Step Reaction Averages

42 shows how to start from the equations of Fig. 3.8 to evaluate the simple equation for the number average molar mass of a step reaction. More difficult is to establish the mass average molecular mass. Starting from the mass-distribution 

The sum in brackets, p , can, however, only easily be summed for large values of X and without the factor x. To achieve the latter, one writes the differential of the integral of x p as shown in the second line of the derivation. The integration constants, c, disappear when ultimately the differentiations are carried out. The first 

---

Multiple covalent bonds are formed in each macromolecule and, in general, statistical, polydispersed structures are obtained. In the case of controlled vinyl polymerizations, the average length of the macromolecule is determined by monomer to initiator ratios. If one views these polymerizations as extraordinarily long sequences of individual reaction steps, the average number of covalent bonds formed/chain may be visualized as shown in Scheme 2 ... 

As done in Chapter 5, the effect of temperature can be determined using average activation of the various steps. Again, the rates of all single step reactions increase as the temperature increases but the overall result may be different for complex reactions. For free radical polymerizations the activation energies are generally of the order Ei>Ei E > El. Remembering that the description of the specific rate constant is... 

The most important factor controlling the morphologies generated is the location of the composition of the initial blend, < )mo with respect to the critical composition, < )M cnt (Figs 8.5 and 8.6). The latter may be calculated from the Flory-Huggins model as applied to a binary blend (step reactions) or a ternary blend (chain reactions), taking into account polydispersity (Kamide, 1990). The size of particles increases with the concentration of the component that forms the dispersed phase. Typically, for < )mo < < >M,crit, an increase in < )M0 will lead to an increase in both the volume fraction and the average size of dispersed phase modifier-rich particles. 

Case 1. In case 1, all of the secondary steps are first-order with respect to active intermediates. Then the rate will not depend on the sector speed, since the rate of all processes depends in the same way on the concentrations of intermediates, namely, in a linear fashion, and the rate of reaction averaged over the light period and the dark period (when the radical concentrations are decreasing) will be the same as the rate obtained by simply using a light source of intensity ala- In a case of this type it will turn out that the over-all rate of reaction will be directly proportional to the first power of the amount of light absorbed. 

Carothers classification (condensation vs. addition) is primarily based on the composition or structure of polymers. The second classification (chainwise vs. stepwise) was proposed by P. J. Floiy, and is based on the kinetic scheme or mechanism governing the polymerization reactions. Step reactions are those in which the chain growth occurs in a slow, stepwise manner. Two monomer molecules react to form a dimer. The dimer can then react with another monomer to form a trimer, or with another dimer to form tetramer. Thus, the average molecular weight of the system increases slowly over a period of time. This is exemplified by the following polyesterification ... 

For step-reaction pol3rmerization, the NACL is also sometimes referred to as the degree of polymerization. It is the average number of muc-tural units per chain and can also be calculated from... 

When dealing with catalysis it is best, however, to classify polymerization reactions according to the mechanism of chain propagation (2). One may distinguish in this way between chain-reaction polymerization and step-reaction (stepwise) polymerization. The essential features of these classes are shown in Table I (15). The diflFerences between the two types of polymerization are also evident from equations of rate (Rp) and average degree of polymerization (DP). For a free-radical polymerization of vinyl compounds (an example of a chain reaction), Rp and DP are functions of monomer and catalyst concentration (Equations 9 and 10) ... 

The kinetic parameters of a polymerization reaction can be correlated with the average degree of poiymerization. For a poiymerization reaction starting with No moiecules of monomer and having N moiecules at a certain point during the reaction, the conversion p for the step-reaction can be defined as the ratio between the number No - N of moiecules that reacted and the initial number of monomer molecules No ... 

In Section 16.1, you learned how to calculate the average rate of a chemical reaction. The word average is important because most chemical reactions slow down as the reactants are consumed and fewer particles are available to collide. Chemists quantify the results of collision theory in an equation called a rate law. A rate law expresses the relationship between the rate of a chemical reaction and the concentration of reactants. For example, the reaction A —> B is a one-step reaction. The rate law for this reaction is expressed as follows. 

The experimental data were fitted with Eqs (36)-(38) by assuming that m = 1, 2, 3, and 4 for Br , which has an average hydration number of 2.1. First, Eq. (38) was used with the initial values of to evaluate [X ] as a function of [H2O] and then an iterative calculation was carried out with Eq. (36) to minimize the standard deviation, cr = [S( caicd ohsd) /(iV — 1)], where aicd obsd te calculated and observed chemical shifts, respectively, and N is the number of data points. In this curve-fitting procedure, the values and were used as adjusting parameters. In Fig. 9, a set of the best regression curves for Br are shown by solid lines. In contrast to the one-step reaction mechanism, the successive reaction mechanism could reproduce the observed behaviors for S reasonably well. 

Fig. 6. Normalized ion intensity curves for ions in moist nitrogen. Fno = 2 Torr, PhjO = 1.6 X 10 Torr, 300°K. Successive intensity maxima indicate sequence N2 - N4 H20 -> H (H20)2 > H (H20)3 H (H20)4. Dashed lines represent theoretical curves calculated from integrated rate equations for consecutive reactions including reversible steps using average rate constants of Table II. In experiments where only position of equilibrium is to be studied, higher water concentrations are used so that equilibrium is established in less than 50 /zsec.

The rate of the first step of the furfural production process, a hydrolysis reaction, averages about 50 times higher than that of the second step, the... 

The top equation in Fig. 3.44 lists the number and mass averages for chain reactions terminated by the combination of two growing molecule-radicals as described in Fig. 3.43. The derivation of the equation must be done in the same way as illustrated for the step reaction in Fig. 3.42. The next example, shown in Fig. 3.44, applies to termination by chain transfer (see Sect. 3.2). In this case the termination... 

---