Number-average degree reaction time

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 last decade has seen quite remarkable advances in our knowledge of the structure and properties of the proanthocyanidins. Viscosity measurements were made of solutions of procyanidins isolated from Theobroma cacao and Chaenomeles speciosa with number-average degrees of polymerization of 6.1 and 11.8, respectively, in water and 1% sodium hydroxide at 25 °C. Procyanidins are apparently completely crosslinked by formaldehyde up to a chain length of 6 units, but few units are crosslinked in polymeric procyanidins. The second order rate constants observed for the formaldehyde reaction with catechin or epicatechin are approximately six times higher than that observed for the C. speciosa polymer. 

In the absence of side reactions the number average degree of polymerization will be c/[M]/r/[I] if initiation is by nucleophilic attack on the monomer or 2 d[M]/c/[I] if initiation is by electron transfer followed by dimerization of the monomeric radical anions (r/fM] and d[I] are the reacted concentrations of monomer and initiator, respectively). If the rale of initiation is very rapid compared to the propagation rate and the initiator is mixed very rapidly and efficiently into the reaction mixture, then all macroions should start growing at almost the same time and should add monomer at equal rates. The active centers can be terminated deliberately and simultaneously since there are no spontaneous termination reactions under appropriate experimental conditions. Polymers made in such reactions have molecular weight distributions which approximate the Poisson... 

The invalidity of the last scenario occurs because in practice it is never possible to have 100.00% complete polymerization. However, the extent of reaction can approach very close to 1, at which time Eq. 20.20 gives a good approximation of the number average degree of polymerization to be expected (cf. Fig. 20.4). 

The number-average degree of polymerization of the reaction mixture, X , is defined as the total number (iVo) of monomer molecules initially present, divided by the total number N ) of molecules present at time t,... 

The number average degree of poymerization at time t during the reaction is given simply by the ratio of the decrease in monomer concentration to the initial concentration of initiator ... 

Solution In radical chain reactions, the overall rate of polymerization, Rp, and the number-average degree of polymerization, X , are functions of the initiator concentration [I], the monomer concentration [M], and also the temperature via the temperature dependence of the individual rate constants. At constant [M] and [I], the Schulz-Flory MWD is produced. However, if [M] and [I] vary with time, a number of Schulz-Flory distributions overlap and thus a broader MWD is produced. In the ideal CSTR [M] and [I] are constant and the temperature is relatively uniform. Consequently, chain polymerizations in CSTR produce the narrowest possible MWD. In the batch reactor, [M] and [I] vary with time (decrease with conversion) while in the tubular reactor [M] and [I] vary with position in the reactor and the temperature increases with tube radius. These variations cause a shift in X with conversion and consequently a broadening of MWD. 

To confirm the predictions from equation 2, a series of model compounds was used to create PF type resin formulations with varying ratios of the two reactive components, phenol functionalities and monofunctional phenol content. The solids content, ratio of phenolic hydroxyl groups to sodium hydroxide catalysts, and reaction temperature were kept constant. This allows for the rapid measurement of the solution viscosity, which can then be equated to the number average degree of polymerization, which is relatively difficult to measure. The extent of the reaction, which is also very difficult to measure accurately in these systems, was equated to the reaction time. 

Table I. Number-average Degree of Polymerization (DP ) of Cellulose Ester Derivatives in Relation to Reaction Time...

Figure 17-9. Variation with time of the viscosity v (in 0.1 Pa s), the number-average degree of polymerization the extent of reaction p, and the branching coefficient in the p-toluene sulfonic-acid-catalyzed multifunctional polycondensation of diethylene glycol with a mixture of succinic acid and tricarballylic acid, ro = 1.002 xa = 0.404at 109° C gel = gel point. (After P. J. Flory.)...

The number-average degree of polymerization is determined by the propagation reaction as well as by both the concentration of true active center [C ] and transfer reactions involving both absorbed monomer and absorbed metal alkyl. Thus, at time t... 

In the absence of an externally added strong acid, the diacid monomer acts as its own catalyst for the esterification reaction. For this case, [HA] is replaced by [-COOH] in Equation 4.1. Assuming that the two functional groups are nearly stoichiometric, and setting [-COOH] = N, the final equations expressing the variation of reaction extent,/ and polymer number average degree of polymerization (NADP), DP,j with time can be written as [34, 35]... 

Figure 17-6. Dependence of the number-average degree of polymerization X on the time t (min) in the polycondensation of 12-hydroxy stearic acid at 152.5 C with 0.01 y p-toluene sulfonic acid as catalyst. (---) Reaction progress after the removal of unused catalyst and renewed addition of the same catalyst/functional group molar ratio as at the start of the polycondensation. y = mol p-toluene sulfonic acid/mol base unit. (After C E. H. Bawn and M. B. Huglin.)...

The number averaged degree of polymerization for this reaction increases linearly in time and is given by... 

To establish the relation—also called the Carothers relation—between the number average degree of polymerization (X ) and the extent of reaction (p), it is necessary first to define these two quantities. The extent of reaction is defined as the fraction of reacted functional groups at that instant (t). If all molecules are bifiuictional and bivalent and if No and Nt are the number of (monomer or polymer) molecules present in the reaction medium at time 0 and t, respectively, the extent of reaction can be written as... 

If purified materials are used and the reaction is carried out under high vacuum there is no termination reaction. Since the formation of the dianions is very rapid most polymer chains are initiated at the same time. Since they all grow at approximately the same rate the polymer chains formed all have about the same degree of polymerization and so the polymer is virtually monodisperse. The reaction can be killed at any time by the addition of impurities. A typical value of MJM would be about 1.1. The degree of polymerization can also be calculated simply. If the initial concentration of sodium naphthalide is [I]o and the initial concentration of monomer is [M]o then the concentration of active centres is [I]o/2 since they are in the form of dianions. In unit volume of the system [M]o moles are distributed between [I]o/2 moles of polymer and so the number average degree of polymerization, is given by... 

---