In this screencast, we're gong to look at a simple example of chain growth polymerization
and then look at applying the steady state approximation to the active centers in the
reaction to determine how the rate of reaction depends on monomer concentration and initiator
concentration. So I've indicated here the first step, the initiation step, where an initiator
breaks up into two radicals with an unpaired electron. And the rate of decomposition of
the initiator is going to be a rate constant for decomposition times the concentration
of initiator. So the symbol in parentheses will indicate concentration. Once we have
this active species, it can react with a monomer, and the monomers are present in high concentration
relative to the concentration of this active center. And we can start the process of initiating
the chain. The rate of initiation is going to be then, rate constant for initiation,
the concentration of these active centers, and the concentration of our monomer. Then
this active center will continue to react with monomers to make longer chains, and in
general what we're going to have is R sub j - where j is the number of monomers - reacts
with a monomer to make R sub j+1, and this is still active. This is propagation
of the reaction, where we have the rate, then, of propagation, is the propagation rate constant, the concentration
of all these active centers, times the monomer concentration. Alright and now the final step
in this simple mechanism is when two of these growing chains, these radicals, Rj and Rk,
collide to make a polymer. So this is a polymer, and it is no longer reactive. This is referred
to as a termination step - terminate the growing polymer chain. The rate of termination then
is a rate constant for termination, concentration of radicals. And what we're going to do is
apply the steady state approximation to this primary radical R sub c that's formed from
our initiator. And so the steady state approximation says the rate, R sub c, is equal to zero.
Namely, we're at stead state, as fast as we make R sub c, we react it away. So we make
it in the first step, initiation step, and because of stoichiometry there's a factor
of two there. And then we use it up in the second step. So, rate of formation, the
rate that we now react it away, and this net rate is zero, which means we can calculate
the concentration of this primary radical. Now what we're going to do is add another
factor in here for the initiator efficiency. So f is the initiator efficiency, some number
less than 1, everything else is the same in the equation. And this initiator efficiency
corresponds to the fact that some of these initiator radicals are lost due to recombination
scavenging, and therefore f is less than 1. So our rate of initiation then - which is
Ki, the primary radical concentration and the monomer concentration - can be written
as 2 times f times the rate constant for decomposition of the initiator and the initiator
concentration. And the steady state approximation says that the rate of initiation equals the
rate of termination. Namely, this says there's no increase or decrease in the number of radicals
that are present and available for the propagation step. As fast as we form them, we lose them
by termination - so we're at steady state. This is the steady state approximation. And
so substituting in the rate of initiation on the left, rate of termination on the right,
I can solve for the concentration of these active centers. Of course what we're interested
in is the rate of polymerization. So the rate of polymerization is going to be the rate
that the monomer reacts since essentially all the monomer is involved in a propagation
step. And this is using what's called the long chain approximation - assuming that these
steps for propagation take place many times for one initiation step. Namely, we make a
long chain with a lot of monomer units in the polymer. And so this means we can write
the rate of propagation as this rate constant for propagation, it's the concentration of
the radicals and the concentration of the monomers. And we just calculated the concentration
of the radicals. And so we calculated here then the rate of polymerization by substituting
in the radical concentration into this equation. And we can see the rate of polymerization's
first order in monomer concentration but it's only half order in the initiator concentration.
And as we increase the rate constant for termination, the rate of polymerization is going to decrease.
We increase the rate constant for decomposition of initiator, the rate of polymerization will
increase.