# Chain Growth Polymerization

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.

## 3 thoughts on “Chain Growth Polymerization”

• #### Rohin Sharda says:

Great help! – well explained

• #### Zahrina Mardina says:

Great! thank you!

• #### Blake M says:

Yes, I think that under the steady state approximation, the rates of formation and depletion of reactive radicals, R•, should necessarily be equal. But I think the right side of the equation at 4:57 should actually be 2k[R•]^2, not k[R•]^2 as it is shown here. Then, the coefficient of 2 would be left out of the final rate law (as I believe it should be).