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which is analogous to Eq. 3-174. The polymerization rate is given by the difference between the rates of the propagation and depropagation reactions:
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Rp d M kp M* M kdp M* dt 7-42
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At equilibrium, the polymerization rate is zero and Eq. 7-42 becomes
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where M c is the equilibrium monomer concentration (as in Eq. 3-178). (The derivations in Sec. 3-9c for M c and Tc as a function of S and H  are applicable to the present system.) Combination of Eqs. 7-42 and 7-43 gives the polymerization rate as
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Fig. 7-1 Determination of the equilibrium monomer concentration [M]c for the (C2 H5 3 O (BF4 initiated polymerization of tetrahydrofuran in dichloroethane at 0 C. After Vofsi and Tobolsky [1965] (by permission of Wiley-Interscience, New York).
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where M 0 is the initial monomer concentration. (A corresponding consideration of the degree of polymerization in an equilibrium polymerization is given in Sec. 7-2b-5-c.) The equilibrium monomer concentration is obtained by direct analysis or as the intercept of a plot of polymerization rate versus initial monomer concentration. Figure 7-1 shows such a plot for the polymerization of THF in dichloroethane solution at 0 C using triethyloxonium tetra uoroborate as the initiator [Vofsi and Tobolsky, 1965]. The polymerization data are then plotted in accordance with Eq. 7-45 as the left side of that equation versus time to yield a straight line (Fig. 7-2) whose slope is kp M* . Since M* for a living polymer can be obtained from measurements of the number-average molecular weight, one can evaluate the propagation rate constant. The concentration of the propagating species and the polymerization rate in a polymerization will be determined by the concentrations of initiator and coinitiator. For the case where the concentration of propagating centers changes with time, integration of Eq. 7-44 yields
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where M 1 and M 2 are the monomer concentrations at times t1 and t2 , respectively.
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Fig. 7-2 Disappearance of monomer in the polymerization of tetrahydrofuran by f2 CH (SbCl6 at 25 C; a plot of Eq. 7-45. After Afshar-Taromi et al. [1978] (by permission of Huthig and Wepf Verlag, Basel and Wiley-VCH, Weinheim).
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7-2b-5-b Values of Kinetic Parameters. The availability of reliable kinetic and thermodynamic data is far less for ring-opening polymerizations than for step and chain polymerizations. However, there is a signi cant trend ROP, although a chain polymerization, has propagation rate constants much more similar to step polymerization rate constants than to the propagation rate constants in the chain polymerizations of carbon carbon double-bond monomers. For various oxirane, oxetane, THF, 1,3-dioxepane, and 1,3,6-trioxocane polymerizations, kp is in the range 10 1 10 3 L mol 1 s 1 [Chien et al., 1988; Dreyfuss et al., 1989; Matyjasewski et al., 1984; Mijangos and Leon, 1983; Penczek and Kubisa, 1989a,b; Saegusa, 1972; Xu et al., 1987]. Oxepane has a kp of 1:3 10 4 L mol 1 s 1 [Matyjasewski et al., 1984]. These values can be compared to the corresponding values for step and chain polymerizations (Tables 2-8, 3-10, 5-3, and 5-11). The kp values for cyclic ether and acetal polymerizations are close to the rate constants for polyesteri cation and similar reactions and much smaller than those for various chain polymerizations. (The concentration of propagating centers in the typical ring-opening polymerization is 10 2 10 3 M, which is comparable to cationic chain polymerizations of alkenes.) There is a glaring exception to this generalization the kp value for 1,3-dioxolane is reported as 102 104 L mol 1 s 1 [Penczek and Kubisa, 1989a,b; Szymanski et al., 1983]. A number of interesting observations have been made for tetrahydrofuran polymeriza tions. The observed rate constants kp and kp for propagation by free ions and ion pairs are the same [Baran et al., 1983; Matyjaszewski et al., 1979; Penczek and Kubisa, 1989a,b]. [The equal reactivity of free ions and ion pairs has been observed in various solvents (CCl4 , CH2 Cl2 , CH3 NO2 ) where the relative concentration of free ions varied from a few percent to over 95%.] This is only slightly different from the situation in cationic polymerizations of alkenes where the ion pair is about an order of magnitude less reactive than the free ion (Sec. 5-2e-3). The ion pair in THF polymerization is apparently even looser than in alkene polymerizations. The equal reactivity of free ions and ion pairs is generally assumed in all cyclic ether (as well as most ring-opening) polymerizations, but there are data for few systems other than tetrahydrofuran [Gandini and Cheradame, 1985]. The value of kp kp decreases with increasing solvent polarity for mixtures of THF with CCl4 and CH3 NO2 [Dreyfuss et al., 1989]. For free ions or their equivalent, loose and
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