to see that for hydroxyacids and for stoichiometric mixtures of diol and diacid, there is an average of one carboxyl per molecule at any state of reaction. So Eq. (5.28) can be written as  

This equation is sometimes referred to as the Carothers equation and is applicable to all step polymerizations represented by equations 

Problem 5.8 Show that in the high conversion region, for uncatalyzed polyesterification is proportional to and that for catalyzed polyesterification is proportional to t, where t is the reaction time. 

The number average molecular weight, is related to Xn by (Odian, 1991)  

For hydroxyacids (which possess a hydroxyl and a carboxyl group on the same molecule) or stoichiometric mixtures of diol and diacid, the number of unreacted carboxyl groups is equal to the total number of molecules present in the system. This is obviously true for hydroxyacids. For stoichiometric mixtures this is true because each molecule larger than a monomer will on the average have a hydroxyl at one end and a carboxyl at the other end, while each diacid monomer molecule contains two carboxyls and each diol monomer contains no carboxyls, which makes an average of one carboxyl per molecule. Thus Eq. (5.35) can be written as 

This equation is sometimes referred to as the Carothers equation and is applicable to all step polymerizations represented by Eqs. (5.3) and (5.4) in systems containing stoichiometric amounts of A and B groups. Note that given by Eq. (5.36) or (5.37) is the number-average degree of polymerization of the reaction mixture and not just of the polymer that has been formed. 

By combining Eq. (5.38) with Eqs. (5.30) and (5.34), the dependence of the molecular weight on reaction time for catalyzed and uncatalyzed polyesterifications, respectively, is obtained as shown by 

The kinetic expressions (5.30) and (5.34) are obeyed for esterifications above 80%. For conversions of this magnitude the values of t are usually sufficiently large that unity in the parentheses of Eqs. (5.40) and (5.41) may be neglected and so also Meg, which has a value of 18. The approximate equations 

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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) ... 

The number average degree of polymerization is given by dividing the number of repeat units by the number of chains, or... 

The above mechanism, together with the assumptions that initiator decomposition is rate controlling and that a steady state in chain radicals exists, results in the classical expressions (eqs. 8 and 9) for polymerization rate, and number-average degree of polymerization, in a homogeneous,... 

Figure I 6.34 Degree of polymerization distribution molar distribution ( ) with number average degree of polymerization (dp = 140 glucose unimers) mass distribution ( ) with weight average degree of polymerization dp = 1242 glucose unimers.

Reaction conditions should usually be chosen such that the fraction of initiator-derived chains (should be greater than or equal to the number of chains formed by radical-radical termination) is negligible. The expressions for number average degree of polymerization and molecular weight (eqs. 13 and 14) then simplify to eqs. 15 and 16 ... 

Assuming that the number average degree of polymerization (DP ) is determined by chain transfer to monomer and assuming unimolecular termination relative to propagation (i.e., chain breaking due to solvent, polymer, impurities are absent), the simple Mayo equation55 ... 

The Carothers equation relates the number-average degree of polymerization to the extent of reaction and average functionality of a step-growth polymer. In the Carothers equation, the number-average degree of polymerization, X , relates to the extent of reaction, p, and average functionality, /avg, of the polymer system ... 

Figure 5. Number average degree of polymerization vs, conversion for hatch-mass styrene polymerization (4j...

To run the residence time distribution experiments under conditions which would simulate the conditions occurring during chemical reaction, solutions of 15 weight percent and 30 percent polystyrene in benzene as well as pure benzene were used as the fluid medium. The polystyrene used in the RTD experiment was prepared in a batch reactor and had a number average degree of polymerization of 320 and a polydispersity index, DI, of 1.17. 

The styrene conversion for the continuous flow stirred tank experiments was determined utilizing the concentration of the polymer in the feed and the number average degrees of polymerization... 

Reactor Conditions for Experimental Runs. Operating conditions for the continuous, stirred tank reactor runs were chosen to study the effects of mixing speed on the monomer conversion and molecular weight distribution at different values for the number average degree of polymerization of the product polymer. 

Our theoretical studies [38] showed that the hyperbranched polymers generated from an SCVP possess a very wide MWD which depends on the reactivity ratio of propagating and initiating groups, r=kjk. For r=l, the polydispersity index where P is the number-average degree of polymerization. 

The number average degree of polymerization Xm of the cyclic species under no circumstances exceeds four for type ii, or two for type i, regardless of the extent of reaction.The weight average Xrw may be shown to increase without limit as p (and hence p ) proceeds to unity. However, it is always very much smaller than the weight average for the open chain species. 

Substituting Eq. (25) for Si in the expression for the number average degree of polymerization... 

The total number of molecules in the sol fraction, neglecting intramolecular cross-linking, will equal the number of primary molecules minus the number of cross-linkages in the sol. Expressing these as numbers of moles per equivalent of structural units, we have N = 1/% primary molecules and p /2 cross-linkages in the sol. The number average degree of polymerization in the sol is then... 

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