It has been reported that the rate of a batch photopolymerization of an aqueous

Question

It has been reported that the rate of a batch photopolymerization of an aqueous acrylamide solution using a light-sensitive dye is proportional to the square of the monomer concentration, [M]^2, and the square root of the absorbed light intensity, I^(1/2). Note that, although this polymerization is free radical, the apparent kinetics appear not to be typical of usual free-radical polymerization for which the rate of polymerization is proportional to the first power of monomer concentration and to the square root of the initiator concentration (eq. (2.25)). The following polymerization mechanism has been proposed to explain the observed kinetics:

Show that this mechanism appears to be correct by deriving an equation for the rate of propagation in terms of [M], I, and the appropriate rate constants. The following assumptions may be made:

1. Equal reactivity in the propagation steps

3. k_2<<k_5

4. The concentration of dye, [D], that has been activated by light and thereby controbutes to the first initiation step is proportional to the absorbed light intensity.

Explanation / Answer

Radical chain polymerization is a chain reaction consisting of a sequence of three steps—

initiation, propagation , and termination.

Homolytic clevage of initial partical

[D] + I ----->2 R*    here kd is the rate constant

chain initiation by addition of monomer

R* + M -------> M1* . here ki is the rate constant

Propagation consists of the growth of M1i by the successive additions of large numbers monomer molecules.

The successive additions can be written as

M1* + M ----> M2*   rate constant is kp

M2* + M----> M3*   rate constant is kp

M3* + M ------> M4* rate constant is kp

or we can write

Mn* + M -----> Mn+1 rate constant is kp

Termination can also occur by a combination of coupling and disproportionation

Mn* + Mm*----> Mn+m   rate constant for coupling is ktc

Mn* + Mm* -----> Mn   + Mm rate constant for disproportionation is ktd

or we can write

Mn* + Mm* -----> dead polymer rate constant is kt

The rate of monomer disappearance which is synonymous with the of polymerization is thus given A/C to question by

-d[M]^2/dt = Ri + Rp   [ Ri= rate of initiation only one times appear , Rp =rate of propogation appear several time]

-d[M]^2/dt = Rp

since equal reactivity in the propagation steps

Rp = kp [M*][M]^2

or

Ri= Rt =2 kt[M*]^2

[M*] = (Ri/2kt)^1/2

putting the value of M*

Rp= kp (Ri/2kt)^1/2 x [M]^2

here the term kp (Ri/2kt)^1/2= (I)^1/2

Rp = (I)^1/2 [M]^2

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