The Extent of Reaction p

If this reaction is indeed third-order, and if Flory s assumption that the intrinsic reactivity of a functional group is independent of chain length is correct, then a plot of 1/c2 versus t should be linear. Because it provides a direct link to the statistics of polymerization, however, it is useful to first follow Flory and define a new parameter, p, the extent of reaction Equation 4-15. 

The fraction of unreacted groups must then, by definition, be (1 - p), so that c, the concentration of groups remaining at time t, must be this fraction times the initial concentration, c0. 

It is hard to find a topic in the field of polymer physical chemistry where Raul Flory has not made seminal contributions, if not the seminal contribution. Not only are his 

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FIGURE 4-10 Graph of 1/(1 -p)1 versus t [redrawn from the data of P. J. Floty, JACS, 61,3334 (1939)]. 

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As with other problems with stoichiometry, it is the less abundant reactant that limits the product. Accordingly, we define the extent of reaction p to be the fraction of A groups that have reacted at any point. Since A and B groups... 

The extent of reaction p is again based on the group present in limiting amount. For the system under consideration, p is the fraction of A groups that have reacted. 

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 ... 

X represents the combined number of both types of units in the polymer chain. Eq. (3) applies also to polymers stabilized (see Chap. Ill) with small amounts of monofunctional units, although here it becomes necessary to replace the extent of reaction p with another quantity, namely, the probability that a given functional group has reacted with a bifunctional monomer. Type ii polymers stabilized with an excess of one or the other ingredient will be discussed later. 

Fig. 54.—Weight fraction Wr of rings vs. the extent of reaction p for a type ii polymer for B Mo/c =0.005 (lowest curve), 0.05 (middle curve), and 0.5 (uppermost curve). The curves correspond to successively increasing dilutions. (Jacobson and Stockmayer." )...

If we regard it as a cross-linked polymer, we require ywj the weight average degree of polymerization if all cross-linkages were severed. This operation would amount to replacing each tetrafunctional unit, or pair of cross-linked units, with two bifunctional units. The extent of reaction p would not be affected. Hence, according to Eq. (VIII-8)... 

The critical value of a being /(/—1) according to Eq. (7), it is apparent from Eq. (15) that a may not reach ac, since the extent of reaction p may approach but never reach unity. These polymers comply with the first two conditions mentioned at the close of the preceding section, they may acquire a multitude of polyfunctional units and the... 

Fig. 70.—Weight fractions of various finite species and of gel in a simple trifunctional condensation as a function of a, which in this case equals the extent of reaction p. Curves have been calculated from Eqs. (36) and (45). ...

The stress depends on the extent of reaction, p(tf), which progresses with time. However, it is not enough to enter the instantaneous value of p(t ). Needed is some integral over the crosslinking history. The solution of the mutation problem would require a constitutive model for the fading memory functional Gf Zflt, t p(t") which is not yet available. This restricts the applicability of dynamic mechanical experiments to slowly crosslinking systems. 

The chemical constraint reduces the number of possible reactions considerably, and consequently it leads to a much narrower molar mass distribution. Furthermore, the extent of reaction a of the A-group can cover all values from zero to unity, but the extent of reaction P of the equally reactive 5-groups cannot become larger than P=a/(f-l). One important consequence of this strict constraint is that gelation can never occur [1,13]. A much higher branching density than by random polycondensation can be achieved. For this reason one nowadays speaks of hyperbranching. 

It is usual to express Equation 4.17 in terms of the extent of reaction p, where p is defined as the fraction of functional groups that have reacted at time t. Thus, 1 - p is the fraction of groups unreacted. At is in turn Ao — p) ... 

Equation 2-78 shows the variation of Xn with the stoichiometric imbalance r and the extent of reaction p. There are two limiting forms of this relationship. When the two bifunctional monomers are present in stoichiometric amounts (r = 1), Eq. 2-78 reduces to the previous discussed Carothers relationship (Eq. 2-27)... 

In a system containing equivalent numbers of A and B groups, the number of monomer molecules present initially is No and the corresponding total number of functional groups is Af0/avg. If IV is the number of molecules after reaction has occurred, then 2(A/ — N) is the number of functional groups that have reacted. The extent of reaction p is the fraction of functional groups lost ... 

The first case is the copolymerization of monomer A with diene BB where all the double bonds (i.e., the A double bond and both B double bonds) have the same reactivity. Methyl methacrylate-ethylene glycol dimethacrylate (EGDM), vinyl acetate-divinyl adipate (DVA), and styrene-p- or m-divinylbenzene (DVB) are examples of this type of copolymerization system [Landin and Macosko, 1988 Li et al., 1989 Storey, 1965 Ulbrich et al., 1977]. Since r = Yi, Fi = f and the extent of reaction p of A double bonds equals that of B double bonds. There are p[A] reacted A double bonds, p[B] reacted B double bonds, and p2[BB] reacted BB monomer units. [A] and [B] are the concentrations of A and B double bonds,... 

The extent of asymmetric enantiomer-differentiating polymerization in an isoselective process is evaluated by measuring the optical activity of unreacted monomer a as a function of the extent of reaction p [Zhong et al., 2003], The rates of reaction of R and S enantiomers are given by... 

The result is that the extent of reaction, p (cf. Section 3.3.1.1), can be related to the weight average molecular weight, for a polyurethane system under equal stoichiometry ... 

Recall also from Section 3.3.1.1 that we introduced the extent of reaction, p, which is related to the degree of polymerization, x . From Eq. (7.91), we can derive the following expressions for the number- and weight-average molecular weights in terms of the extent of reaction ... 

Derive an expression for the relationship between the extent of reaction, p, and time, t, in an uncatalyzed polyesterification. Show all the integrations explicitly ... 

Hint calculate the extent of reaction, p, from this data and proceed from there. 

There is an important difference between the distributions calculated for equilibrium, bifunctional step-growth polymerization in Chapter 5 and for the free-radical polymerizations with termination by disproportionation or chain transfer that are being considered here. The distribution functions in the step-growth case apply to the whole reaction mixture in the free-radical polymerization this distribution describes only the polymer which has been formed. There is obviously a strong parallel between the probability S of this section and the extent of reaction p used in the step-growth calculations in Chapter 5. Many authors use the same symbol for both parameters. Different notations are used here, however, for clarity. 

Here [COOH] = [OH] = C at any time, barring side reactions which could consume one or the other functional group. Then, if Co is the initial concentration and p is the extent of reaction (p. 170) at any instant during the polymerization. 

In the statistical analysis of polycondensation the extent of reaction, p, is defined as the probability that a functional group has reacted at time t. (In kinetic treatments p is often used as the equivalent expression, the fraction of functional groups which has reacted.) It follows that (1—p) is the probability of finding a functional group unreacted. 

This last equation is the statistical number-distribution function for a linear polycondensation reaction at the extent of reaction p. 

This is the statistical weight-distribution function for a linear polycondensation reaction at the extent of reaction p. The number-distribution and weight-distribution functions are illustrated in Figs. 1 and 2 for values of p. 

Equations have been developed relating a to the extent of reaction, P. In general, the observed results of e slightly higher than those calculated indicating that some of the reactive sites are lost in intramolecular connections. Flory cites the case of pentaerythritol (/ = 4) and adipic acid. For this case, a = P2 and = 1/3 = 0.577. The experimental result was Pc = 0.63. 

Let us consider a system in a state P characterized by physical variables x and y (e.g. T and p), together with the extent of reaction p. Suppose a change takes place such that the system moves to a state P characterized by Then the uncompensated heat corresponding to this change (c/. 3.21) is... 

That is, the number average degree of polymerization becomes dependent only on the extent of reaction, p, as in the first version of the Carothers equation discussed. 

The ratio of the number of formed bonds to the maximum possible number of bonds in a reaction is called the extent of reaction p. If we select any group (A or B) randomly,/ is the probability that the group has reacted. In linear condensation polymerization, each chain has one unreacted A group at one end of the chain and one unreacted B group at the other end. Flory... 

The weight-average molar mass for linear condensation polymers also diverges as the extent of reaction p—> 1 (see Fig. 1.19). The polydispersity index of the linear condensation polymers,... 

Gelation is a connectivity transition that can be described by a bond percolation model. Imagine that we start with a container full of monomers, which occupy the sites of a lattice (as sketched in Fig. 6.14). In a simple bond percolation model, all sites of the lattice are assumed to be occupied by monomers. The chemical reaction between monomers is modelled by randomly connecting monomers on neighbouring sites by bonds. The fraction of all possible bonds that are formed at any point in the reaction is called the extent of reaction p, which increases from zero to unity as the reaction proceeds. A polymer in this model is represented by a cluster of monomers (sites) connected by bonds. When all possible bonds are formed (all monomers are connected into one macroscopic polymer) the reaction is completed (/> = 1) and the polymer is a fully developed network. Such fully developed networks will be the subject of Chapter 7, while in this chapter we focus on the gelation transition. 

The extent of reaction p can, in principle, be measured by molecular spectroscopy methods such as FTIR and NMR. Howevei, these methods always have some small relative error (typically of order a few percent). Since the same error has to apply to any determination of the extent of reaction at the gel point p, the relative error in the relative extent of reaction e = p — Pc)IPc diverges at the gel point. As a result, plots such as Fig. 6.34,... 

It is convenient to introduce the extent of reaction, p, as the fraction of the reagent that has reacted at a time, t, so that... 

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