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Steps to convert excel document to PDF program matically:
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Equation 2-65, which applies equally to A polymerizations, indicates that [H2O] must be B greatly reduced to obtain high X n values. [H2O] is inversely dependent on essentially the square of X n since X n 1 is close to X n for large values of X n . It is also seen that the level to which the water concentration must be lowered to achieve a particular degree of polymerization increases with increasing K and increasing initial concentration of the reactants. Table 2-6 shows the calculated [H2O] values for selected values of K and X n at



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For the time tp between the exit of the ith atom from the cavity and the arrival of the (i + 1)th atom at time ti+ I, the field dynamics is governed by the Lindblad master equation for cavity damping introduced in 6:





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TABLE 2-6 Effect of Water Concentration on Degree of Polymerization in Open, Driven System K 0.1 Xn 1.32b 20 50 100 200 500 2b 20 50 100 200 500 5b 20 50 100 200 500 10 b 20 50 100 200 500 20 b 50 100 200 500 [H2O]a (mol L 1) 1.18b 1:32 10 3 2:04 10 4 5:05 10 5 1:26 10 5 2:00 10 6 2.50b 1:32 10 2 2:04 10 3 5:05 10 4 1:26 10 4 2:01 10 5 4.00b 0.211 3:27 10 2 8:10 10 3 2:01 10 3 3:21 10 4 4.50b 1.07 0.166 4:09 10 2 1:02 10 2 1:63 10 3 4.75b 0.735 0.183 4:54 10 2 7:25 10 3

d LA dtP = p,

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M 0 5 M. A concentration of 5 M is fairly typical of a step polymerization that is often carried out with only the reactant(s) present (without solvent). The lowering of [H2O] to achieve a particular X n is less the more favorable the equilibrium (that is, the larger the K value). Thus the synthesis of polyamides (with typical K values > 102) is clearly easier from the equilibrium viewpoint than polyester synthesis (K $ 0:1-1). Polyesteri cation requires a greater lowering of [H2O] than does polyamidation. It should be understood that simply to lower [H2O] as much as possible is not the desired approach. One needs to control the [H2O] so as to obtain the desired degree of polymerization.

(8.115)

Although reversible or equilibrium polymerizations would almost always be carried out in an irreversible manner, it is interesting to consider the kinetics of polymerization for the case in which the reaction was allowed to proceed in a reversible manner. (The kinetics of reversible ring-opening polymerizations are discussed in Sec. 7-2b-5). Consider the polyesteri cation of Eq. 2-58 under stoichiometric conditions, where k1 and k2 are the rate constants for the forward and back reactions. The initial carboxyl and hydroxyl group concentrations are [M]0. The values at any time are [M], which is given by (1 p) [M]0. The concentrations of the products, [COO] and [H2O], at any time are equal because of the stoichiometry and are given by p M 0. The polymerization rate is the difference between the rates of the forward and back reactions

p in the following way24:

d M M 0 d 1 p k1 1 p 2 M 2 k2 p2 M 2 0 0 dt dt 2-66

dp M 0 k1 1 p 2 k2 p2 dt 2-67

AtA A AAtA} (8.116) L P == -1- {2A AAt - a ap - pa a . apa 2tcav We can integrate (8.115) formally and write the reduced-density matrix for the field at the time of the arrival of the (i + l)th atom at the cavity as

Click in the field by which you want to filter. Click the that appears and choose a criterion. Repeat Steps 5 and 6 to add more criteria to the filter.

ln ! pe p 2pe 1 1 1 M 0 t 2k1 pe p pe 2-68

etpLp(ti + tint).

Equation 2-68 can be used to calculate the extent of conversion at any time if k1 and pe are known. pe is experimentally determined, or calculated from Eq. 2-61 if K is known. From experimental values of p as a function of time, Eq. 2-68 yields a straight line when the left side of the equation is plotted against time. The slope of the line then allows one to calculate k1 .

(8.117)

m m m m r (mr) = 2 (rr)

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