Steady State | MATSE 202: Introduction to Polymer Materials

We have now written rate equations for initiation (generation of radicals), propagation (growth of polymer), and termination (consumption of radicals). At the very beginning of the reaction, the rate of formation of radicals is faster than consumption of monomer. But the number of radicals cannot keep growing forever! Eventually, the rate of formation of radicals and consumption of radical monomer/active centers reaches a steady state — radicals are formed as quickly as they are consumed/propagated in the reaction. Radicals are consumed just as fast as they are created. This is called steady state. We work under this assumption for the rest of our derivation of steady state polymerization rates.

The rate of production of radicals is the same as the consumption of radicals:

R_{i} = R_{t}

Eq 5.9

Substitution of our previous expressions:

\frac{d [R \cdot]}{d t} = - 2 k_{t} {[M \cdot]}^{2}

Eq 5.10

Rearrange to solve for the steady state concentration of radicals:

[M \cdot] = {(\frac{R_{i}}{2 k_{t}})}^{1 / 2}

Eq 5.11

OK great, why do we care? Well, we ultimately want to figure out the rate of polymerization, which recall that we defined at the very beginning of the section. To refresh your memory, we defined rate of polymerization as rate of consumption of monomer. Where else have we seen rate of consumption of monomer? We saw it in the expression for the rate of propagation!

R_{p} = R_{p r o p} = - \frac{d [M]}{d t} = k_{p} [M] [M \cdot]

Eq 5.12

While we can measure the concentration of monomer, it's not easy to measure the concentration of radical. But if we are under steady state conditions, we can substitute for $[M \cdot]$ using $[M \cdot] = {(\frac{R_{i}}{2 k_{t}})}^{1 / 2}$

R_{p} = k_{p} [M] {(\frac{R_{i}}{2 k_{t}})}^{1 / 2}

Eq 5.13

Note: the upper limit for $[M]$ is pure monomer. When pure monomer is used, it's called bulk polymerization

PROBLEM 1

What happens to the total radical concentration at steady state if we double the concentration of initiator?

$[M \cdot]$ doesn't depend on $[I]$ so nothing changes
$[M \cdot]$ doubles
$[M \cdot]$ increases by a factor of $\sqrt{2}$
$[M \cdot]$ decreases by half

Click here for the answer to Problem 1.

ANSWER 1

C. $[M \cdot]$ increases by a factor of $\sqrt{2}$

We substitute for $R_{i}$ in the expression for $[M \cdot]$ and find $[M \cdot] = {(\frac{2 k_{d} [I]}{2 k_{t}})}^{1 / 2}$ so $[M \cdot] \propto {[I]}^{1 / 2}$

PROBLEM 2

What happens to the total rate of polymerization at steady state if we double the concentration of monomer?

$R_{p}$ doesn't depend on $[M]$ so nothing changes
$R_{p}$ doubles
$R_{p}$ increases by a factor of $\sqrt{2}$
$R_{p}$ decreases by half

Click here for the answer to Problem 2

ANSWER 2

B. $R_{p}$ doubles.

$R_{p} \propto [M]$