polymer chemistry

Please derive the rate of polymerization ( equation (35)).

Theoretical Rates of Polymerization Applying the mass action law to the mechanism given on page 334 for the rate of disappearance of monomer leads to the expressions shown in (32) and (33) for the theoretical rate of polymerization , = k'[R'][M] + k[R][M] + k2,[R][M] + (32) OT Sky.} Tp = K'[R'] + [R] (33) According to the "assumption of long chains" made earlier, k'[R') is very much less than 2km/R.) and may be neglected in the bracketed term of (33). Simi- larly, according to the assumption made regarding radical size and reactivity, kip = k = kmp = k . Incorporation of these two approximations into (33) yields T, = k[M] [R] =k,[M[R] (34) where [R] is the total concentration of propagating radicals. We have already evaluated the steady-state radical concentration (19), in terms of the rate of ini- tiation, and substitution of (19) into (34) gives the steady-state rate of poly- merization: 1/2 1/2 P [M] (2fk) [M] (35) 2k, 2k where the last term in (35) refers specifically to initiation by the thermal disso- ciation of an initiator. Equation (35) predicts that the rate of polymerization at a given temperature with the square root of the initiation rate or initiator concentration and with the first power of the monomer concentration. The validity of this equa- tion has been verified many times. An example of the dependence of r, on the ini- tiator concentration is shown in Figure 12.3, while an example of the dependence ofr, on monomer concentration is given in Figure 12.4. should vary