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So. as its square is -1. U*i1 ut cannot be 1 but rather. must be one of the quatemion-imaginary units it. h. or h. The same holds for the transformation of the other two quatemion-imaginary units. hand h.

All data are for decompositions in benzene solution. Data from Brandrup et al. [1999]. The units of kd are s 1 ; the units of Ed are kJ mol 1 .



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where D, Ct. C 2 , and C 3 are real, is a four-vector, then 1)2 is a six-vector (i.e., the scalar component of 1) 2 is the same in all Lorentz frames). This scalar component is just i {(1/c)8D/8t + \1. C}, where C is the vector formed by ct. c 2 , and C3 . So if S (1)2) = 0, the time and spatial components of the Hermitian four-vector 2 will satisfy the continuity equation (1/c)8D/at + \1 . C = 0 in every Lorentz frame. If we are looking for something bilinear in the wavefunction that forms a fourvector, as ;:: is a six-vector and A is a four-vector, the only combinations that would obviously lead to a four-vector are At;::, n, and their Hermitian conjugates. Unfortunately, none of them leads to a positive-definite time component. 24 So we must look for something else. Consider the biquaternion





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3-9a-2

(3.46)

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To determine the effect of temperature on the molecular weight of the polymer produced in a thermally catalyzed polymerization where transfer reactions are negligible, one must consider the ratio kp = kd kt 1=2 , since it determines the degree of polymerization (Eq. 3-98). The variation of this ratio with temperature is given by

As A and 0 are four-vectors and;:: is a six-vector, U is a four-vector. Moreover, as nO is the quaternionic four-momentum operator, U has dimensions of momentum in units of n. From (3.42) and (3.43), we find that

3-170

;::to;:: = (;::to);:: +;::t (0;::)

where the energy term Ep Ed =2 Et =2 is the composite or overall activation energy for the degree of polymerization EXn . For bimolecular termination, X n is governed by Eqs. 3-98 and 3-99a and one can write

(3.47)

3-171

(At1)) A+At (1)A) -;::tA + At;::,

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EX n has a value of about 60 kJ mol 1 for thermal initiator decomposition, and X n decreases rapidly with increasing temperature. EXn is about the same for a purely thermal, self-initiated polymerization (Fig. 3-16). For a pure photochemical polymerization EXn is positive by approximately 20 kJ mol 1 , since Ed is zero and X n increases moderately with temperature. For a redox polymerization, EXn is close to zero, since Ed is 40 60 kJ mol 1 , and there is almost no effect of temperature on polymer molecular weight. For all other cases, X n decreases with temperature. When chain transfer occurs in the polymerization, X n is given by an appropriate form of Eq. 3-108. The temperature dependence of X n can be quite complex depending on the relative importance of the various terms in Eq. 3-108. For the case where chain transfer to compound S is controlling (Eq. 3-119), one obtains

ln ! M 1 1 kp Ap Ep Etr;S ln ln S X n X n 0 RT ktr;S Atr;S 3-172

(3.48)

The quantity Ep Etr;S is now EXn and can be obtained from a plot of either of the two forms of the left side of Eq. 3-172 versus 1=T. Etr;S usually exceeds Ep by 20 65 kJ mol 1 , with the more active transfer agents having lower values. The term (Ep Etr;S ) is usually 20 to 65 kJ mol 1 (Table 3-14) and the molecular weight decreases with

where the parentheses denote the limits beyond which the differential operator no longer acts [e.g., in (At 1) ) A 1) operates only on At, not on A]. Using (3.47) and (3.48) in (3.46), we find that U always vanishes:

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TABLE 3-14 Activation Parameters for Chain Transfer in Styrene Polymerization (60 C)a Transfer Agent Cyclohexane Benzene Toluene Ethylbenzene Isopropylbenzene t-Butylbenzene n-Butyl chloride n-Butyl bromide n-Butyl iodide Carbon tetrachloride

(3.49)

log (Atr;S =Ap ) 3.1 3.9 1.7 0.55 0.47 3.8 4 2 1 1

Now we will use the quaternion version of the Gauss integral theorem (see [146] or Appendix C)

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