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where [M]0 is the initial (at t 0) concentration of hydroxyl or carboxyl groups. It is convenient at this point to write Eq. 2-20 in terms of the extent or fraction of reaction p de ned as the fraction of the hydroxyl or carboxyl functional groups that has reacted at time t. p is also referred to as the extent or fraction of conversion. (The value of p is calculated from a determination of the amount of unreacted carboxyl groups.) The concentration [M] at time t of either hydroxyl or carboxyl groups is then given by



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M M 0 M 0 p M 0 1 p 2-21

(5.122)

1 1 p 2 2 M 2 kt 1 0 2-22

where X is any given function of rand t, the observable quantities should not change. To get any further, we must make this physical requirement more precise by translating it into equations. Let us denote by unprimed quantities the quantum-mechanical description in the old gauge and, by primed quantities, the description in the new gauge. If the two descriptions are equivalent, they must be related by a unitary transformation 1'. This way, both descriptions will yield the same values for any observable quantity. Suppose that 6 is an observable; then its expectation value in the new gauge is given by (5.123) as 1't1' = 1. The unitarity of l' also allows us to write l' as

2-2a-1

(ihC') ,





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Equation 2-22 indicates that a plot of 1= 1 p 2 versus t should be linear. This behavior has been generally observed in polyesteri cations. Figure 2-1 shows the results for the polymerization of diethylene glycol, (HOCH2CH2)2O, and adipic acid [Flory, 1939; Solomon, 1967, 1972]. The results are typical of the behavior observed for polyesteri cations. Although various authors agree on the experimental data, there has been considerable disagreement on the interpretation of the results. At rst glance the plot does not appear to exactly follow the relationship. The experimental points deviate from the third-order plot in the initial region below 80% conversion and in the later stages above 93% conversion. These deviations led various workers [Amass, 1979; Fradet and Marechal, 1982a,b; Solomon, 1967, 1972] to suggest alternate kinetic expressions based on either 1- or 3-order dependencies of 2

(5.124)

Fig. 2-1 Third-order plot of the self-catalyzed polyesteri cation of adipic acid with diethylene glycol at 166 C. After Solomon [1967] by permission of Marcel Dekker, New York) from the data of Flory [1939] (by permission of American Chemical Society, Washington, DC).

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where 6 is a Hermitian operator. Topnd out how the Hamiltonian in the new gauge iI' is related to the Hamiltonian in the old gauge iI, we write down the Schrodinger equation in the new gauge and use the fact that 11/1') = 1'11/1). Then, as

the reaction rate on the carboxyl concentration, that is, 2- and 21-order dependencies 2 according to

Click in the cell to which you want to assign a function. Click the Formulas tab on the Ribbon. Click the Insert Function button.

(86) 1'11/1)

However, a critical evaluation shows that both kinetic possibilities leave much to be desired. The experimental rate data ts Eq. 2-23a well only in the region between 50 and 86% conversion with an excessively poor t above 86% conversion. On the other hand, a plot according to Eq. 2-23b ts reasonably well up to about 80% conversion but deviates badly above that point. Neither of the two alternate kinetic plots comes close to being as useful as the third-order plot in Fig. 2-1. The third-order plot ts the experimental data much better than does either of the others at the higher conversions. The t of the data to the third-order plot is reasonably good over a much greater range of the higher conversion region. The region of high conversion is of prime importance, since, as will be shown in Sec. 2-2a-3, high-molecular-weight polymer is obtained only at high conversions. From the practical viewpoint, the low conversion region of the kinetic plot is of little signi cance. 2-2a-2 Reasons for Nonlinearity in Third-Order Plot

(5.125)

The failure to t the data over the complete conversion range from 0 to 100% to a third-order plot has sometimes been ascribed to failure of the assumption of equal functional group reactivity, but this is an invalid conclusion The nonlinearities are not inherent characteristics of the polymerization reaction Similar nonlinearities have been observed for nonpolymerization esteri cation reactions such as esteri cations of lauryl alcohol with lauric or adipic acid and diethylene glycol with caproic acid [Flory, 1939; Fradet and Marechal, 1982b] 2-2a-2-a Low-Conversion Region Flory attributed the nonlinearity in the low conversion region to the large changes that take place in the reaction medium The solvent for the reaction changes from an initial mixture of carboxylic acid and alcohol to an ester.

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