Statistical Derivation of Copolymerization Equation

Although the derivation above involves the steady-state assumption, the copolymerization equation can also be obtained by a statistical approach without invoking steady-state conditions [Farina, 1990 Goldfinger and Kane, 1948 Melville et al., 1947 Tirrell, 1986 Vollmert, 1973]. We proceed to determine the number-average sequence lengths, h and h2, of monomers 1 and 2, respectively. h is the average number of Mj monomer units that follow each other consecutively in a sequence uninterrupted by M2 units but bounded on each end of the sequence by M2 units. h2 is the average number of M2 monomer units in a sequence uninterrupted by Mj units but bounded on each end by Mi units. 

The transition or conditional probability pn of forming a MiMi dyad in the copolymer chain is given by the ratio of the rate for Mf adding Mx to the sum of the rates for M adding Mi and M2, that is 

Similarly, the transition probabilities p 2, p2, and p22 for forming the dyads, MiM2, M2Mi, and M2M2, respectively, are given by 

The sum of the transition probabilities of addition to M and M are each, separately, equal to 1  

The meaning of Eq. 6-23 can be seen by considering, for example, the probability of forming a sequence M ]. The probability of forming such a sequence is the probability p of Mf adding M] multiplied by the probability p of a second addition of M, multiplied by the probabihty pn of addition of M2 or p2upn- 

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