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.

## 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).