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[Bin (x) , Ein(X')] = -i2h :x {8(x - x') - 8(x [Bin(x),Bin(x')] = [Ein(X),Ein(x')] =0,



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Reactants of the A (or B type are not likely to undergo direct cyclization instead of A B) linear polymerization. A groups do not react with each other and B groups do not react with each other under the conditions of step polymerization. Thus there is usually no possibility of anhydride formation from reaction of the carboxyl groups of a diacid reactant under the reaction conditions of a polyesteri cation. Similarly, cyclization does not occur between hydroxyl groups of a diol, amine groups of a diamine, isocyanate groups of a diisocyanate, and so on. Once linear polymerization has reached the dimer size, intramolecular cyclization is a possibility throughout any A B

+ x' + 2L)},

2-71

in the Ribbon and then click Relationships. (If you created your database from a template, certain relationships are predefined.)

9.7 Using the canonical commutation relations for the radiation field outside the cavity,

2-72





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A,] A [Bout(X), Eout(x)

The extent to which cyclization occurs during polymerization depends on whether the polymerization proceeds under equilibrium control or kinetic control, the ring sizes of the possible cyclic products, and the speci c reaction conditions. 2-5b Cyclization Tendency versus Ring Size

= -~2h ax 8(x - x),

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Whether cyclization is competitive with linear polymerization for a particular reactant or pair of reactants depends on thermodynamic and kinetic considerations of the size of the ring structure that may be formed. An understanding of the relative ease of cyclization or linear polymerization comes from a variety of sources. These include direct studies with various bifunctional monomers in cyclization reactions (such as those in Eqs. 2-69 through 2-72) as well as ring-opening polymerizations (Chap. 7) and data such as the heats of combustion of cyclic compounds [Carothers and Hill, 1933; Eliel, 1962; Sawada, 1976]. Consider rst the thermodynamics stability of different sized ring structures. Some of the most useful data on the effect of ring size on thermodynamic stability is that on the heats of combustion of cycloalkanes (Table 2-7) (1 kJ 0.2388 kcal). A comparison of the heats of combustion per methylene group in these ring compounds with that in an open-chain alkane yields a general measure of the thermodynamic stabilities of different-sized rings. More precisely, thermodynamic stability decreases with increasing strain in the ring structure as measured by the differences in the heats of combustion per methylene group of the cycloalkane and the n-alkane. The strain in cyclic structures is very high for the 3- and 4-membered rings, decreases sharply for 5-, 6-, and 7-membered rings, increases for 8 13-membered rings, and then decreases again for larger rings. The strain in ring structures is of two types angle strain and conformational strain. Ring structures of less than ve atoms are highly strained due to the high degree of angle strain,

TABLE 2-7 Heats of Combustion and Strains of Cycloalkanes per Methylene Groupa Heat of Combustion per Methylene Group (kJ mol 1) 697.6 686.7 664.5 659.0 662.8 664.1 664.9 664.1 663.2 660.3 660.7 659.0 659.5 659.5 658.2 659.0 Strain per Methylene Groupb (kJ mol 1) 38.6 27.7 5.5 0.0 3.8 5.1 5.9 5.1 4.2 1.3 1.7 0.0 0.5 0.5 0.8 0.0

[Bout(x),Bout(x')] = [Eout(x),Eout(x')] = 0,

Data from Eliel [1962]. Calculated as the heat of combustion per methylene group minus the value (659.0) for the n-alkane methylene group.

9.8 Another way to test (9.39) is to substitute it in (9.2) and see if the resulting intracavity electric field agrees with (9.24). Substituting (9.39) in (9.2), we find that Ein(X) = Ein,D(X) + Ein,c(X), where

that is, the large distortion of their bond angles from the normal tetrahedral bond angle Bond angle distortion is virtually absent in rings of ve or more members For rings larger than ve atoms the strain due to bond angle distortion would be excessive for planar rings For this reason rings larger than ve atoms exist in more stable, nonplanar (puckered) froms The differences in strain among rings of ve members and larger are due to differences in conformational strain The 5- and 7-membered rings are somewhat strained in comparison to the 6-membered ring because of the torsional strain arising from eclipsed conformations on adjacent atoms of the ring Rings of 8 or more members have transannular strain arising from repulsive interactions between hydrogens or other groups which are forced to crowd positions in the interior of the ring structure.

Dn(x)

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