Page 232 - Carrahers_Polymer_Chemistry,_Eighth_Edition

Free Radical Chain Polymerization 195

.
Solving for [M] from Equation 6.23 gives
•  1/2
•  k i [R ][M]
[M ] =   (6.25)
 2k t 
and [R] from Equation 6.24 gives

• 2kf [I]
d
[R ] = (6.26)
k [M]
i
•
•
Substituting in Equation 6.25 the expression for [R ], we obtain an expression for [M ], Equation
6.27, which contains more easily experimentally determinable terms.
•  kf [I]  1/2
d
[M ] =     (6.27)
k t
.
Using this relationship for [M] we get more useful rate (Equation 6.28) and kinetic chain length
.
(Equation 6.30) equations that are free from the hard to measure [M].
2
•  kf [I]  1/2  kk f  1/2 1/2 1/2
pd
d
R = k [M][M ] = k [M] =   [M][I] = k [M][I]
p p p   k     k   (6.28)
t t
Thus, the rate of propagation, or polymerization, is directly proportional to the concentration of
the monomer and square root concentration of initiator.
In preparation of describing the kinetic chain length, we can also describe the rate of termination
.
using the new description for [M].

2kk f [I]
• 2
td
R t = 2[M ]k t = = 2k f [I] (6.29)
d
k t
R p  (kf [I]/ ) 1/2  k p [M] k [M]
k
DP = = k p [M]  d t  = = (6.30)
R i  2k f [I]  2(kk f [I]) 1/2 [I] 1/2
t
dt
Thus, chain length is directly proportional to monomer concentration and inversely proportional
to the square root of initiator concentration.
Typical energies of activation for propagation and termination are given in Table 6.2 and typical
free radical kinetic values in Table 6.3.
As done in Chapter 5, the effect of temperature can be determined using average activation of
the various steps. Again, the rates of all single step reactions increase as the temperature increases
but the overall result may be different for complex reactions. For free radical polymerizations, the
activation energies are generally of the order E > E ≅E > E . Remembering that the description of
i
d
t
p
the specific rate constant is

k = A e −Ea/RT (6.31)
the overall or “net” activation energy is
E (overall) = E + E + E + E (6.32)
d
t
i
p

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